XI.—On the Sums of the Digits of Numbers. By the Right Reverend Bisnor TERROT. 
[Read 2d December 1845. ] 
Tue general properties of numbers, considered without reference to the nota- 
tion in which they are expressed, have been very fully investigated by several of 
the most distinguished mathematicians. Little attention, however, has been paid 
to the particular properties resulting from the principle of the modern notation, 
which is the expression of every number in a series, a + bn + cn’, &c. where a, b,c, 
are the digits, and n the local value or root of the notation. Having been led to 
examine some of these results, and to account for them, I am now desirous of 
laying them before the Society. I do not flatter myself that they possess any 
great practical importance; but as I have reason to believe that they are new, I 
_ trust the Society will not think them entirely unworthy of their attention. 
If, then, we look at the multiplication table, and examine, in the first place, 
the multiples of seven, we find them— 
7,14, 21, 28, 35, 42, 49, 56, 63, 70, 77, &e. 
Sums 7, 5, d,s, 6,04, 2° 9) “7, 5, tee 
If we also take, as above, the ultimate sums of the digits of these multiples, that 
is to say, the sum of the digits of each if that sum be a single digit, or, if not, the 
sum of the sum of those digits, till in each case we arrive at a single digit, it ap- 
pears, that, for the first nine places, these sums range through all the digits of our 
notation, without any recurrence, and then commence over again in the same 
sequence as before. 
On looking at the adjacent line of the multiples of stz, we find the case very 
different. The multiples are, 
6, 12, 18, 24, 30, 86, 42, 48, 54, 60, 66, 72, '&e. 
and their sums 6, 3, 9, 6, 3, 9, 6, 3, 9, &e. &e. 
Here only three digits occur in the series of sums, and these repeated over and 
over in the same order. Farther, we may observe, that what is true of seven is 
true of five, eight, and all numbers which are prime to nine; and that what is 
observed of the multiples of six, occurs also in the multiples of three, the only 
other digit which has a common divisor with nine. 
I began with accounting for these facts; and, proceeding from simple mul- 
tiples to the consideration of other integer series, such as the series of squares, 
cubes, &c., the successive powers of a given root, the polygonal and figurate num- 
bers, I found that wherever there is a fixed law of relation between the succes- 
VOL. XVI. PART II. Z 
