88 BISHOP TERROT ON THE SUMS OF THE DIGITS OF NUMBERS. 
sive numbers, there is also a definite sequence and recurrence in the sums of the 
digits which express them ; and the results of these inquiries, with the requisite 
demonstrations, I will now, as briefly as possible, lay before the Society. 
Prop. I. 
If m and n are prime to one another, am cannot equal bn, unless a and b be 
equimultiples of and m respectively. For, if am=bn, = = But by hyp. - is a 
- fraction in its lowest terms, therefore b=pm, and a=pn. 
Prop. II. 
If N=P .»—1+r,,. being the local value of the notation, and P . n—1 being the 
greatest multiple of x—1, which is less than N; then ,,, is the ultimate sum of the 
digits of N. 
Let N=a+bn+cn?+dn’, &e. 
NT bake Ledeen 2 wee eee.) a eee 
n—1 Fat 
N=pn—l+atb+te, &e. = pn—1+r, 
Again, let r,=¢.n—1+r,, where r, is the sum of the digits of r,, or the 
second sum of the digits of N. 
Then N=p+gn—1+7,. Let this operation be continued till 7,, becomes a 
single digit, we have N=P . n—1+7,, where 7,, is the ultimate sum of the digits 
of N. 
Ex. In our notation »=10, and »—1=9. 
Let N=567434=63068 x 9+2 
here Ist sum =29 
2d do. =11 
3d do. = 2 
Cor. If + =nx—1, then N is a multiple of n—1. And, conversely, if N be a 
multiple of n—1, r,=n—1. 
Prop. III. 
If a be a number prime to x—T; and p, g be two numbers, whose difference is 
neither x—1, nor a multiple of x—1, then pa and ga cannot have the same ulti- 
mate sum. 
If possible let pa=mn—1+r and ga=m,n—1+r, and let s=g—p, then 
sa=g a—pa=m,—m+n—1; but by hyp. a is prime to x—I, and s is neither »—1, 
nor a multiple of it; therefore, by Prop. I. sa cannot equal m,—mxn—1, and 
therefore pa and ga cannot have same ultimate sum. 
