SOME PROPERTIES OF PROJECTED SOLIDS. 225 



From these equations we obtain 



/ m .'""" n 



The locus of the centres is therefore a straight line parallel 

 to the primitive axis ; we might have drawn any arbitrary- 

 curve, either plane or of double curvature, terminated by 

 the bounding planes parallel to plane of x, y, and piled up 

 the successive slices, so that their centres lay on the curve ; 

 the solid so generated would also fulfil the condition of a 

 projected solid. 



As another example, consider the projection of the 

 elliptic paraboloid 



z=Ax 2 + 'By z . 



The equation to the derived solid is 



z=n{(x— a)l+m(y — b)—cn} + n 1 (A# a + By 2 ) . 



The volume of the primitive solid included between the 

 curved surface and the plane 



will be 



Ax2+B#2 



The plane, when projected, will give an equation of the 

 form 



z=n{l(x—a) +m(y—b) —nc} +n z h. 



The volume intercepted between this plane and the pro- 

 jected solid will be 



% n{l(x— a)+ra(y— 5)— «c}+n 2 A 



dx dy dzj 



