284 
Proceedings of Royal Society of Edinburgh. [sess. 
In the next place, four equations having been found in a, a > 
2 
a" . viz., 
mn - V 
A 2 
a 
+ 
a 
+ 
a = 
n 2 2 
tjr a 
+ 
.2 .2 
G a 
+ 
p /,2 //2 
lx a — 
1 2 
1 ,2 
1 „2 
~~2 a 
G 
+ 
g /2<x 
+ 
2 a = 
G" 
n4 2 
br a 
+ 
4 .2 
G' a 
+ 
4 2 
G" a" - 
if the first three he taken there is obtained for a 2 the value 
(G 2 - m)(G 2 - n) - V 2 ' 
(G 2 - G ,2 )(G 2 - G' /2 ) 
and, if the 1st, 2nd and 4th, the alternative form 
(/ - G /2 )(Z - CP 2 ) + m ' 1 + v/ 2 
(G — G )(G — G" ) 
where the identity of the two numerators is readily verifiable. In 
the same way the expressions for the squares of the six other 
coefficients of the first substitution may he obtained. The 
difficulty of the double sign resulting from the extraction of the 
square root is readily got over, because rational expressions similar 
2 2 
to those for a , a , . . . are given for the nine binary products 
a/3 , a/3', a"/3", ay , , from which, when the sign of one of the 
coefficients is fixed, the signs of the others at once follow. It is 
not noticed, however, that the numerators of these eighteen values 
are the principal minors of the three eliminants, 
l- G 2 
n 
in 
n 
m - G 2 V 
m 
V 
n-G ‘ 
2 f 
l — G' n m 
»' m - G' 2 V 
m V n - G' , 
l — G" n' m 
n m - G" 2 V 
m V n- G" 2 
above referred to, the corresponding unknowns being 
( 2 
a a p ay 
) ( 
; - 
a a p 
, . : 
a y 
) ( 
: > 
a a fj ay 
f P 7 
/3' 2 
Pi 
7 2 
/r 2 p'Y 
7 
5 
7 
3 
y" 
and the corresponding denominators, 
(G 2 -G' 2 )(G 2 -G" 2 ), (G' 2 -G" 2 )(G /2 -G 2 ), (G" 2 -G 2 )(G" 2 -G' 2 ) 
