136 Proceedings of Royal Society of Edinburgh. [sess. 
write out in ttie usual triangular form the \n(n - 1) differences of 
the elements, view the differences thus arranged as being the 
elements of a Pfaffian, and then take the terms of this Pfaffian, 
For example, in the case of if (abed) we form the Pfaffian 
\b - a c-a d — a 
c-b d — b 
d-c 
the expansion of which is 
(i b -a) (d-c) - (c - a) (d -b) + (d-a) (c- b) : 
and this is exactly Jacobi’s expansion with the non-linear factors 
left out. Again, in the case of if(abcdef) we form the Pfaffian 
e 
1 
e 
i 
rO 
d — a 
e - a 
f-a 
c-b 
d-b 
e-b 
f-h 
d-c 
e - c 
f-o 
e - d 
f-d 
take the expansion of it 
f-e 
(& - a){d - c)(/ - e) - ( b-a)(e - 
c)(f-d) 
+ (6- 
a) ( c - d) 
and all that remains in order to obtain Jacobi’s expansion is to 
annex to each term the appropriate non-linear factor. 
No separate rule of signs, it will be observed, is necessary, the 
signs of the expansion of the auxiliary Pfaffian being exactly the 
signs of Jacobi’s expansion of the difference-product. 
(4) Let us now look at the mode of formation of the non-linear 
factor. 
In the case of if (abed) and the case of if(abcdef) the types are 
c 2 d 2 + a 2 b 2 , 
c 2 c? 2 e 4 / 4 + c 4 c7 4 e 2 / 2 + . . . . ; 
and these, we fortunately observe, resemble determinants, and are, 
in fact, found to be the permanents 
+ + + + 
1 a% 2 
1 a 2 b 2 aW 
1 cH 2 
1 c 2 d 2 c 4 # 
1 e 2 / 2 e 4 / 4 
so that the complete first term of Jacobi’s expansion may be accur- 
ately written 
