74 
ON A CERTAIN ENVELOPE DEPENDING ON A 
[396 
and I find also that the coefficient of b 5 c (the factor — 192 being omitted) is 
= ZX (;3X 2 + 3Z 2 + 3YZ-6ZX + 5X Y), 
whence those of the terms c s a, &c. are also known. 
I denote the result as follows : 
(:YZ(Y-Z)\ ZX(Z-X) 2 , 17(1- 7) 2 , ...$a, b, c) 6 = 0; 
this equation divides as we have seen by (a + b + c) 2 , and the quotient is 
(YZ(Y-Z) 2 , ZX(Z-X) 2 , XY(X — Y) 2 , ...$a, b, c) 4 = 0; 
and it may be remarked that the coefficient of b 3 c in this quartic function of (a, b, c) is 
= £X(X 2 + X 2 + 3YZ- 2ZX + 5X7. 
The last mentioned equation, if the calculation were completed, would be analytically 
the best form for the equation of the envelope; but in view of what follows, I will 
change it by writing ax, by, cz in place of (X, Y, Z)\ x is therefore =--^X, that 
uclll 
is, it is = perpendicular distance x sin A; or, what is the same thing, the new 
coordinates {x, y, z) are proportional to the perpendicular distances from the sides, each 
distance divided by the perpendicular distance of the side from the opposite angle, 
the equation of the line infinity is thus x+y+z= 0. I write also (a, b, c) = 
that is, we have (a, b, c) = (cot^l, cot B, cot G). The system of equations is therefore 
CG '1/ 2 
giving for the envelope the equation 
bcyz (cy — bz) 2 + cazx (az — ex) 2 + abxy (bx — ay) 2 + &c. = 0 ; 
and in this function, corresponding to the term 
b 3 cZX (X 2 + Z 2 + 3YZ- 2ZX + 5X 7), 
we have the term 
azx (bc 2 x 2 + a 2 bz 2 + 3 a 2 cyz — 2 abezx + 5ac 2 xy). 
It may be noticed that, arranging in powers of (x, y, z), the several portions of 
each coefficient are distinct literal functions ; thus we see that the coefficient of z 3 x 
is = a 3 c + a 3 b + other combinations of (a, b, c) : this is material in order to the 
comparison of the foregoing equation of the envelope in a different form which will 
be presently mentioned.