386 
NOTE ON THE THEORY OF INVARIANTS. 
[522 
To deduce this result from the theory of the sextic function, I observe that 
denoting by A, B, C, A, the values of the quadrinvariant, the sextinvariant, and the 
discriminant, as given in Salmon’s Higher Algebra, Ed. 2, pp. 202—211, then in the 
particular case a = 0, 5 = 0, we have 
A = -10<B B = d 4 
+ 15 ce, — 3 cd 2 e 
+ c 2 e 2 , 
and hence forming the new invariants 
B= 100 B-A\ 
T= 1000 G -120)0 AB + 4A 3 , 
the values of these in the same particular case a = b = 0 are 
B = 25 c 2 (8 df — 5e 2 ) T = — 2500 c 3 ( 8 d 2 g — 12 def+ 5e 3 ) 
- 100 c 3 g, + 3000 c 4 (10 eg - 9/ 2 ). 
Taking now A, B, C, A as the invariants of the sextic, one of the conditions for the 
transformation is B 3 : O 2 = B' 3 : C'~. 
In the particular case a = b = c — 0 and a =b' — c = 0, the invariants vanish and 
the equation is satisfied identically. But if we assume in the first instance only 
a = b= 0, a = b' = 0, then the terms contain the common factors c 6 and c' 6 respectively ; 
and throwing these out, and then writing c = 0, c' = 0, we obtain the condition 
previously found in a different manner. 
It will be observed that the condition is of the original form P : Q=P' : Q', 
but with the difference that P, Q and the corresponding functions P', Q', are not 
invariants. As possessing the foregoing property these functions may however be called 
“ imperfect invariants,” it being understood that an imperfect invariant is not an 
invariant, and is not in any case included in the term “ invariant ” used without 
qualification. 
And we may now establish the general theory as follows: Consider the similarly 
constituted special forms (a,..)(x, y, z, . .) n and (a',..)(x\ y', z,..) n : to fix the ideas 
the coefficients (a, . .) may be regarded as homogeneous functions of the elements 
(a, /3, . .) which are either independent, or homogeneously connected together in any 
manner; and then the coefficients (a\ . .) will be the like functions of the elements 
(a/3', . .) which are either independent or (as the case may be) homogeneously connected 
in the like manner. 
The entire series of functions P, Q,... of (a, /3,..), which are such that P, Q being 
of the same degree, and P', Q' being the like functions of (a', /3'), we have for the 
linearly transformable functions (a,..) (%, y, z,. .) n and (a',...) {x, y, z’,. .) n the relation