Memoir on the Theory of the Partition of Numbers. 225 

 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 larger class of sets which involve 

 the symbol. We may have the single condition 



wherein A 15 A 2 , A 3 ...... A s are integers +, zero, or - , of which at 



least one must be positive, or we may have the set of conditions 



>a * 2= 



a, ^ 



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, and lead to results of the same 

 general character. 



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 func- 

 tions. I seek the summation 



for every set of values (integers) 



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. : 



