234 PEOCEEDINGS OP THE AMERICAN ACADEMY 



c = ^QcpQ = go^ -\- ^qVXqh 



^^ 9Q^ ~\~ ^q{ — pSAfi -|- X'^nQ -j- h^Xq) 

 =. Q'^{g — SXfi) -f- 2SX(jSiiQ ; 



and if we cut the surface by planes perpendicular to I or ^, — i.e., by 



the planes 



SXq = c', 



we find 



SjUp = c", 



(g — S).^)()^ -f- 2 c'Si^Q = c, 



(g — SXh)q- + 2 c"SIq = c, 



either of which is a sphere whose intersection .with the plane is a cir- 

 cle. This would still be true if c should vanish, and the quadric 

 become a cone ; the only difference being that in this case the origin 

 would be on the surface of the sphere. 



Tangent Cone. 



The plane passing through all the points of contact of tangents 

 from a to the quadric 



is the polar plane of «, as we have seen, and its equation is 



Snq)a = 1. 



A surface of the second order tangent to the quadric along its inter- 

 section with this plane will be represented by 



S(>cp(i — 1 -|~ ^(St'fjP« — 1)" = 0. 



If this surface pass through a, its equation must be satisfied by «, and 

 therefore 



Saqia — 1 -f- x(S«gpa — l)*^ = 0, 



and this gives 



^ -1 



X 



(So^o— !)■ 

 Substituting this value of x, we obtain the equation 



(SgcpQ — 1) (S«g:« — 1) — {SQ(fa — 1)^ = 0. 



