134 Proceedings of Royal Society of Edinburgh. [sess. 
the number of terms on the right being 15. To each of these, how- 
ever, when we have removed a monomial factor of the 8th degree, 
we can employ the preceding case of the theorem, e.g., 
1 a 
I 1 b 
L|cW/ 5 | 
1 a 
1 b 
.c 2 d 2 e 2 f 2 &(cdef), 
= (b - a) • c 2 d 2 e 2 f 2 . [ (d - c) (/ - e) (c 2 d 2 + e 2 / 2 ) 
- (e - c) (/ - d) ( c 2 e 2 + d 2 f 2 ) 
+ (f-c) (e-d) (c 2 f + d 2 e 2 )l 
= {b- a) (d.— c) (/ - e) (cWf + cWef 4 ) - 
and in this way we shall obtain for tf(abcdef) an expression consist- 
ing of 45 terms. But when this has been done it will be found 
that the number can be reduced again to 15 by combining the 45 
in a different way into sets of three, viz., by selecting those which 
have three binomial factors in common. Thus just as the first of 
the original 15 terms gives rise to the term 
(5 - a ) ( d-c ) (f - e ) (c 4 #e 2 / 2 -f c 2 d 2 ef 4 ) 
the tenth term, | c°d 1 1 • | a 2 5 3 e 4 / 5 1 , gives rise to 
(5 - a) (d - c) (/- e) (aW/ 2 + dW/ 4 ) , 
and the fifteenth, | e 0 / 1 1 • | a 2 5 3 c 4 c£ 5 1 , 
to 
(b - a) (d - c) (f-e) (aWc 2 d 2 +a 2 b 2 eW) ; 
so that one of the resulting 15 terms is 
(b-a) (d-c) (f- e) (c 4 d 4 e 2 f 2 + c 2 d 2 e 4 f 4 + a 4 b 4 e 2 f 2 + a 2 b 2 e 4 / 4 + a 4 b 4 c 2 d 2 + a 2 b 2 c 4 d 4 ). 
Further than this we do not need to go : it is this 15-termed ex- 
pression which according to Jacobi is the equivalent of if(abcdef). 
(2) The two cases may thus be written — 
&(abcd) = 2(5 -a) (d- c).(a?b 2 + tfd?) , 
Q(abcdef) = 2(5 -a) (d- e) (/ - e).{aPb^dd + . ) ; 
and the question which naturally arises to the mind of one who 
looks at them is as to the law of formation of the terms under the 
symbol of summation and the mode of determining the sign of each. 
Jacobi’s answer to this is to the effect that he would prefer to 
