522] 
385 
522. 
NOTE ON THE THEORY OF INVARIANTS. 
[From the Mathematische Annalen, vol. ill. (1871), pp. 268—271.] 
If two binary quantics {a,..) (x, y) n , (a,..) {pc', y') n are linearly transformable the 
one into the other, and if for the first of them P, Q are any two invariants whatever 
of the same degree, and P', Q' are the like invariants for the second of them, then 
we have 
P : Q = P' : Q', 
(or, what is the same thing, the absolute invariants have the same values for the two 
functions respectively); and the entire system of these equations constitutes only a 
(n — 3) fold relation between the two sets of coefficients. But the converse theorem, 
viz. that if the entire system of equations is satisfied, the two functions are linearly 
transformable the one into the other, is only true sub modo. 
For instance, considering the two binary sextics 
(0, 0, 0, d, e, f g)(x, y) 6 and (0, 0, 0, d!, e, f, g')(x', yj, 
or, what is the same thing, 
(20 d, be, 2/, g) (x, yf y 3 and (20 d', be', 2/', g) (x, yjy' 3 , 
the invariants of the two functions respectively are each and all of them =0, and yet 
the two functions are not in general linearly transformable the one into the other. 
For they can be transformable only by the substitution 
« = + M'> V ~ py f 5 
or, what is the same thing, only if the cubic functions are transformable by the substitution 
co = x' + ay', y = y'\ and forming for these the seminvariants ac — b 2 and a?d — 3abc + 26 s 
for the cubic (a, b, c, d) (x, y) 3 , we have as the necessary condition for the transformability 
(8df- bej : (8d?g - 12def+ be 3 ) 2 = (8d'f - be' 2 ) 3 : {8d' 2 g -12d'e'f + be' 3 ) 2 . 
C. VIII. 
49