94 BISHOP TERROT ON THE SUMS'OF THE DIGITS OF NUMBERS. 
Prop. XIII.—Of Polygonal Numbers. 
In any series of polygonal numbers, » the root of notation being even, the 
sum of the digits of the (s+—1)™ term = sum of s‘* term. 
m— 2.8? —m—A4.s 
9 , where m is 
For every polygonal number is of the form P= 
the number of the order, and s that of the term. 
For s substitute s+n—1, 
pata +A HG+n=D) 

_ 

m—4.s m—2.(2sn—1+n—1\")—m—4.n—1 
2 
5 pee pecs Rae 
But 7 being even, the fractional expression is integer, whether m be even or odd. 
Therefore P and P’ have same ultimate sum. (Prop. VI. Cor.) 
If n be odd, the fractional expression is integer only when m is even. 
The same inference might at once be drawn from the consideration, that the 
s* term of any order of polygonals is the sum of s terms of an arithmetical series. 
Prop. XIV. 
If, as in our notation, m be even, the s®, (s+yp.n—1)™ and (p.2—1—s+1)™ terms 
of a triangular series have all the same ultimate sum. 
In this case, m=8, and s™ term = sst1 
2sp oo 1+P 5-7 a Here the co-efficient 

Therefore, (s+p.n—1)® = 
of n—1 is integer, whether p be odd or even: and therefore sum of (s+p.n—1)® 
term = sum of s™. 
pn—1—s+1).(pn—1—s) 
2 

Again, (pn—1—s+1)™ term = 


ieee ee tate 
and as the coefficient of n—1 is again integer, whether p be odd or even. Sum of 
(pn—1—s+1)*= sum of s™. . 
Ex. The triangular numbers are,— ’ 
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, &e. 
Sums, Le’ oe OS aie Oy os A eos eeseeo.. iece: 

