434 Prof. Hanumanta Rao, On a property of 



On a property of focal conies and ofbicircular quartics. By C. V. 

 Hanumanta Rao, University Professor, Lahore. (Communicated 

 by Prof. H. F. Bakek.) 



[Received 20 April. Read 2 May 1921.] 



The present note arises directly out of a note with the same 

 title by Prof. H. F. Baker in the Proceedings (vol. xx. pp. 122-130). 

 The property referred to in the title is that a varying circle, of one 

 mode of generation, makes with two fixed circles, of a second mode, 

 angles with a constant sum; and this result is here deduced from 

 a particular case of it, viz. that where the curve consists of two 

 circles. A prehminary series of results is inserted of some interest 

 in themselves. By distances and angles will be meant throughout 

 the Cayley separations, and the quadric or conic with respect to 

 which the homographies are considered is indicated in each case. 



1. Given two conies a, ^ in a plane, they have four common 

 tangents meeting in three pairs of points. Indicate by V^ , V^ one 

 such pair of points. Then the sum of the distances, with respect to 

 a, of 7i P and V^ P, is constant as P moves on ^; and this constant 

 remains the same if a, ^ are interchanged. 



The proof of this result depends on the existence of two fixed 

 points on the fine 7i V^, and this fact in turn is an easy consequence 

 of the space figure of two conies with two common points. 



Conversely, given a conic a and two fixed points Fj , Fg in its 

 plane, the locus of a point P, which moves so that the sum of 

 the distances with respect to a of Vj^P and Fg P is constant, is 

 a conic ^ touching the four tangents from V^ , Fg to «• 



In particular, when one of the three pairs of points such as 

 Fi, Fa is projected into the circular points at infinity, the other 

 two pairs are the foci and the conies are confocal. This leads to a 

 shghtly more general definition of confocal conies than the usual 

 one, namely taking a fundamental conic S the system of conies 

 touching any four arbitrary fixed tangents of E may be called a 

 confocal system. Or again when the conic S is made to degenerate 

 tangentially into the two circular points at infinity, we have the 

 usual definition of confocal conies; it was this idea in fact which 

 suggested the theorems of this note. 



2. Precisely similar results hold in space of three dimensions. 

 Given a quadric a and two points Fj , Fg , the locus of a point P 

 such that the sum of the distances with respect to oc of Fj P and 

 V^P is constant, is a quadric jS which is enveloped by the enveloping 



