414 Proceedings of the Koyal Society of Edinburgh. [Sess. 
stricted to the second degree, are not so far apart : for, according to Hesse, 
the determinants of the same two forms are 
2 a 2b 
2 cq 2 b 3 2b 2 
2b 2c 
3 
26 3 2a 2 2 \ 
2b 2 2b Y 2 a z 
i.e. - 2 2 (6 2 - ac) , - 2 3 (a 1 b 1 2 + a 2 b 2 2 
The elements of Hesse’s “determinant of /” being evidently the second 
differential-quotients of /, those in conjugate places must be equal — that 
is to say, the determinant is axisymmetric. 
The first result enunciated is (p. 85) — Die Determinante der Deter- 
minant# einer gegebenen homogenen Function dritten Grades von drei 
Variabeln ist gleich der Summe der gegebenen Function und ihre 
Determinante, jede mit einem passenden constanten Factor multiplicirt. 
In symbols at a later date this would have been written 
H{HK)} = cu S3 + c'H(w 33 ) . 
Following thereupon is a theorem of like type 
H | CjWgg + C 2 H(w 33 ) j- = C 3 W 33 + c 4^(%3) 5 
and this is used to solve the equation 
33 ) = M 33 j 
where u 33 and u' S3 stand for ternary cubics, and u' S3 is the unknown. 
The effect of linear transformation, so strikingly brought to the front 
by Boole three years before, is then (§19) entered on, / being no longer 
a ternary cubic, but any function whatever of x v x 2 , , x n , and supposed 
to be expressed also as a function of the variables y v y 2 , ... , y n by means 
of the equations 
a^ ) x 1 + </\ 2 + • • • • + a^ ] x n = y 1 ,^ 
a 2 "t a 2 ■** • • • • + ///i = y < 2 . ? 
a n X l + a n X 2+ 
Denoting the determinant of / when viewed as a function of the x’s by <p, 
and when viewed as a function of the y’s by (p Hesse affirms that 
</> = r 2 <£', 
where r is the determinant formed from the coefficients of the x’s in the 
