1824.] Mr. Rerapath on the Solution of^"x — x. 325 



(i r+ , = a, b r + a r c, = a,.b t + a t c r (8) 



b r + t = b t b r + a,, d, = b,.b, + a t d T (9) 



c T+ , = c, c r + a,, d, =i c,. c t + a t d T (10) 



d r+t = b,d r + c> d, = b r d t + c,d r (11) 



By (9) or (10) a,. d t — a, d r ; therefore, when a,. = o, d r — o, 

 and by (8) or (11) at the same time b T — c r . This circumstance 

 has been noticed by Mr. Horner. If also a t = oo , d t = oc • ; for 



j = ^' = -° = . — °- = some finite positive or negative quantity. 



Supposing t = o, equas. (8) and (11) give a„ = o, d — o, and 

 b = c = 1, which brings out the obvious case 



\|/° x = x. 



Because when a r + t = o, b r+t = c r+ „ we have (9) and (10) 

 b r b, = c r c t . That is the product of any two, and, therefore, of 

 any other number of the component values of b which give a 

 resultant value for a = o is equal to the product of two, or of the 

 same number of corresponding component values of c ; conse- 

 quently if b, = oc, c, = oc ; for -'=- = - = 1. 



(Vr 



Assuming A v — —and t — (v — 1) r, we obtain by (8) 



A„ = A,_, b, + c (lT _„ r 

 or, 



A r =A„_ I 6,+d r a ( „_ 9Jr +^c i ^. s)r =A._ 1 b r + a r d r A„_ 2 + r r c (c _ 3)r 



by taking t = (v — 2) >• in (10). Again substituting v — 1 for 

 v in the preceding value of A r and putting the value of c (p _, )r 

 thus derived for its equal in the second value of A r , we shall 

 have 



A„ -p A,._, - q A,_„ = o (12) 



supposing p = b r + c, and f/ = a,. d r — b r c r . In like manner 



it is found that 



B, - p B„_, - q B, _ , = o (13) 



C,-pC r ., _ ? C._ 9 = o (14) 



».-i»D._,r!?^-.= o ( 15 ) 



where B, = 6 rr ,C„ = c,. r , and D„ = ^J-. Each equa. (12), (13), 



(14), (15), is evidently an equa. of differences of the second 

 order with respect to v, the coefficients p, q, being constant in 

 relation to this v. The solution of either of them (13), for 

 instance, by the usual methods, is 



„ _ « (S * s/^h * (5 - y/R 



