252 Mr. JL W. L. Glaislier on [Feb. 3, 



one element =3, the number involving two elements = 17, and the 

 number involving three = 10 ; and I. and II. become 



30-67+47-10=0, 

 30 + 67+47+10=3. 2 + 17. 2 2 +10.2 3 . 

 Combining the formulae I. and II., we have 



S(l+^+^2;ty + &c.)=2(^+^2 + &c.)=p2 , •=22 , - 1 , 



where 22 r_1 may be written 22 , s denoting the number of changes in a 

 partition. The two formulae taken together, as just written, form of 

 course a much better verification than either singly. 



There are two subsidiary verifications connected with I. and II. which 

 are also worth attention. From Jacobi's equation (' Fundainenta Nova/ 

 p. 185), 



\+ t ; i ;? : l+f : : ; -!-»+»-? +*»-*■. 



we see that 



III. 2±2 y ~ 1 = l, -l,or 0, 



according as n is an even square, an uneven square, or not a square. 

 The sign + is to be taken if the partition contains an even number of 

 terms, and the sign — if the number is uneven. 

 Also from the same identity inverted, viz. from 



1+*.1+*M+* 3 ... 1 



1—t . 1-f .1-f... l-2t + 2t i -2t 9 + 2t 16 -&c, 



we see that 



IV. 22 1, =(-) W (E-B'), 



where R=the number of representations of n as the sum of an even 



number of squares, and E' the number of representations as an uneven 



number of squares. 



In the case of n=9, of the three partitions involving only one element, 



all consist of an uneven number of terms ; of the seventeen partitions 



involving two elements, in nine the number of terms is even, and in eight 



uneven ; and of the ten partitions involving three elements, in five the 



number is even and in five uneven ; so that since 9 is an uneven square, 



III. gives 



• jK-3.2 + 9.2 2 -8.2 2 + 5.2 3 -5.2 3 )=-1. 



The partitions of 9 into squares are four in number, viz. 9, 4+4+1, 

 4 + 1 + 1 + 1 + 1 + 1, l + 1 + l + l + l + l + l + l + l; and these give rise 

 respectively to 2, 3 . 2 3 , 6 . 2 6 , 2 9 representations ; so that substituting in IY. 



30 + 67+47 + 10=(-) 9 (384-2-24-512). 



The following are also formulae of verification : — 



