82 ON A CERTAIN ENVELOPE DEPENDING ON A TRIANGLE &C. [396
The function on the left-hand side is the quotient by — 48 (a + b + c) 2 of the
sextic function, Table 67 of my third Memoir on Quantics, [144]; the foregoing quotient
was calculated without using the coefficient of the term in a 2 b 2 c 2 (£ y 2 £ 2 ) of the table,
but by way of verification, I calculated from the table the term in question, and found
it to be
(x 4- y + zf
— 2 (x + y + zf (yz -f zx 4- xy)
-f 296 (x + y + z) xyz
— 8 {y 2 z 2 + z 2 x 2 + x 2 y 2 ),
and this should consequently be equal to the coefficient of a 2 b 2 c 2 in the product of
(a + b + c) 2 into the foregoing quartic function of (a, b, c) that is, it should be
= x (x 3 — 2x 2 y — 2x 2 z + xy 2 + 38xyz -f xz 2 + 12y 2 ^ + 12yz 2 )
+ y (y 3 — 2y 2 z — 2y 2 x + yz 2 + 38xyz + yx 2 + 12z 2 x + 12zee?)
+ z (z 3 — 2z 2 x — 2z 2 y + zx 2 + 38xyz + zy 3 + 12x 2 y + 12xy 2 )
+ 4>yz (11 xi 2 + y 2 + z 2 — Zyz + 24xy + 24^)
+ 4zx (lli/ 2 + z 2 + oc 2 — 2zx + 24yz + 24xy)
-1- 4ixy (11 z 2 + x? + y 2 - 2xy + 24zx + 24yz),
which is accordingly found to be the case.
