266 
Mr. Woodhouse on the Integration 
and A»=a 2m + I . (i+e') 2 ”'+' ./(i— «■ . +— r, 
— a 2m+1 . (i-\-e') 2m+l .JR 2 m+ 1 d 9 , which, by what has preceded, 
can always be determined by means of JRd 9 ,J-~. To determine B, 
a 2m+l . (i+e') 2m + l . R 2m+I . cos. 9 = A . cos. 9 + B . (cos. 0 )*+ 
&c. 
a ™+ ’ . ( 1 +e') 2m + i ,fR 2m + ' . cos. 6 . di=A . sin. $ + + 
— + &c. 
2 1 
making 0 = 7r, sin. 0, sin. 20, &c. = o ; 
consequently, (a . (i+e')) 2 ”^ 1 ./R 2 ™+ 1 . cos. 0 . J 0 . = B-j ; 
which integral can always be expressed by finite algebraic forms, 
and the integrals of R<i0,-~- ; for, putting x=cos. y , 
cos. 0=2x 2 — l, we have R 2m+I . cos. 0 . d 9 =(2^ a — ■ i).R 2m+I J0 
= 2jr*. R 2m + 1 . d& — R 2WZ + 1 <^ 0 , 
and, generally, to determine the coefficient (N) belonging to cos. 720 , 
<J+.(i+/)} 2ni+I .R 2m+I cos. 720 =Acos. 720 +Bcos. 720 . cos,0+&c, 
+ N cos. 720 }*+N<. cos. ?z0 . cos. [n -j- 1 ) 9 -f- &c. 
multiply each side by d 9 , and integrate, making 0=7r ; then, since 
f cos. m 9 . cos. (mz±zp)QdQ=f± ; cos. (2 nv=±zp)^-\-J^ cos. 
— J-sin. 
;■£. Sin, 
2m±.p 
(2 m ±p) it 
2tn±p 
4. i sl ? -- ~ == o, and, since 
NS 
2 
/N .(cos. 720) »^O=i/N^0. (cos. 2720 + 1) =-7^- . N sin. 2720 + 
= N^ = {a. (i+^ / )} 2m + I . /R 2m + 1 d 0 . cos. 720 , and the integral 
of R 2m+1 cos. 720 . dl can always be determined in terms consisting 
of finite algebraic quantities, and of the integrals fRdQ, for. 
