42 



Trans. Acad. Sci. of St. Louis. 



We may proceed in this way indefinitely and thus obtain 

 successively each of the series 



Table 6. 



which are the series commonly known as figurate num- 

 bers,^ defined by the property evident from inspection that 

 the n*^ term in any series is equal to the sum of the first n 

 terms of the series immediately preceding it. The n*^ 

 term, therefore, of each of these series is indicated in the 

 formula for the sum of the first n terms of the series imme- 

 diately preceding, by which formula its structure is dis- 

 closed ; so that these series may be written in the form 



1 



1 



V2 



V2 



1-2-3 



1-2-3 



1 

 2 



2;3 

 1-2 

 2-34 



1 



3 



M 



1-2 



3-4-5 



1 

 4 



4;5 

 1-2 



4-5-6 



1-2-3 1-2-3 1-2-3 

 1-2-3-4 2-34-5 3-4-5'6 45-6-7 



1-2-3-4 1-2-3-4 1-2-3-4 l-2-3'4 1-2-3-4 



1-23-4 



Table 7. 



Inspection of Table 6 shows that the first column is 

 identical with the first row; that the second column Is 

 identical with the second row; that the third column is 



2 See Archiv der Mathematik und Physik, I. 5:82-89, also "Figurate 

 Series " B. B. Smyth, Trans. Kansas Acad. Sci. 14: 29. 



