I 



OF ARTS AND SCIENCES. 341 



which is equivalent to the condition that trr shall be the ordinate par- 

 allel to a for any assumed point of the surface. SnT^),'(>./da;d?/, 

 being always the elementary parallelopiped four of whose edges are 

 parallel and equal to «Shic, will therefore represent the element of the 

 volume which we are seeking. By easy transformations, we find 



Dj.Dj, F= 8or(^>/(>2' = Sh-u;Chi;eSr/:(ja' 



= Sh-icChxS«^j' i=: aAcSh'^aiClia;, 



whence 



r= ahc C Csii^xChx = $ '^[sh^xl^ 



and if we take zero as the lower limit, 



ahc I 



K=^f[8..v], 



which gives an expression for the volume contained between the sur- 

 face and the" elliptical right cylinder whose axis is a and whose base is 

 the section determined by the vector oChx. 



Cambridge, June 1, 1877. 



