MR. R. LACHLAN ON SYSTEMS OF CIRCLES AND SPHERES. 
625 
and then it must be possible so to choose h that the equation 
lc(ax*+hi/+ez~+dw~ + ev~) — (aLX J r fiy J r yz-\-hiv J r ev)({;x-\-r)y J r lz-\-uiv-\-Txv), . (310) 
shall be identical with any one of the cubic cvclides given by (308). We have at once 
then 
(3r] v £ 8co esr ? 
ccbcde 
Consequently through the feet of the twelve normals from (x'y'z'io v) can be drawn two 
CC J) c cl c 
spheres, such that (fn^rrr) being the coordinates of one of them, - - must be 
£ Tj f CO C7 
proportional to the coordinates of the other. 
Also equation (310) must represent a cubic : hence we must have 
. v . £ co . m\f a b c d e \ 
+ r+r s + ; + 7j[ i r+^+h + ^ + W 
So that if the given surface is a cubic cyclide, one of these spheres on which the 
feet of the normals lie must be of infinite radius. 
4 L 
mdccclxxxvi. 
