and of bicircular quartics 127 



§ 8. The proof of the general theorem of § 7 is, analytically, 

 identical with that of the following theorem, of three dimensions, 

 which leads, in § 9, to the theorem of § 4, and may thus be regarded 

 as summarising all the analogous theorems here obtained: — If two 

 quadrics V, W both have ring contact with a quadric U, and also 

 both have ring contact with a quadric K, and PX, PF be tangents 

 respectively to V and W from a point P of K, the sum (or difference) 

 of the Cayley separations PX, PY, in regard to U, is independent 

 of the position of P upon K. When V and W coincide the difference 

 of the separations is zero for all positions of P and the quadric K 

 is unnecessary. 



The theorem is easy to prove. In order that two quadrics 

 V = 0,W = should both have ring contact with another quadric 

 U = 0, they must, if P = 0, Q = be suitable planes, be capable 

 of the forms V = U - P^, W ^ U - Q^ and thus F, W must have 

 two points of contact, there being an identity of the form 



V-W = pq, 



where p = 0, q = are two planes. Any quadric having ring 

 contact with both V and W is then capable of either of the 

 identical forms 



V +1 {a-'^p - aqf =0, W +1 {a-'^p + aqf = 0, 



wherein a is a constant, and two such quadrics can be drawn 

 through an arbitrary point. We. may then suppose 



U=V + l {a-^p - aq)\ K=V + 1 (b'^p - hq)\ 



where ii = is the quadric of the enunciation, and 6 is a constant. 

 Thus we have the identity 



U - K = l (a-2 - 6-2) (^2 _ a^j^Y), 



involving in particular that U, K have two points of contact on 

 the line joining the points of contact of V and W. Putting 



P = I [a-^p — aq), ^ == | {a-'^p + aq), 

 this is the same as 



(1 - a2) (C/ - A') = P^+Q^+ 2aPQ, 



where a = {a^ + b^)/{a'^ — 62). This again, if U is not zero, is the 

 same as 



(P2- U) (Q2 - U)- {PQ f aUf = (1 - (t2) UK. 



We remarked however above (§ 4), that if 6, (f> be the Cayley 

 separations PX, P Y, taken in regard to V, 



P Q 



cos 8 = — 7 , cos (b = —- 



