Expression of Certain Symmetric Functions. 



315 



= &{a,b) . £{c,d) . £(e,f) . { a s b s c r d y + c s d s a r b r + [ 



- p(d,b) . ?(c,e) . p(dj) . { tf&W + f 



+ 



= S?(a,6) • ?(M) • *W) 



+ 

 1 &''&'" a*6* 



1 c r d'' c s d s 



1 <f e s / ? 



where i?(a,b) . i?(c,d) . (fi(e,f) is any one of the fifteen terms of the 

 Pfaffian 



\b — a c — a cl — a e — a f—ct 



c—b d—b e—b f—b 



d—c e—c f—c 



e—d f—d 



f-e 



When r=2 and s = 4 this degenerates into Jacobi's theorem regard- 

 ing the difference-product. 



