144] 
A THIRD MEMOIR UPON QUANTICS. 
331 
S(6aPU + /3QU) 
' + IT 2 +192 8 s , r ) 
+ 1287 1 S 2 , 
< + 18T*S +384 S\ 6a> ^4 
+ IT* + Q4>TS 3 , 
y 5T 2 S 2 — 64 S 5 
T(6clPU+/3QU) = 
'- 8 T 3 + 4608TS 3 , ^ 
+ 1920T 2 $ 2 + 73728 8 
+ 360T 3 £ + 384007 7 /Si 4 , 
+ 20T 4 + 8960M 3 , 6;«, /3) 6 . 
+ 840T 3 £ 2 + 7680TS S , 
+ 36™ + 384™* + 24576 &, 
+ IT 5 - 407 T3 /S' 2 + 2560TS 6 j 
R (6aPZ7 + /3QU) = [(48$, 87 7 , -96/S' 2 , - 2477?, - T 72 - 16$ 3 /3) 4 ] 3 7?. 
F(GaPU +/3QU) = ( 192 <8, 327 T ,-384 $ 2 967 7 >8 
+ ( 0 , 512 S s , 192T S' 1 , 24<T‘ 2 S 
+ (1344$ 2 , 352T8, 247' 2 —1152$ 3 , -288T8 2 
+ ( 48 7\ 0 , 288T8 , 24T 72 + 1536$ 3 , 
47’ 2 - 64/S 3 J«, /3) 4 © 17 
7 73 /3) 4 . U- 
20™ + 64$ 4 %«, /3) 4 77. 7777 
1447 7 /Si 2 \a, $Y{HU)\ 
The tables for the ternary cubic become much more simple if we suppose that 
the cubic is expressed in Hesse’s canonical form; we have then the following- 
table : 
No. 70. 
U = P + y 3 + z 3 + Qlxyz. 
S =-l+l\ 
T =1-201 3 - 81*. 
R = - (1 + 81 3 ) 3 . 
HU = I s (x 3 + y 3 + P) — (1 + 2I s ) xyz. 
SU = (1 + 8 ?) 2 (y 3 z 3 + z 3 P + x 3 y 3 ) 
+ (— 91 6 ) [P + y 3 + z 3 ) 2 
+ (— 21 — 5Z 4 — 201 7 ) (P + y 3 + z 3 ) xyz 
+ (- 151 3 - 781 3 + 121 8 ) x*y 2 z\ 
(S) y 77 = 4 (1 + 8Z 3 ) 2 (jfz 3 + z 3 P + Py 3 ) 
+ (- 1 - 4Z 3 - 4?) {P + y 3 + z 3 f 
+ (41 + 100Z 4 + 112/7) (,x 3 + y 3 + z 3 ) xyz 
+ (48? + 552? + 48?) xyz 2 . 
42—2