406 
A MEMOIR ON CURVES OF THE THIRD ORDER. 
[146 
The equation of the three tangents is 
II = [(a?! 2 + 2ly x z^) x + (i/* 2 + 2IzjOBj) y + (z 3 2 + 2Ix^) z] j = 0, 
x [(x? + 2ly. 2 z 2 ) X + (yi + 2lz& a ) y + Oa 2 + 2lx 2 y 2 ) z] 
x [(x 3 2 + 2ly 3 z 3 ) X + (y./ + 2lz 3 x 3 ) y + (z 2 + 2lx 3 y 3 ) z] j 
and if we put 
F= (P + V 3 + P) 2 - 24Z 2 (p + y 3 + p) £??£+(- 24£ - 48£ 4 ) p^p + (- 4 + 32£ 3 ) (y 3 £ 3 + pp + p»f), 
(F 7 is the reciprocant .FZ7 of my Third Memoir), then we have identically 
F . U — II = (%x + yy + p) 2 (g'x + yy + gz), 
and the equation of the satellite line is %x + y'y + %z — 0. In fact the geometrical 
theory shows that we must have 
F. U — iVTI = (pc + yy + p) 2 (J;'x + yy + £'z), 
and it is then clear that JST is a mere number. To determine its value in the most 
simple manner, write 1 = 0, y = 0, x = f, z = — we have then F. U—N 11 = 0, where 
F = p + rf + p - 2?fp - 2pp - 2% 3 y 3 , H = p - p. 
The value of II is U = F. U, and we thus obtain IV= 1. For, substituting the above 
values, 
II = (xtt-ztf) (xtt-z 2 £) (xtf- ztf) 
— f*3/y» 2/yi 2/v) 2 
L, tAsi 1^2 O/g 
— p£ (oc 2 x 2 x 3 + &c.) 
+ £P («AV + &C.) 
— % 3 Zl% 2 Z 3 2 , 
and we have 
XyXcJK 3 = y s P, 
x x x& 3 + &c. = 3pp 
x x z 2 z 3 + &c. = - 3£p, 
■W 3 = I 3 - y 3 , 
and thence 
xfxizi + &c. = 9pp + 6p 2 (y 3 - p) = 3pp + 6£pTf, 
x 2 zfz 3 2 + &c. = 9pp - 6pf(f - y 3 ) = 3pp 4- 6pp7 3 , 
and consequently 
n = p (v 3 - P) 2 
-?£.S&(? + y s ) 
+ SF.3p£(P+^) 
- P(P-«?*)* 
= (P - P) (P + ?? 6 + P ~ 2p77 3 - 2p| 3 - 2% 3 y 3 ).