516 Proceedings of Boy al Society of Edinhurgh. 
It occupies, however, only fifteen pages (pp. 70-84), and of these 
only nine are devoted to alternating functions and the solution of 
simultaneous linear equations. Of course, in so limited a space, the 
merest sketch of a theory is all that is possible. An alternating 
function is first defined, the word “alternee” being now set in 
contrast with “ symetrique,” and not, as formerly, with ‘‘per- 
manente.” Functions other than those that are rational and 
integral being left aside, the latter, if alternating, are shown (1) 
to consist of as many positive as negative terms, in each of which 
all the variables occur with different indices, and (2) to be 
divisible by the simplest of all alternating functions of the variables, 
viz., the difference-product. The set of equations 
a^^-h^y + c^z-^. . . + g^u-\-\v = (r = 0, 1, . . .,?^-l) 
is then attacked, the method being — to take the difference-product 
of cq &, . . . /q — denote by D what the expansion of this becomes 
when exponents are changed into suffixes, — denote by A,, the 
co-factor of in D, — then obtain the equations 
Aq(^ 0 d" •'^1^1 h A2(^2 • • • • "b ~ ) 
Aq&o + -f A2?^2 + • • * • + = 0 , 
Ao^o -1- AjCj + A 2 C 2 + ....+ = 0 , 
AJiq + A^/q + + • • • • + A„ _ j/q^ _ 1 = 0 , 
— and thereafter proceed as Laplace had taught. As in the memoir 
of 1812, the “symbolic” form of the values of ir, y, . . . is 
unfailingly given. 
A note is added (pp. 521-524) on the development of the 
difference-product, showing how all the terms may be got from 
one by interchanging one exponent with another, how the signs 
depend on the number of said interchanges, and how it may be 
ascertained whether any two given terms have like or unlike signs. 
It will thus be seen that not only is the name “determinant” 
never mentioned in the chapter, and the notation S±ao^iC 2 ••• ^^n-i 
never used, but that the subject is scarcely so much as touched 
upon. Although, therefore, Cauchy’s text-book went through 
a considerable number of editions, and had a widespread influence, 
it gave no such impulse as it might have done to the study of 
the theory of determinants. 
