SOME PROPERTIES OF PROJECTED SOLIDS. 219 



Let ABC be the primitive plane, which is fixed in the 

 solid, and DE an axis perpendicular to this plane, and 

 which may he called the primitive axis. Let P be a point 

 situated on the intersection of the solid by a plane parallel 

 to the primitive plane. Draw PF perpendicular to that 

 plane. 



In deducing the equations to the derived solid we might 

 have taken a system of axes in the primitive solid as 

 follows : take DE as one axis, and two straight lines per- 

 pendicular to this, lying in the plane ABC and passing 

 through the point D, as the other two axes. Then, if the 

 equation to the solid be given, referred to these movable 

 axes, we may deduce by geometry the coordinates of any 

 point on the derived solid referred to the axes fixed in 

 space Ox , Oy, Oz. We may, however, refer both the 

 derived and the primitive solid to the same system of fixed 

 axes. 



Draw PG perpendicular to the plane xy ; on PG take 

 a length LG, so that 



LG = PFcos 7 , 



y being the inclination of the primitive axis to the axis of 



z. Then L will be a point on the derived solid. Also we 



have 



PF = PDcosDPF. 



DPF is the angle between PF and PD. PF is parallel to 

 the primitive axis, and its direction cosines will therefore 

 be cos a, cos /3, cos 7. Let a, b, c be the coordinates of 

 the point D ; then the direction cosines of the line DP 



will be 



x— a y — b z — c 



~PD~' "PD"' "PD"' 

 Therefore we have 



(x — a) cos« + {y—b) cos ft H- (z— c) cosy # 



cos DPF- pD 



q2 



