62 
Proceedings of Royal Society of Edinburgh. [sess. 
et quatre autres nombres aussi quelconques 
faites 
vous aurez 
al , a.2 , a3 , a4 ; 
al a2 - al a2 = aU , 
a2 a3 — a2 a3 •'= a l 2 , 
a3 a4 — a3 a4 = a 1 3 , 
al a3 - al a3 = a 2 l , 
a2 a4 - a2 a4 = a 2 2 , 
al ai - a\ a4 = a 3 l , 
a 3 l a 1 2 - a 2 l a 2 2 + a*l a*3 = 0 
Manifestly this is the identity which in later times came to be 
written 
I a A I • I I - I a A I • I °A I + I a A I • I a A I = 0 ? 
and which, so far as we know, appeared first in its proper connec- 
tion in the writings of Bezout. 
It is curious to note that Fontaine was not satisfied with the 
lemma in this form, but proceeded to take “ autant de nombres 
quelconques que Ton voudra al, a2, . . . . , alO, ” and wrote 
the identity one hundred and twenty-six times before he appended 
et cetera,” the 126tli being 
a 3 6 a 1 7 - a 2 6 a 2 7 + ^6 a l S = 0 . 
Cauchy (1829). 
[Sur l’equation a l’aide de laquelle on determine les inegalites 
seculaires des mouvements des planetes. Exercices de 
Math., iv. ; or CEuvres (2), ix. pp. 172-195.] 
As the title would lead one to expect, the determinants which 
occur in this important memoir belong to the class afterwards 
distinguished by the name “ axisymmetric,” and thus fall to be 
considered along with others of that class. Since, however, the 
proof employed for one of the theorems therein enunciated is 
equally applicable to all kinds of determinants, it would be 
scarcely fair to omit here all mention of the said theorem. 
