222 DR. J. BOTTOMLEY ON THE EQUATIONS AND ON 



In a similar manner we might show that 



V x = COS z aV, 



V y =cos*/3V, 



and so establish the relation given in a previous paper, 



v,+v,+v x =v. 



In equation (2) suppose that the limits of the inte- 

 gration are ^ and £ a , so that the volume of the solid is 

 u{£i — &)d%dr}. Let z x and z % be the corresponding values 

 of the points on the primitive surface. Now, in integra- 

 ting with respect to z, we regard x and y as constant ; 

 hence 



f z = cos7 j(#— a) cosa+ {y—b) cos/3+ {z z —c) COS7J-, 

 f I= =cos7J (x—a) cosa-f {y — b) cos/3+ (z l — c) COS7J-; 



by subtraction we obtain 



?» — £ = 0z— ^i)cos 2 y; 

 hence, by substitution, we obtain 



jj(?a — ?0 d% drj = cos 2 7 ${z z —z x ) dx dy. 



As a particular case, consider the projected solid derived 

 from a sphere 



(x-ay+(y-by+(z-cy=r\ 



Take as primitive plane a plane passing through the centre 

 of the sphere, and as primitive axis a line perpendicular 

 to this passing through the centre of the sphere ; then the 

 equation of the projected solid will be 



n*{(x-aY+(y-by} + {z-n(l(x-a) + m{y--b))y=n*r % i (3) 



/ being written for cos «, m for cos /3, n for cos 7. 



To simplify the equation to this surface remove the 



