362 Prof. Cayley on the Locus of the Foci of the 



other. The effect pointed out by Professor Thomson and the 

 Astronomer Royal does not, however, in the least degree pre- 

 vent the consumption of the vis viva of the earth's motion round 

 the common centre of gravity, although to a certain extent, at 

 least, it must prevent this consumption from diminishing the 

 moon's distance and increasing her angular motion. But as 

 this consumption of vis viva will go on throughout indefinite 

 ages, if the present order of things remains unchanged, the 

 earth and the moon must therefore ultimately come together. 



XL VIII. On the Locus of the Foci of the Conies which pass through 

 four given Points. By Professor Cayley, F.R.S.* 



THE curve which is the locus of the foci of the conies which 

 pass through four given points is, as appears from a ge- 

 neral theorem of M. Chasles, a sextic curve having a double point 

 at each of the circular points at infinity ; and Professor Sylves- 

 ter, in his " Supplemental Note on the Analogues in Space to 

 the Cartesian Ovals in piano" (Phil. Mag. May 1866), has fur- 

 ther remarked that the lines (eight in all) joining the circular 

 points at infinity with any one of the four points are all of them 

 double tangents of the curve ; whence each of these points is a 

 focus (more accurately a quadruple focus) of the curve. It is to 

 be added that, besides the circular points at infinity, the curve 

 has 6 double points (3 of these are the centres of the quadrangles 

 formed by the 4 points), in all 8 double points; the class is 

 therefore = 14. Hence also the number of tangents to the 

 curve from a circular point at infinity is =10; viz. these are the 

 4 double tangents each reckoned twice, and 2 single tangents ; 

 and the theoretical number of foci is =100; viz. we have 



16 quadruple foci or intersections of a double K ~ v 4 — 64 

 tangent by a double tangent .... J 



16 double foci, or intersections of a double"! -■ ~ <, _ oo 

 tangent by a single tangent .... J 



4 single foci, or intersections of a single tan-T \ -. _ ^ 

 gent by a single tangent J 



100 



To verify the foregoing results, consider any two given points 

 I, J, and the series of conies which pass through four given 

 points A, B, C, D ; we have thus a curve the locus of the inter- 

 sections of the tangents from I and the tangents from J to any 

 conic of the series ; which curve, if I, J are the circular points at 



* Communicated by the Author. 



