278 Mr. S. Butterworth on a Method for 



or a possible solution of (4) is 



y = lt x=0 (A + B^{X^G(« + \, /3+X, y + K, 1 + X, «)}, 



. . . (5) 



When 7 is anything but a negative integer or zero (5) 

 reduces to 



y=BF(«, /3, 7 , *j, 



the usual first solution of (4), but when y is a negative 

 integer (say — n), all the terms in 



G(« + X, /3 + X, y + \, 1 + X, *) 



beyond the (n+l)st contain X as a factor in the denomi- 

 nator, so that both the particular solutions implied in (5) 

 will remain finite when X vanishes. Hecce, in this case 

 (5) yields the complete primitive. 



Moreover, if 7 is a positive integer greater than unity, 

 the substitution y=uar~y applied to (4) will convert that 

 equation into one of the same type in which the new 7 is 

 a negative integer or zero. 



Finally, if 7 is unity, the complete primitive of (4) is 



y = ltx-o(A + B^){^G(« + X, /3 + X, 1 + X, 1 + X, at)}. 



V OX/ # m m ( 5a) 



4. Using (5) and (5 a) to obtain the primitives of (3), we 

 find that 



^^=^x=0^(A + B^{X,t' X - 2 G(X~i X-i, X-l, X + l, *)} . 

 = ltA = o^(A 1 + B 1 ^-){^ 1 X G(X + f, X + l, X + l, X + l, x x )} . 

 = lt^o^--(A 2 + B 2 ^{ t r 2 x G(^-i X-i.X + 1, X + l, x 2 )}. (66) 

 = ltx = o^A 3 + B 3 | x ){X^ x -- 2 G(X-i X-i X-l, X + l, 4)} (60) 



=lt A=0 ^A 4 + B 4 ^-){^ x G(X + f ? X-i,X+l,X + l, *J}, . (6d) 

 in which the constants A„, B n have still to be determined. 



