77 2 
MR. J. W. L. GLA.ISHER ON - RICCATI’S 
and the expression in brackets is equal to the coefficient of t m in 
( 1 +^ *— (m —i) t (i+ ty -*+ 1 »V ( i+• • •+(—)* ^ m ^ " V (1+ *)*-»- 
t I m 
1+t 
=(i+ty~ m . 
The coefficient of t m in the expansion of (1 is equal to 
(p—m)(p—m — 1) . . . (p — 2m + l) 
m! 
and therefore the coefficient of ci^x* 1 ™ in xP.Tie ^ is equal to 
1 (p—m)(p — m— 1) . . . (p — 2m+l) 
(2m)! (p—i)(p—i) • • • (p-m + i) 
which is the coefficient of a 2m x 2m in xP. P when the factors p(p — l)(p — 2) . . . (p — m-j-1) 
are divided out from the numerator and denominator. 
Similarly, if w=2m+l, the coefficient of a 2 " l+1 x 2m+l in x p XJe~ aj: is found to be equal to 
1 (p—m — l)(p—m — 2) . . . (p — 2m) 
(2m + 1)! (p-i)(p—i) ■ ■ • (p—m> + i) 
which is the coefficient of a 2m+1 x Zm+1 in jc^.P when the factors p(p —l) . . . (p— m ) are 
divided out from the numerator and denominator. 
Thus, if p=i, the coefficients of the terms involving x l+l , x t+2 , . . . x 2t in the series 
in P vanish, and we have U=P'-fpQ. 
(3°) If p = a negative integer =— i — 1, then Q and S involve zero terms, and, 
denoting by Q' and S' the values of Q and S when the series are supposed to terminate 
at the term preceding the first term involving a zero factor in the numerator, Y, Q, 
and S become equal to U, P, and II when p is put equal to i, that is, to the U, P, 
and It of (2°) and vice versd. In this case, therefore, 
and 
Q = Q' + 0P=V = S = S'-</R, 
P=lt=u. 
The relations between the particular integrals in the three cases are therefore 
(1°) p not = an integer, 
P=P=U, 
Q=S=V. 
