522] 
NOTE ON THE THEORY OF INVARIANTS. 
387 
may be called the “ perfect and imperfect invariants ” of (a,..) (x, y, z,. ,) n ; and the 
relation in question be briefly referred to by the expression that the perfect and 
imperfect invariants are proportional. 
We have then the theorem that if the two functions (a,..) (x, y, z,. ,) n and 
(o',..) (x, y, z',..)” are linearly transformable the one into the other, the two functions 
have their perfect and imperfect invariants proportional ; and conversely the theorem,, 
that two functions which have their perfect and imperfect invariants proportional, are 
linearly transformable the one into the other. 
There is thus a wide field of inquiry in regard to the imperfect invariants, even 
of a binary function, but still more so as to those of a ternary or quaternary function 
representing a curve or surface possessed of singularities. 
We have in what precedes the explanation of an error into which I fell in my 
paper “ On the transformation of plane curves,” Proc. Loud. Math. Soc., vol. I. No. 3, 
Oct. 1865, [384], see Arts. Nos. 27—30. Considering a given curve of deficiency D and, 
by means of a system of D — 3 points chosen at pleasure on the curve, transforming this 
into a curve of the order D + 1 with deficiency 2) ; then for any two of the transformed 
curves (that is, two curves obtained by means of different systems of the D — 3 points) 
I showed that these had the same absolute invariants—or in the language of the 
present paper, that they had their invariants proportional, and I thence inferred that 
the two transformed curves were linearly transformable the one into the other—whereas, 
to sustain this conclusion, it is necessary that the two curves should have their perfect 
and imperfect invariants proportional; and this was in no wise proved. That the two 
transformed curves are not in fact linearly transformable ’ the one into the other has 
since been shown a posteriori by Dr Brill in the particular case D = 4. Riemann’s 
conclusions, with which my own were at variance, are thus correct. 
I remark that if a binary function of an odd or even degree n = 2p +1 or 
= 2p, has p + 1 equal factors, then the invariants all of them vanish ; but the equality 
of the p + 1 factors implies only a _p-fold relation between the coefficients; that is, 
the vanishing of all the invariants gives only a j^-fold relation between the coefficients, 
viz. the relation is % (n — 1) fold or |%-fold according as n is odd or even. Thus for 
a sextic function the equations A = 0, B = 0, 0=0, A = 0 constitute only a 3-fold 
relation between the coefficients. 
Similarly if the function has p equal factors, then every invariant is a mere 
numerical multiple of a power of one and the same function © ; so that the vanishing 
invariants can be at once formed. And we have thus only a (p— 1) fold relation 
between the coefficients, viz. the relation is ^ (n — 3) fold or ^ (n — 2) fold according as n 
is odd or even. 
Cambridge, 4 August, 1870. 
49—2