882] 
58 7 
882. 
A CORRESPONDENCE OF CONFOCAL CARTESIANS WITH THE 
RIGHT LINES OF A HYPERBOLOID. 
[From the Messenger of Mathematics, vol. xviii. (1889), pp. 128—180.] 
Take a, /3, y arbitrary, A, B, G=/3 — 7, 7—a, a — ¡3 (so that A+B + C= 0), and 
writing p, a, t for rectangular coordinates, consider the hyperboloid 
Ap 2 + Ba 2 + Pt 2 + ABC = 0. 
Let p 0 , a 0 , t 0 be the coordinates of a point on the surface (Ap 0 2 + Bar 0 2 + Ct 0 2 + ABC = 0). 
The equations of a line through this point are p, a, r = p 0 +f£l, a 0 +gfl, r 0 + hCl (il 
indeterminate); and if this lies on the surface, we have 
Ap 0 f Ba 0 g + Cr 0 h = 0, 
A/ 2 +Bg 2 + Ch 2 =0, 
which equations determine the ratios f : g : h; the equations give 
(Ap 0 f + Ba 0 g) 2 = Ct 2 . Cli 1 , = - Pt 0 2 (Af 2 + Bg 2 ) ; 
(A 2 p 2 + ^.Pt 0 2 )/ 2 + 2ABp 0 a 0 fg + (B 2 a 0 2 + BCt 0 2 ) g 2 = 0, 
whence 
that is, 
{(B 2 a 0 2 + BCt 0 2 ) g + ABp 0 a 0 f } 2 
= {A 2 B 2 p 0 2 a 0 2 — (A 2 p 0 2 + ACt 0 2 ) (B 2 a 0 2 + BCr 0 2 ))f 2 , 
= - ABC (Ap 0 2 + Ba 0 2 + Crf) t 0 2 / 2 , 
= A 2 B 2 C 2 T 2 f 2 ; 
{(Ba o 2 + Pt 0 2 )# + Ap 0 a 0 f} 2 = A 2 C 2 T 0 2 f 2 , 
or say 
(iL-0 2 + Pt 0 2 ) g + A (p 0 a 0 ± Pt 0 )/= 0, 
which equation, together with Ap 0 f+ Ba 0 g + Cr 0 h = 0, determines the ratios f : g 
We have thus the two lines through the point (p 0 , cr 0 , t 0 ). 
: /¿. 
74—2