Engler — Figurate Numbers. 



43 



identical with the third row; and so on indefinitely; there- 

 fore, if we designate the general row by r as we have 

 already designated the general column by n, we have as 

 the formula for the general row expressed in terms of r 

 and based upon its analogy with the nS'^ column, the 

 following: 



r(r+l) r (r+1) (r + 2) r (r+1) (r + 2) r + 3) 



hr, 



1-2 1-2-3 1-2-3-4 



r(r+l) (r+2)....(r + n-2) 



(7) 



1-2-3 {71-1) 



in which formula, by giving r the successive values 



1, 2, 3, we can obtain each of the rows of 



Table 6 in succession; or, by giving r any integral value 

 whatever, we can obtain the series corresponding to the 

 value of r chosen. The value of r, or the second number 

 in anj' of the series may therefore be regarded as the 

 Index of the series, and indicates the Order of the series. 

 As formula for the sum of any number of terms of the 

 j.th YQ^;Y, we have 



or, 



S = 



S = 



l'2-3 {n-1) 



If we now write the general row r a number of times, 

 making a rectangle with its zigzag diagonal, two triangles, 

 their respective rows and columns, as before. 



Table 8. 



