[mM‘ponaLD] NUMBER OF REPRESENTATIONS OF A NUMBER 79 
9° = in accordance with the notation of Weierstrass we have 
6 
(y dy) 
mee az) 
ae Wi AU VU") 
quadratic transformation any elliptic differential into the Weierstrass 
normal form. Since the absolute invariants of the quantities under the 
radicals are the same, it is not a true transformation but a multiplication 
by 2. 
The transforming involution J, is the involution of a form f and its 
Hessian H. The question arises whether every transformation of this 
type where the involution contains 3 forms which are squares, is the 
involution of a form and its Hessian, 7 ¢., whether Hermite’s transforma- 
tion is the general one of this kind. The answer is in the affirmative. 
According to Jacobi, Fundamenta Nova, Werke, L, p. 57, we must 
have equations as follows: 



ie, we transform by a bi- 
v—fPpu=p 
v— fu = T/ 
v— fu = T} 
v—f,u= TP 
(v, u) being the J,, and B, f,, 6,, 2, the roots of the radical in the trans- 
formed differentiai. 
The six double elements of the J, are given by T, 7, T, = 0. But 
I, = (T} T,?) and if we denote the Jacobian of p, p, by 2e p we have 
0: pr 
aps T2= ARRETE 
but op: T= Gale oS: TT, where 6, is a constant factor. 
ee re D 
likewise T, = 6, 9 
ikewise 7’, TT. 
Se TU 
Hence the 3 forms 7, T, T, have the property that each is the Jacobian 
of the other two up to a factor. 
It is possible to transform any two quadratic forms which are non- 
singular and whose resultant does not vanish into the forms 
ery me + my 
Let this be done to JT, and 7, 
so that T° = 2’ + y° 
T, = 1% %+ mY? 
then 7, = 6, (x y). 
But since T, = 6, S 7, 7, we must have 
th mace 230 (y° RES a") 

