BISHOP TERROT ON THE SUMS OF THE DIGITS OF NUMBERS. 97 



term of the 1st, 2d, 3d, &c., order of figurates, the whole may be reduced to a 

 common denominator, and represented thus : — 



2.3.4 ■ ■ ■ . m-l.m 3.4.5 m. m + 1 „ 



1.2.3 m-1' 1.2.3 m-V ' 



of which the numerators follow the law of the series treated in the last pro- 

 position. If, therefore, in the series of figurates, the successive sums be taken, 

 and each multiphed by 1 . 2 . 3 . . . m—1, the products wOl form a series recur- 

 ring after n~l terms. 



Ex 1 I 4-5, 4.5.6 ^ 

 Jix. 4 + 172+1.2.3' '^^ 



=4 + 10 + 20 + 35 + 56 + 84 + 120 + 165 + 220+286 + 364 + 455 

 Sums, =4, 1, 2, 8, 2, 3, 3, 3, 4, 7, 4, 5, 

 Miiltiplying by 1 . 2 . 3=6, the sums of products of sums become 



6, 6, 3, 3, 3, 9, 9, 9, 6, \ 6, 6, 3. 



Pbop. XIX. — 0/the Ultimate Difference of Digits. 



It is a wen knoAvn property of digits, that the remainder, when any number 

 is divided by the root of the scale employed + 1, is equal to the ultimate remain- 

 der of the even digits subtracted from the odd ; or, using a notation similar to that 

 we have before employed, that 'N=p.n + l + dn. As, however, <?„must always be 

 + , if at any step the sum of the even digits be greater than that of the odd, n + l, 

 or such a multiple of n + l as will make it the greater, must be added to the latter. 



From this fundamental proposition, a series of propositions analogous to the 

 preceding may be deduced, relating, not to the sums, but to the differences of the 

 digits. The demonstrations are so similar to those already given, that I shall 

 merely illustrate the matter by examining the succession of differences in the 

 series treated in Prop. XV; 



The general term was ax^ + bx + c. 



Let X become x+p.n + 1, the term becomes 



ax^ +2 a p x.n + 1 + a.p^ .n + \}^ + bx + bp.n + '\. + c = ax^ +bx + c-\-q.n+\. 



Hence the remainder, after dividing by w + 1, wUl be the same in both cases, 

 or the ultimate difference will recur after n + l terms. 

 Next, let x+x^—y, then 



ax\ + bx-^ +c=z^ + bx + c + ay + b.y — 2x. 



Assume ay + b=n + l, then the two terms wiU have the same ultimate differ- 

 ence. 



Ex. Take as before for the general term z" + 3 « + 1, 

 here o=l, 6=3, re + l=ll, therefore y + 3=11, ory=8. 



