254 Mr. J. W. L. Glaisher on [Feb. S, 



For ?i=9, the first formula gives 



30-12-10-5-2-1=0. 



xii. Sl 2"=, + pi+p^+P^;§+&c., 



where \ refers only to partitions in which all the elements are uneven, 

 v denotes the number of partitions of n into uneven elements (so that 

 2 1 l=y); and, for example, P^'e d en °tes the number of partitions of n 



into the elements 2, 4, 6, 1, 3, 5, 7 (2, 4, 6 being the only admissible 



even elements, but all uneven elements being admissible), and in which 

 1, 3, 5 must each occur at least once ; so that in the partitions all the 

 uneven numbers and the (even) suffixes may appear, and the (uneven) 

 exponents must all appear. (Thus we might have written Pg 4 6 . &c. for 



*i£t &c -) 



For m=9, we have 



30 = 8 + 14 + 7 + 1. 



The notation being as in XII., 



XIII. 0=v-P^+P^-Pj;^+&c. 



Forn=9, 0=8-14 + 7-1. 



Combining XII. with XIII., we have 



-+^;!+ p 2;t'6;8+&c.=pj+pj; 4 3 ; 5 6 +&c.=i2^. 



XIV. P-P 2 +P 2 ' 4 -P 2 ' 4 > 6 + &c.=l or 0, 



according as n is or is not a triangular number. 



Here the notation is practically the same as in XII. and XIII. ; P de- 

 notes the number of partitions into uneven elements, P 2 the number of 

 partitions into the elements 2, 1, 3, 5 . . ., in which 2 must appear, P 2 ' 4 

 into the elements 2, 4, 1, 3, 5 . . ., in which 2 and 4 must both appear, 

 and so on. 



For n=9, the formula gives 



8-11+3 = 0. 



The formulae V. to XIV., which have just been written down, are merely 

 translations into analytical language of the identities : — 



OQ r^+i^+izp+ ^i.^.iJ^. !_<»... 



2f 3? 



(i - t)\i - ff . i - f . . . T (i - *) 2 (i - 1 ) 2 (i - ty . 



™ (H(rVr^ +&c - h-*.i-'-i-*°... 



-&c. 



2f J- Jim 



+ &C 



l-t.(l-f) 2 .l-t 



