ON CUBIC CONES AND CURVES. 
415 
351] 
34. But we have 
¿(1 + 6P + 21% 
4 (1 - 2V + 4i' ! ) = («' -1) 2 + 3 = 9 ~ |(2 + W + №f + 27/j = gt (1 + 1 + Vf (1 -21 + 4P), 
and thence 
* (1 + £ + ¿ 2 ) 2 (1 — 21 + 4£ 2 ) ' 
(1 + 61 2 + 2/ 3 ) 3 
But the equation 
gives 
and we thence have 
which determines k' in terms of k. 
35. It may be observed that the value k = — 6 corresponds to l'= 1, that is, the 
Hessian is here a 3 + y 3 + z 3 4- 6xyz — 0, a simplex neutral form not transformable into 
(X'+ Y'+ Z') 3 + 6k'X'Y'Z'= 0; the corresponding value of l is of course given by the 
equation 1 + 6l 2 + 2l 3 = 0 ; the only speciality of the cone x 3 + y 3 z 3 + 6lxyz = 0, or 
(X + Y+ Z) 3 — S6XYZ= 0, consequently is that the Hessian is a simplex neutral cone. 
The value k = — 4 corresponds to 1 = 0, V = oo , k' = oo ; hence X' : Y' : Z' = x : y : z 
and the transformation of the Hessian x 3 + y 3 + z 3 + 6l'xyz = 0 into the new form 
(X' + Y' + Z') 3 + OkX'Y'Z' = 0 degenerates into the mere identity xyz = xyz. 
Cambridge, 19th Feb. 1865.