697J 
ON THE DOUBLE ^-FUNCTIONS. 
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Expanding, and observing that 
(adbiCj + a^bc) 2 = (adh^ — a^bc) 2 + 4abcda 1 b 1 c 1 d 1 = (be, ad) 2 (x - y) 2 + 4abcda 1 b 1 c 1 d 1 , 
the whole equation becomes divisible by (be, ad) 2 , and omitting this factor, the 
equation is 
(be, ad) 2 (a — z) 2 — 2 (a — b) (a — c)(a — z) (d — z) (adh^ + a I d 1 bc) 
4- (a — b) 2 (a — c) 2 (d — z) 2 (x — y) 2 = 0, 
or, as this may also be written, 
z 2 {(be, ad) 2 —2 (a — b)(a — c) (adbjCj + ajdffic) + (a — b) 2 (a — c) 2 (x — y) 2 } 
— 2 z {(be, ad) a— (a-b)(a — c) (adb^x + ajdjbc) (a + d) + (a — b) 2 (a — c) 2 (x — y) 2 d } 
4- {(be, ad) a 2 — 2 (a — b) (a — c) (adb^ 4- a^Jbc) ad +(a — b) 2 (a — c) 2 (x — y) 2 d 2 } = 0. 
This is really a symmetrical equation in x, y, z of the form 
A 
+ 2 \B(x + y + z) 
4- G (x 2 + y 2 + z 2 ) 
4- 2D (yz -f zx 4- xy) 
4- 2 E (y 2 z + yz 2 + z 2 x + zx 2 + x 2 y + xy 2 ) 
+ 4 Fxyz 
+ 2 G (x 2 yz + xy 2 z + xyz 2 ) 
+ H (y 2 z 2 + z 2 x 2 + x 2 y 2 ) 
+ 21 (xy 2 z 2 + x 2 yz 2 + x 2 y 2 z) 
+ Jx 2 y 2 z 2 = 0 ; 
the several coefficients being symmetrical as regards b, c, d, but the a entering un- 
symmetrically: the actual values are 
A = a- 4 {b 2 c 2 + b 2 d 2 + c 2 d 2 — 2bed (b + c + cZ)} 4- 2a s bcd (be + bd + cd) — 3a 2 b 2 c 2 d 2 , 
B = 2a i bcd — a 3 (b 2 c 2 + b 2 d 2 + c 2 d 2 ) + ab 2 c 2 d 2 , 
G = — 4a 3 bcd + a 2 (be + bd+ cd) 2 - 2abcd (be + bd+ cd) + b 2 c 2 d 2 , 
B = — a i (be + bd + cd) + a 3 (b 2 c + be 2 + b 2 d + bd 2 + c 2 d + cd 2 — 2bed) 
4- a 2 {b 2 c 2 + b 2 d 2 + e 2 d 2 — bed (b + c + d)} — b 2 c 2 d 2 , 
E = a 3 (be + bd + cd) - a 2 (b 2 c + be 2 + b 2 d + bd 2 + c 2 d + cd 2 ) + abed (b + c + d), 
F = a 4 (b + c +d) - a 3 (b 2 + c 2 + d 2 +be+ bd + cd) + 6a 2 bcd 
- a {b 2 c 2 + b 2 d 2 + c 2 d 2 + bed (b + e + rf)} + bed (be + bd + cd), 
G = — a 4 + a 2 (b 2 + c 2 + d 2 — be — bd — cd) + a (b 2 c + be 2 + b 2 d 4- bd 2 4- c 2 d 4- cd 2 - 2 bed) 
— bed (b +c 4- d), 
H = a 4 — 2ft 3 (b 4* c 4" d) 4" ft 2 (b -H c 4" d)~ 4abed, 
I— a 3 — a(b 2 +c 2 + d 2 ) 4- 2bed, 
J = - 3ft 2 4- 2ft (b + c 4- d) 4- b 2 + c 2 4- dr - 2 (be + bd + cd). 
C. X. 
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