44 



Trans. Acad. Sci. 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 procedure 

 shows the structure of the several terms of each series, 

 as follows: 



1 



1 2 



1 3 



1-2 

 ^2 



1-2 



34 

 1-2 



1-2-3 



123 

 2-3-4 



r¥3 



345 



1-2-34 

 1-2-34 



2-3-4-5 

 123-4 



345G 

 1-2-3-4 



1 (r-4) 



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



1-2 



1-2-3 



1 (r-3) 



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



1-2 



1-2-3 



1 (r-2) 



(r-2)(r-l ) 

 1-2 



(r-2) (r-1)r 

 12-3 



1 (r-1) 



m 



(r-1 )r 

 12 



r (r+1) 

 1-2 



(r-1) r (r + 1) 

 1-2-3 



r (r + 1) (r-1- 2) 

 1-2-3 



Table 9. 



* Published by Pascal in his Traits du triangle arithm^tiQuc, 1665. 



