MAJOR P. A. MACMAHON ON THE THEORY OP PARTITIONS OP NUMBERS. 401 
For O&SG 7j rGdnciiig by Ji J 2 — Jsj wg find 
VI 
5 “h ^ 
which is right. 
For CasG 8, rodacing by Ji + J., = J 3 , wg find 
s + ^ 
which is satisfied. 
For CasG 9, reducing by Ji + J 2 = J 3 — 4, the ratio is one of equality. 
For Case 10 , reducing by Jj + Jg = J 3 , we find 
5 -j- ^ ^ 0, 
which is right. 
Hence the relation is universally satisfied, and we have proved that the expression 
X V2 lY Y Y \i Y'-2 Y 
l+n-i 
is in every case finite and integral. 
Art. 123. In Part 3 of this Memoir I hope to treat of other systems of algebraical 
and arithmetical functions which fall within the domain of partition analysis and the 
theory of the linear composition of integers ; also to take np the general theories of 
partition analysis and linear Diophantine analysis, with possible extensions to his’her 
degrees. 
VOL. CXCII.—A. 
3 F 
