449 
702] 
ON THE TRIPLE ^-FUNCTIONS. 
a, 
h, 
g> 
l, 
a 
a, 
h, 
g> 
l, 
a' 
- 
a, 
h, 
g> 
h 
a 
2 _ 
a, 
h, 
g> 
l 
h, 
b, 
f 
m, 
ß 
h, 
b, 
f. 
m, 
P 
h, 
b, 
y 
in, 
ß 
K 
b, 
y 
m 
g> 
f 
G, 
n, 
7 
g> 
f 
C, 
n, 
i 
g> 
/ 
C, 
n, 
7 
g> 
y 
c, 
n 
l, 
m, 
n, 
d, 
8 
1, 
VI, 
n, 
d, 
8' 
1, 
m, 
n, 
d, 
8 
1, 
m, 
11, 
d 
a, 
ß, 
7> 
8 
ff, 
/ 
7> 
8' 
a > 
ß', 
f 
7 > 
8' 
where U is an easily calculated function of the second order in a, b, c, d, f g, h, 
l, m, n, and also of the second order in the determinants a/3' — a.'/3, etc. 
We can obtain such a form of the equation of the quartic, from the before- 
mentioned equation 
\/af+ \/bg -f- V ch = 0, 
viz. this equation gives 
*, h, g, a 
= 0, 
h, *, f b 
g, f, *, c 
a, b, c, * 
which is of the required form, symmetrical determinant = 0; the equation is, in fact, 
a 2 /' 2 + b 2 g 2 + c 2 h 2 — 2 bcgh — 2 calif — Zabfg = 0, 
which is the rationalised form of 
and we hence have the cubic 
\i af+ \/bg + *Jch = 0, 
*, k g, a, a 
h, *, /, b, ß 
= 0, 
g, f *, c, 7 
a, b, c, *, 8 
a, ß, 7, 8, * 
the developed form of which is 
a 2 bcf+ /3 2 cag + 7 2 abh + 8 2 fgh 
— (a/3y + fa.8) (— af+bg + ch) 
— (bya + g/38) ( af— bg + ch) 
— (ca/3 + hy8) ( af+bg- ch) = 0. 
Considering the intersections with the quartic 
Va/’-f Vbg + *lch = 0, 
we have 
— af+bg+ ch, af—bg + ch, af +bg — ch = — 2 \/bcgh, — 2 V calif, — 2 V abfg, 
and the equation thus becomes 
(a fbcf+ /3 V cag + 7 V abh + 8 f fgh) 2 = 0 ; 
C. X. 
57