540 
Proceedings of the Boyal Society 
By the values of G', g\ we get 
( X 2 = ay - G' = c 2 x 2 4- {a 2 - h) 2 , 
l Y 2 = &V - G' = - cY + Y ~ Ilf, 
where we introduce the letters X 2 , Y 2 , as abbreviations. 
We have now 
c^z 2 = c 2 X 2 + a 2 Y 2 ± 2a5XY . 
The second member being a square we deduce 
(39) c%=±5X±aY, 
the double signs being independent from each other, and the four 
values of the second member constitute the four roots 2 as functions 
of x , y and the parameter h, satisfying (36). 
§ 7. 
Bor the construction of p we introduce 
p = ix + hz\ 
p"=jy + kz", 
where z\ z represent the two terms of z respectively, say 
c 2 z'=±bX, c 2 z"=±aY; 
the double signs showing the possibility for z' and z" to represent 
positive and negative values like any other coordinate. 
We have then 
t . H 
P = P +p . 
If, for a given value of h (leaving out for the present the values 
h = a 2 , and h = b 2 ), we construct p and p" separately we get by the 
former a hyperbola, and by the second an ellipse. The planes of 
these curves will he at right angles. In ordinary coordinates the 
equations of these curves will he 
z 
li — b 2 \ a 
x 
c c 
+ 
y 
h - b 2 
= 1 . 
If we take the point p of the hyperbola for the centre of the 
