44 



Trans, Acad. ScL of St, Louis. 



it will be found in general that the sum of all the terms 

 in each column of the right-hand triangle is equal to r 

 times the sum of all the terms in the corresponding row 

 of the left-hand triangle. 



It will be noticed that in Table 6 the diagonals ascend- 

 ing from left to right give the coefficients of the ordinary 

 binomial expansion with positive integral exponents, thus 

 forming Pascal's Triangle.^ 



The explanation of this fact is at once seen from the 

 structure of the terms of the different series as derived 

 from the general expression given in (7) . This procedlire 

 shows the structure of the several terms of each series, 

 as follows: 



1 1 



1 2 



1 3 



1 (r-4) 



1 (r-3) 



1 (r-2) 



1 (r-1) 



1;2 

 f2 



2;3 

 1-2 



34 

 12 



1-2-3 



2-34 

 1-2-3 



3-45 

 1-23 



1-2-3-4 

 1-2-3-4 



2-3-4-5 

 l-2-3'4 



3-4-5-6 

 1-2-34 



(r-4)(r-3) (r-4) (r-3) (r-2) 

 1-2 1-2-3 



(r-3) (r-2) (r-3) (r-2) (r-1) 



1-2 



(r-2) (r-1) 

 1-2 



1-2-3 



(r-2) (r-1)r 

 12 3 



(r-1)r 

 12 



(r-1) r(r+l) 

 1-2-3 



1 IT] -^ 

 ' * 1-2 



m 



r (r + 1) r (r+1) (r-h2) 



1-2-3 



TaUe 9. 



''published by Pascal in his Traite du triangle arithm6tiQue, 1665. 



