1826.] Solutions of the Function ^' x. 243 



A 



Equation (32) gives, putting — r^ = s 



^-x = <p-'\- - ...._!_ (34) 



the fraction extending to z terms. Now if <p x be neglected in 

 the last term, the law of the numerators in the converging series 

 is well known to be 



N, = 5N_, -N._, 



and the same formula serves for the denominators also. Where- 

 fore the roots of 



y- - sy + 1 = 

 being R and r, we have 



K = a 11= + /3 r" 

 D, = a, R= + &, r- 



and the values of the constants being determined from the 

 known values of the fractions when z = and z = — 1. viz 



1 



J- and -, the general value of the fraction is readily found to be 



N^ _ R= -r-- 



D= ~ R- + 1 _ r= + ' 



Whence, restoring ip x to its place. 



These formulae accommodate themselves to all conditions, by- 

 making s, that is, — , = 2 cot. 9, when the lower sign appears, 



or = 2 cosec. ^, or 2 cos. 9 with the upper sign ; the values of 

 R, r, being in the first case cot. s. S and — tan ^ 9 ; in the 

 se cond, cot -i- 5 and tan. J- 9; and in the third, cos. 9 + sin. 9 



^y — I- The last alone appertains to periodical functions. 

 Hence the solution of^/"*' = .r, is 



^ ^ 6in. (j + 1J& _5in.s&.*a;> • • • • V.'^OJ 



which is reducible to 



sin. (« — !)& 



1^2 COS. a i -^ ^xj 



sin. I ^ 



(36)* 



which, when z = I, gives 



4. X = ^-' ^2 cos. li: - <p xj" ' (37) 



This solution, while, through the efficacy of the arbitrary 

 function, it possesses all the generality of formula (28), has the 

 great advantage of being convenient for differential purposes. 

 All the partial solutions of functional equations which I have 

 ever njet with are contained in it ; and being quite as simple in 



ii2 



