BISHOP TERROT ON THE SUMS OF THE DIGITS OF NUMBERS. 



Prop. IV. 

 If a have a common divisor with n—\ as v, then p a and q a will have the same 

 n-\ 



V 



ultimate sum \i q-p=- 



Let B«=P.w-l + r, therefore qa=pa + ^ — a=P.M— l + r + ^ — a. But v is a 



V V 



divisor of a ; therefore ga= P.n—1 + r + b.n— 1 = P, .n—1 + r, that is, /> a and q a have 

 same ultimate sum. 



Prop. V. 



If a be a divisor of »— 1, or '^^—=v, then p a and a a will have the same ul- 



a 



timate sum if 9— ^=e. 



Let pa='P.n — l + r,qa — pa + va = 'P.n — l + r + n — \ = (^ + Y).n — l+r. 



Prop. VI. 

 If P=Q + R. The ultimate sum of P = ultimate sum of (sum Q + simi R). 



Let Q=»j.»— l + y, R=»«j.w— 1 + r^ 



P = Q + R=»« + »fj.w — l + r + r. 



But r and r^ being single digits, their aggregate is either a single digit, or 

 »-l + a single digit. In the former case, the ultimate sum of P = sum of Q + 

 sum of R. In the latter, sum of P = sum (sum of Q + sum of R). 

 CoR. If R be a multiple of n—1, or r^=n~l, sum of P = sum of Q. 



Prop. VII. 



From these propositions it follows, that in any arithmetical series, whose 

 common difference is prime to w— 1, the ultimate sum of any term (the p^^) = the 

 ultimate sum of {p + gM—lf^ ; but that no two terms at any other interval can 

 have the same ultimate sum ; and hence, that all the terms from the jf^ to the 

 (jO + »—!)"' range, as to their ultimate sums, through aU the digits of the scale. 

 For if the p"' term — s.n-l + r, then the (p + g.n-iy^ term -s.n-l + r + q.n-l.b 



=z s+b q.n — \-\-r. 



Again, let jo"" term = a, q^^=a + q—p.b ; but by Prop. I., since h is now taken 

 prime to n—\, and q-p is neither »-l, nor a multiple of it, the §'"' term must 

 have an ultimate sum different from the />"'. 



Ex. I. The successive multiples of any number prime to 9, are an arithmetic 

 series whose common difference is that number. Thus, the multiples of 5 are, 



5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, &c. 

 Sums 5, 1, 6, 2, 7, 3, 8, 4, 9, 5, 1, 6, «&c. 



