312 DR THOMAS MUIR ON A CLASS OF ALTERNATING FUNCTIONS, 

 the results take the form 



Fo,l,2 = | S 1 O- | , 

 Fo, 1, 3 = i S 2 (T | , 



1^2,3,4 = I s 4 cr 3 I , 



where the suffixes on the right are got by subtracting from 4 each of those 

 omitted on the left. 



With the help of this notation, also, we can combine all the results in one 

 statement, viz. : — - 



Ifm, n, r co-ranged in order of magnitude be any three of the values 0, 1, 2, 

 3, 4, and u, v arranged in order of magnitude be the remaining two, then 



\b m c"d r \ (a-a){a-fi)(a-y)(a-$) \ a m Pd r I _ (b-a)(b- /3)(b-y)(l— 8) 



Jb°cW\ (a-b)(a-c)(a-d) I a°c l d 2 I (6 - a)(b - c)(b - d) 



| a m b"d r | (c-a)(c-/3)(c-y)(c-S) \ a m b n c>- 1 (d-g)(d-jB){d-y)(d-8) 



+ \a b 1 d 2 \' ' (c-a)(c-b)(c-d) + | a°¥c 2 | ' (d-a){d-b){d-c) 



where s and a are explained by the examples s. 2 = lab, a 3 = 2<x/3y. 



The case where # = 4 has been given by Sylvester, being the subject of his 

 unsolved problem No. 2810 in Mathematics from Educational Times, vol. xlv. 

 p. 129. 



Of course the foregoing results are not at all confined to two sets of four 

 variables [a, b, c, d), (a, /3, y, 8). Two sets of w variables have not been taken 

 merely on account of inconvenience in writing. The typical term for the next 

 case (two sets offve variables) is 



I b°c l d*e? | ' (a - b)(a - c)(a - d)(a - e) 

 where (m, n, r, s) is a set of four values taken from 0, 1,2, 3, 4, 5. 



