3HE. W. H. L. EUSSELL ON THE CALCULUS OF FUNCTIONS. 
269 
the equation becomes 
Let 
and the equation may then be written 
(ai+^)(a2 + ^)(«3 + r) . . . 
, r(«j +2')r(«2+^)r(a3+2’) . . . ^ g+i r(flti+g+ i)r(«2+.g+ i )r(«3+.g’ + 1) • » » 
r (^1 + 2') r(/32 + ^) r(/33 + ^) 
r(/3] + 2+ i)r(/32 +2 + i)r(/33+2 + 1) . . . 
- r(«j+ 2 )r(ao+ 2 )r(« 3 + 2 ) . . . 
Hence 
zj zU- (/3 i + 2)(/32 + 2){/33 + 2) ... + ^r) (/Sj + 0 + 1) (/S 2 + 2 ) (/32 + 2 + 1 ) . . . 
C T{B, + z)T{^,+z)... 
r(«i+2)r(a2 + ^) . . • 
Yu. is a rational function of (2:), and may therefore be decomposed into a series of 
terms of the form 
1 . 1 . 1 . 
Yu. 
Hence 
<PUz 
' h^ + k^z'^ + k<^z' h^-\- k^z 
(«i +2) (a2 + 2)(«3 + 2) 
^1 + ^,2 {Bi + z){B 2 + z){B 3 + 2) 
1 , («1 + -^)(«2 + '^)(“3 + '^) 
7^2 F k^z 
1 
fa 
Ol + ^){^2 + (^3 + 
(«1 4-2')(«2 + -g)(«3 + 'g) 
^3 + ^ 3 ^ (.^ 1 + i^2 + ■ 2 ^) (^3 + -S') 
^ r(/3i +2)r(/32+2)r(/33q-2) . . . 
a® r(aj+ 2 )r(« 2 + 2 )r(« 3 + 2 ) . . . 
1 
/q + X:,(2+ 1) 
1 
/t^ + k^iz+l) 
1 
&C. 
&C. 
^^3 + + 
.+ !)+ '^C. 
We may obtain a multiple integral which shall be equivalent to any of the above 
series, by remembering that 
a(«4- l)(a + 2) . . . (ci + n — l) 
also 
B{B + m + 2)...{B+n 
1 
/3(/3 + 1)(/3 + 2)...(|3+h 
— ^ — f 
-i)-r«r(/ 3 -a)J/ 
r^-r“ 
-1)~ 2. J-0 
dv 
(1 
and summing accordingly. We may hence immediately deduce the value of 
It is evident that the functional equation 
^{x) 
(p{a-\-bx-{-C.T^) — Ci 
(a + cx -{- cx^) 
2 0 
:2. ®(^)=F(.r), 
MDCCCLXII. 
