12 REPORT — 1875. 



and (2) gives 



15+30+17+2 = 2 . 2+11 . 2-+2 . 2=. 



The two formulae (1) and (2) taken together form a much better verification than 

 either singly ; viz. we have 



2(l + ay+a;i/2M?+&c.) = 2(x+xyz+Scc.) = 22'-i, 



in which we may replace 22''~i by 22', s denoting the number of cliaiu/es in any 

 partition. 



The following formulae afford additional verifications : — 



II, From the identity 



we have 



2+2'-i = 0,1, or -1, 



according as n is not a square, is an even square, or is an uneven square : the sign+ 

 is to be used if the partition contains an even number of terms, and the sign — if 

 the number is uneven. 



in. From the same identity inverted, viz. from 



l+M + i!M+i!^. ■ . ^ 1 



l—t.l-f.l-f . . . l~2t+2t^-2f+&c: 

 we have 



22'= (-)»(E-R'), 



■where II denotes the number of representations* of « as the sum of an even number 

 of squares, and R' the number of representations as an uneven number of squares. 

 _ To verify these results in the case w=7 we have^ for II., considering the parti- 

 tions with an even number of terms, 



22'-i=6x 2+1x2'= 16; 



and for the partitions with an uneven number of parts, 



22'-i=2x 1 + 5x2+1x2^=16, 



thus verifying the theorem, since 7 is not a square. 



For III. the partitions of 7 as a sum of squares are two, viz. 



1+1+1+4 and 1 + 1+1+1+1+1+1. 



The former gives rise to 4 X 16 representations and the latter to 1 x 128 representa- 

 tions, and the formula becomes 



64=(-)-{4x 10-128}. 



The four theorems taken together, viz. 



2(l+a?y+&c.) = 2(,x+xyz+&c.) = 22'-i = i(-)''(R-Il'), 

 with 



2+2'-i=0, 1, -1, 



form a striking system of mutually related formnlje of verification. 



The author had investigated other systems, but this was the most satlsfactorv he 

 had met with. 



* See Professor H. J. S. Smith's "Report on the Theory of Numbers," Brit. Assoc. 

 Rep. 186.5, p. 337. } ' 



