SOME PROPERTIES OF PROJECTED SOLIDS. 223 



origin to the point a, b, o, and let the new axis of x make 

 an angle 6 with the old axis, so that 



tan 0= -j i 

 then the equation becomes 



z z — 2znx \/i—n z -\- n z x z + n 4 y z = w 4 r\ 



Now transform to new axis in the plane of x, z } the new 

 axis of x making, with the old, an angle (/>, so that 



tan 20 = 



in 



Vi—n z> 



then we get the equation 





x z 



z z 



2n 4 r z 



2n*r z 



+ *- = ! 



i+ft 2 — </\-\-2n z —yrf i+n z + </i + 2n z — 3/i 4 r z ' ™ 



Thus the surface is an ellipsoid ; its volume may readily 



be shown to be -7rftV 3 , that is ri 1 xvol. of sphere. If 



ti= + 1, that is if the primitive axis be parallel to the axis 

 of z, the equation to the ellipsoid becomes 



x z +y z + z z =r z , 



which is the equation to the primitive solid referred to 

 coordinates passing through its centre. 



If, in equation (4), we make ra=o, the principal axis of 

 the ellipsoid parallel to the axis of x will assume the 



indeterminate form -. On examination it will be found 



o 



to be r*, but when n=o the projected solid vanishes; 

 hence, just as it vanishes the ellipsoid becomes a spheroid, 

 of which one of the axes is indefinitely small. 



In a former communication I stated that a sphere when 



