226 UNIVERSITY OF COLORADO STUDIES 



Let the segment be turned round an axis through O, perpen- 

 dicular to the plane of the paper, through a small Z a, bringing the 

 Center of Mass to G ' , and giving the section A ' CB ' , then Z AM A ' = 

 AOA'=GOG'=a and the result is the same as if the wedse whose 

 section is AM A' were transferred on to the other half of the face of 

 the segment giving the section BMB' and carrying its Center of 

 Mass from g io g' , then 



GG ' wedge 

 gg' segment 



Now the radius of the wedge is in the limit r sin 9, 20 being the 

 angle of the segment, .-. by (11) gg'=2 — ^^ ■ , the volume of 



4 a 



the wedge is - 7r(r sin ^)^X — =^ar^ sin^ 6, and the volume of the 



segment is found by considering it as the difference between the 

 sector and cone having their vertices at O, to be — 



^^ (l_cos^)X2+cos 0) 



3 

 GG ' wedge 



5 becomes in the limit — 



gg' segment 

 OGXa |a/'^ sin^^ 



t'^^ ^^^ ^~^\l-cos^)X2 + cos^) 



4(1— cos 6')\2+cos 0) 

 3/'(l+cos 0y 



(13) 

 2+cos ^ ^ 



If h is the height of the segment this result may be written 



. ^ ^ ^r. S(2a—hy 



in the form UG=^ —-- ^4.^ 



4(3a-A) ^^^> 



