54 NOTE ON THE GEOMETRICAL REPRESENTATION OF [l 13 
tion of the terms) let the tangent at P of the conic x 2 + y 2 + z 2 = 0 be also touched 
by the conic 6 (x 2 + y 2 + z 2 ) + ax 2 + by 2 + cz 2 — 0, and similarly for &. The coordinates 
of the point P are given by the equations 
x\y\z=-*Jb — cVa + 0:Vc — a^b + B :*Ja — b*/c + 0 ; 
and substituting these values in the equation of the line PP', we have 
(b — e)Va + k l 'Ja + p'\/a + 0 + (c — a ) *Jb + k *Jb + p *Jb + 0 + (a — b)\/c + k\/c+p^/c + 0 
= 0. ..(*), 
an equation connecting the quantities p, 0. To rationalize this equation, write 
V(a + k) {a + p) (a + 0) = A + ya, 
\/(b + k) (b +p) (b + 0) = A + yb, 
V(c + k) (c +p) (c + 6) = A + yc, 
values which evidently satisfy the equation in question. Squaring these equations, we 
have equations from which A 2 , Ay, y 2 may be linearly determined ; and making the 
necessary reductions, we find 
A 2 = abc + kp0, 
— 2A y = be + ca + ab — {p6 + kp + k6), 
yr = a + b + c + k + p + 6 ] 
or, eliminating A, y, 
{be + ca + ab — (p6 + kp + k0)) 2 —4<(a + b + c + k+p + 0) (abc + kpd) = 0, (*), 
which is the rational form of the former equation marked (*). It is clear from the 
symmetry of the formula, that the same equation would have been obtained by the 
elimination of L, M from the equations 
V (k + a) (k 4- b) (k + c) = L -f Mk, 
V (p + a) (p + b) (p + c) = L + Mp, 
\/(0 + a)(d + b)(6 + c) = L + Me- 
and it follows from Abel’s theorem (but the result may be verified by means of 
Euler’s fundamental integral in the theory of elliptic functions), that if 
dx 
V (x -f a) (x + b) (x + c) ’ 
then the algebraical equations (*) are equivalent to the transcendental equation 
+ Tlk ± Yip ± 110 = 0 ;