80 
ON A CERTAIN ENVELOPE DEPENDING ON A 
[396 
or writing x = z — Y, y = z + X, this is 
HU+3MU-*U= -%z(aX* + bY 2 + cZ 2 ) 
+ X 2 {(/' + aX) Y - F'Z} 
4 Y*{J'Z-(G' + b*)X] 
4 Z 2 {K'X — H'Y }, 
we may determine so that the cubic function of X, Y, Z contains the factor 
aX 2 + bY 2 4- cZ 2 ; writing Z = — X — Y, then 
Contains the factor 
(a 4 c) X 2 
+ 
¿c 
XY 
+ (b + c) F 2 
X 3 ( K'+ F' ) 
4 X 2 F (— H' + 2 K' + F' + I' + a*) 
+ X F 2 ( K , -2H'-G'-J'-b%) 
+ Y 3 H ) 
We have seen that 
K + F' = — 3 (a + c) (ac + M), 
J' +H' = -3(b+c) (bc+M), 
whence the quotient is, as above stated, 
= — 3 (ac + M) X + 3 (be + M) Y. 
Comparing the coefficients of X 2 F, we have 
Quotient is 
— 3 (ac + M) X 
+ 3 (be + M) F. 
a& = -(- H' + 2K' + F' + F) + 3 (a 4- c) (be + M) — 6c (ac + M), 
= 9a (a 4- c) (b 4 c) 4- 3 (a 4- c) (ab + ac 4- 2be) — 6c (ab + 2ac 4- be), 
= 12a (be + ca + ab) = 12aM, 
that is ^r = 12M; and the same value would have been obtained by comparing the 
coefficients of XY-. Hence HU — ^MU divides by aX 2 + bY 2 + cZ-, the quotient being 
— 12ili^ — 3 (ac + M) X 4" 3 (be 4 M) F, 
which is 
= — 12Mz — 3 (ac + M) (y — z) + 3 (be + M) (z — x), 
or, finally it is 
= — 3 {(be 4- M)x+ (ca + M) y 4- (ab + M)z\, 
and we thus have 
HU — U= — 3 (aX 2 4 bY 2 4 cZ 2 ) x {(be + M) x + (ca + M) y + (ab + M)z], 
so that the three inflexions are the intersections of the cubic curve by the line 
(be + M) x 4 (ca + M) y 4- (ab 4- M) z = 0. 
It may be noticed, that if we write 
x 4- y + z = u, 
hex + cay 4- abz = — Mu, 
ax 4- by 4- cz — 
v,