EQUATIONS IN FINITE DIFFERENCES. 101 



and for its expansion, 



U., ^ A + By- + C, {By-'Y + cABy-y &C., 

 the quantities A, B, y, c,, c,, &e. being the same as before. 



Before leaving this example, we may observe, that the most rapid 

 way of finding A, is this, let log. fi = h, i. e. a'' = (i; and since 



A = ^. «"*, .•. A = h + log. A ; take b then as a first approximation, 



take its log. and add to h for a second; take the log. of the second 

 approximation, and add to h for a third; and when b>\, we shall get 

 a very converging series of values for A. 



If we had applied Lagrange's theorem in this case to the equation 

 A = a'^~\ we should have A expressed in a divergent series; it is 

 necessary therefore to limit the announcement, that this general series 

 gives the least root, to the case of real roots, for when there are some 

 imaginary, we see that it may express one of these instead of tlie 

 least real root. 



IV. By a similar process we easily obtain 



m- (Bt, 



i' - 1' m 



m" (in- + 3^) {Bm') 



sm.msm.7)ism....7nsin.miL, = Bm' . -'^ ^- 



m' - 1 1.2.3 



&c. 



(m' - 1)(»«' - 1) ■ 1 .2.3.4 .5 



Isin.-lsin.-lsin.- ^sm.-^u, = Bm- - , '"' { Bm-f 



m m m m " {itr — 1) 1.2.3 



vi' jm- + 3') {Bnr'J 



^ {nf - 1) (»«' - 1) ■ 1.2.3.4.5' 



the value of B being the same in both, and found as before. 



CASE OF FAILURE OF THE GENERAL SOLUTION. 



When the equations (^ {A) = A, (p' (A) = 1 are simultaneously true, 

 the terms in the expansion of u^ become infinite, as before remarked, 

 we shall therefore give a solution in a different form, for this case. 



