376 
ON THE EXPANSION OF INTEGRAL FUNCTIONS IN A SERIES &C. [60 
or, expanding this last equation in powers of 8, 
(2r + l)^ r f 1 ,lS 2 ,(2r + 7) 
2 r °V2r + l + 2 2 2 2.4 
S 4 (2r + 9)(2r + ll) S 4 
2 4 + 2.4.6 2 t: ' ' 
V • • (6) ; 
or, replacing the successive terms of the form 8 q . C by their respective values, 
/a , T N f2.4 ... 2r A , 4 . 6 ... (2r + 4)_ ^ 
a *“ r - (2r + ^ | 3 . 5 _ (2r + 1) 2 r s ~ r + 3.5 ... (2 r + 3) 2 r + 2 
t (2k+2)(2k + 4)... (2r + 4k) A \ 
+ 3 . 5 ...(2r + 2k + 1) 2 r+ * s_r_2i: 
a). 
Thus, if S = /n 8 , so that M s _ r = 0, except in the particular case M = 1, 
or 
«*-i = 0, 
2s - 4k + 1 (2fc + 2) (2k + 4) 
2s 
a* 
^ = {(2s-4Z; + l) 
3 
(2&+2) (2& + 4) 
(28-2k 4-V, 
2s 
3 
5 ... (2s — 2& + 1) 
Q S -2, 
* j 
(8), 
(9), 
which of course includes the preceding case. By substituting the expanded values of 
the coefficients Q, or again, by determining the value of (1 — y) s in terms of these 
coefficients, and equating it with that given in Murphy’s Electricity, [8°. Cambridge, 1833], 
p. 10, or in a variety of other ways, a series of identical equations involving sums of 
factorials may readily be obtained. The mode of employing the general theory of the 
separation of symbols made use of in the preceding example, may easily be applied 
to the solution of analogous questions.