396] OX A CERTAIN ENVELOPE DEPENDING ON A TRIANGLE &C. 73 
We have a cubic equation in (A,, g, v) with coefficients which are linear functions of 
(X, Y, Z), and the required equation is that obtained by equating to zero the 
reciprocant of this cubic function, the facients of the reciprocant being the (a, b, c) 
of the linear relation; the reciprocant is of the degree 6 in (a, b, c) and of the 
degree 4 in the coefficients of the cubic function, that is in (X, F, Z). But I remark 
that the equation in (X, g, v), regarding these quantities as coordinates, is that of a 
cubic curve having a node at the point \ = g — V} or say the point (1, 1, 1); the 
corresponding value of Xa + gb + vc is = a + b + c, and the reciprocant consequently 
contains the factor (a + b + c) 2 , or dividing this out, the equation is only of the degree 
4 in (a, b, c). The equation of the curve thus is 
( a + b + c) 2 recip ’ ^-/*) (\-v) + Yu (g - v )(g-\) + Zv (v -X)(v- g)} = 0, 
being of the degree 4 in (a, b, c), and also of the degree 4 in (X, Y, Z), that is, 
treating (X, Y, Z) as current coordinates, the envelope is as above stated a curve of 
the fourth order. 
A symmetrical method for finding the reciprocant of a cubic function was given 
by Hesse, see my paper “ On Homogeneous Functions of the Third Order with Three 
Variables,” Camb. and Dubl. Math. Jour., vol. i. (1846), pp. 97—104, [35]; the 
developed expression there given for the reciprocant is however erroneous; the correct 
value is given in my “Third Memoir on Qualities,” Phil. Trans., vol. cxlvi. (1856), see 
the Table 67, p. 644, [144] and we have only in the table to substitute for (f, rj, £) 
the quantities (a, b, c), and for (a, b, c, f, g, h, i, j, k, l) the coefficients of the 
cubic function of (X,, g, v), viz. multiplying by 6 in order to avoid fractions, these are 
(a, b, c, f, g, h, i, j, k, l ) 
= (6X, 6F, bZ, -2Y, -2Z, — 2X, -2£, -2X, -2Y, X+Y + Z) 
respectively. The substitution might be performed as follows, viz. for the coefficient of 
a 6 , we have 
5 2 c 2 
+ 
1. 
1296 
Y 2 Z n - + 
1296 
befi 
6. 
144 
Y 2 Z- - 
864 
bi 3 
+ 
4. 
,-48 
YZ 3 - 
192 
cf 3 
+ 
4. 
,-48 
Y S Z - 
192 
fH 2 
- 
3. 
16 
YZ 2 - 
48 
192YZ(Y-Zy, 
and so for the other coefficients; but I have not gone through the labour of per 
forming the calculation. Omitting the numerical factoi 192, the coefficients of 
a 6 , b 6 , c 6 are of course 
C. VI. 
YZ(Y-Z) 2 , ZX(Z-X) 2 , XF(Z-F) 2 ; 
10