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 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 fr-1) 



1-2 



r2 



2;3 

 1-2 



3-4 

 12 



1-2-3 



2-3-4 

 1^3 



v¥6 



l-2o-4 

 1-2-3-4 



2-3-4-5 

 1-2-3-4 



34-5-6 

 V2¥4 



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

 T^ 1-2-3 



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



1-2 



(r-2)(r-l) 

 1-2 



1-2-3 



(r-2) (r-1)r 

 12 3 



(r-1) r 

 12 



(r-1) r (r + 1) 

 1-2-3 





r (r+1) r (r-f 1) (r-{-2) 



1-2-3 



Table 9. 



•Published by Pascal in his Traite du triangle arithmetiquc, 1665. 



