SOME PROPERTIES OF PROJECTED SOLIDS. 221 



As an example of the use of the above given relation- 

 ships between the coordinates of the primitive and the 

 derived solid, suppose we find the relation between their 

 volumes. Then using V* to denote the volume of the 

 derived solid between limits o and £, we have 



V.-JS&ME; (2) 



by substitution this becomes 



Vs=cos 7 jj { (w — a) cos « + (y — b) cos /3 + (z — c) cos 7} dy dx, 



a result which may also be written in the form 



V*=cos* 7 (j*(*- C + ( *~ a)C0S " t (y ~ &)C ° S/3 )<fr'fo> 



or finally in the form 



Vs = cos 2 <y ln&%^. 



1)1)1) (x— a) cos a.+(y—b) cob fi 

 cosy 



If, in equation (1), we make f=o, we obtain 



(x— a) cos a + (y — b) cos/3+ (z—c) cos 7=0. 



This, being an equation of the first degree in x, y, z } will 

 represent a plane ; it is, in fact, the equation to the primi- 

 tive plane. Hence the lower limit in the above integral 

 reminds us that the integration is to extend from the 

 primitive plane to points upwards ; but if V be the volume 

 of the primitive solid between these limits we shall have 



V= 1 1 \ dx dy dz. 



JjJ c (x-a)coaa+Q/-b)cosp 

 cosy 



Hence 



v*=cosyv\ 



