212 



DR. THOMAS MTJIR ON 



= (cd + 1) (ab+l) - hd.ac + (be + 1) ad, 

 = abed + ab + cd + 1 - abed + abed + ad , 

 = ~K(a,b,c,d) . 



Again, taking the case where the originating function is 



K (a,b,c,d,e,f,) 

 we have our Hessian-like Pfaffian 



3 2 K 3-^ &k_ &K 3 2 K 

 da.db da.de da.dd da.de da.df 

 3 2 K 3 2 K 3 2 K 3 2 K 

 dbFc Wdd, db.de db.df 

 3 2 K 3 2 K 3 2 K 

 3c.3c2 dc.de dc.df 

 3 2 K 3 2 K 

 3d.3e 3d.3/ 

 3 2 K 

 3e.3/ 



= ,(c,d,e,/) b(d,e,f) (b,c) (e,f) (b,e,d)f (b,e,d,e) 



a(d,e,f) ac(e,f) a(e,d)f a(c,d,e) 



(a,b)(e,f) (a,b)df (a,b){d,e) 



(a,b,e)f (a,b,c)e 



(a,b,e,d) 



= (c,d,e,f)da,b){e,f) (a,b)df (a,b) (d,e) 



(a,b,c)f (a,b,c)e 



(a,b,c,d) 



- b(d,e,f) I ac(e,f) a(c,d)f a(c,d,e) 



I (a,b,c)f (a,b,c)e 



(a,b,e,d) 



+ 



+ (b,c,d,e) I a(d,e,f) ac(e,f) a(e,d)f 



I (a,b) (ej) (a,b)df 



(a,b,c)f 



Now it is easily verified that the five minor Pfaffians here are equal to 



(a,6)-K, ac-'K, ad-~K, ae-K, af-~K; 



consequently the original Pfaffian 



= K{ (a,b) (c,d,e,f) - abc(d,e,f) + a(b,e)d(e,f) - a(b,c,d)ef+ a(b,c,d,e)f ] , 



= K-[ (a,b) (e,d,e,f) - {abcd(e,f) + abef) + {abcd(ej) + ad(e,f)} - a(b,e,d)ef + {a(b,e,d)ef+ a(b,c)f} ] , 



= K-[ (a,b) (c,d,e,f) - abcf+ ad(e,f) + a(b,e)f] , 



= K-[ (a,b) (c,d,e,f) + ad(e,f) + of] , 



= Kia,b,c,d,e,f) , 



= K 2 . 



(10) These two cases raise the presumption that when the originating function is 



K («l» «2> a 3> ' a ln) 





