395 
1888-89.] Dr T. Muir on the Theory of Determinants. 
The analytical portion of the essay is to a considerable extent 
identical with the original memoir. In so far as there is a differ- 
ence, the change is towards greater simplicity, less seemingly aimless 
plunging into widely extensive theorems, and in general a better 
and more attractive style of exposition. Less space too is given to 
it, — not even half what is occupied by the portion on the tetrahedron, 
the main aim now being to urge on mathematicians the capabilities 
of the analysis in its application to geometry. 
The matters falling to be noted as not having been given in the 
original memoir are few in number and of little importance. In 
restating the theorem 
( abc ...?*, a/3y . . . p) = ( abc . . . r, aj3y . . . p) 
the remark is incidentally made that the order of the terms on the 
one side is never the same as that on the other except when the 
number of bases is 1, 2, or 3 ; for example, the number of bases 
being 4, we have 
(abed, 1 234) = af^eft^ - af 2 cft^ - af> z cfl± 
+ a A C 4^2+ • • • > 
whereas 
(abed, 1234 ) = a 1 b 2 c s d^ — a 1 b 2 d 3 c 4 ^ — a l c 2 b 3 d^ + . . . 
+ a Y c 2 df>± + . . . , 
the difference first appearing at the fourth term. (ix. 6) 
Bezout’s recurrent law of formation, formerly merely enunciated, 
is now accompanied by a demonstration. This is not without its 
weak point, the cause of which, as might be expected, is the 
awkwardness of Reiss’s rule of signs. The first paragraph, which 
will suffice to show its character, is as follows (p. 233) : — 
“ Portons notre attention d’abord, seulement sur la fonction 
(abc . . . r, a f3y . . . p). Si l’on se represente la maniere 
dont on fait les permutations des n elemens a,/3,y, ... p, on 
verra qu’a partir de la premiere, il y aura 1.2.3 . . . (n-1) 
complexions qui commencent par a, et que, si Ton separe cet 
element par un trait vertical des autres, on aura a droite toutes 
les permutations des elemens (3,y, . . . p. Les 1.2.3. . . (n- 1) 
premiers termes de (abc . . . r, a/3y . . . p) commencent 
