( 13 ) 
Substituting these values in (7), (8), (9), we obtain 
Y z COSyV 
y = l 1 EH 
Y^ cos/3 Y 
Y “ L 
V* COSaV 
X = ~L~' 
(14) 
(15) 
where Y has been written for J AdL and denotes the 
volume of the primitive solid which is, of course, a constant. 
Since X, Y, Z are the projections of L on the coordinate 
axes we shall have 
X - Lcosa 
Y = Lcos/3 
Z = Lcosy 
Substituting in (13), (14), (15) the resulting equations will be 
Y* = COS 2 yY 
Y y = cos 2 /3 Y 
V. = cos*aV 
adding and remembering that cos 2 a + cos 2 /3 + cos 2 y = 1, we 
obtain 
Y. + V^ + V.-V 
Hence if we suppose the original solid to move in any way, 
the volume of each projected solid will vary, but the sum 
of these volumes will be always constant and equal to the 
volume of the original solid. 
If the primitive axis at any time should be perpendicular 
to one of the coordinate planes, then of the three quantities 
V*>V„Y* two will vanish, and the remaining one will be a 
solid equal and similar to the original solid. 
