1899-1900.] Dr Muir on the Theory of Alternants. 
117 
The meaning will be made quite apparent by taking a case other 
than Jacobi’s above referred to, say the case where there are six 
elements, a 0 , a lt a 2 , . . ., a 5 . According to the rule, what we 
have got to do at the outset is to form the term 
( a i — tf 0 )(«3 — a 2)( a b — tt 4)^( a 0 C/ 'l)°( t< 2 a 3) 2 ( a 4 a 5) 4 i 
then derive from it two others by the cyclical substitution 
/«3 «4 « 5 \ 
W 4 «5 (Jtj ; 
and finally, from each of these three derive four others by the 
■cyclical substitution 
'This being done, the sum of the fifteen terms so obtained 
■can be taken as an expansion of the difference-product of 
•Qf 0 ) • • • •» 
Although, as has been said, the theorem is given without proof, 
it has to be noted that Jacobi draws attention to the fact that the 
number of ultimate terms in the expansion of the compound term 
(a 1 - a 0 )(a 3 - a 2 ). . . (a n - %-i)2( Vi) 0 (%%) 2 (a 4 a 5 ) 4 . . . (a n -ia n ) n ~ l 
is 
n+l / 
2 2 . f 1.2.3 
that the number of ultimate terms obtainable from all the compound 
terms of this form is 
2T(l.2.3 .... 'ffj ■ (3.5 . .. . n): 
and finally that this is equal to 
1.2.3 . . . (n+l), 
a result which agrees with what we know of the difference- pro- 
duct from its determinant form. 
From this general theorem regarding the difference-product of 
an even number of elements, an advance is made to a theorem of 
•still greater generality, the means employed in obtaining it being 
