1888-89.] Dr T. Muir on the Theory of Determinants. 42 7 
found him, indeed, in almost constant employment of the functions. 
In the memoir now reached, however, we have still stronger evidence 
of his interest in the subject, and of his opinion as to its importance. 
Knowing of no succinct and logically arranged exposition of their 
properties readily accessible to mathematicians, he deliberately set 
himself the task of preparing a memoir to supply the want. In his 
few words of preface he says : — 
“Sunt quidem notissimi Algorithmi, qui aequationum lin- 
earium litteralium resolutioni inserviunt. Neque tamen video 
eorum proprietates praecipuas, ita breviter enarratas atque in 
conspectum positas esse, quantum optare debemus propter 
earum in gravissimis quaestionibus Analyticis usum. Scilicet 
illae proprietates quamvis elementares non omnes ita tritae sunt, 
ut quas indemonstratas relinquere deceat, et valde molestum est 
earum demonstrationibus altiorum ratiociniorum decursum 
interrumpere. Cui defectui hie supplere volo quo commodius 
in aliis commentationibus ad hanc recurrere possim ; neutiquam 
vero mihi propono totam illam materiam absolvere.” 
While Jacobi was aware, as we have already partly seen, of the 
labours of Cramer, Eezout, Yandermonde, Laplace, Gauss, and Binet, 
his main source of inspiration is Cauchy. Of all the writers 
since Cauchy’s time, indeed, he is the first who gives evidence of 
having read and mastered the famous memoir of 1812. It scarcely 
needs be said, however, that his own individuality and powerful 
grasp are manifest throughout the whole exposition. 
At the outset there is a reversal of former orders of things ; 
Cramer’s rule of signs for a permutation and Cauchy’s rule being 
led up to by a series of propositions instead of one of them being 
made an initial convention or definition. This implies, of course, 
that a new definition of a signed permutation is adopted, and that 
conversely this definition must have appeared as a deduced theorem 
in any exposition having either of these rules as its starting 
point. 
The new definition has its source in Cauchy, and rests on the 
well-known agreement as to a definite mode of forming the pro- 
duct P of the differences of an ordered series of quantities. This 
being settled to be 
