MR. W. H. L. RUSSELL ON THE THEORY OF DEFINITE INTEGRALS. 
161 
whence we have 
j j dddv v{ 1 —v)i cos 0 cos {[j^v sin 9 cos ^+3^— tan 9) 
v'o c/ — — 
2 
It is to be particularly remarked, that we may in many cases simplify the final 
results, which we obtain by means of these summations, by the use of the theorem 
r- r- r — r — =( 25 r) 2 n~^. 
Again, let (D— 1)(D — 3)(D— 5)w — 2)(D— 4 )£®m=0, 
and assume as the transformed equation 
(D — 1 )?; — = 0. 
Then m=(D — 2)(D — 4)i; 
0=(D — 2)(D — 4)V, 
whence Y — Ax^ , 
and v = Ci(fj(jX^ — x)s'^''-\-C 2 X^-\-C 3 X, 
whence u = C^ifjuV — 3f^x^ + 3^) Cga?® + C 3 X, 
where the constants must be determined by comparison of this expression with the 
series to be summed. Thus we have 
2.4 
3 . 5 ^ 
2. 3.4. 5 jx' 
Q 24 4 
j^ + &c.|=^,£''(,<.V-3fox»+3^)— . . (VI. 
f.f: 
vzh^^'^^dvdz =—^ - 3^ + 3) — j^+^- 
3. 4. 5. 6 
Hence 
Moreover we shall find 
gf, . 2.5 , 2. 3. 5. 6 fAV ] 30, , , , , . . x . 10/ ^ , 3^ 120 , 
■3^ |l +46 /‘''^4*4.5.6,7 ■ 1 .2 “I" ^O.j — +4^)+ 
whence 
1 1 +?)-?• 
We shall also find 
^-{l+|;..x+f^-^+|§;|-J^+&c.}~(f<,:c»-2^)r+^{^.t“+2*). (VIII.) 
Hence 
z;(l— 2)£'^+^(^+2). 
These three last integrals can be obtained by ordinary integration. I have intro- 
duced them here partly for the sake of system, and partly because we shall require 
the series which they represent on other occasions. 
