﻿190 ON SOME APPLICATIONS OF THE METHOD 



continued addition, A t , A 2 A n . And, in this manner, in the case supposed, that 



is, of J 3 , J\ J 3 , being insensible, we shall obtain the total change of A between 



the limits t = and t = n r. 



Before correcting this process, in order to take account of the third and higher orders 

 of differences, it will be well to observe, that the effect of increasing n, or the number 

 of intervals into which the whole time is divided, is to diminish J 2 , z/ 3 , &c, relatively 

 to z/ 1 , nearly in the proportion of the corresponding powers of n. Thus, when the 

 number of intervals is doubled, J' has about one half its previous value, z/ 2 one quarter, 

 z/ 3 one eighth, and so on. The intervals, therefore, can always be diminished so as to 

 make z/ 3 insensible compared with z/'. But it is better, in most cases, to allow z/ 3 to 

 be a small quantity, and to correct A for its influence, and for that of higher orders of 

 differences. 



Arranging the numerical values of the first differential coefficients * of A, represented 

 by V, with their differences, D\ Z) 2 , &c, in vertical columns, as has already been done 

 with A, we have, taking for convenience t for the unit of time, or t= 1 : — 



t = — | V_ 3 



(2) =+i V+l 2 D-i Dlj 



^+1 -p Dl l 





D+l 



If we put a, b, c, Sec, for the first, second, &c. differential coefficients of A , we 

 shall have at any time t = n z = », 



(3) A n = A -\-at + ^ 2 e+, &c. 



(4) F.= a+»«+i^+,te. 



By making successively t=£, f—f, &c, in the expression for V n , we obtain by 

 elimination a, b, c, &c, in terms of V and numerical coefficients, and these substituted 

 in the value of A n give 



(5) JI=A„ + 1 -A„= V„ +i + ^ Dl^-^y Dl_i + ^y^ Dl^-, &c. 

 And thence 



(6) A„ — A B =mm of all the quantities Fi F n _,-f ^ (D\_, — Z>1,) —-gihs (-D 3 -|— -D 3 -' ) + 



rtVftw (^-| — D_ s ) — , &c. 



* The usual notation for first and second differential coefficients is not employed in this paper, from its 

 resemblance to that here used for finite differences. 



