1888 - 89 .] Dr T. Muir on the Theory of Determinants. 
393 
first five integers have been got, and that the altering of the sign 
simultaneously with the shifting of the 5 is in accordance with 
Cramer’s rule, because every time that the 5 is moved one place to 
the left the number of inversions is increased by unity. The only 
question remaining is as to whether all the permutations are thus 
obtainable ; and as it is seen that each of the 24 permutations of 
the first four integers gives rise to 5 permutations of the first five, 
we have at once grounds for a satisfactory answer. (iii. 27) 
LEBESGUE (1837). 
[Theses de Mecanique et d’Astronomie. Premiere Partie : For- 
mules pour la transformation des fonctions homogenes du 
second degre a plusieurs inconnues. Lioumllds Journal de 
Math., ii. pp. 337-355.] 
This simply-worded and clear exposition is a natural outcome of 
a study of Jacobi’s memoirs on the subject. Like these it mainly 
concerns determinants of the special form afterwards individualised 
by the term axisymmetric ; and, indeed, it is notable as being the 
first memoir in which a special name is given to a special form, the 
expression u determinants symetriques” being repeatedly used for the 
particular determinants referred to. 
His general definition is (p. 343) : — 
“ Si l’on considere le systeme d’equations 
T ^ 1,2^2 + + A lt J n = m i > 
i 
j ^-2,1^1 + ^2,2^2 + + A 2n t n = W 2 5 
I * 
[ A n f i + A n f 2 + + A n J n = m n , 
le denominateur commun des inconnues t ly t 2 , ... , t n est ce 
que l’on nomme le determinant du systeme des nombres 
' A u 
A h 2 • • • 
A^,! 
A 2)2 • • ■ 
i 1 
^ A n> i 
A n> 2 • • 
A 
