210 Mr. Arry on the Figure of a Fluid Mass 
(13.) Multiplying this by cos. 9)’, and integrating with respect 
to @ through a circumference, since the definite integral of cos. |’ 
ST Pee oe Sel 
Ee Se Bee wwii’ 27, we have 
p-p—2....4.2 
- 5 5 2m—p—1.2m—p—3....3.1 
cos. d\?. Saleh" LE) SSS SE 
Solfo 08. St ae oN 7 2m+1.2m—1....p+3.p+1 
and the definite integral of the p +1)\" term of 
m—1,...m—-p+t 
1.2....p 
(c cos. 0+4@ Cos. @ sin. 0)”. cos. 6)”. sm. 0 1s 27 x 
2m — p—1 .2m—p— 3. 301 
2 Nie wn oe pP+3.p+i1 
Hence, we have the following rules for finding the value of 
Sofy Sim. 9 (x (¢-v cos. #) — x (c) ). 
1. Expand c+ a)”, and select those terms 
(0 PE ro) 
in which the index of a is even or 0. 
2. Multiply the term involving a’ by 
2m—p—1.2m—p—3....3.1 
2m+1.2m—1......p+3.p+1 
3. Collect the terms for different values of p with the same 
Tale dx (0) 
value of m, and multiply their sum by ae ee 
x ct Bear. 
x 
4. Collect the series found by giving to m the values 1, 2, 3, &c. 
call the sum ¥(c): then 27.W (c) is the integral required. 
(14.) The expression for ¥ (c) may be put under the follow- 
ing form, which will probably be found more convenient, ; 
~ — 2c, x’(c) 20? x" (c) 2c\° 
tO) = ali d.>: 6) me heme OS: 
(Bi sieessl:. 1 5.7 
