1888.] Dr Muir on a Class of Sim'ple Alternants. 299 
It would probably be more convenient to extend the definition to 
any determinant loliich is an alternating function ; and it would 
certainly be advantageous to include within the scope of the term 
determinants of the form about to be considered in the present 
paper. They differ from the above alternant proper in having one 
or more columns such as 
/(< 2 ).r(all variables except a) 
/(6).F(all variables except V) 
/(c).F(all variables except c ) , 
where F, unlike the other functional symbols, denotes a symmetric 
function. It is readily seen that such a determinant possesses the 
alternating property ; indeed, the effect produced by interchanging 
any two of its variables is exactly of the same kind as that pro- 
duced in the narrower class of determinants which up till now have 
monopolised the name “ alternant.” A priori, therefore, the term is 
as appropriately applicable to the one class as to the other. 
(2) Very few of this extended class of determinants have as yet 
been investigated, and such of them as have turned up have been 
dealt with by a process which becomes exceedingly lengthy when 
complicated cases are taken, or when any attempt at generalisation 
is made. The nature of it will be understood by observing its 
application to a particular case, say the case of the determinant 
1 a a‘^ a{ b^c^ + -f 
1 c c2 c{d^a^^d%^^a%^) 
\ d d?' d{cAb^ -1- a^d^ -1- b^c^) . 
By using '%pdb'^ as a contraction for 
+ a'^d‘^ + , 
the last column is written 
a{%a%^ - a'^b^ - a'^d^ - a"^d^) 
b{%a^b^-b^c^ - bH^-b^a^) 
c{%a‘^b^ - c‘^a^ - c%^ - c^d^) 
d{%a^^-d^a^~d^b^-d^c^), 
