Centre of Gravity in particular Bodies. 65 



lhat is, 



h : e :: dx : \/dx*-{-dy* = r . 



n> 



We have also, on account of the similar triangles, ACN and 

 APM, 



AC : CAT : : AP : PM, 



that is, 



ax 

 h : a : : x : y = r- $ 



therefore 



2 TT (/i a?) a/ -v/dz* +dy* 



becomes f 2 w x to art x a * x e dar - 



j ^ 7f x {ft #; x --X -- , 



-sr 



of which the integral is 



r-^-> or '-^(3 ft- 2*; 

 Now, the surface of the portion AM'LMA, or tf, is equal to Geom. 

 X circum PM, and we have 



n-p 



AC : AP :: AN : AM, = 



therefore, 



AP X AJV . a; x 



therefore the distance of the centre of gravity of the surface 

 AMLMA, from the point C is 



' or 





Therefore, when a? = ft, or AP = ^C, we shall have the distance 

 CG, of the centre of gravity of the whole curved surface of the cone 



that is, the centre of gravity is found in the same manner as 

 that of the surface of the triangle ANN 9 . 



