878 
MAJOR P. A. MACMAHON ON THE THEORY 
in the former series is 
2-“ (2== - 2) - (gj 2-* (2= - 2) + ... + (-r ■ (*) (2- - 2), 
which has the value 2'' — 2. 
52. Hence the numerator of the typical fraction may, to a factor . . . s^ pres 
be thrown into the form ; — 
^ {-^K, + • • • (A,, + + 2a^J + 2aJ, 
which is of great service. 
Of the series of quantities 
^l> • • • ^11) 
the typical fraction contains only the t quantities 
If it can be shown that the numerator consists of terms each of which contains 
quantities of the series 
^l> ^-2) • • • 
which are not included in the set 
it will follow at once that the fraction can, when expanded, contain no term which is 
a function of 
It suffices to show that the numerator vanishes when all the quantities of the set 
not included in the set 
are put equal to zero. 
Under these circumstances, remembering that Ko . . . are in ascending order of 
magnitude, we have 
A„^ = + . . . + 
A«3 = 2a^^ + + • . • + 
A«3 = -f- + “^v 
Ak,_^ — 2 + • • • + 
A<, = 2 . -j- j 
