430 
Proceedings of the Royal Society of Edinburgh. [Sess. 
calling it also “ il determinante delle derivate-seconde.” He confines 
himself to three of the main theorems. Hesse’s theorem of 1851 (March) 
he amplifies, his enunciation being: — If u , a homogeneous integral function 
of the variables x 1} x 2 , . . x n , be transformed by means of the substitution 
(*) , w , & 
S* = Vl + a 2 V2+ ‘ • + a n Vn 
into v , and one of the new variables, say y 1 , be absent from v , then (1) the 
Hessian H of u must vanish identically, (2) the cof actor of the elements of 
any row of H must be proportional to the coefficients of y 1 in the sub- 
stitution, (3) the product of the first differential-quotients of u by the said 
column of coefficients is equal to 0. The third of these Bella vitis reaches 
very easily, because generally we have 
dv 
3 Vr 
du dxj + du dx 2 
~ dx 1 dy r dx 2 dy r 
+ dudXn 
dx n dy r 
and therefore when 
r = 1 
0 = *#+»§+ . . 
dx 1 1 dx 2 L 
du ( n ) 
• + r fl 
dx n 
or 
0 = u 1 af + u 2 a ( f+ • • • 
1 n 
+ u «, . 
n 1 
As regards the second he notes that on account of the vanishing of H we 
have in the first place 
B”n : U 12 : • • • : U ln == U sl : U s2 : • • • : U s?i , 
and in the second place * the set of equations 
^1^11 + ^2^12+ ’* 
* +^nLL = °,1 
“l^21 + “2^22 + ' ' 
■ • + u n XJ 2n = 0 , j 
from which there is the evident deduction that the said set reduces to a 
single equation : the identity of this equation with (71-) is then assumed. 
Hesse’s converse theorem he treats with a wise caution, deducing as 
before from the vanishing of H the existence of a single equation of the 
form 
du 
a dXj 
0 du 
+ % + 
= 0, 
but then adding, “ ma rimane da dimostrare che le a , f3 , . • . sieno quantita 
costanti.” 
Lastly, he notes that if there be two such equations with constant 
coefficients, the function is transformable into one with two fewer variables, 
and all the primary minors of H vanish. 
* See (a) in Jacobi’s proof of 1849. 
