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 



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 



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

 Series " B, B. Smyth, Trans. Ka7isas Acad. Sci. 14: 29. 



