44 



anonymously by Professor De Morgan in a paper printed in the 

 fourth volume, p. 87, of the Cambridge Mathematical Journal. 



4. In (II.), for [N, p n +{] substitute [N— pj,£>i], and transpose 

 the terms. Then 



[N^l-CN-^-l^EN-p},^]; . . (IV.) 



n 

 and this leads to 



[N-5j,p-l]-[N-2vj,p-2] = [N-p?,7)-l] ; 



w n n 



and that leads to the summation 



[N,p,]=S^[N-^,0 I ] (V.) 



z 

 The lower limit of z in (V.) is made 0, in order that the formula 

 may comprehend the extreme case [0, Uj] =1, analogous to the ex- 

 treme case in Combinations. 



5. After substituting 1 for tj, the author applies formula (IV.) to 

 determining in how many different ways N can be resolved into p 

 parts not less than 1 . Let [N, p Y ~] be the term in a table of double 

 entry corresponding to column N, line p, in the table. From the 

 head, in line 0, of each of the columns 0, 1, 2, 3, &c, draw a diagonal, 

 advancing one column and one line at a time. Take these diagonals 

 one after another, and in each of them compute by formula (IV.) the 

 terms situate on lines 0, 1, 2, 3, &c, one by one in succession. If N 

 be the number at the head of the column from which any diagonal 

 takes its departure, there wiil be only N terms to compute on that 

 diagonal, the further terms being only repetitions of the term on the 

 line N. For the diagonal in question intersects line N in column 

 2N; and, by formula V, 



[2N,N 1 ] = S?[N,* 1 ] 



z 

 = the sum of all the terms in column N. But, moreover, 



[2N+y,N 1 4-y] = S^[N,z 1 ] 

 z 



= the same constant. The leading property of the table, indicated 

 by the formula 



z 

 is, that the term [N, p^~\ = the sum of all the terms in column N — p, 

 from line to line p inclusive. After the publication of the anony- 

 mous paper before referred to, Professor De Morgan discovered this 

 theorem also, but he did not announce it*. 



II. On Combinations. 



1. In ordinary Combinations, the combining elements are of differ- 

 ent kinds, and there is but one element of a kind : in the case here 

 considered, there are different kinds of elements, and there may be 

 many elements of a kind ; and more than one element of a kind may 

 enter into the same combination. 



* The author has recently discovered an equivalent formula in p. 264 

 of Euler's Int. in An. Infinitorum ; but investigated by a totally different 

 method, and not applied as the author has applied it. 



