340 PROCEEDINGS OF THE AMERICAN ACADEMY 



The ellipsoidal volume will therefore be 



F= $ — - — ■ I (sin X — m sin x cos x") 



X 



= $ £^ fcos a; 4- "I sill- xl ''°. (55) 



If m be determined by the condition m = cos x, where x is the supe- 

 rior limit of integration, then cb- = c sin x at that limit, showing that 

 the volume obtained will be limited by a plane perpendicular to a ; 

 and if the lower limit is 0, we have for the volume of a segment : — 



V^-^—ll — cos a: — ^ cos a; sin" a: . (56) 



30°. Similarly, the parted hyperboloid gives 



(/j = «Sha: -|- cfCha;, 



q'o = (T'Shx, 



"^QQ'i'j'z = abcShx, 



mSuQ^Q'^ = »iS.a(«Shx -j- oChx)o'Shx 



= mabcShxChx ; 

 and 



SnTQ\Q'^ = Spp'ip'j — mSuQ^o'^, 



the origin being changed to any assumed point on « ; whence easily 

 V=^ -^ j (Shx — mShxCha:) 



X 



z=^"Mchx — '!L Sh^a:] \ (57) 



Determining m by the condition m = Clix, where x is the superior 

 and the lower limit, this formula becomes 



V=^— fchx — I- Cha: Sh^ar — l] ; (58) 



and gives a portion of the interior volume, determined by a plane sec- 

 tion perpendicular to a. 



The unparted hyperboloid can be treated in a somewhat similar 

 manner. In this case, we may assume 



vT = Q — crCho; := «Sha;, 



