CLASS OF ALTERNATING FUNCTIONS. 311 



F °' 2 ' 4 = &(a, b, c, d) [ " ' a ° bW ' 2a ~ I a%2c4dl I ZaPy] , 

 F °' 3 ' 4 = f *(g, 6, c, a 1 ) [ I a ° b3cid * I 2a _ I a ° 53c ^ 1 1 2 «£y] ' 



Fl ''' ,= gK»,Urf) [ ' al5W ' + ' aWd ° ' Sa8y< Q ' 



Fl ' 2 ' 4 = ftfg b c d) \_ ~ ! aWd? ' ' 2a + ' a ^ Vrf ° ! 2a/3y 5~1 , 



Fl ' 3 ' 4 = g*(g, &, e, d) [ I al&3c ^ 2 1 2a/3 + I ftl&3c4rf ° ' ^Wl - 

 F 2 , 3 , 4 = ^ ^ c> rf) [ ~ I aWcW | 2a/3y + |aWd«|2aSy<f] . 



Now, from the theory of alternants it is known that 



\aWd 3 \ = ^(a,b, c,d), 

 | a ojW | = £K a > h > c > d ) x 2a , 

 | aObhW | - £*(a, &, c, d) x 2a& , 

 | a°5W 4 1 = f Ha, b, c, d) x Zabc , 

 | aWoP 1 = g*(a, b, c, d) x "Edbed ; 

 and thus it follows that 



F , 1,2= 2a — 2a 



Fo, i, s = 2a& — 2a/3 , 



F ,i,4 = 2«&, 2a-2a/3.2a. 



F , 2, 3 = 2aZ>c — 2a/3y , 



F 0)2 ,4=2a&c . 2a — 2a/3y . 2a,- 



F a ,3,4=2a5c . 2a J 8-2a i 8y . 2a&, 



Fi, 2 ,3— 2a&ca'— 2a/3y<5 , 



Fi, 2, 4 = 2afcrt > . 2a — 2a/5yo* . 2a , 



F li3)4 = 2a&ca 7 . 2a/3-2a£y3 . 2a&, 



F 2 ,3,4 = 2a&ca' . 2a/3y — "EafiyS . 2,abc , 



all the expressions on the right being manifestly alternating functions with 

 respect to the interchange 



/a b c d\ 



l« /? 7 Sj' 



These expressions are seen to be the ten determinants of the matrix 



1 2a 2a/3 2a£y 2a/3yo 

 1 2a 2a& 2a&c ~Zabcd 



and consequently, if we represent this matrix by 



o" a - ! (T 2 <r 3 <r A 



