419 
1907-8.] Dr Muir on the Theory of Hessians. 
results are reached, such, for example, as the theorem that it is 'possible to' 
determine constants c x , c 2 so that 
^1^24 ^^(^24) 
may be an exact square. Most of the matter, however, more directly 
concerns the quartic than its determinant. 
Aronhold, S. (1849, July). 
[Zur Theorie der homogenen Functionen dritten Grades von drei 
Variabeln. Crelle’s Journal , xxxix. pp. 140-159.] 
This is an inspiration from, and a striking development of, the latter 
part of Hesse’s paper of the year 1844, and like that paper may be said to 
concern itself more with the ternary cubic than with the so-called 
determinant of that function. In regard to the latter, however, there is 
one very noteworthy result : for, just as Cayley in 1845 established the 
two invariants I and J of a binary quartic u 24 , and used them for the 
expression of H {a-u 2i + AH(u 24 )} in the form A -u 2i 4- B-H(^ 24 ), so 
Aronhold here announces the two invariants S and T of a ternary cubic, 
and gives the similar expression for 
H{a.w 33 + 6.H(w 33 )}. 
Obtained from this by putting a = 0 is the result 
H { H(m S8 ) } = 3S 2 -« 33 - 2T-H(m 33 ) 
— the longed-for definite form of Hesse’s theorem of the year 1844. 
We may also note in passing that the result of eliminating x, y, z from 
the equations 
_ n ^33 _ 0 du 33 _ 
dx ’ dy ’ dz 
is expressed in terms of S and T, namely, 
T 2 - S 3 = 0 ; 
or, in later phraseology, that the discriminant of u 33 is T 2 — S 3 . 
Hesse and Jacobi (1849, December). 
[Auszug zweier Schreiben des Prof. Hesse an den Herrn Prof. Jacobi 
und eines Schreibens des Prof. Jacobi an Herrn Prof. Hesse. 
Crelle’s Journal, xl. pp. 316-318.] 
Hesse having communicated to Jacobi a theorem regarding a homo- 
geneous function of three variables, Jacobi sent back a proof showing that 
