40 



Trans. Acad. Sci. of St. Louis. 



Tliis expression, therefore, represents the sum of the 

 series of numbers contained in the left-hand triangle, 

 namely of the series 



1, 3, 6, 10, 15, 21, 28 



If now we write a number of rows of the series 

 1, 3, 6 . . . 



Table 3. 



in the form of a rectangle, draw the zigzag diagonal, and 

 divide the rectangle of numbers into two triangles, as be- 

 fore, it becomes evident on inspection that the sum of the 

 numbers in each column of the right-hand triangle is equal 

 to three times the sum of the numbers in the correspond- 

 ing row of the left-hand triangle, and that, therefore, the 

 sum of all the numbers in the left-hand triangle in this case 

 is equal to one-third the sum of all the numbers in the 

 right-hand triangle, or equal to one-fourth the sum of all 

 the numbers in the whole rectangle. 



Again, by a process similar to the one used in the pre- 

 ceding case, the sum of all the numbers in the left-hand 

 triangle becomes 



n{n-\-l){n-h2){n+'S) 



S = 



1-2 



o 

 O 



(4) 



This expression, therefore, represents the sum of the 

 series of numbers contained in the left-hand triangle, 

 namely of the series 



1, 4, 10, 20, 35, 56, 84 ... . 



