Engler — Figurate Numbers. 57 



in which the coefficients of the different letters are col- 

 lected, series for series, and inclosed in brackets. It 

 will be seen that the sets of series in the brackets are 

 identical in form with Table 6, and differ from each other 

 only in the number and position of the significant columns. 

 Formulae for the sums of these series may therefore at 

 once be written by the methods already given, and the 

 relation of the general case exhibited in Table 31 to the 

 ordinary figurate numbers of Table 6 becomes clear. 



By assigning special values to a, h, c, d, . . . .in 

 Table 31 or Table 32 all the preceding sets of series may 

 be obtained. 



Thus for a=l, and b = c=d= . . . = 0, Table 31 becomes 

 Table 6. 



For a=l, and h = c=d= . . . =l+d, Table 31 becomes 

 Table 13. 



For ^=1, and b = c = d= . . . =d, Table 31 becomes Table 

 21. 



For a=3, 6 = 4, c=5, d=6, e = 5, . . . Table 31 becomes 

 Table 30. 



Issued June 23, 1911. 



