80 ROYAL SOCIETY OF CANADA 
Hence we must have 7, + 7, = 0, 
or) = —h = 1 
2 11 — x? a5 OP 
T, = 74 (@—y’) 
Ts = 0,7 xy 
The J, containing 7, T?, T,? contains also $7 — 5— T2 = xt yt. 
Henco =a a: 7). 
But 2° y’ is proportional to the Hessian of x‘ + y', 
.. 1, = Involution of a form and its Hessian q. e. d. 
From Jacobi, loc. cit., we see that the equations 
v—f u=f,t- 
v—f,u=f, t? 
v— f,u = fie ff: =f 
DOUTE 
represent a transformation ; f,, f, being quadratics, t,, t, linear forms. 
It is possible to find all the elements of such transformation among 
the irrational co-variants of f. 
For if we start from a decomposition of f into two quadratic factors, 
we introduce ¢, #, x where 6, 7 y= —27, Clebsch, Binire Formen, and 
we are able to equate linear combinations of 7, T, to ¢, and y say, which 
gives an invariantive determination of the involution 7,2 7,2 
There are no other biquadratic transformations possible than those 
indicated. 
Although an J, which contains only one square is not suitable for 
the transformation of ee integrals, it enables us to determine hyper- 
elliptic integrals of genus 2, which are reducible to elliptic. A theorem 
about the conjugate system to such an involution may fe some interest. 
Let J, = (at 4?) where g = b,?. 
Let (xo) be a double element of Z,, not one of the factors of b,2, 
There are four such (x6). Then we know (Stephanos : Sur les faisceaux 
des formes binaires ayant une même jacobienne) that (x6)* (xo) belongs 
to the system apolar to Z,. The theorem in question determines (x 6). 
Theorem: (xo) (x6) is conjugate to b,”. 
We know that (x6) is uniquely determined by this condition. Sup- 
pose the theorem true. Let (xo) = x, (xo) =y. Then (xo) (ac) = ay, 
and because of conjugacy to b,? we have b,? = b, x? + b, y Let us take 
for a,‘ the form which contains (ao)? as factor, which is allowed. 
Then af = 2? (a, x? + 2a, xy + a, y). 
We see at once that x y is apolar to x? (a, x? + 2a,, xy + a, y*), and 
(b, x? + b, y*)*, since the term x y is wanting in both. Hence the theo- 
rem is true. go. id, 
N.B.—Two forms, a,” and b," are apolar or conjugate if (a b)" = o. 
