FUNCTIONS OF REAL VARIABLES

Exercise X, problem 1, p. 39

Let $y = x$ or $2x$ or $\frac{1}{2}x$ or $x^{2} +1$. Nothing further need be said at present about cases such as these.

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Exercise X, problem 2, p. 39

Let $y = 0$ whatever be the value of $x$. Then $y$ is a function of $x$, for we can give $x$ any value, and the corresponding value of $y$ (viz. $0$) is known. In this case the functional relation makes the same value of $y$ correspond to all values of $x$. The same would be true were $y$ equal to $1$ or $-\frac{1}{2}$ or $\sqrt{2}$ instead of $0$. Such a function of $x$ is called *a constant*.

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Exercise X, problem 3, p. 39

Let $y^{2} = x$. Then if $x$ is positive this equation defines *two* values of $y$ corresponding to each value of $x$, viz. $±\sqrt{x}$. If $x = 0$, $y = 0$. Hence to the particular value $0$ of $x$ corresponds *one* and only one value of $y$. But if $x$ is negative there is *no* value of $y$ which satisfies the equation. That is to say, the function $y$ is not defined for negative values of $x$. This function therefore possesses the characteristic (3), but neither (1) nor (2).

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Exercise X, problem 4, p. 39

Consider a volume of gas maintained at a constant temperature and contained in a cylinder closed by a sliding piston. I borrow this instructive example from Prof. H. S. Carslaw’s *Introduction to the Calculus.* Let $A$ be the area of the cross section of the piston and $W$ its weight. The gas, held in a state of compression by the piston, exerts a certain pressure $p_{0}$ per unit of area on the piston, which balances the weight $W$, so that W = Ap_0. Let $v_{0}$ be the volume of the gas when the system is thus in equilibrium. If additional weight is placed upon the piston the latter is forced downwards. The volume ($v$) of the gas diminishes; the pressure ($p$) which it exerts upon unit area of the piston increases. Boyle’s experimental law asserts that the product of $p$ and $v$ is very nearly constant, a correspondence which, if exact, would be represented by an equation of the type pv = a, (i) where $a$ is a number which can be determined approximately by experiment. Boyle’s law, however, only gives a reasonable approximation to the facts provided the gas is not compressed too much. When $v$ is decreased and $p$ increased beyond a certain point, the relation between them is no longer expressed with tolerable exactness by the equation (i). It is known that a [pg]40 much better approximation to the true relation can then be found by means of what is known as ‘van der Waals’ law’, expressed by the equation (p + v^2)(v - ) = , (ii) where $\alpha$, $\beta$, $\gamma$ are numbers which can also be determined approximately by experiment. Of course the two equations, even taken together, do not give anything like a complete account of the relation between $p$ and $v$. This relation is no doubt in reality much more complicated, and its form changes, as $v$ varies, from a form nearly equivalent to (i) to a form nearly equivalent to (ii). But, from a mathematical point of view, there is nothing to prevent us from contemplating an ideal state of things in which, for all values of $v$ not less than a certain value $V$, (i) would be exactly true, and (ii) exactly true for all values of $v$ less than $V$. And then we might regard the two equations as together defining $p$ as a function of $v$. It is an example of a function which for some values of $v$ is defined by one formula and for other values of $v$ is defined by another. This function possesses the characteristic (2). to any value of $v$ only one value of $p$ corresponds: but it does not possess (1). For $p$ is not defined as a function of $v$ for negative values of $v$; a ‘negative volume’ means nothing, and so negative values of $v$ do not present themselves for consideration at all.

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Exercise X, problem 5, p. 39

Suppose that a perfectly elastic ball is dropped (without rotation) from a height $\frac{1}{2}g\tau^{2}$ on to a fixed horizontal plane, and rebounds continually. The ordinary formulae of elementary dynamics, with which the reader is probably familiar, show that $h = \frac{1}{2}gt^{2}$ if $0 \leq t \leq \tau$, $h = \frac{1}{2}g(2\tau - t)^{2}$ if $\tau \leq t \leq 3\tau$, and generally h = 12g(2n- t)^2 if $(2n - 1)\tau \leq t \leq (2n + 1)\tau$, $h$ being the depth of the ball, at time $t$, below its original position. Obviously $h$ is a function of $t$ which is only defined for positive values of $t$.

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Exercise X, problem 6, p. 39

Suppose that $y$ is defined as being *the largest prime factor of $x$*. This is an instance of a definition which only applies to a particular class of values of $x$, viz. *integral* values. ‘The largest prime factor of $\frac{11}{3}$ or of $\sqrt{2}$ or of $\pi$’ means nothing, and so our defining relation fails to define for such values of $x$ as these. Thus this function does not possess the characteristic (1). It does possess (2), but not (3), as there is no simple formula which expresses $y$ in terms of $x$.

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Exercise X, problem 7, p. 39

Let $y$ be defined as *the denominator of $x$ when $x$ is expressed in its lowest terms*. This is an example of a function which is defined if and only if $x$ is *rational*. Thus $y = 7$ if $x = -11/7$: but $y$ is not defined for $x = \sqrt{2}$, ‘the denominator of $\sqrt{2}$’ being a meaningless form of words.

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Exercise X, problem 8, p. 39

Let $y$ be defined as *the height in inches of policeman $Cx$, in the Metropolitan Police, at 5.30 p.m.p.m. on 8 Aug. 1907*. Then $y$ is defined for a certain number of integral values of $x$, viz. $1$, $2$, …, $N$, where $N$ is the total number of policemen in division $C$ at that particular moment of time.

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Exercise XI, problem 1, p. 46

Trace the curves $y = 7x^{4}$, $y = 3x^{5}$, $y = x^{10}$.

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Exercise XI, problem 2, p. 46

Compare the relative magnitudes of $x^{12}$, $1,000,000x^{6}$, $1,000,000,000,000x$ when $x = 1$, $10$, $100$, etc.

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Exercise XI, problem 3, p. 46

Draw the graph of $ax^{2} + 2bx + c$.

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Exercise XI, problem 4, p. 46

Trace the curves $y = x^{3} - 3x + 1$, $y = x^{2}(x - 1)$, $y = x(x - 1)^{2}$.

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Exercise XII, problem 1, p. 48

Draw the graphs of $y = 1/x$, $y = 1/x^{2}$, $y = 1/x^{3}$, ….

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Exercise XII, problem 2, p. 48

Trace $y = x + (1/x)$, $x - (1/x)$, $x^{2} + (1/x^{2})$, $x^{2} - (1/x^{2})$ and $ax + (b/x)$ taking various values, positive and negative, for $a$ and $b$.

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Exercise XII, problem 3, p. 48

Trace y = x + 1x - 1,0pt minus 3pt(x + 1x - 1)^2,0pt minus 3pt1(x - 1)^2,0pt minus 3ptx^2 + 1x^2 - 1.

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Exercise XII, problem 4, p. 48

Trace $y = 1/(x - a)(x - b)$, $1/(x - a)(x - b)(x - c)$, where $a < b < c$.

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Exercise XII, problem 5, p. 48

Sketch the general form assumed by the curves $y = 1/x^{m}$ as $m$ becomes larger and larger, considering separately the cases in which $m$ is odd or even.

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Exercise XIII, problem 1, p. 50

$\sqrtb{(x - a)(b - x)}$, where $a < b$, is defined only for $a \leq x \leq b$. If $a < x < b$ it has two values: if $x = a$ or $b$ only one, viz. $0$.

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Exercise XIII, problem 2a, p. 50

Consider similarly gather* (x - a)(x - b)(x - c) 0pt minus 3pt(a < b < c), x(x^2 - a^2),0pt minus 3pt[3](x - a)^2(b - x)0pt minus 3pt(a < b), 1 + x - 1 - x 1 + x + 1 - x,0pt minus 3ptx + x. gather*

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Exercise XIII, problem 2b, p. 50

Consider similarly gather* (x - a)(x - b)(x - c) 0pt minus 3pt(a < b < c), x(x^2 - a^2),0pt minus 3pt[3](x - a)^2(b - x)0pt minus 3pt(a < b), 1 + x - 1 - x 1 + x + 1 - x,0pt minus 3ptx + x. gather*

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Exercise XIII, problem 2c, p. 50

Consider similarly gather* (x - a)(x - b)(x - c) 0pt minus 3pt(a < b < c), x(x^2 - a^2),0pt minus 3pt[3](x - a)^2(b - x)0pt minus 3pt(a < b), 1 + x - 1 - x 1 + x + 1 - x,0pt minus 3ptx + x. gather*

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Exercise XIII, problem 2d, p. 50

Consider similarly gather* (x - a)(x - b)(x - c) 0pt minus 3pt(a < b < c), x(x^2 - a^2),0pt minus 3pt[3](x - a)^2(b - x)0pt minus 3pt(a < b), 1 + x - 1 - x 1 + x + 1 - x,0pt minus 3ptx + x. gather*

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Exercise XIII, problem 2e, p. 50

Consider similarly gather* (x - a)(x - b)(x - c) 0pt minus 3pt(a < b < c), x(x^2 - a^2),0pt minus 3pt[3](x - a)^2(b - x)0pt minus 3pt(a < b), 1 + x - 1 - x 1 + x + 1 - x,0pt minus 3ptx + x. gather*

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Exercise XIII, problem 3a, p. 50

Trace the curves $y^{2} = x$, $y^{3} = x$, $y^{2} = x^{3}$.

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Exercise XIII, problem 3b, p. 50

Trace the curves $y^{2} = x$, $y^{3} = x$, $y^{2} = x^{3}$.

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Exercise XIII, problem 3c, p. 50

Trace the curves $y^{2} = x$, $y^{3} = x$, $y^{2} = x^{3}$.

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Exercise XIII, problem 4a, p. 50

Draw the graphs of the functions %[** TN: Not displayed in the original] y = a^2 - x^2,0pt minus 3pty = b1 - (x^2/a^2).

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Exercise XIII, problem 4b, p. 50

Draw the graphs of the functions %[** TN: Not displayed in the original] y = a^2 - x^2,0pt minus 3pty = b1 - (x^2/a^2).

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Exercise Misc-II, problem 1, p. 65

Show that if $y = f(x) = (ax + b)/(cx - a)$ then $x = f(y)$.

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Exercise Misc-II, problem 10, p. 65

Discuss the graphical solution of the equation x^m + ax^2 + bx + c = 0 by means of the curves $y = x^{m}$, $y = -ax^{2} - bx - c$. Draw up a table of the various possible numbers of roots.

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Exercise Misc-II, problem 11a, p. 65

Solve the equation $\sec\theta + \cosec\theta = 2\sqrt{2}$; and show that the equation $\sec\theta + \cosec\theta = c$ has two roots between $0$ and $2\pi$ if $c^{2} < 8$ and four if $c^{2} > 8$. [pg]66

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Exercise Misc-II, problem 11b, p. 65

Solve the equation $\sec\theta + \cosec\theta = 2\sqrt{2}$; and show that the equation $\sec\theta + \cosec\theta = c$ has two roots between $0$ and $2\pi$ if $c^{2} < 8$ and four if $c^{2} > 8$. [pg]66

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Exercise Misc-II, problem 12, p. 65

Show that the equation 2x = (2n + 1)(1 - x), where $n$ is a positive integer, has $2n + 3$ roots and no more, indicating their localities roughly. % [0]% (*Math. Trip.* 1896.)% [1]%

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Exercise Misc-II, problem 13, p. 65

Show that the equation $\frac{2}{3}x\sin x = 1$ has four roots between $-\pi$ and $\pi$.

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Exercise Misc-II, problem 14(1), p. 65

Discuss the number and values of the roots of the equations %[** TN: Items in multiple columns in the original] 0pt minus 3pt% [2.25em][l](1)% [2.25em][l](1)% % $\cot x + x - \frac{3}{2}\pi = 0$, 0pt minus 3pt% [2.25em][l](2)% [2.25em][l](2)% % $x^{2} + \sin^{2} x = 1$, 0pt minus 3pt% [2.25em][l](3)% [2.25em][l](3)% % $\tan x = 2x/(1 + x^{2})$, 0pt minus 3pt% [2.25em][l](4)% [2.25em][l](4)% % $\sin x - x + \frac{1}{6}x^{3} = 0$, 0pt minus 3pt% [2.25em][l](5)% [2.25em][l](5)% % $(1 - \cos x)\tan\alpha - x + \sin x = 0$.

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Exercise Misc-II, problem 14(2), p. 65

Discuss the number and values of the roots of the equations %[** TN: Items in multiple columns in the original] 0pt minus 3pt% [2.25em][l](1)% [2.25em][l](1)% % $\cot x + x - \frac{3}{2}\pi = 0$, 0pt minus 3pt% [2.25em][l](2)% [2.25em][l](2)% % $x^{2} + \sin^{2} x = 1$, 0pt minus 3pt% [2.25em][l](3)% [2.25em][l](3)% % $\tan x = 2x/(1 + x^{2})$, 0pt minus 3pt% [2.25em][l](4)% [2.25em][l](4)% % $\sin x - x + \frac{1}{6}x^{3} = 0$, 0pt minus 3pt% [2.25em][l](5)% [2.25em][l](5)% % $(1 - \cos x)\tan\alpha - x + \sin x = 0$.

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Exercise Misc-II, problem 14(3), p. 65

Discuss the number and values of the roots of the equations %[** TN: Items in multiple columns in the original] 0pt minus 3pt% [2.25em][l](1)% [2.25em][l](1)% % $\cot x + x - \frac{3}{2}\pi = 0$, 0pt minus 3pt% [2.25em][l](2)% [2.25em][l](2)% % $x^{2} + \sin^{2} x = 1$, 0pt minus 3pt% [2.25em][l](3)% [2.25em][l](3)% % $\tan x = 2x/(1 + x^{2})$, 0pt minus 3pt% [2.25em][l](4)% [2.25em][l](4)% % $\sin x - x + \frac{1}{6}x^{3} = 0$, 0pt minus 3pt% [2.25em][l](5)% [2.25em][l](5)% % $(1 - \cos x)\tan\alpha - x + \sin x = 0$.

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Exercise Misc-II, problem 14(4), p. 65

Discuss the number and values of the roots of the equations %[** TN: Items in multiple columns in the original] 0pt minus 3pt% [2.25em][l](1)% [2.25em][l](1)% % $\cot x + x - \frac{3}{2}\pi = 0$, 0pt minus 3pt% [2.25em][l](2)% [2.25em][l](2)% % $x^{2} + \sin^{2} x = 1$, 0pt minus 3pt% [2.25em][l](3)% [2.25em][l](3)% % $\tan x = 2x/(1 + x^{2})$, 0pt minus 3pt% [2.25em][l](4)% [2.25em][l](4)% % $\sin x - x + \frac{1}{6}x^{3} = 0$, 0pt minus 3pt% [2.25em][l](5)% [2.25em][l](5)% % $(1 - \cos x)\tan\alpha - x + \sin x = 0$.

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Exercise Misc-II, problem 14(5), p. 65

Discuss the number and values of the roots of the equations %[** TN: Items in multiple columns in the original] 0pt minus 3pt% [2.25em][l](1)% [2.25em][l](1)% % $\cot x + x - \frac{3}{2}\pi = 0$, 0pt minus 3pt% [2.25em][l](2)% [2.25em][l](2)% % $x^{2} + \sin^{2} x = 1$, 0pt minus 3pt% [2.25em][l](3)% [2.25em][l](3)% % $\tan x = 2x/(1 + x^{2})$, 0pt minus 3pt% [2.25em][l](4)% [2.25em][l](4)% % $\sin x - x + \frac{1}{6}x^{3} = 0$, 0pt minus 3pt% [2.25em][l](5)% [2.25em][l](5)% % $(1 - \cos x)\tan\alpha - x + \sin x = 0$.

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Exercise Misc-II, problem 15, p. 65

The polynomial of the second degree which assumes, when $x = a$, $b$, $c$ the values $\alpha$, $\beta$, $\gamma$ is (x - b)(x - c)(a - b)(a - c) + (x - c)(x - a)(b - c)(b - a) + (x - a)(x - b)(c - a)(c - b). Give a similar formula for the polynomial of the $(n - 1)$th degree which assumes, when $x = a_{1}$, $a_{2}$, … $a_{n}$, the values $\alpha_{1}$, $\alpha_{2}$, … $\alpha_{n}$.

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Exercise Misc-II, problem 16, p. 65

Find a polynomial in $x$ of the second degree which for the values $0$, $1$, $2$ of $x$ takes the values $1/c$, $1/(c + 1)$, $1/(c + 2)$; and show that when $x = c + 2$ its value is $1/(c + 1)$. % [0]% (*Math. Trip.* 1911.)% [1]%

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Exercise Misc-II, problem 17, p. 65

Show that if $x$ is a rational function of $y$, and $y$ is a rational function of $x$, then $Axy + Bx + Cy + D = 0$.

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Exercise Misc-II, problem 18, p. 65

If $y$ is an algebraical function of $x$, then $x$ is an algebraical function of $y$.

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Exercise Misc-II, problem 19, p. 65

Verify that the equation 12x = 1 - x^2x + (x - 1)2 - x3 is approximately true for all values of $x$ between $0$ and $1$. [Take $x = 0$, $\frac{1}{6}$, $\frac{1}{3}$, $\tfrac{1}{2}$, $\frac{2}{3}$, $\frac{5}{6}$, $1$, and use tables. For which of these values is the formula exact?]

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Exercise Misc-II, problem 2, p. 65

If $f(x) = f(-x)$ for all values of $x$, $f(x)$ is called an *even* function. If $f(x) = -f(-x)$, it is called an *odd* function. Show that any function of $x$, defined for all values of $x$, is the sum of an even and an odd function of $x$. [Use the identity $f(x) = \frac{1}{2}\{f(x) + f(-x)\} + \frac{1}{2}\{f(x) - f(-x)\}$.]

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Exercise Misc-II, problem 20, p. 65

What is the form of the graph of the functions z = [x] + [y],0pt minus 3ptz = x + y - [x] - [y]?

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Exercise Misc-II, problem 21, p. 65

What is the form of the graph of the functions $z = \sin x + \sin y$, $z = \sin x\sin y$, $z = \sin xy$, $z = \sin(x^{2} + y^{2})$?

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Exercise Misc-II, problem 22, p. 65

**constructions for irrational numbers.** In ChapterI we indicated one or two simple geometrical constructions for a length equal to $\sqrt{2}$, starting from a given unit length. We also showed how to construct the roots of any quadratic equation $ax^{2} + 2bx + c = 0$, it being supposed that we can construct lines whose lengths are equal to any of the ratios of the coefficients $a$, $b$, $c$, as is certainly the case if $a$, $b$, $c$ are rational. All these constructions were what may be called Euclidean constructions; they depended on the ruler and compasses only. [pg]67 It is fairly obvious that we can construct by these methods the length measured by any irrational number which is defined by any combination of square roots, however complicated. Thus [4]17 + 31117 - 311 - 17 - 31117 + 311 is a case in point. This expression contains a fourth root, but this is of course the square root of a square root. We should begin by constructing $\sqrt{11}$, *e.g.* as the mean between $1$ and $11$: then $17 + 3\sqrt{11}$ and $17 - 3\sqrt{11}$, and so on. Or these two mixed surds might be constructed directly as the roots of $x^{2} - 34x + 190 = 0$. Conversely, *only* irrationals of this kind can be constructed by Euclidean methods. Starting from a unit length we can construct any *rational* length. And hence we can construct the line $Ax + By + C = 0$, provided that the ratios of $A$, $B$, $C$ are rational, and the circle (x - )^2 + (y - )^2 = ^2 (or $x^{2} + y^{2} + 2gx + 2fy + c = 0$), provided that $\alpha$, $\beta$, $\rho$ are rational, a condition which implies that $g$, $f$, $c$ are rational. Now in any Euclidean construction each new point introduced into the figure is determined as the intersection of two lines or circles, or a line and a circle. But if the coefficients are rational, such a pair of equations as Ax + By + C = 0,0pt minus 3ptx^2 + y^2 + 2gx + 2fy + c = 0 give, on solution, values of $x$ and $y$ of the form $m + n\sqrt{p}$, where $m$, $n$, $p$ are rational: for if we substitute for $x$ in terms of $y$ in the second equation we obtain a quadratic in $y$ with rational coefficients. Hence the coordinates of all points obtained by means of lines and circles with rational coefficients are expressible by rational numbers and quadratic surds. And so the same is true of the distance $\sqrtb{(x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2}}$ between any two points so obtained. With the irrational distances thus constructed we may proceed to construct a number of lines and circles whose coefficients may now themselves involve quadratic surds. It is evident, however, that all the lengths which we can construct by the use of such lines and circles are still expressible by square roots only, though our surd expressions may now be of a more complicated form. And this remains true however often our constructions are repeated. Hence *Euclidean methods will construct any surd expression involving square roots only, and no others*. One of the famous problems of antiquity was that of the duplication of the cube, that is to say of the construction by Euclidean methods of a length measured by $\sqrt[3]{2}$. It can be shown that $\sqrt[3]{2}$ cannot be expressed by means of any finite combination of rational numbers and square roots, and so that the problem is an impossible one. See Hobson, *Squaring the Circle*, pp. 47 *et seq.*; the first stage of the proof, viz. the proof that $\sqrt[3]{2}$ cannot be a root of a quadratic equation $ax^{2} + 2bx + c = 0$ with rational coefficients, was given in Ch.I ([misc:I]Misc. Exs. 24). [pg]68

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Exercise Misc-II, problem 23, p. 65

**quadrature of the circle.** Let $O$ be the centre of a circle of radius $R$. On the tangent at $A$ take $AP = \frac{11}{5}R$ and $AQ = \frac{13}{5}R$, in the same direction. On $AO$ take $AN = OP$ and draw $NM$ parallel to $OQ$ and cutting $AP$ in $M$. Show that AM/R = 1325146, and that to take $AM$ as being equal to the circumference of the circle would lead to a value of $\pi$ correct to five places of decimals. If $R$ is the earth’s radius, the error in supposing $AM$ to be its circumference is less than $11$ yards.

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Exercise Misc-II, problem 24, p. 65

Show that the only lengths which can be constructed with the ruler only, starting from a given unit length, are rational lengths.

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Exercise Misc-II, problem 25, p. 65

**for $\sqrt[3]{2}$.** $O$ is the vertex and $S$ the focus of the parabola $y^{2} = 4x$, and $P$ is one of its points of intersection with the parabola $x^{2} = 2y$. Show that $OP$ meets the latus rectum of the first parabola in a point $Q$ such that $SQ = \sqrt[3]{2}$.

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Exercise Misc-II, problem 26, p. 65

Take a circle of unit diameter, a diameter $OA$ and the tangent at $A$. Draw a chord $OBC$ cutting the circle at $B$ and the tangent at $C$. On this line take $OM = BC$. Taking $O$ as origin and $OA$ as axis of $x$, show that the locus of $M$ is the curve (x^2 + y^2)x - y^2 = 0 (the *Cissoid of Diocles*). Sketch the curve. Take along the axis of $y$ a length $OD = 2$. Let $AD$ cut the curve in $P$ and $OP$ cut the tangent to the circle at $A$ in $Q$. Show that $AQ = \sqrt[3]{2}$.

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Exercise Misc-II, problem 3, p. 65

Draw the graphs of the functions 3x + 4x,0pt minus 3pt(2 x). % [0]% (*Math. Trip.* 1896.)% [1]%

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Exercise Misc-II, problem 4, p. 65

Draw the graphs of the functions x(a^2 x + b^2 x),0pt minus 3ptxx(a^2 x + b^2 x),0pt minus 3pt(xx)^2.

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Exercise Misc-II, problem 5, p. 65

Draw the graphs of the functions $x[1/x]$, $[x]/x$.

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Exercise Misc-II, problem 6(i), p. 65

Draw the graphs of the functions align* % [2.25em][l](i)% [2.25em][l](i)% % & (2x^2 - 1) - 2 x, % [2.25em][l](ii)% [2.25em][l](ii)% % & a + x1 - ax - a - x, align* where the symbols $\arccos a$, $\arctan a$ denote, for any value of $a$, the least positive (or zero) angle, whose cosine or tangent is $a$.

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Exercise Misc-II, problem 6(ii), p. 65

Draw the graphs of the functions align* % [2.25em][l](i)% [2.25em][l](i)% % & (2x^2 - 1) - 2 x, % [2.25em][l](ii)% [2.25em][l](ii)% % & a + x1 - ax - a - x, align* where the symbols $\arccos a$, $\arctan a$ denote, for any value of $a$, the least positive (or zero) angle, whose cosine or tangent is $a$.

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Exercise Misc-II, problem 7, p. 65

Verify the following method of constructing the graph of $f\{\phi(x)\}$ by means of the line $y = x$ and the graphs of $f(x)$ and $\phi(x)$: take $OA = x$ along $OX$, draw $AB$ parallel to $OY$ to meet $y = \phi(x)$ in $B$, $BC$ parallel to $OX$ to meet $y = x$ in $C$, $CD$ parallel to $OY$ to meet $y = f(x)$ in $D$, and $DP$ parallel to $OX$ to meet $AB$ in $P$; then $P$ is a point on the graph required.

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Exercise Misc-II, problem 8, p. 65

Show that the roots of $x^{3} + px + q = 0$ are the abscissae of the points of intersection (other than the origin) of the parabola $y = x^{2}$ and the circle x^2 + y^2 + (p - 1)y + qx = 0.

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Exercise Misc-II, problem 9, p. 65

The roots of $x^{4} + nx^{3} + px^{2} + qx + r = 0$ are the abscissae of the points of intersection of the parabola $x^{2} = y - \frac{1}{2}nx$ and the circle x^2 + y^2 + (18n^2 - 12pn + 12n + q)x + (p - 1 - 14n^2)y + r = 0.

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Exercise XIV, problem 1, p. 51

If $m = 1$, $y$ is a rational function.

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Exercise XIV, problem 2, p. 51

If $m = 2$, the equation is $y^{2} + R_{1}y + R_{2} = 0$, so that y = 12-R_1 ± R_1^2 - 4R_2. This function is defined for all values of $x$ for which $R_{1}^{2} \geq 4R_{2}$. It has two values if $R_{1}^{2} > 4R_{2}$ and one if $R_{1}^{2} = 4R_{2}$. If $m = 3$ or $4$, we can use the methods explained in treatises on Algebra for the solution of cubic and biquadratic equations. But as a rule the process is complicated and the results inconvenient in form, and we can generally study the properties of the function better by means of the original equation.

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Exercise XIV, problem 3a, p. 51

Consider the functions defined by the equations y^2 - 2y - x^2 = 0,0pt minus 3pty^2 - 2y + x^2 = 0,0pt minus 3pty^4 - 2y^2 + x^2 = 0, in each case obtaining $y$ as an explicit function of $x$, and stating for what values of $x$ it is defined.

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Exercise XIV, problem 3b, p. 51

Consider the functions defined by the equations y^2 - 2y - x^2 = 0,0pt minus 3pty^2 - 2y + x^2 = 0,0pt minus 3pty^4 - 2y^2 + x^2 = 0, in each case obtaining $y$ as an explicit function of $x$, and stating for what values of $x$ it is defined.

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Exercise XIV, problem 3c, p. 51

Consider the functions defined by the equations y^2 - 2y - x^2 = 0,0pt minus 3pty^2 - 2y + x^2 = 0,0pt minus 3pty^4 - 2y^2 + x^2 = 0, in each case obtaining $y$ as an explicit function of $x$, and stating for what values of $x$ it is defined.

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Exercise XIV, problem 4a, p. 51

Find algebraical equations, with coefficients rational in $x$, satisfied by each of the functions x + 1/x,0pt minus 3pt[3]x + [3]1/x,0pt minus 3ptx + x,0pt minus 3ptx + x + x.

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Exercise XIV, problem 4b, p. 51

Find algebraical equations, with coefficients rational in $x$, satisfied by each of the functions x + 1/x,0pt minus 3pt[3]x + [3]1/x,0pt minus 3ptx + x,0pt minus 3ptx + x + x.

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Exercise XIV, problem 4c, p. 51

Find algebraical equations, with coefficients rational in $x$, satisfied by each of the functions x + 1/x,0pt minus 3pt[3]x + [3]1/x,0pt minus 3ptx + x,0pt minus 3ptx + x + x.

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Exercise XIV, problem 4d, p. 51

Find algebraical equations, with coefficients rational in $x$, satisfied by each of the functions x + 1/x,0pt minus 3pt[3]x + [3]1/x,0pt minus 3ptx + x,0pt minus 3ptx + x + x.

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Exercise XIV, problem 5, p. 51

Consider the equation $y^{4} = x^{2}$. [Here $y^{2} = ±x$. If $x$ is positive, $y = \sqrt{x}$: if negative, $y = \sqrtp{-x}$. Thus the function has two values for all values of $x$ save $x = 0$.]

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Exercise XIV, problem 6, p. 51

An algebraical function of an algebraical function of $x$ is itself an algebraical function of $x$. [For we have alignat*4 y^m &+ R_1(z)y^m-1 &&+ …&&+ R_m(z) &&= 0, where z^n &+ S_1(x)z^n-1 &&+ …&&+ S_n(x) &&= 0. Eliminating $z$ we find an equation of the form y^p &+ T_1(x)y^p-1 &&+ …&&+ T_p(x) &&= 0. alignat* Here all the capital letters denote rational functions.]

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Exercise XIV, problem 7, p. 51

An example should perhaps be given of an algebraical function which cannot be expressed in an explicit algebraical form. Such an example is the function $y$ defined by the equation y^5 - y - x = 0. But the proof that we cannot find an explicit algebraical expression for $y$ in terms of $x$ is difficult, and cannot be attempted here.

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Exercise XVI, problem 1, p. 55

Let $y = [x]$, where $[x]$ denotes the greatest integer not greater than $x$. The graph is shown in [fig:15a]Fig. 15a. The left-hand end points of the thick lines, but not the right-hand ones, belong to the graph.

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Exercise XVI, problem 10, p. 55

Let $y = 1$ when $x$ is rational, but $y = 0$ when $x$ is irrational. The graph consists of two series of points arranged upon the lines $y = 1$ and $y = 0$. To the eye it is not distinguishable from two continuous straight lines, but in reality an infinite number of points are missing from each line.

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Exercise XVI, problem 11, p. 55

Let $y = x$ when $x$ is irrational and $y = \sqrtb{(1 + p^{2})/(1 + q^{2})}$ when $x$ is a57 rational fraction $p/q$.

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Exercise XVI, problem 2, p. 55

$y = x - [x]$. ([fig:15b]Fig. 15b.)

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Exercise XVI, problem 3, p. 55

$y = \sqrtb{x - [x]}$. ([fig:15c]Fig. 15c.)

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Exercise XVI, problem 4, p. 55

$y = [x] + \sqrtb{x - [x]}$. ([fig:15d]Fig. 15d.)

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Exercise XVI, problem 5a, p. 55

$y = (x - [x])^{2}$, $[x] + (x - [x])^{2}$.

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Exercise XVI, problem 5b, p. 55

$y = (x - [x])^{2}$, $[x] + (x - [x])^{2}$.

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Exercise XVI, problem 6a, p. 55

$y = [\sqrt{x}]$, $[x^{2}]$, $\sqrt{x} - [\sqrt{x}]$, $x^{2} - [x^{2}]$, $[1 - x^{2}]$.

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Exercise XVI, problem 6b, p. 55

$y = [\sqrt{x}]$, $[x^{2}]$, $\sqrt{x} - [\sqrt{x}]$, $x^{2} - [x^{2}]$, $[1 - x^{2}]$.

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Exercise XVI, problem 6c, p. 55

$y = [\sqrt{x}]$, $[x^{2}]$, $\sqrt{x} - [\sqrt{x}]$, $x^{2} - [x^{2}]$, $[1 - x^{2}]$.

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Exercise XVI, problem 6d, p. 55

$y = [\sqrt{x}]$, $[x^{2}]$, $\sqrt{x} - [\sqrt{x}]$, $x^{2} - [x^{2}]$, $[1 - x^{2}]$.

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Exercise XVI, problem 6e, p. 55

$y = [\sqrt{x}]$, $[x^{2}]$, $\sqrt{x} - [\sqrt{x}]$, $x^{2} - [x^{2}]$, $[1 - x^{2}]$.

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Exercise XVI, problem 7, p. 55

Let $y$ be defined as *the largest prime factor of $x$* (cf. x. 6). Then $y$ is defined only for integral values of $x$. If alignat*3 x &= 1, 2, 3, 4, 5, 6, 7, 8, 9, &10,& 11, &12,& 13, …, then y &= 1, 2, 3, 2, 5, 3, 7, 2, 3, & 5,& 11, & 3,& 13, …. alignat* The graph consists of a number of isolated points.

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Exercise XVI, problem 8, p. 55

Let $y$ be *the denominator of $x$* (x. 7). In this case $y$ is defined only for rational values of $x$. We can mark off as many points on the graph as we please, but the result is not in any ordinary sense of the word a curve, and there are no points corresponding to any irrational values of $x$. Draw the straight line joining the points $(N - 1, N)$, $(N, N)$, where $N$ is a positive integer. Show that the number of points of the locus which lie on this line is equal to the number of positive integers less than and prime to $N$.

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Exercise XVI, problem 9, p. 55

Let $y = 0$ when $x$ is an integer, $y = x$ when $x$ is not an integer. The graph is derived from the straight line $y = x$ by taking out the points … (-1, -1),0pt minus 3pt(0, 0),0pt minus 3pt(1, 1),0pt minus 3pt(2, 2), … and adding the points $(-1, 0)$, $(0, 0)$, $(1, 0)$, … on the axis of $x$.

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Exercise XVII, problem 1, p. 58

**quadratic equation $ax^{2} + 2bx + c = 0$.** This may be solved graphically in a variety of ways. For instance we may draw the graphs of y = ax + 2b,0pt minus 3pty = -c/x, whose intersections, if any, give the roots. Or we may take y = x^2,0pt minus 3pty = -(2bx + c)/a. But the most elementary method is probably to draw the circle a(x^2 + y^2) + 2bx + c = 0, whose centre is $(-b/a, 0)$ and radius $\{\sqrtp{b^{2} - ac}\}/a$. The abscissae of its intersections with the axis of $x$ are the roots of the equation.

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Exercise XVII, problem 2a, p. 58

Solve by any of these methods x^2 + 2x - 3 = 0,0pt minus 3ptx^2 - 7x + 4 = 0,0pt minus 3pt3x^2 + 2x - 2 = 0.

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Exercise XVII, problem 2b, p. 58

Solve by any of these methods x^2 + 2x - 3 = 0,0pt minus 3ptx^2 - 7x + 4 = 0,0pt minus 3pt3x^2 + 2x - 2 = 0.

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Exercise XVII, problem 2c, p. 58

Solve by any of these methods x^2 + 2x - 3 = 0,0pt minus 3ptx^2 - 7x + 4 = 0,0pt minus 3pt3x^2 + 2x - 2 = 0.

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Exercise XVII, problem 3, p. 58

**equation $x^{m} + ax + b = 0$.** This may be solved by constructing the curves $y = x^{m}$, $y = -ax - b$. Verify the following table for the number of roots of gather* x^m + ax + b = 0: alignedat3 &[1.5em][l](*a*) &&m *even* && aligned &$b$ positive, *two or none*, &$b$ negative, *two* aligned . &[1.5em][l](*b*) &&m *odd* && aligned &$a$ positive, *one*, &$a$ negative, *three or one*. aligned . alignedat gather* Construct numerical examples to illustrate all possible cases.

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Exercise XVII, problem 4, p. 58

Show that the equation $\tan x = ax + b$ has always an infinite number of roots.

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Exercise XVII, problem 5a, p. 58

Determine the number of roots of x = x,0pt minus 3ptx = 13 x,0pt minus 3ptx = 18 x,0pt minus 3ptx = 1120 x.

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Exercise XVII, problem 5b, p. 58

Determine the number of roots of x = x,0pt minus 3ptx = 13 x,0pt minus 3ptx = 18 x,0pt minus 3ptx = 1120 x.

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Exercise XVII, problem 5c, p. 58

Determine the number of roots of x = x,0pt minus 3ptx = 13 x,0pt minus 3ptx = 18 x,0pt minus 3ptx = 1120 x.

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Exercise XVII, problem 5d, p. 58

Determine the number of roots of x = x,0pt minus 3ptx = 13 x,0pt minus 3ptx = 18 x,0pt minus 3ptx = 1120 x.

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Exercise XVII, problem 6, p. 58

Show that if $a$ is small and positive (*e.g.* $a = .01$), the equation x - a = 12^2 x has three roots. Consider also the case in which $a$ is small and negative. Explain how the number of roots varies as $a$ varies.

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Exercise XVIII, problem 1, p. 61

The points of intersection of the two curves whose equations are $f(x, y) = 0$, $\phi(x, y) = 0$, where $f$ and $\phi$ are polynomials, can be determined if these equations can be solved as a pair of simultaneous equations in $x$ and $y$. The solution generally consists of a finite number of pairs of values of $x$ and $y$. The two equations therefore generally represent a finite number of isolated points.

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Exercise XVIII, problem 2, p. 61

Trace the curves $(x + y)^{2} = 1$, $xy = 1$, $x^{2} - y^{2} = 1$.

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Exercise XVIII, problem 3, p. 61

The curve $f(x, y) + \lambda\phi(x, y) = 0$ represents a curve passing through the points of intersection of $f = 0$ and $\phi = 0$.

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Exercise XVIII, problem 4a, p. 61

What loci are represented by [1.5em][l]$(\alpha)$ x = at + b,0pt minus 3pty = ct + d, [1.5em][l]$(\beta)$ x/a = 2t/(1 + t^2),0pt minus 3pty/a = (1 - t^2)/(1 + t^2), when $t$ varies through all real values?

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Exercise XVIII, problem 4b, p. 61

What loci are represented by [1.5em][l]$(\alpha)$ x = at + b,0pt minus 3pty = ct + d, [1.5em][l]$(\beta)$ x/a = 2t/(1 + t^2),0pt minus 3pty/a = (1 - t^2)/(1 + t^2), when $t$ varies through all real values?

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Exercise XV, problem 10a, p. 53

Draw the graphs of $\arccos x$ and $\arcsin x$.

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Exercise XV, problem 10b, p. 53

Draw the graphs of $\arccos x$ and $\arcsin x$.

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Exercise XV, problem 11a, p. 53

Draw the graphs of x,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x.

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Exercise XV, problem 11b, p. 53

Draw the graphs of x,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x.

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Exercise XV, problem 11c, p. 53

Draw the graphs of x,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x.

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Exercise XV, problem 11d, p. 53

Draw the graphs of x,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x.

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Exercise XV, problem 11e, p. 53

Draw the graphs of x,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x.

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Exercise XV, problem 11f, p. 53

Draw the graphs of x,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x.

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Exercise XV, problem 11g, p. 53

Draw the graphs of x,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x.

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Exercise XV, problem 11h, p. 53

Draw the graphs of x,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x.

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Exercise XV, problem 12, p. 53

Draw the graphs of $\arctan x$, $\arccot x$, $\arcsec x$, $\arccosec x$. Give formulae (as in Ex. 10) expressing all the values of each of these functions in terms of any particular value.

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Exercise XV, problem 13a, p. 53

Draw the graphs of $\tan(1/x)$, $\cot(1/x)$, $\sec(1/x)$, $\cosec(1/x)$.

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Exercise XV, problem 13b, p. 53

Draw the graphs of $\tan(1/x)$, $\cot(1/x)$, $\sec(1/x)$, $\cosec(1/x)$.

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Exercise XV, problem 13c, p. 53

Draw the graphs of $\tan(1/x)$, $\cot(1/x)$, $\sec(1/x)$, $\cosec(1/x)$.

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Exercise XV, problem 13d, p. 53

Draw the graphs of $\tan(1/x)$, $\cot(1/x)$, $\sec(1/x)$, $\cosec(1/x)$.

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Exercise XV, problem 14, p. 53

Show that $\cos x$ and $\sin x$ are not rational functions of $x$.

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Exercise XV, problem 15, p. 53

Show, more generally, that no function with a period can be an algebraical function of $x$.

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Exercise XV, problem 16, p. 53

The inverse sine and inverse cosine are not rational or algebraical functions.

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Exercise XV, problem 1a, p. 53

Draw the graphs of $\cos x$, $\sin x$, and $a\cos x + b\sin x$.

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Exercise XV, problem 1b, p. 53

Draw the graphs of $\cos x$, $\sin x$, and $a\cos x + b\sin x$.

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Exercise XV, problem 1c, p. 53

Draw the graphs of $\cos x$, $\sin x$, and $a\cos x + b\sin x$.

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Exercise XV, problem 2a, p. 53

Draw the graphs of $\cos^{2} x$, $\sin^{2} x$, $a\cos^{2} x + b\sin^{2} x$.

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Exercise XV, problem 2b, p. 53

Draw the graphs of $\cos^{2} x$, $\sin^{2} x$, $a\cos^{2} x + b\sin^{2} x$.

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Exercise XV, problem 2c, p. 53

Draw the graphs of $\cos^{2} x$, $\sin^{2} x$, $a\cos^{2} x + b\sin^{2} x$.

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Exercise XV, problem 3, p. 53

Suppose the graphs of $f(x)$ and $F(x)$ drawn. Then the graph of f(x)^2 x + F(x)^2 x is a wavy curve which oscillates between the curves $y = f(x)$, $y = F(x)$. Draw the graph when $f(x) = x$, $F(x) = x^{2}$.

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Exercise XV, problem 4, p. 53

Show that the graph of $\cos px + \cos qx$ lies between those of $2\cos\frac{1}{2}(p - q)x$ and $-2\cos\frac{1}{2}(p + q)x$, touching each in turn. Sketch the graph when $(p - q)/(p + q)$ is small. % [0]% (*Math. Trip.* 1908.)% [1]%

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Exercise XV, problem 5a, p. 53

Draw the graphs of $x + \sin x$, $(1/x) + \sin x$, $x\sin x$, $(\sin x)/x$.

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Exercise XV, problem 5b, p. 53

Draw the graphs of $x + \sin x$, $(1/x) + \sin x$, $x\sin x$, $(\sin x)/x$.

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Exercise XV, problem 5c, p. 53

Draw the graphs of $x + \sin x$, $(1/x) + \sin x$, $x\sin x$, $(\sin x)/x$.

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Exercise XV, problem 5d, p. 53

Draw the graphs of $x + \sin x$, $(1/x) + \sin x$, $x\sin x$, $(\sin x)/x$.

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Exercise XV, problem 6, p. 53

Draw the graph of $\sin(1/x)$.

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Exercise XV, problem 7, p. 53

Draw the graph of $x\sin(1/x)$.

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Exercise XV, problem 8a, p. 53

Draw the graphs of $x^{2}\sin(1/x)$, $(1/x)\sin(1/x)$, $\sin^{2}(1/x)$, $\{x\sin(1/x)\}^{2}$, $a\cos^{2}(1/x) + b\sin^{2}(1/x)$, $\sin x + \sin(1/x)$, $\sin x\sin(1/x)$.

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Exercise XV, problem 8b, p. 53

Draw the graphs of $x^{2}\sin(1/x)$, $(1/x)\sin(1/x)$, $\sin^{2}(1/x)$, $\{x\sin(1/x)\}^{2}$, $a\cos^{2}(1/x) + b\sin^{2}(1/x)$, $\sin x + \sin(1/x)$, $\sin x\sin(1/x)$.

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Exercise XV, problem 8c, p. 53

Draw the graphs of $x^{2}\sin(1/x)$, $(1/x)\sin(1/x)$, $\sin^{2}(1/x)$, $\{x\sin(1/x)\}^{2}$, $a\cos^{2}(1/x) + b\sin^{2}(1/x)$, $\sin x + \sin(1/x)$, $\sin x\sin(1/x)$.

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Exercise XV, problem 8d, p. 53

Draw the graphs of $x^{2}\sin(1/x)$, $(1/x)\sin(1/x)$, $\sin^{2}(1/x)$, $\{x\sin(1/x)\}^{2}$, $a\cos^{2}(1/x) + b\sin^{2}(1/x)$, $\sin x + \sin(1/x)$, $\sin x\sin(1/x)$.

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Exercise XV, problem 8e, p. 53

Draw the graphs of $x^{2}\sin(1/x)$, $(1/x)\sin(1/x)$, $\sin^{2}(1/x)$, $\{x\sin(1/x)\}^{2}$, $a\cos^{2}(1/x) + b\sin^{2}(1/x)$, $\sin x + \sin(1/x)$, $\sin x\sin(1/x)$.

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Exercise XV, problem 8f, p. 53

Draw the graphs of $x^{2}\sin(1/x)$, $(1/x)\sin(1/x)$, $\sin^{2}(1/x)$, $\{x\sin(1/x)\}^{2}$, $a\cos^{2}(1/x) + b\sin^{2}(1/x)$, $\sin x + \sin(1/x)$, $\sin x\sin(1/x)$.

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Exercise XV, problem 8g, p. 53

Draw the graphs of $x^{2}\sin(1/x)$, $(1/x)\sin(1/x)$, $\sin^{2}(1/x)$, $\{x\sin(1/x)\}^{2}$, $a\cos^{2}(1/x) + b\sin^{2}(1/x)$, $\sin x + \sin(1/x)$, $\sin x\sin(1/x)$.

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Exercise XV, problem 9a, p. 53

Draw the graphs of $\cos x^{2}$, $\sin x^{2}$, $a\cos x^{2} + b\sin x^{2}$.

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Exercise XV, problem 9b, p. 53

Draw the graphs of $\cos x^{2}$, $\sin x^{2}$, $a\cos x^{2} + b\sin x^{2}$.

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Exercise XV, problem 9c, p. 53

Draw the graphs of $\cos x^{2}$, $\sin x^{2}$, $a\cos x^{2} + b\sin x^{2}$.

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Exercise XIX, problem 1, p. 62

What is represented by *three* equations of the type $f(x, y, z) = 0$?

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Exercise XIX, problem 10, p. 62

**surfaces.** Cylinders and cones are special cases of *surfaces composed of straight lines*. Such surfaces are called *ruled surfaces*. The two equations x = az + b,0pt minus 3pty = cz + d, (1) represent the intersection of two planes, *i.e.* a straight line. Now suppose that $a$, $b$, $c$, $d$ instead of being fixed are *functions of an auxiliary variable $t$*. For any particular value of $t$ the equations (1) give a line. As $t$ varies, this line moves and generates a surface, whose equation may be found by eliminating $t$ between the two equations (1). For instance, in Ex. 7 the equations of the line which generates the cone are x = zt,0pt minus 3pty = zt, where $t$ is the angle between the plane $XOZ$ and a plane through the line and the axis of $z$. Another simple example of a ruled surface may be constructed as follows. Take two sections of a right circular cylinder perpendicular to the axis and at a distance $l$ apart ([fig:18a]Fig. 18a). We can imagine the surface of the cylinder to be made up of a number of thin parallel rigid rods of length $l$, such as $PQ$, the ends of the rods being fastened to two circular rods of radius $a$. Now let us take a third circular rod of the same radius and place it round the surface of the cylinder at a distance $h$ from one of the first two rods (see [fig:18a]Fig. 18a, where $Pq = h$). Unfasten the end $Q$ of the rod $PQ$ and turn $PQ$ about $P$ until $Q$ can be fastened to the third circular rod in the position $Q'$. The angle $qOQ' = \alpha$ in the figure is evidently given by l^2 - h^2 = qQ’^2 = (2a12 )^2. Let all the other rods of which the cylinder was composed be treated in the same way. We obtain a ruled surface whose form is indicated in [fig:18b]Fig. 18b. It is entirely built up of straight lines; but the surface is curved everywhere, and is in general shape not unlike certain forms of table-napkin rings ([fig:18c]Fig. 18c). %[Illustration: Fig. 18a.] %[Illustration: Fig. 18b.] %[Illustration: Fig. 18c.] figure[hbt!] minipage0.3 1.5inp064a Fig. 18a. minipage minipage0.3 1.5inp064b Fig. 18b. minipage minipage0.3 1.5inp064c Fig. 18c. minipage figure

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Exercise XIX, problem 2, p. 62

Three linear equations in general represent a single point. What are the exceptional cases?

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Exercise XIX, problem 3, p. 62

What are the equations of a plane curve $f(x, y) = 0$ in the plane $XOY$, when regarded as a curve in space? [$f(x, y) = 0$, $z = 0$.]

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Exercise XIX, problem 4, p. 62

**.** What is the meaning of a single equation $f(x, y) = 0$, considered as a locus in space of three dimensions? [All points on the surface satisfy $f(x, y) = 0$, whatever be the value of $z$. The curve $f(x, y) = 0$, $z = 0$ is the curve in which the locus cuts the plane $XOY$. The locus is the surface formed by drawing lines parallel to $OZ$ through all points of this curve. Such a surface is called a *cylinder*.]

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Exercise XIX, problem 5, p. 62

**representation of a surface on a plane. Contour Maps.** It might seem to be impossible to represent a surface adequately by a drawing on a plane; and so indeed it is: but a very fair notion of the nature of the surface may often be obtained as follows. Let the equation of the surface be $z = f(x, y)$. If we give $z$ a particular value $a$, we have an equation $f(x, y) = a$, which we may regard as determining a plane curve on the paper. We trace this curve and mark it $(a)$. Actually the curve $(a)$ is the projection on the plane $XOY$ [pg]63 of the section of the surface by the plane $z = a$. We do this for all values of $a$ (practically, of course, for a selection of values of $a$). We obtain some such figure as is shown in [fig:17]Fig. 17. It will at once suggest a contoured Ordnance Survey map: and in fact this is the principle on which such maps are constructed. The contour line $1000$ is the projection, on the plane of the sea level, of the section of the surface of the land by the plane parallel to the plane of the sea level and $1000$ ft. above it. We assume that the effects of the earth’s curvature may be neglected. %[Illustration: Fig. 17.] 17p063

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Exercise XIX, problem 6, p. 62

0.375em plus 0.75em minus 0.25emDraw a series of contour lines to illustrate the form of the surface $2z = 3xy$.

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Exercise XIX, problem 7, p. 62

**circular cones.** Take the origin of coordinates at the vertex of the cone and the axis of $z$ along the axis of the cone; and let $\alpha$ be the semi-vertical angle of the cone. The equation of the cone (which must be regarded as extending both ways from its vertex) is $x^{2} + y^{2} - z^{2}\tan^{2} \alpha = 0$.

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Exercise XIX, problem 8, p. 62

**of revolution in general.** The cone of Ex. 7 cuts $ZOX$ in two lines whose equations may be combined in the equation $x^{2} = z^{2}\tan^{2}\alpha$. That is to say, the equation of the surface generated by the revolution of the curve $y = 0$, $x^{2} = z^{2}\tan^{2}\alpha$ round the axis of $z$ is derived from the second of these equations by changing $x^{2}$ into $x^{2} + y^{2}$. Show generally that the equation of the surface generated by the revolution of the curve $y = 0$, $x = f(z)$, round the axis of $z$, is x^2 + y^2 = f(z).

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Exercise XIX, problem 9, p. 62

**in general.** A surface formed by straight lines passing through a fixed point is called a *cone*: the point is called the *vertex*. A particular case is given by the right circular cone of Ex. 7. Show that the equation of a cone whose vertex is $O$ is of the form $f(z/x, z/y) = 0$, and that any equation of this form represents a cone. [If $(x, y, z)$ lies on the cone, so must $(\lambda x, \lambda y, \lambda z)$, for any value of $\lambda$.]

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