490 
MAJOR MACMAHON ON SYMMETRIC FUNCTIONS 
nncl hence 
/+ {lMzlh<h • • •) + • • •) + IMz •■•) + ••• 
= (1 + + ^'Gfoi + f^~^2o + “1" ^'^02 ’ 
and equating coefficients of like products /x^V'q we find 
^ih'/S Vi^h PzHz Pi'h • ■ ■) — {PzHiPPh • • •)■> 
^p/hiPiPPzpPPh • • ') — {piPPPL?, • • •)’ 
^pAPiP P2<ll P2,% • • •) = {PiPPPlz • • •)> 
^pApPi) = 
^Pi'Ap^'h • • • ^P.^'u{ PP[l P-2^Z • • • Pnp^ k 
and Grsf — 0, unless the bipart rs is involved in the expression of f. 
From the above we gather the very important fact that the eftect of the operation 
Gp,i u]Jon a partition is to obliterate one bipart pq when such bipart is present, and to 
annihilate the partition if it contains no bipart pq. 
14. I return to the result 
I + + t-Ggi + . . . + + . . . = exp ipgiQ + vg^^, 
wherein he it remembered the multiplication of operators in the right hand expression 
is symbolic. I seek to replace exp (pPio + ^^/oi) expression containing products 
of linear partial differential operations in which the multiplication is not symbolic. 
We have by definition 
i/io = + «io + ffin + • • -5 
9qI — ^<■'01 + ‘ : 
let further 
a definition which includes the former. 
15. I will establish the relation 
exp (wio^'io + + • • • + '^^Pj9pq + • • •) 
= exp IMioPiQ + MqiPoi + . . . + Mj„jgp,j +...), 
wliere on the left and right hand sides the multiplications of operators are respectively 
symbolic and not symbolic, and 
