187 



face, and let #, y, z be the projective coordinates of the point of con- 

 tact of the tangent plane ; then 



_ 

 dv 



__- 



dc, dv 



Z 





~ 



dv 



(15) 



By the help of these three equations and the original equation 

 4>=0(|, v, ) = 0, we may eliminate , v, , and obtain the final equa- 

 tion in x, y, z. 



Again, let F=f(x, y,z) = Q be the projective equation of a curved 

 surface. The tangential coordinates I, v, of the tangent plane 

 drawn through the point (xyz) may be found from the following 

 expressions : 



~dx 



dJ? 



dy 



dF . d ,dF 



X+ y+g 

 dx dy dz 



dz 



(16) 



dF , dF . dF 



As an application of this method, let it be required to find the ex- 

 pressions for the projective coordinates of the surface of the centres 

 of curvature. 



If we apply the general expressions (15) to the particular equa- 

 tion (9), we shall have 



