405] 
AN EIGHTH MEMOIR ON QUANTICS. 
187 
different real equations which correspond to these data must be derivable one from 
another by real transformations, and must consequently have a determinate character; 
that is, the Absolute Invariants, and J, constitute a system of auxiliars. 
Annex. Analytical Theorem in relation to a Binary Quantic of any Order. 
The foregoing theory of the superimaginary transformation led me to a somewhat 
remarkable theorem. Take for example the function 
or, as this may be written, 
(a, b, cfx + k, 1 — kx) 2 , 
k 2 
k 
1 
X 2 
c, 
26, 
a 
or ( 
o, 
26, 
a j 
X 
2 b, 
2a — 2c, 
- 26 
26, 
2a — 2c, 
- 26 
1 
a, 
-26, 
c, 
a, 
-26, 
c 
determinant 
c, 
26, 
a 
26, 
2a — 2c, 
-26 
a, 
-26, 
c 
a \k, 1 ) 2 (x, l) 2 , 
is a product of linear functions of the coefficients (a, b, c); its value in fact is 
= — 2 (a + c) (a + 2bi + ci 2 ) (a — 2bi + ci 2 ), = - 2 (a + c) [(a — c) 2 + 46 2 ]. 
To prove this directly, I write 
and we then have 
a' 
= a —2bi+ ci 2 , 
6' 
= a 
- ci 2 , 
c' 
= a + 26t + a 2 , 
C, 
26, 
a 
1, 
2 
26, 
2a — 2c, 
-26 
o 
a, 
-26, 
c 
i 2 , 
- 2r’, 
(], t, if (2, 0, -2if (1, ~i i 2 ) 
= ( c, 2b, 
(2b, 2a— 2c, 
(a, — 2b, 
a) 
2b) 
c) 
i 2 a', 
— 2i 2 b', 
i 2 c' 
, = a'b'c' 
i\ 
— 2r’, i 2 
2 ia!, 
Ob', 
- 2ic' 
2 », 
0 , 
- 2 i 
a', 
26', 
c' 
1, 
2 , 
1