220 DR. J. BOTTOMLEY ON THE EQUATIONS AND ON 



also we have 



#=KG, 



y=HG, 



* = PG. 



Let f, 77, £ denote the coordinates of the corresponding 

 point on the derived solid, thus 



? = KG, 

 *7 = HG, 

 J = LG = PD cos DPF cos 7. 



Hence we obtain 



%=cosy{(x—a) cosa+(y— b) eos/3+(z — c) COS7}. 

 Hence the equation to the derived solid will be 

 fit * ? (g-a) cos *+ (77-6) cos /3 \ 



f V> V >COS Z ry ~ COS 7 +^-0. 



If z be given as an explicit function of x and y, say 



*=<£(#, y), 



then the derived surface will be 



?=cos7{(|— a) cos u + (i}—b) cos/3— ccos7} + cos*7<£(£, rj), 



Hence the equation of the derived solid will be of the 

 same degree as the primitive solid. 



In the figure we have taken as primitive plane, a plane 

 intersecting the solid, and its projection as falling on the 

 plane x, y ; in this case that portion of the solid lying 

 below the primitive plane will, when projected, lie below 

 the plane x, y. We might have taken as the primitive 

 plane the tangent plane at the lower extremity of the 

 primitive axis ; in this case the whole of the solid, when 

 projected, would be above the plane x> y. 



