501 
1887.] Dr. T. Muir on the Theory of Determinants. 
seront des fonctions symetriques permanentes, la premiere du 
second ordre et les autres du troisieme ; et au contraire, 
cif>2 “H af) 3 -t- af>Y — af>^ — cq5 3 — af> 2 > 
sin (a 1 — a 2 ) sin (cq - af sin ( a 2 — a 3 ) 
seront des fonctions symetriques alternees du troisieme ordre.” 
The question of nomenclature being settled there next arises the 
question of notation. This also is decided on the ground of the 
resemblance of the functions to symmetric functions. It being 
known that any symmetric function is representable by a typical 
term preceded by a symbol indicating permutation of the variables, 
e.g. 
S( a 1 b 2 ) or S*(aA) standing for a 1 b 2 -{-a 2 b 1 
and S 3 (a 1 b 9 ) standing for a 1 b 2 + a 2 b s + af l + a 2 b 1 + ajb 2 + a 1 b B ; 
also, that any non-symmetric function may be taken as the typical 
term of a symmetric function, the question arises whether the like 
may not be true of alternating functions. A lengthy examination 
of the latter point leads to the conclusion that any non-symmetric 
function K cannot be the originating or typical term of an alter- 
nating function unless it satisfies a certain condition, viz., that it be 
such that any value of it obtained by an even number of transposi- 
tions of indices will be different from any other value obtained by 
an odd number of transpositions. Should, however, this condition 
be satisfied, and K a , K^, K y , .... be all the values of the former 
kind, and K x , K M , K^, .... all the values of the latter kind, then 
(K a + K^ + K y + . . . . ) — (Ka + K^ + K„ + ....) 
is an alternating function and is appropriately representable by 
S(±K) 
if the indices appearing in K alone are to be permuted, and by 
S w ( ± K) 
if the indices to be permuted be 1, 2, 3, . . . , n. For example, 
taking the typical term a x b 2 we have 
S ( ± cq5 2 ) = af) 2 - a 2 b 1 , 
and S 3 ( ± of 2 ) = af 2 + a 2 b B + ajb Y - - a B b 0 - a 1 b B , 
= S 3 ( + afj = S 3 ( + afz) = . . . . 
