﻿OF MECHANICAL QUADRATURES. 191 



It will be noticed, on reference to (2), that the differences Dl-\, D',,-*, &.C., are on the 

 same horizontal line with F„ + >. The intervals may usually be taken small enough to 

 enable the computer to neglect differences of V of a higher order than the second. 

 Then the total variation of A between the limits for which V is known is the sum of 

 all the values of F-f ^ (D»+j — #-§)' 



Where the intermediate values of A are required, d\, J\, J\ //„'_„ are first 



found, and these, by continual addition, give A„ A. 2 A n \ and in this case (6) serves 



to test the correctness of A n — A„, which should be the same by both processes. 



If, instead of the first, we have the second differential coefficients of A, denoted by 



F_ s , F_ F F n , for t — — 2, I = — 1 , t = t = n, these may be arranged 



as before, with their differences d\ d', &c. : — 



/ = — 2 F_ , 



dl, 

 = —1 F_, dl 2 



d'_, d 3 _ 2 



= F dl t rii, ( 7) 



<*J ii, 



= 1 P, d\ 



d\ 

 = 2 F a 



By a process similar to that used in finding z/„', we obtain 



Al = F„+ , + A *l - ?io *i_i + **&* *:_.—, *C (8) 



^J— j; = sum of all the quantities F l F„ + T ' 5 « — <) — zhi d l-> — d*-i) + *Afew {dl-.-dl,)-, &c; (9) 



and ^ — A= sum °f a ^ trie quantities z/ ' <4!_i- 



When, therefore, ^ and J\ are known, with the values of F for each interval, we 



obtain by (8) d\, J\ ^I_ 25 thence by successive additions J' , A\ ^,!_„ 



and finally A u , A x A,. 



From the values of F„, F { F H _ lf t' ie nrst differential coefficients of A may 



also be found by using (5), and considering .Fas the first differential coefficient of V \ 

 V_ i being first known, by (5) we find V + i, V +? V+n-lr 



It will be observed, that when V, V n _ x are given, and it is required to find A„ 



for any time, we must first know A , and then we have A n = (A n — A ) -\- A , (A n — A u ) 



being derived from (6). But when F x F n _ l are known, and it is required to find 



A n , we must know z/ ' as well as A . 



The following expressions derived from (3) and (4) give J l in terms of F and its 

 differences, when l\ or V x are known : — 



4 = V + i F + £ d\ - s-V dl + ^ d\ - & d\ + T J3^ dl - riff, d\ +, &c. ; ( io ) 



A= v l + A < - ,*, dl, + „*§„ di, - &c. ; < u ) 



which will be afterwards referred to. 



