( 309 ) 



XIV.— On a Class of Alternating Functions. By Thomas Muir, LL.D. 



(Read 7th March 1887.) 



A glance at the expression 



(a — a)(a — 8) (a — y)(a ■ 



S) , (b-a)(b-8)(b-y)(b-S) 



(a — b)(a — c)(a — d) 



+ 



(e- a)(c-8)(c-y)(c-S) 

 ~*" (c-a)(c-b)(c-d) " 1 " 



(b-a)(b-c)(b-d) 



(d-aXd- 8)(d- y)(d-8) 

 {d-a)(d-b)(d-c) 



is sufficient to verify the fact that it is symmetric with respect to a, b, c, d, and 

 also with respect to a, fi, y, 8. It is likewise, although not quite so evidently, 

 an alternating function with respect to the interchange 



(a b c d\ m 

 a /3 y S } ; 



that is to say, if a and a be interchanged, and at the same time b and ft, c and 

 y, d and S, the function is not altered in magnitude, but merely changes sign. 

 With a little trouble, indeed, the expression can be transformed into 



(a + b + c + d) - (a + /3 + y + S), 



or say 2« — 2a 



This alternating function is only one of a large class to which it is proposed 

 here to direct attention. It may be looked upon as in a certain sense the 

 generator of the other members of the class, because they are derivable from it 

 by prefixing to each of its component fractions a symmetric function of the three 

 variables which occur only once in the corresponding denominator, e.g., the 

 symmetric function bc + bd + cd prefixed as a factor to the first fraction, the like 

 function ac + ad + cd prefixed to the second fraction, and so on. The various 

 kinds of symmetric functions which may be used in this way as prefixed factors 

 are best expressed in the form 



b m b n b r 





1 b b 2 



c m c n c r 



-^ 



1 c c 2 



d' n d n d r 





1 d d 2 



or 



\b m c n d r \ 



m, n, r, having any three values chosen from 0, 1, 2, 3, 4; for example, 

 the above-mentioned instance 



bc + bd + cd 

 is, in this form, 



I b°c 2 d 3 1 



, b chl 2 1 



VOL. XXXIII. PART II. 



2z 



