352 MAJOR P. A. MACMAHOX ON THE THEORY OF PARTITIONS OF NUMBERS. 
We thus obtain 7"“^ different sets of conditions that may be assigned; these are not 
all essentially different and in many cases they overlap. 
Art. 65. For the moment I concentrate attention upon the symbol 
and remark that the s — 1 conditions, which involve this symbol, set forth above, 
constitute one set of a large class of sets which involve the symbol. We may have 
the single condition 
+ Ai^’ao + A'^^^as + . . . + Af^a, > 0, 
Avherein Aj, Ao, A 3 ... A^ are integers zero or —, of which at least one must 
be positive, or we may have the set of conditions 
A^aj + A^'a., + + . . . + A^a, > 0 ^ 
Af a, + APa, + Af «3 + . . . + Af a, > 0 
Af a, + A^a, + Ar)a3 + • • • + Af > 0 
Ai+ Ao^a., -f- As^tts -f- . . . Aj^a, — b ^ 
as the definition of the partitions considered. If the symbol be = instead of > 
the solution of the equations falls into the province of linear Diophantine analysis. 
The problem before us may be regarded as being one of linear partition analysis. 
There is much in common between the two theories; the problems may be treated 
by somewhat similar methods. 
The partition analysis of degree higher than the first, like the Diophantine, is of a 
more recondite nature, and is left for the present out of consideration. 
I treat the partition conditions by the method of generating functions. I seek the 
summation 
. . . X 7 
for every set of values (integers) 
«!, ao, as) • • • a. 
which satisfy the assigned conditions. 
It appears that there are, in every case, a finite number of ground or fundamental 
solutions of the conditions, viz.:— 
a[^\ ot^P, aT . . . af^ 
af\ a.f, af 
. a' 
( 2 ) 
a^ . . . af‘> 
