138 CENTER OF GRAVITY 



Similarly we can find the center of gravity of a solid in three- 

 dimensional polars by supposing the polar coordinates r, 0, <f> con- 

 nected with x y y, z by the usual transformation 



x = r sin 6 cos $, y = r sin 6 sin <, z = r cos 6. 

 Using this transformation, the first of formulae (31) becomes 



CCCp (r sin cos 0) (r 2 sin 6 drd0d<f>) 

 r sin 6 cos (f> = 



CCCp(r 2 sm0drd0d<t>) 

 I I I pr s sin 2 cos $ di 



(32) 



pr 2 sin drd0d(f> 

 while similarly we have, from the remaining two formulae, 



fffpr* sin 2 sin < drdOdj 

 rsni<9sin0=^^_ _, 



/ / Ipr 2 sin 6> drdOdj* 



sin ^ cos drd0d<f> 



P r 2 s 



(33) 



(34) 



108. An exactly similar method will lead to formulae giving 

 the position of the center of gravity in any system of coordinates. 



The methods which have already been employed, or a combina- 

 tion of them, will suffice to determine any center of gravity. As 

 illustrations of the use and combination of these methods, we 

 shall find the center of gravity of the same solid figure in three 

 different ways. 



