1902-3 ] Dr Muir on Axisymmetric Determinants. 
567 
where the integral is extended to all real values of x 1 , x 2 , . . . , x n _ x 
for which 
[Theses de Mecanique et d’ Astronomic. Premiere Partie : Pormules 
pour la transformation des fonctions homogenes du second 
degre a plusieurs inconnues. Liouville's Journ. de Math., ii. 
pp. 337-355.] 
Lebesgue’s subject is exactly that dealt with in the first part 
of Jacobi’s memoir of 1833, viz., the transformation of a general 
homogeneous function of the second degree into one containing 
only squares of the variables. Indebtedness to Jacobi, Cauchy, 
and Sturm is indirectly intimated at the outset, and the paper is 
modestly offered as being new in manner rather than in matter. 
Like Cauchy and Jacobi, the author of course is led to the set 
of equations from which by elimination there is deduced the 
equation for the determination of the coefficients of the new 
variables ; and recognising that “ le premier membre de cette 
equation n’est qu’une de ces fonctions nommees determinants,” 
he devotes his second section of five pages to the properties of 
these functions. Throughout this section prominence is notably 
given to determinants having the elements A a p , Ap a equal ; and 
such determinants are spoken of as “ symetriques,” — a noteworthy 
fact, since up to this time no separate name had been applied to 
any specific form. “ On peut dire alors,” Lebesgue says, 
“ que le systeme est symetrique, puisque les nombres qui 
le forment sont places symetriquement par rapport aux 
nombres a indices egaux A n , A 22 , . . . , A nn qui forme la 
diagonale du systeme.” 
The first proposition is that in a symmetric determinant 
[ff, i] = [i, g] , where [g, i] is used to denote the determinant got 
aq 2 + x 2 2 + • • • + x n i x < 1 , 
and where S stands for 
according as n is even or odd. 
Lebesgue (1837). 
