122 Professor Baker, On a projperty of focal conies 



On a property of focal conies and of bicircular quartics. By 

 Professor H. F. Baker, 



[Read 9 February 1920.] 



The property of focal conies referred to in the title is the well- 

 known one that if P, R be any two points of the focal hyperbola 

 of a system of confocal quadrics, and Q, S be any two points of the 

 focal ellipse, then the distances PQ, PS have the same difference 

 as the distances RQ, RS. The theorem remains true if every one 

 of the distances be replaced by the Cayley separation of its end 

 points in regard to an arbitrary quadric of the confocal system, 

 and the original theorem is then obtainable by making the para- 

 meter of this arbitrary quadric increase without limit. It is shown 

 that the generalised theorem is equivalent to the geometrical 

 theorem that two enveloping cones of the arbitrary confocal exist, 

 each of which touches the four lines PQ, QR, RS, SP. The theorem 

 that the sum of the two focal distances of a point of an ellipse is 

 constant may similarly be replaced by the theorem that the sum 

 of the Cayley separations of a point of the ellipse from the foci is 

 constant, in regard to an arbitrary confocal conic; a theorem is 

 obtained which includes both this last result and the former. It 

 is unnecessary to point out that this last result is equivalent \^'ith 

 Chasles's theorem that a variable tangent plane of a quadric cone 

 makes angles with the planes of circular section whose sum is 

 constant (Chasles, Geom. Super., 1880, § 812, p. 517). 



The property of bicircular quartics referred to is that the angles 

 which a variable bitangent circle of one mode of generation makes 

 with two fixed bitangent circles of another mode of generation, have 

 a constant sum (Jessop, Quart. Journ., xxiii, 1889, 375). This is 

 shown to be equivalent to the former theorem. 



There exist much more general theorems in regard to the 

 generation of a quadric with the help of a thread of constant 

 length, whose systematic investigation is in connexion with the 

 theory of hyperelliptic functions (Chasles, Liouville, xi, 1846, 15; 

 Darboux, Theorie des surfaces, Livre iv, Ch. xiv, 296-312; Staude, 

 Math. Ann., xx and xxii, 1883; Finsterwalder, Math Ann., xxvi, 

 1886; Maxwell, Works, ii, 156 or Quart. Journ., 1867). I have added 

 some lines in regard to this general point of view. 



§ 1. Ii P, Q, R, S be four coplanar points of a quadric, and 

 through the lines SP, PQ, QR, RS be drawn four arbitrary planes, 

 respectively, a, ^, y, 8, the lines a^, ^y, yh, 8a meeting the quadric 



