Application of the Theorem for Total Differentials to the Determination of Total Resultant Errors

Excerpts

• Application of the Theorem for Total Differentials to the Determination of Total Resultant Errors

  For if any result be obtained from a set of *data*, no one of which is exactly correct, the error in the result would be a very complicated function of the errors in the *data*, if the latter were considerable.

• Application of the Theorem for Total Differentials to the Determination of Total Resultant Errors

  When they are small, the error in the results is very nearly the sum of the errors which would arise from the error in each *datum*, if all the others were correct.

• Application of the Theorem for Total Differentials to the Determination of Total Resultant Errors

  Next suppose only the second error, and then only the third to exist, and calculate the effect of each separately, all which may be done by simple formulæ.

• Application of the Theorem for Total Differentials to the Determination of Total Resultant Errors

  The effect of all the errors will then be the sum of the effects of each separate error, at least with sufficient accuracy for practical purposes.

• Application of the Theorem for Total Differentials to the Determination of Total Resultant Errors

  The formulæ employed, like the equations in 28, are not actually true in any case, but approach more near to the truth as the errors are diminished.

Equations

• Application of the Theorem for Total Differentials to the Determination of Total Resultant Errors

  \dfrac{dz}{dp}\, dp + \dfrac{dz}{dq}\, dq + \dfrac{dz}{dr}\, dr + \dfrac{dz}{ds}\, ds

  The function z, when its data p, q, r, s each carry a small error dp, dq, dr, ds, is corrected very nearly by increasing z by the sum of the separate effects of each error, taken one at a time.

Problems

No exercises in this chapter.