774 
MR. J. W. L. GLAISHER ON RICCATI’S 
Putting 2 p——m — 1 in this identity, we have 
/ m + 1 , (m + l)(m + 3) cdx 2 (m + l)(m + 3)(m + 5) cdx 3 „ \ aJc 
\ m + 1 1 (m+l)(m + 2) 2! (m + l)(m + 2)(m + 3) 3! °/ 
1 o?,t? 1 abt 1 a 6 * 6 ^ 
~ m + 2 AT + (m + 2)(m + 4) 2®2! + (m + 2)(m + 4) (m + 6) 213! + C ' 
The right-hand side of this equation is unaltered by a change of sign of x, and 
therefore, putting a — 1, 
/ m+1 (m + l)(m + 3) a 3 (m + l)(m + 3)(m + 5) x 3 p \ 
\ “m + l X+ (m+l)(m + 2) 2! _ (m +l)(m + 2)(m +3) 3! +&C 'P 
_/ m + 1 ( m + l)(m + 3) jc 3 (m + l)(m + 3)(m + 5) a- 3 ^ \ _ r 
~\ " t "m + i a?+ (m + l)(m + 2) 2!"*~(m + l)(m + 2)(m+~3) 3! + C ' + ' 
which is true for all values of m, except m — a negative even integer. 
Writing n in place of m+1, it follows that 
1 +x+ 
e M = 
n+ 2 
n +1 
1 —+ 
n + 2 
n+ 1 
£ 3 (w + 2)(% + 4) a 3 „ 
2! + (?t+ l)(n + 2) 3! + &C ~ 
a 3 (?i + 2)(?i + 4) ar 3 „ 
2! (n + l)(n+2) 3 + 
which is true for all values of n, except n = a negative uneven integer. Several deduc¬ 
tions from this formula are given in a paper “Generalised Form of Certain Series” 
(‘Proceedings of the London Mathematical Society,’ vol. ix., pp. 197-204, 1878). 
§ II. 
Integration of the differential equation ivhen p = an integer. Arts. 8-15. 
8. A particular integral of the equation 
dhi 2 _ Id ddu 
dfd~ a ' a ~dd did 
^ £ __ qO. V(x^+.r/i) 
for, from this value of u we find at once by differentiation 
d~u _ 9 (x+^h) 2 _ \ld 
dx 2 ~ U ~ U x*+xh ~ CtU (x 2 + xhf 
