OP TURBINATED AND DISCOID SHELLS. 367 



where y is any ordinate of the generating curve at right angles to the axis A x, and 5 



is taken to represent the length of the generating curve. Assuming ih^njy'^ ds or 



the moment of inertia of the perimeter of the curve to be represented by K, the mo- 

 ment of tliis imaginary surface about the plane z y \& represented by 



^ sin A S, 



and, similarly, that about the plane z x\s represented by 



^ cos A S. 



Conceiving now a similar elementary surface to be generated by the curve Q C 

 without changing its dimensions, the two moments of that surface will be repre- 

 sented by 



|±4|cos(0 + A0)AS 

 and 



|^sin(e + A0)AS. 



Moreover, the moment of the actual element of the conchoidal surface evidently lies 



between the moments of these imaginary elements ; as before, therefore, the whole 



moments of the conchoidal surface about the planes z x and z y, being represented 



by N^ and N2, 



tZNi _ K . c?^ 



d® — R^^^^ d® 



Similarly, if N3 represent the moment of the surface about a vertical plane perpendi- 

 cular to the axis A z, and passing through the point V; and if x be an abscissa to any 

 point of the generating curve measured along the axis from that point ; and if H re- 

 present the integral / .r ?/ d s, taken in respect to the whole perimeter of the genera- 

 ting curve ; then 



c?N3 _ H d^ 



d® -~ R ' d® 



Tlie distances of the centre of gravity from the planes zy, z x, and x y, are then re- 

 spectively 



^ sin (/ S 

 § cos f/ S 



""n • • • • ^^■> 



/ 



/ 



/ 



and y-H^g 





