522 
MAJOR MACMAHON ON SYMMETRIC FUNCTIONS 
Hence any symmetrical function 
f{CtiQ, Oqi, . . . Clpq . . .) ~f, 
is transformed into 
/{^lo + + ^0 • • • cCpq + 1 + s-i) ^ . }> 
or writing 
9m — 4 " + • • • 5 
^ n 'P +^,9 ^ p,Q + i 
this is 
^ { { 9 w + 9 oi) + (S'oo + 2^ij + ^oa) dj + • • • }/> 
the bar over exp denoting that the multiplications of operators, which arise, are 
symbolic. 
Now, by the theorem of Art. 15 , this is 
exp {M^qP^q d- MqiPoi + . . . + ...}/, 
the multiplication denoting successive operations, and identically 
exp -}-•••+ + • • •) 
= 1 + 1 + -»? + ^ . 
= exp (^-1-77). 
Hence M^q = = 1 and the other coefficients M are zero. 
Hence the symmetric function f is converted into 
exp (P'10 + 9 oi) ■ f> 
and, if fhe a function of the differences 
exp {9io + Poi) • /=/ 
Hence the necessary and sufficient condition, that f may be a function of the 
differences, is the satisfaction of the linear partial differential ec^uation 
9 io + 9 oi = 0. 
to. These operators ^iq and ^qi have been previously met with in the discussion of 
the symmetric functions connected with the fundamental identity 
(1 + a^x + ^ip) (1 + a^x + / 3 oy) ...” 1 + a^^x + + . . . + a^^^xY + • • • > 
but then they played a different role.'" 
* Two simple cases of this important transformation should be verified by the reader. For7i = 2, (II) 
is transfornied into i (ctj — ^ 2 ) (di ~ /h) connected with the ternary quadric. For n = 3, (21) is trans¬ 
formed into tV - (‘=‘1 ~ Y {(“1 “ “ 3 ) + (“3 ‘='^ 3 )} (9i ~ /h) connected with the ternary cubic. 
