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 1 1 1 1 



1 2 3 4 5 n 



r2 2;3 34 4-5 5-6 n(n+l) 



^2 V2 V2 1-2 V2 12 



1-2-3 2-34 345 4-5-6 5-6-7 n{n + l){n + 2) 



\¥^ V2% 12^ 1^3 l^ ' ' ' ' V2/6 



1-234 234-5 3-4-5-6 45-67 5-6-7-8 n(7i + l)(^ + 2)(n + 3) 



1-234 f2^4 1-2-34 1-2-3-4 12-34 " ' " 12-3-4 



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 



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

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



