1899-1900.] Dr Muir on the Theory of Skew Determinants. 185 
wenn man die Horizontalreihen und Yerticalreihen der Co- 
efficienten mit einander vertauscht. Fur unsern besondern 
Fall nun wird, wenn wir die Determinante mit A bezeichnen, 
hieraus folgen : A = ( - 1) 2J+1 A , da jedes Glied der Deter- 
minante ein Product aus p+ 1 Coefficienten ist, von denen 
jeder durch Y ertauschung der Horizontal- und Yerticalreihen 
sich in sein Negatives verwandelt. Diese Gleichung 
A = (-l) J5+1 A aber kann nur bestehen, wenn jp + 1 eine 
gerade Zahl ist, wofern nicht A = 0 sein soil.” 
Thus, besides being the originator of the functions which came 
long afterwards to be known and are still known as ‘ Pfaffians,’ 
Jacobi was the first to discover and prove the now familiar- worded 
theorem “ A zero-axial skew determinant of odd order vanishes 
JACOBI (1845). 
[Theoria novi multiplicatoris systemati sequationum differen- 
tialium vulgarium applicandi. Grelle’s Journ ., xxvii. pp. 
199-268, xxix. pp. 213-279, 333-376.] 
As is well known, this long and exhaustive memoir of Jaccbi’s 
is broken up into three chapters, — the first giving the definition 
and various properties of the new multiplier, the second explaining 
the application of it to the integration of differential equations, and 
the third illustrating this application by means of particular 
differential equations of historical interest. One of the latter is 
the equation associated then, and still more since, with the name 
of Pfaff, the discussion of it occupying §§ 20, 21 on pp. 236-253 of 
Yol. xxix. We are thus prepared to find the function, defined by 
Jacobi eighteen years before, again referred to. 
The old definition of the function, which he here denotes by R, 
is practically repeated, the initial and originating term being now 
of the form a 12 a 34 . . . a 2m _ li2m * an( l then he makes the pregnant 
general remark that the properties of R are analogous to those of 
determinants. Prominence is given to the theorem regarding the 
effect of interchanging two indices. This is followed by the twin 
pair of identities 
