850 
MAJOR P. A. MACMAHON ON THE THEORY 
y 
Hence ope.ration on the relation 
H" = r).{p^2X2P^ . . .) 
ields the relation 
(1 + H)'^ 
* ^2• • • • 
+ 
^hlhlh . . ., r - /r + 2) 4- . . . \ .{Pilhn- • ■) 
^hlhlh • • • 1% ^’) -- {p - '^)f{Pilhlh . ■ . 2P ^ r) 
ffM-2 
1 
iPzlh • • • \ r) + 
— 2 
'hVzlh ■ • • 2“H-^r) 
+ • • • \ -{Pilhlh ■ ■ •)> 
or, equating coefficients and inverting 
filhlhlh . . . H, r) - (/r - l)/(pii^ 2 P 3 • • • 2P-q r) 
+ (/^ - ‘^)f{lhP-2lh • • • 3F-',r) + . . . 
= 0 . . ., P - p) + i^^fiPilhV-i . . .,r- p^-l) 
+ \!Pjf{PiPzlh‘ . .,7'-P+2) + . . 
IIH’ < p the dexter vanishes and 
fipiPzPs . • . H, r) -ip - l)/(PiP3P3 • . • 2H qr) 
+ (P - ^)fipiPzPs • • • r) + . . . = 0. 
If r = p the dexter is 
fiPiPzPz • • •> 0) + 
^hPzPz 
1 ) + 
PiPiP-i . . ., 2) + . . . 
where J i2^i2hJ’-i ■ • in general vanishes, but 
./■(O, 0) = 1. 
