366 THE REV. H. MOSELEY ON THE GEOMETRICAL FORMS 



Mg =y*I COS ^ 

 .*. distance of centre of gravity from plane zy —~~f ^^% > • • (4.) 



.*. distance of centre of gravity irom plane %x •=. r^^% — • • \^-) 



The generating curve has here been supposed to revolve about the axis A z^ other- 

 wise retaining its position upon it. 



If we suppose P C to slide along the axis as it revolves (fig. 4.), the moment of the 

 elementary solid P C Q about A z, and therefore the moments M^ and Mg of the whole 

 solid about the planes z x and zy will remain unaltered. 



Another dimension will however now have become necessary to determine the 

 position of the centre of gravity ; viz. its distance from a given point in the axis A z^ 

 measured along that axis. 



Let V (fig. 4.) be the point where the generating curve intersects the axis Kz\ :. by 

 equation (3.) the momentum of the element yn r (fig. 5.) about a plane passing through 

 V perpendicular to the axis A 2 is represented by 



-QVu{um-\-u6) mq . sin A . cos A ; 



and assuming V i< = x, and u m (figs. 4 and 5.) = y, the momentum of the whole ele- 

 mentary solid generated by the revolution of P C through the angle A is repre- 

 sented by 



J J X y dx dy .f^wvC^Q . cos A 0. 



And representing /yj:^ 3/ dx dy by L, and reasoning as before, the moment of the 

 whole solid about a plane perpendicular to A 2 passing through V is represented by 



f\.d(d. 



And if A V = ^, the distance of the centre of gravity from A measured along the axis 

 is represented by 



" ^fWd% 



, r\^d% (tK\ 



To find the Centre of Gravity of a ConcJioidal Surface. 



Imagine the generating curve to describe, without altering its dimensions, an angle 



about the axis A z (fig. 3.), such that the circular arc described on this supposition 



by the point P may equal the element P Q of the length of the curve or A S ; this 



AS 

 angle will be represented by-^^- Moreover, the moment of the elementary surface 



thus generated about the plane z y will be represented by 



^ y^ sin -w- d s, 



