of Edinburgh, Session 1881-82. 
539 
and 
(37) 
( - rik' = ix(li - b 2 ) +jy(h - a 2 ) 
\ + k\z(h - M 2 ) + gab ] , 
where now (by (27)) 
(37 bis) 
n = gh - abz ; 
g 2 being expressed by (25), becomes 
(36 bis) g 2 = M 2 — 2 h + x 2 + y 2 + z 2 . 
We abandon the problem of discussing the roots h as expressed 
in function of xyz, when the elements of p are given alone. But 
we will consider wbat surfaces the equation (36) represents when h 
being considered as a parameter receives any value between - go 
and + oo . 
The equation (36) when freed of the radical represented by g will 
be biquadratic in x , y, z. We resolve in respect to z. Putting 
b 2 x 2 + a 2 y 2 + M x - li 2 = G' , 
x 2 + y 2 + M 2 - 2 h = g ' , 
we get the equation under the form 
(M 2 2 2 + G') 2 - ^M^ 2 + g') = 0 . 
The coefficient of z 4 being 
M| - 4M 2 = (a 2 - b 2 ) 2 , 
we introduce c by 
a 2 -b 2 — c 2 , 
and, without loss of generality, we assume a > b once for all , so that 
c be real. 
We have now, in resolving in respect to z 2 
ch 2 = (2M x y' - M 2 G') ± R , 
where we put 
R 2 = (2M 1 ^-M 2 G') 2 -c 4 G' 2 . 
This being the difference of squares, and as in virtue of (32) 
M 2 - c 2 = 2b 2 
M 2 + c 2 = 2a 2 , 
we transform R 2 into 
R 2 = 4:a 2 b 2 (a 2 g f - G ')(%' - G') . 
Also 
2M^ - M 2 G' = b\a 2 g' - b’) + a\b 2 g' - G') . 
