445 
1888-89.] Dr T. Muir on the Theory of Determinants. 
siquidem pro indicibus inferioribus m, m' &c. sumuntur 
quilibet n+1 diversi e numeris 0, 1, 2 ... . p.” (xviii. 7) 
The two remaining sections (15 and 16) deal with a special 
system of simultaneous linear equations, interesting application 
being made to the theory of the Method of Least Squares. 
It is important to note, in conclusion, that from one point of 
view Jacobi’s memoir was but the introduction to two others of 
really greater importance, both treating of a special class of deter- 
minants. The first concerns determinants of the kind afterwards 
deservedly associated with his name, and bears the title “De deter- 
minantibus functionalibus .” It occupies the forty-one pages (pp. 
319-359) immediately following the general memoir. The other, 
with the title “ De functionibus alternantibus earumque divisione per 
production e differ entiis elementorum conflatum,” treats of those deter- 
minants, first considered by Cauchy, in which the members of one 
set of indices represent powers, and to which the name alternants 
afterwards came to be assigned. It extends to twelve pages (pp. 
360-371). The three memoirs together constitute an excellent 
treatise on the subject, and are known to have been markedly 
influential in spreading a knowledge of it among mathematicians. 
CAUCHY (1841). 
[Note sur les diverses suites que l’on peut former avec des termes 
donnes. Exercices d’ analyse et de phys. math ., ii. pp. 
145-150.] 
[Memoire sur les fonctions alternees et sur les sommes alternees. 
Exercices d? analyse et de phys. math., ii. pp. 151-159.] 
[Memoire sur les sommes alternees, connues sous le nom de resul- 
tantes. . Exercices d’ analyse et de phys. math., ii. pp. 160-176.] 
[Memoire sur les fonctions differentielles alternees. Exercices 
di analyse et de phys. math., ii. pp. 176-187.] 
From internal evidence there can be little doubt that this series 
of papers, containing the fundamental conceptions and salient pro- 
