BISHOP TERROT ON THE SUMS OF THE DIGITS OF NUMBERS. 97 
term of the Ist, 2d, 8d, &c., order of figurates, the whole may be reduced to a 
common denominator, and represented thus :— 
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 multiplied by 1.2.3 .. . m—1, the products will form a series recur- 
ring after »—1 terms. 
bag + ee &c. 
—4410+204+35 +56 +844 120 +165 + 220 +286 +3644 455 
Soret 2 8 2, 3. 3 8 4 7%, 4 5, 
Multiplying by 1 . 2. 3=6, the sums of products of sums become 
6, Grromeso5 9, 0, 9,6, |,6, 6, 3. 
By Aix 
Prop. XIX.—Of the Ultimate Diference of Digits. 
It is a well known 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+1+d,. As, however, d, must always be 
+, if at any step the sum of the even digits be greater than that of the odd, »+1, 
or such a multiple of »+1 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 a2?+62+c. 
Let w# become z+p.x+1, the term becomes 
ax2?+2Q2aprn+l+ap%n+12+b2+bpnt1+c=ar?+bat+ct+gnt+l. 
Hence the remainder, after dividing by +1, will be the same in both cases, 
or the ultimate difference will recur after »+1 terms. 
Next, let +2, =y, then 
a4 +6%,+c=23 +boet+cet+ayt+by—2Qz-. 
Assume ay+6=n+1, then the two terms will have the same ultimate differ- 
ence. 
Ex. Take as before for the general term z2+32+1, 
here a=1, 6=3, n+1=11, therefore y +3=11, or y=8. 
