96 BISHOP TERROT ON THE SUMS OF THE DIGITS OF NUMBERS. 
numbers substituted for x are together equal to siz, will have the same ulti- 
mate sum. Thus, in the above series, the Ist and 7th terms, in which O and 6 
are respectively substituted for 2, have the same sum; so also the 2d and 6th, in 
which 1 and 5 are substituted, and so on. 
Prop. XVI. 
In the series whose general term is mm+1 .. .. m+r—1,ifm+n—1 be 
substituted for m, the ultimate sum of digits will remain as before. 
If to each factor we add a, the term becomes 
atmxatm+1xatmt2 . . . atmt+r—1 
=ar+par—+g.av2 . . . . +mm+tilm+2 .. . . m+r—i1, 
where a is a factor of every term except the last. Let a=n—1 then term 
m+n—lm+n, &c. =mm+l1, &e. +sn—1, whose ultimate sum = that of 
mm+1.m+2, &e. 
Taking the same general term, if m+m,=n—r, m1=n—1—m+r-—1. 

Therefore m,.m,+1.m,+2 . . . m+r—1l=n—1-m+r—1 
xn—l—m+r—2 
m GC: 
xn—1l—m 
In this product, n—1 will enter as a factor into every term except the last, 
which is m.m+1 . . . m+r—1 with the sign + according asr is even or odd. 
If be even, the m** and m‘ terms will have the same ultimate sum; but 
if 7 be odd, the sums will be complemental. 
All the terms from the n—r* to the n—1" must have x—1 for their sum; be- 
cause x—1 must manifestly be a factor in each of them. 
Ex. Let r=2. Series is 1.2, 2.3, &e. 
=2, 6, 12, 20, 30, 42, 56, 72, 90, 110 
Sums are, 2,.6,.3,. 2) 3, 6.2) 99) 2, ce 
Let 7=3, or series 1.2.3, 2.3.4, &e. 
=6, 24, 60, 120, 210, 336, 504, 720, 990 
Sums, 6, 6,6; 18, 7 “8, 3 eas 19; de: 
In the Ist example, 7 being even =2, m+m,=10—2=8; 
therefore the Ist and 7th, 2d and 6th sums ought to be identical. 
In the 2d, 7 being odd =3, m+m,=10—3=7 ; 
therefore the Ist and 6th, 2d and 5th, &c. sums are complemental. 

Prov. X VIII.—Series of Figurate Numbers. 
mm +1 mm+1.m+2 
J : iF th 
1.2 i 33 &c., where each term is the m 

* If the series be m, 
a 
