THEORY OP THE PARTITIONS OP NUMBERS. 157 



The quantity 2 is at once eliminated and we obtain 



fl 



1- X 



a, 

 a 3 is now easily eliminated and we obtain 



i 

 To eliminate i we require the easily established theorem 



1 1 x . lxy 



11 J 



c 



Thence we reach the final result 



\.i\. 1 .7X2- 1 .'VV:/ 



which shows in the clearest possible manner how the partitions are constructed. The 

 denominator factors indicate that we may write do\vn any partition composed of three 

 parts ; the numerator terms xf, .i\ 2 .c 2 show that we may add either '2 to the first part 

 or simultaneously 2 to the first and 1 to the -second part. We in that manner obtain 

 every partition satisfying the conditions, but the numerator term .f.yV 2 shows that 

 certain partitions are in this manner obtained twice over. 



10. As we are only concerned with the magnitude of the highest part and not at 

 all with the weight of the partition, we may for the present purpose put Xi = x, 

 %2 = 3 = 1, and consider the result 



(I-*) 3 

 as the one to generalise. 



11. I write down the expression 



~' 2 ~ 3 



y 1 ( * 2 1 a * 1 ( ^ 1 ^2'-4 1. 



. . ,a 2n _ 3 x . i . i . i . ... i . i 



as the crude expression for the sum. 



We can immediately eliminate all the auxiliaries a which have an even suffix and 

 reach the expression 



. 11 



1 a t 3 a 6 . . .a 9H - 9 c .1 .1 -.1 



