TRANSACTIONS OF THE SECTIONS. 11 



On ihe Theory of Linear Transformations : I. The Graphical Representation 

 of Invariants ; II. The Ex^iansion of Uns>/mmetrical Functions in Symme- 

 trical Functions and Determinants; III. The Notation of Matrices. By- 

 Professor Clifford, F.R.S. 



On the Calculus of Motors. By Professor J. D. Everett, F.R.S.E. 



See tlii-ee articles, entitled " On a new Method in Statics and Kinematics," in the 

 'Messenger of Mathematics ' for 1874 and 1875. 



FormulcB of Verification in Partitions. By J. W. L. Gi,aisher, M.A., F.R.S. 



At the Edinburgh Meeting (Report, 1871, Transactions of the Sections, pp. 23- 

 25) Sylvester gave a formula for verifying, in writing down all the partitions of a 

 given number w, that none had been omitted. The formula in question was that 



2(l—x+xy—xyz+.ryzw—&C.)—0, (1) 



where in any partition x denotes the number of I's present, i/ the number of 2's, s 

 the number "of 3's, and the 2 extends to all the partitions ; so that 2l=N, the total 

 number of partitions of «. 



In this very elegant formula, however, as the terms are alternately positive and 

 negative, an omission may easily cancel itself ; ex. gr. if the omitted partition con- 

 tains one 1 and no 2, it would appear as 1 in the first term, as 1 in the second term, 

 and as zero in the succeeding terms, so that its omission would not be pointed out. 

 It becomes therefore a matter of interest to examine what the formula (1) becomes 

 if all the terms are taken with the positive sign. 



I. Starting from the identity 



and dividing throughout by 1 - < . 1 - i* . 1 — <' . . . , we have 



1 I t 1 _ t' 



1-t .l-tW-t''^ (l-tf .!-('. 1-t' .. .^ Q.-tfQ.-fy .1-f' 



l+t.\-{-t\\-\-f .. 



&c. 



~ l-t.l-f'-l-t^ . . . 

 = (l+2<+2i!=+2i!'+&c.) (l+2<=+2<'+&e.) (\+2t^-\-2f+kQ.) . . . ; 



whence, equating the coefficients of <", 



•S,{l-\-x+xy-^xyz-\-xyzw+kc.) = 'S.% (2) 



where r is the number of different elements contained in any partition. Thus, take 

 as an example w=7 : the partitions are 



so that N, the number of partitions, =15, 2a? = 30, 2j-y = 17, 2a-ys=2. Also there 

 are 2 partitions in which only one element occurs, 11 in which two elements occur, 

 and 2 in which three elements occur. Thus Sylvester's formula (1) gives 



15-30+17-2 = 0; 



2* 



