332 
A THIRD MEMOIR UPON QUANTICS. 
[144 
£ V , ? 
® /y JJ = — 2 (1 4- 8 l 3 ) 2 (y 3 z 3 + z 3 x? + ocPy 3 ) 
+ (1 - 10 l 3 ) (x 3 + y 3 + z 3 ) 2 
+ (61 — 180£ 4 + 06P) (x 3 + y 3 + z 3 ) xyz 
+ (6l 2 - 624Z 5 -1922«) x 2 y 2 z 2 . 
PU = — l (I 3 + y 3 + + (— 1 + 4Z 3 ) gyZ. 
QTJ = (1 - 10Z 3 ) (| a + y 3 + f 3 ) - (5 + 4l 3 ) gy£. 
FU = - 4 (1 + 8Z 3 ) (y 3 £ 3 + + f t? 3 ) 
+ (f+ ?f + £ 3 ) 2 
— 24£ 2 (f 3 + y 3 + £ 3 ) 
- 24Z (1 4- 2l 3 ) fV£ 2 , 
to which it is proper to join the following transformed expressions for ©H, % t U, ©„Z7, 
viz. © U = (1 + 81 3 ) 2 (y 3 z 3 + z 3 x 3 + x?y 3 ) 
+ (- 21 3 - l 6 ) U 2 
+ (2l-5P) U .HU 
+ (-3Z 2 )(HU). 
©, U =4(1+ 8Z 3 ) 2 (y 3 z 3 + z 3 x 3 + x^y 3 ) 
+ (- 1 + 12Z 3 + 4>P) U 2 
+ (-161 + 4,1* )U.HU 
+ (-12Z 2 )(HUy. 
©„?7 =—2(1 + 8£ 3 ) 2 (y 3 z 3 + z 3 x 3 + spy 3 ) 
+ (1 —161 3 — 6l 6 ) U 2 
+ (61 )U.HU 
+ (6Z 2 ) (HU) 2 . 
The last preceding table affords a complete solution of the problem to reduce a 
ternary cubic to its canonical form. 
[I add to the present Memoir, in the notation hereof (a, b, c, f g, h, i, j, k, l~$cc, y, z) 3 
for the ternary cubic, some formulge originally contained in the paper “ On Homo 
geneous Functions of the third order with three variables,” (1846), but which on account 
of the difference of notation were omitted from the reprint, 35, of that paper. 
Representing the determinant 
- ax + liy + jz, hx + ky + bz, jx +ly + gz, £ 
hx + ky + bz, kx + by + fz, lx +fy + iz, y 
jx +ly + gz, lx + fy + iz, gx + iy + cz, £