1908-9.] Dr Muir on the Theory of Jacobians. 
505 
Cayley, A. (1847, February). 
[On the differential equations which occur in dynamical problems. 
Cambridge and Dub. Math. Journ., ii. pp. 210-219 : or Collected 
Math. Papers, i. pp. 276-284.] 
This is a short exposition of Jacobi’s elaborate memoir of 1844 with 
considerable variation in the details. The portion (§ 1) which concerns 
us is of course that referring to the “ fundamental lemma.” This is 
o 
established in its third form, the proof, like that originally given by Jacobi, 
being dependent on the theorem 
0R ^ SXr 
dx k 
but differing in appearance, mainly because of the use of differentials. 
Bertrand, J. (1851, February). 
[Memoire sur le determinant d’un systeme de fonctions. Journ. (de 
Liouville) de Math., xvi. pp. 212-227 : abstract in Comptes 
Renclus .... Acad. des. Sci. (Paris), xxxii. pp. 134-135.] 
Recalling how Jacobi had insisted on the marked analogy between a 
functional-determinant and a differential-coefficient, Bertrand at once 
intimates the adoption of a new definition of the former which in his 
opinion makes the analogy still more striking, and from which the 
properties of the determinant are deducible like mere corollaries. 
Save that A and S are used where Bertrand without distinction uses d, 
the following is the definition: — If f \ , f 2 , . . . , f n be functions of x ly x 2 , 
. . . , x n , and the latter receive n distinct sets of increments 
^jJi .... 
Ag&i AgX., .... Aq.X-,^ 
A rP^\ A,,.^ 2 .... A n X n 
with the result that the corresponding increments of the functions are 
1 • • • • A 1 _/ n 
^2/l ^ 2/2 • • • • ^2 ‘bn 
^nf\ “At f 9 .... A n f n , 
