498 
MAJOR MACMAHOR ON SYMMETRIC FUNCTIONS 
and performing tlie operation 
we have 
gpq — 4- ^10 + + ^01 + i + • ■ -[-^rs + + , + 
■ • • + Ua,p, + • . 
Moreover, 
hence 
gjjrj'U = 
and replacing U by its value, we have 
^ , gM<^j-j = (m) ; 
while in general 
gjiq^rs — {jP^l) Cr—j),s — r/- 
Now regarding the coefficients hp^ as functions of the coefficients Cp,^ only, we have 
gpi ~ {gpi + •••"!“ {gjpq Oi-s) “h • •" 
= ( 2 ^ 7 ) (^6^^ + Cio 9o;+i.r/ + <^01 + . . . + C,_p^s-q^crs + • • •) 
Thus 
gpi — ( 7 ^ 7 ) gpi • 
But the assumed relation is symmetrical as regards the quantities in the first two 
identities; hence also 
g^q ^pg)igp'i > 
and thence, since = {p<]) {p^l)v^ we have 
{p'l)-2gpq ~{p'l)igpq ~ ipg) Ppr 
If we then regard the assumed relation as defining a transformation of the 
quantities occurring in the identity III. into either ot the sets of quantities 
associated with I. or IL, the operation 
{pg^gpq" 
is an invariant. 
28. Since Ppq ^ {p<l) Qpq' 
