Manchester Memoirs, Vol. hi. (191 2), No. 4. 



1 1 



i. " The points along a generator of a skew surface at 

 which the tangent planes arc inclined at a given angle 

 trace out two homographic rows of points." 



ii. " The tangent planes also trace out homographic 

 pencils." 



It is well known that the anharmonic ratio of any 

 four points on a generator of a skew surface is equal to 

 the anharmonic ratio of the tangent planes to the surface 

 at these points, so that the second theorem follows at once 

 if the first is established. This can be done by proving it 

 to be true for a hyperboloid, since a hyperboloid can be 

 drawn to touch any skew surface along a generating line. 

 If the generator G be taken as axis of x its equation 

 will be 



by"' -\- c z"- \ 2fyz + 2g zx + 2/1 xy ■{- 2W z= O. 



Let G' be the consecutive generator and 00' the shortest 

 distance between G and G'. Let O, the point where the 

 line of striction crosses the generator, be the origin, 00' 



Fig. 4. 



the axis of y and the normal to the surface at O the axis 

 of z. 



The generators through any point are 



and 



\{cz + ifv + 2^a; + 271') = 2hx + by 

 z=-\y 



(viii), 



\'{cs + 2jy + 2gx 4- 2w) ^y , .. . 



z= -}^{2hx-^by)] ■ ^^""l- 



