208 Mr. Airy on the Figure of a Fluid Mass 
2 2 
(9.) The first part of a sin. @ is 5 sin. @, or 
2sin. 6 {c’ cos.? @ + 2ac cos. p.sin.6.cos. 6 + a cos.’ d.sin.’ 6}. 
The general integral with respect to @ is 
2c 4ac : 2a° : 
Fagn cos.° 6 + =z COs: - sin.’ 6— “3 COS.” p cos. 6 (sin.? 6+ 2). 
Upon giving to @ the value ©, it will be found that each term 
involves an odd power of cos. ¢, divided by an odd power of 
V{acos. p\? + e—F x’ (c)\*}. 
Now the sign with which this radical is taken, can never alter, 
for the radical never becomes = 0; if then we put 7+ for ¢, 
we shall have the same expression with a different sign. Hence, on 
performing the next integration, these terms will disappear by 
the opposition of signs, and we may, therefore, reject them at 
once. On giving to @ the value 0, the expression becomes - 
the value of the integral is therefore 
20 4a : 
ce + a cos. d. 
(10.) Integrating this with respect to ¢ through a circum- 
ference, we have, for the first part of the required expression, 
+e). 
(11.) The other part to be integrated is 
sin. 6.{x(c—v cos. 0)—x (c)}. 
If we expand y(c—vcos.@) by Taylor’s Series, the m term 
a” .x(c) 
7 dc™ m m ; m = 
will be Ma el Pikes \™ sin. 6 
2)" d™, x (c) 
Ta... Mm’ dem (€COS. 8 +4 Cos. > sin. 8)”.cos. 0)". sin. 6. 
