500 
MAJOR MACMAHON GIST SYMMETRIC FUNCTION’S 
are exchanged with those occurring in the second identity 
1 d- + &oi3/ + . . . + hjjrpcPy'i = (1 + (1 + .. . 
each with each. (Art. 26.) 
(ii.) That we can always proceed to a relation between the operations by Avriting 
Gj,g foY Cj,rj and G^;/'for 5^,^. (Art. 30.) 
I will refer to these facts as the first and second properties of the relation 
respectively. 
§ 5. The First Law of Symmetry. 
32. By means of the equality 
(M)z={m) (m)i 
which has been established ante (Art. 26), it is clear that any symmetric function 
expressed in a bracket ( ).2 can be expressed as a linear function of products of 
symmetric functions of the form ( ) ( )^ ; it is also clear from the first propeidy 
above defined, that such expression will remain unaltered when the brackets ( ) and 
( )j are interchanged ; it must, therefore, be a symmetric function in regard to these 
brackets. 
We may, therefore, suppose an equation 
r.^s.f . . fz 
. . . + J (cq^]“' • • •) {'PiQT'^ 'Pi'jF • • - ji 
+ J {aff . . .)i IhPF ...)+••• 
(A) 
Moreover, we can express any product of the coefficients c^q, Cq^, . . . Cp^j, ... as a 
linear function of expressions each of which contains a monomial symmetric function 
of the quantities a^, ; a.i, ; ... and a product of coefficients hiQ, 6q^, . . . hpq, . . . 
Assume then 
TTj TT^ 
Gr/i ^'p/h ■ • • 
ai a-i 
• • • 
+ L {afp^ afp ^-. . .) 5,^,^ 6,^,^. . (B) 
+ M {pyiP'pyiF • • •) • • (C) 
From equation (B) is derived by the second property the operator relation 
n' r.v 
^ p\<ii ^ vm • 
= ^. . . -h L • • •) 
