1907-8.] Dr Muir on the Theory of Skew Determinants. 305 
Scheibner, W. (1859, July). 
[Ueber Halbdeterminanten. Berichte . . . Gess. d. Wise, zu Leipzig : 
Math.-phys. Cl., xi. pp. 151-159.] 
This paper is not put forward by its author as containing new matter, 
being in fact such an exposition of the theory of Pfaffians as would suitably 
have formed a chapter, and a good one, of a text-book like Brioschi’s or 
Baltzer’s. 
From the vanishing of a zero-axial skew determinant of odd order 
Scheibner reaches the already known fact that the product of two of its 
coaxial primary minors is equal to the square of a non-coaxial primary 
minor. In a quite fresh manner it is then shown that the square root of 
this is a rational and integral expression (Pfaffian), whose law of formation 
is thereafter established. Naturally following on this comes the proof 
(p. 156) that each of the non-coaxial primary minors is the product of two 
Pfaffians, the result being written in the form 
Lpq {p 1> * • • 5 ^rn> 6j . . . , p — 1)(^ “f” 1 J • • * 3 ^rn 3 9, ... , 2 1 )> 
where the suffixes of the elements of the original determinant are 0, 1, . . . , 
2m. On a later page (p. 158) it is shown that a similar proposition holds 
when the original determinant is of even order, namely, 
= (-iy(0, 1,2,. .. 3 2 m+1 )(g + l,:..3 p-l>p + l,...,q-l). 
Cayley’s theorem regarding a “ bordered ” skew symmetric determinant 
thus appears broken up into two parts. 
The paper concludes with the suggestions that a skew symmetric 
determinant should be called a Weehseldeterminante, that its square root 
should be called a Halbdeterminante , and that the latter should be denoted 
b y 
a 0\ a 02 ft 03 • • • • a 0p 
a l2 • • • • A&i p 
et 23 .... et~2p 
® p—\,p > 
an expression which would thus be an alternative for (0, 1, 2, ... , p), and 
which would vanish when p was even. 
VOL. XXVIII. 
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