703] 
ON THE ADDITION OF THE DOUBLE ^-FUNCTIONS. 
457 
as serving to determine the ratios a : /3 : y : B : e in terms of x, y, z, w; and we 
have then for the determination of p, q the remaining two equations 
cap 3 + /3p* + yp + 8 = e VP, 
aq 3 + ¡3q 2 + yq + 8 = e V Q, 
which two equations may be replaced by the identity 
(a6 3 + && 2 -f-y 0 + 8) 2 — e 2 © = a 2 - e 2 .0 — x. 6 - y .6 — z .6 — w . 6 — p .6 — q. 
Writing herein 0=any one of the values a, b, c, d, e, f, for instance 0 = a, and 
taking the square root of each side, we have 
a a 3 + /3a 2 + ya + 8 = Va 2 — e 2 Va — x .a — y '/a — z .a — w \/a — p. a — q, 
or as this may be written 
olo? + /3a 2 + ya + 8 = Va 2 — e- A 12 . . J. g6 , 
which equation when reduced gives the proportional value of .¿l 56 . 
For the reduction we require the value of aa 3 + /3a 2 + ya + 8: calling this for the 
moment il, we join to the four equations a fifth equation 
aa 3 + /3a 2 + ya -f B = fi. 
Eliminating a, /3, y, B, we find 
= 0, 
X 3 , 
x 2 , 
x, 
1, 
eVX 
y 3 > 
y-. 
y> 
1, 
e VP 
z 3 , 
z 2 , 
z, 
1, 
Z 
w 3 , 
W-, 
w, 
1, 
eVF 
a 3 , 
a 2 , 
a, 
1, 
n 
or, what is the same thing, 
O 
/y>3 rp2 rp ! 
Iv y tv y it y JL 
+ e 
\/X, x 3 , x 3 , x, 1 
y\ y\ y. 1 
V F, y 3 , y-, y, 1 
z 3 , z~, z, 1 
^ Z, z 3 , z-, z, 1 
w 3 , w-, w, 1 
VTF, w 3 , w 2 , w, 1 
a 3 , a 2 , a, 1 
= 0; 
viz. this is 
fi. x—y.x— z .x — w. y—z.y—w. z — w = — e{VX .y — z .y—w.y — a.z—w.z — a . w — a 
+ V F .z—w.z —a.z —x.w — a. w — x.a - x 
+ \iZ .w — a. w — x.w — y . a — x.a — y. x — y 
+ V W .a — x .a — y . a — z . x — y . x — z .y — z), 
58 
C. X.