429 
1907-8.] Dr Muir on the Theory of Hessians. 
“ et l’equation 
u(x 1 , x 2 , . . . , x r ) = 0 
est elle-meme homogene” — a sort of converse of Euler’s theorem above 
o 
referred to. 
Brioschi, Fr. (1854, March). 
[La Teorica dei Determinant!, e le sue Principali Applicazioni ; 
del Dr Francesco Brioschi: viii-j-116 pp. ; Pavia. Translation 
into French, by Combescure, ix + 216 pp. ; Paris, 1856. Transla- 
tion into German, by Schellbach : vii + 102 pp. ; Berlin, 1856.] 
The last section (§ 11, pp. 106-116) of Brioschi’s text-book is headed 
“ Del determinante di Hesse.” Opening with the definition of “ l’Hessiano,” 
it gives a clear and orderly exposition of a goodly number of the main 
theorems up till then discovered, with geometrical applications.* 
Separated altogether, however, from these is a demonstration (pp. 20, 
21) which strictly belongs to this section. Recognising that Hesse’s expres- 
sion (1847, August) for the product of u and its Hessian A is in reality 
obtained by eliminating x lf x 2 , x 3 from four equations, Brioschi performs 
this elimination openly, with the result : 
0 
m 
-u u, . . . u n 
m - 1 
U \ U \\ U Y2 
u ln 
U n ^nl 
5 
m 
m-X 
uA 
+ 
U l 
u 2 
• • • “Un 
U 1 
u n 
Wi2 
. . . U ln 
U 2 
^21 
u 22 
. . . u 2n 
U n 
Uni 
U n 2 ■ 
• • u nn 
Bellavitis, G. (1857, June). 
[Sposizione elementare della teorica dei determinanti. Veneto, 
Memorie . . . Istituto, vii. pp. 67-144.] 
Bellavitis (§§ 79, 80) denotes “YHessiano delle funzione <p” by 
I DJA D^Dy I </>, 
* The reason given for the deduction U rs = Navx s , which occurs in his presentation of 
Jacobi’s proof of the year 1849, is disappointing. 
