OF TURBINATED AND DISCOID SHELLS. 365 



or by 



-g- (mw + op) nr 

 or by 



-^ {um -\- u o) .mq . sin A . cos A . (3.) 



The MOMENTUM of the element about a plane passing througb A z, and perpendi- 

 cular to P C, (fig. 3.) is therefore represented by 



— {um -\- u o)2 7w ^ . sin A . cos^ A 



or by 



(the momentum of inertia of the plane m <^) sin A . cos^ A 0. 



Assuming then to be measured from the plane z y, tbe momentum of the element 

 m r about the plane z y vi represented by 



(momentum of inertia of elementary plane m q) sin sin A cos^ A 0, 



and the momentum of the same element about the plane z x \?> represented by 



(momentum of inertia of elementary plane m q) cos sin A cos^ A 0. 



Hence if we imagine two solids to be generated, one by the revolution of P C, with- 

 out altering its dimensions, through the angle P C Q, and the other by the revolution 

 of Q C through the same angle ; and if we take I to represent the momentum of 

 inertia of the plane P C ; then will the momentum of the first solid about the plane z y, 



be represented by 



I sin sin A cos^ A 0, 



and that of the second by 



(l + ^ A + ) sin (0 + A 0) sin A cos2 A 0. 



Now the momentum of the elementary solid P C Q evidently lies between those of 

 these elementary solids. Calling then the momentum of the whole solid, of which 

 P C Q is an element, Mj, when estimated in respect to the plane z y, we have 



I sin cos2 A sin A < ^' A + &c. . . . < I sin cos^ A sin A + &c. 



.-. I sin cos2 A < '^ . -^^ + &c, . . . < I sin cos2 A + &c. 

 And this is true for all values of A 0. 



.•.-^=Ism0. 

 Similarly calling Mg the moment of the whole solid about the plane z x 



-^^=Icos0 



.-.Ml =J^lsmQdQ 



