422 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Hesse, 0. (1851, March). 
[Ueber die Bedingung, unter welcher eine homogene ganze Function 
von n unabhangigen Variabeln durch lineare Substitutionen von 
n andern unabhangigen Variabeln auf eine homogene Function 
sich zurtick fiihren lasst, die eine Variabel weniger enthalt. 
Crelle’s Journal , xlii. pp. 117-124 : or Werke, pp. 289-296]. 
Hesse here returns to the subject of § 19 of his original paper, calling 
2 ± u n u 22 . . . n nn or A the determinant of u with respect to the variables 
x 1 , x 2 , . . . x n , and 2 ± u n u 22 . . . u nn or y the determinant of u with respect 
to y 1: y 2 , . . . , y n , and proving once more his theorem that 
V = r 2 A . 
He then supposes that in the result of the transformation y n does not 
appear, and says that as this implies that u ln , u 2n , . . . , u nn all vanish, it 
follows that V = 0, and that therefore from the said theorem A also must 
vanish. There is thus obtained the result that, “ Wenn eine homogene 
ganze Function der n unabhangigen Variabeln x x , x 2 , . . . , x n , durch 
x k = a\y x + aXy^+ ••• +a k n y n 
in eine Function der Variabeln y x , y 2 , . . ., y n ubergeht , in welcher eine 
dieser Variabeln fehlt, so ist die Beterminante dieser Function in Rilck- 
sicht auf die Variabeln x r , x 2 , . . . , x n , identisch gleich 0.” 
The rest of the paper is occupied with the converse theorem ; but as 
the author himself came to be dissatisfied with his attempt at a proof and 
returned to the subject seven years later, it need not be entered on here. 
Sylvester, J. J. (1851, April). 
[Sketch of a memoir on elimination, transformation, and canonical 
forms. Cambridge and Bub. Math. Journ., vi. pp. 186-200: or 
Collected Math. Papers, i. pp. 184-197.] 
The expression “ determinant of a function ” or, more definitely, 
“ determinant of a function in respect to certain variables ” occurs re- 
peatedly in Sylvester’s writings of the year 1850, the accompanying 
notation being 
B(u); 
for example, when dealing with ternary quadrics U and V, expressions like 
LJ L j (AU+/XV) 
A fx xyz 
