400 
MR. W. D. RIVER OR CERTAIR DEFIRITE IRTEGRALS’ 
The expression upon which the operator has to operate mav he found by expressing 
Q^, &c.. by Laplace’s Formula, and using the series D: it is as follows :— 
9-y 
y f - " -, {(€i a +Vi a )(p> a )<-j{£i a -Vi a )[p> a ll¥ a 
x 
2-y. 
/3! q-/3l 
X .+fllr—J fli ((^+>73 a+ 0(^ «+/3)<—i(4 a+ '-% a+ ')[L 
—a.—? 
If we omit irrelevant terms this becomes 
2 8 p! g'! r! 
a! /3! « + /3! p—ocl q—/3l r—0.-/3 
x {(p> a )«(q> P)i( r > a +/3), 
~{p, «)[(/, /3][?’, a+/3] c — \_p, cP] a {q, P) b [r, « + /3] c — [p> ci] a [q, fi] b (r, a + /3) 
x{£i a &v3 a+ *+vi a v/& + *)zr a *p~% r ~ a ~* 
The result of the operation is accordingly 
2 q\ r \ K {(p, a) a (q, p) 6 (r,a+P) c — &c. . . . } 
Turning now to the integral J JQ y ,Q ? Q^S, let us expand each of the Q’s according to 
Laplace’s Formula. We thus find for the general term 
2i(*+P) s p\ p\ q \ q \ r\ r\ 
p + o\ p — o\ q + /3l q—f3l r + o + /3\ r — a — /3l 
x fj{(,b rj )a{p, «)+[>, “1 [p, cP]}{{q, f3) b (q , P) + q, (P] b [cq, /3]}{(r, a +/3) c (r, a + /3) 
fi- jr, a+/3] c [r, « + /3]}<^S 
Now, by the formula of § 10, -when we come to equate the two results just found, we 
shall get 
22(a+/3)-l p] g \ r \ 
p + ol q + /3l r—o—[3 
jj(^ a )(?> ^)( r ’ a + /^ S = ^ 
47tR 2 i! 2 XI 2ft\ 2vl 
+T; \! /jl ! v \ 
w 
or finally 
{{(>, a )(P «+/3)c?S=^ 
47rR 2 i\ 2X! 2/d 2 v\p-\-o\ g + /3! r—«—/ 3! 
i+1! 
X! /i! v! j>! S'! r! 
W. . (29) 
where W is defined above (28) as the coefficient of a term in a certain product, or 
as a series whose terms depend upon a single parameter. 
