256 Mr. J. W. L. Glaisher on [Feb. 3, 



(r) n Z7-i ATi ^ = * + 



u 



-tu.l-fu.l-fu. .. ' 1-t 1-tu ' 1-f.l-f 



^ 2 , __J!__ ^ 3 



Tl J 1 j2 T ^3 ' T ^.. T j2„. 1 _i3 . . I **C« 



l_te.l_f« ' 1-t. 1 — <M-« J 1-*M.1-< 2 M.1-« 3 M 



(a) ilfaIfa-?... i=1+<+i3+i6+ '' 0+&c - ; 



o£ which (a) and (/3) are Euler's well known formula3, and (y) and (£) 

 are given by Jacobi (' Fundamenta Nova,' pp. 180, 185). 



Of the fourteen verification formula) the first two are generally most 

 convenient ; and, taken in connexion with III. and IV., the four equations 

 form an interesting system of mutually related verifications. All the 

 formulae can be used with great facility after a little practice, but some 

 are evidently far preferable to others. Nos. II., III. and IV. were com- 

 municated to the Bristol Meeting of the British Association (1875). 



The following formulas, XV. to XX., involve the consideration of the 

 partitions of a number in which repetitions are excluded. If by Q (a, 6, 

 c . . .) n be denoted the number of partitions without repetitions of n 

 into the elements a, b, c . .., and by P (a, b, c . . .) n the number of par- 

 titions into the same elements with repetitions, then, from the identity 



which is derivable at once from (a), we have 



XV. 2Q(l,2,3...>i = l + P 1 (l 5 2>i + P 1 ' 2 (l,2,3> + P 1 - 2 . 3 (l,2,3,4> 



+ &c, 



where P 1,2 --- r (l, 2 . . . r-{-Y)n denotes the number of partitions of n into 

 the elements 1, 2 . . . r + 1, in which all except the highest must appear 

 at least once. 



For n = 9, Q,(l, 2, 3 . . .)n=8, and the formula gives 



2.8=1 + 5 + 7+3. 

 From the identity 



1 



1 + M + *M + * S .. 

 we have 



= l-t.l-f.l-t* 



XVI. P even w-P uneven n = (-)%>(1, 3, 5 . . .>, 



where P even ft denotes the number of partitions of n that contain an even 

 number of terms, and P uneven n the number that contain an uneven number. 

 If n be even, all the partitions in Q(l, 3, 5 . . .)n are included in P even n ; 

 and if n be uneven, all are included in P uneven ?i; so that we have the 

 theorem that ii' all the partitions of n into uneven elements without 

 repetitions be left out of consideration, then of the rest the number of 

 partitions in which the number of terms is even is equal to the number 



