focal conies and of bicircular quartics 435 



cones of a from V^, V^- In fact taking a as Sa;^ = 0, the equation 

 to j8 is of the form 



SXi^ -V/l^^l ) (^^2 ) ^^2 



where C is a constant, and this is clearly a quadric enveloped by 

 the two cones like (Ex^) (Lx^^) = (LxXi)^. 



Conversely, given two qiiadrics a, jS having two common en- 

 veloping cones from V^, V2, either quadric may be thought of as 

 the locus of a point P such that the sum of the distances with 

 respect to the other quadric, of V^ P and V2 P, is constant; and the 

 constant is the same whichever quadric is considered as the locus. 



In particular, for some definite value of the constant the latter 

 quadric will degenerate into a pair of planes, viz. the two planes 

 of intersection of the two enveloping cones. Thus, given two conies 

 |Si, ^2 intersecting in two points, V^, V^ indicating the vertices of 

 the two cones through both of them, the locus of a point P such 

 that the sum of the distances with respect to ^1 , ^2 , of Fj P and 

 V2 P, is constant, is a quadric enveloped by the two cones. 



Taking the two points 7i , Fg in the first result of this article 

 to coincide, we find that if two quadrics have ring contact with V 

 for the pole of the plane of contact, and if P be an arbitrary point 

 on either quadric, then the distance VP with respect to the other 

 quadric is a constant. 



3. Reciprocally, take two quadrics having two common conies. 

 Then an arbitrary tangent plane to either quadric makes with the 

 planes of the two conies angles (measured with respect to the other 

 quadric) whose sum is constant. Indicating the quadrics by 



Sa;2 + 2Xzt = 0, Sx^ + 2ixzt = 0, 



V(l- A2)(a2- 1) 

 the constant referred to is found to be arc tan TZTx • 



In particular taking the two quadrics as a cone and a sphere, 

 we have the well-known theorem that a varying tangent plane to 

 a cone makes, with two circular sections of opposite systems, angles 

 with §1 constant sum. 



Conversely, given a quadric a and two arbitrary planes z, t, the 

 envelope of a plane which makes with them two planes angles with 

 a constant sum, is a quadric /3 intersecting the given quadric along 

 the two given planes z, t. 



We observe that in the above results the second quadric consists 

 of two planes, and two conies having two common points represent 

 the elliptic quartic curve which by projection is to yield the bi- 

 circular quartic. From this particular result we shall deduce the 

 theorem in the general case. 



