770 
MR. J. W. L. GLAISHER ON RICCATI’S 
(l+£)^— + 2H 2 (l+t)ty~ 2 . . . 
,(\n P(P~ 1 ) • • • {p-(«-l)} 
’ n\ 
that is, in 
If, therefore, n is uneven the coefficient of a n x n in P is zero, and if n is even the 
coefficient 
_1_ p(p ~ 
2p(2p-l) . . . [2p-{n-l)} X ^ > {\n)\ 
r 
which is the coefficient of a n x n in U. 
Since If differs from P only in having the sign of a changed, and since U is a 
function of « 3 only, it follows that P=It=U. Also, since Q, S, Y differ from P, R, U 
only in having —(_p+l) in place of p, it follows that Q=S = V. 
(2°). Suppose p a positive integer, —i, say. 
In this case the (i +1 ) th term of the series in P, including the factor x~ [ , is 
_ f /_y i(i — l)(i — 2) . . . — 1)} apd 
X i(i—%){i— t) . . . {i—i(i —!)} i\ : 
and the next term vanishes owing to the presence of the factor i—i or 0 in the 
numerator. 
For the same reason all the succeeding terms vanish until the factor i—i appears in 
the denominator also, when the zero factors cancel one another and the series 
recommences, the first term of the new series being 
i(i- 1) . . . 1.0. —1. — 2 . . . (i-2i) « 2i+ Y 2i 
i(W) • • • F-R2f-1)}.0 (2i + l)! 
l2i+l/y.»+l 
/ tty 
=gx l+1 , where g =(— ) ?+1 
2i +1 
{1.3.5 . . . (2m)} 2 ' 
