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