Differential Coefficients

Excerpts

• Differential Coefficients

  Let there be any function of $x$, which we call $\phi x$, in which $x$ is increased by an increment $h$; the function then becomes

• Differential Coefficients

  Therefore to find the coefficient of $h$ in the development of $\phi(x + h)$, find $\phi(x + h) - \phi x$, divide it by $h$, and find the limit towards which it tends as $h$ is diminished.

• Differential Coefficients

  Hence the ratio of the increments of $\phi x$ and $x$, produced by changing $x$ into $x + h$, though never equal to $\phi' x$, approaches towards it as $h$ is diminished, and may be brought as near as we please to it, by sufficiently diminishing $h$.

• Differential Coefficients

  It follows, therefore, that if, instead of the full development of $\phi(x + h)$, we use only its two first terms $\phi x + \phi' x\, h$, the error thereby introduced may, by taking $h$ sufficiently small, be made as small a portion as we please of the small term $\phi' x\, h$.

Equations

• Differential Coefficients

  \phi x + \phi' x\, h + \phi'' x\, \frac{h^{2}}{2} + \phi''' x\, \frac{h^{3}}{2·3} + \etc.

  The function of x+h, expanded in powers of the increment h, has as its coefficients the function, its first, second and third differential coefficients divided by the factorials 1, 2, 2·3 (the displayed right-hand side; the chapter does not write the left-hand side).

• Differential Coefficients

  \phi' x\, h + \phi'' x\, \frac{h^{2}}{2} x + \phi''' x\, \frac{h^{3}}{2·3} + \etc.

  The increment of the function produced by changing x into x+h is the series in h whose first term is φ'x·h. Note: the chapter prints a stray letter x after h²/2 in this line; this is recorded as printed and flagged as a probable typesetting error in the book.

• Differential Coefficients

  \frac{\emph{increment of } \phi x}{\emph{increment of } x} = \phi' x + \phi'' x\, \frac{h}{2} x + \phi''' x\, \frac{h^{2}}{2·3} + \etc.

  The ratio of the increment of the function to the increment of its variable equals φ'x plus a series in h that vanishes as h diminishes. Note: the chapter prints a stray letter x after h/2 in this line; recorded as printed and flagged as a probable typesetting error in the book.

• Differential Coefficients

  h\left(\phi'' x\, \frac{1}{2} + \phi''' x\, \frac{h}{2·3} + \etc.\right)

  The part of the ratio other than its first term φ'x is written as h times a series; this is an expression for that remainder (no equality sign in the chapter).

• Differential Coefficients

  kh^{n} : lh^{n+1} + mh^{n+2} + \etc.,\ ::\ k : lh + mh^{2} + \etc.,

  The ratio of the term kh^n to the tail lh^(n+1)+mh^(n+2)+… is the same as the ratio of k to lh+mh²+…; a proportion used to show the tail can be made a small part of the term kh^n.

Problems

No exercises in this chapter.