160 
MR. W. H. L. RUSSELL ON THE THEORY OF DEFINITE INTEGRALS. 
Also 
a?®-! 
1-f 
1 
1 
7 8 
3 3^ 3 3 *3 3 
+ &C. 
40 , „ , 4007 , v'3 
=—,{(j.x^—2xy^-^{[MX-2)s 2 COS— 
)>. 
Sfji.' 
40 V3x 
Whence 
TT TT 
2 
TT J TT 
’¥ 
3j«.' 
cos^ ^ cos=^ (p 
. , ,,, _ef . -v/s 
^-(i«/ 7C+2)£ 2 sm — 
(in.) 
f , 70 8® , 
coss [jb cos ^ cos <p — (tail ^-j- tan (p) 
2 'k 
3^5 3 \/3 ^|U. 
(3^fjf:-2y^'^-{3^(jt.-2)s 2 cos 
3 ^3e2 
-‘l-'y/^S |M/-j- 2 )£ 2 sin 
Again, let the symbolical equation be 
. 3 v' S ^ 
(D-l)(D-2)(D-5)M-po^(D-3)£"“M = 0, 
and let the transformed equation be 
whence 
(D— 1 )(D — 2)i;— po¥“v=(D— 1)(D — 2)V, 
■u^{I)-3')v, 0 = (D-3)V. 
Hence we find 
V=C¥, 
and 
= Cl -\-C^XZ~'^ , 
whence 
u— — 2Cia7+C2 — 2^17 ) + C 3 ( — (J!jX^ — 2x)£ 
we determine Ci, 
C 2 , C 3 according to the series we have to sum. 
x^< 1 
2 j«,^07^ 
2.3 /iV 
1 24 o 7 . 6 
-.3.1 
2 
02 
3. 4. 1.2 
2 2 
2^* 
f- &c. 2 o 7 )£''^— - 4 (ia, 7 r^+ 2 .r)£ (IV.) 
Hence we have 
TT 
1 ^;(l — y )“3 COS cos (/t/oy sin ^cos^+S^— tan 0 ) 
/o.^£ fJ- e 
By a similai’ method we find 
2 ju.^0?^ 
r' 14 
7 2"^ ' 7 9 
3.1.1 3. 4.1. -.1.2 
2.3 
24 -b&c. 
2 2 
•1 30 30 
• -44 — b 775 - 307 ) ^ (f^x^- -{- 3 x)s-^%{\.) 
fl ^ 
