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XL— On the Sums of the Digits of Numbers. By the Right Reverend Bishop Terrot. 



[Read 2d December 1845.] 



The general properties of numbers, considered without reference to the nota- 

 tion in which they are expressed, have been very fuUy 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 + cri', &c. where a, 6, 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 beheve 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, &c. 

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



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, tiU 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 six, we find the case very 

 diflferent. The multiples are, 



6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, &c. 



and their sums 6, 3, 9, 6, 3, 9, 6, 3, 9, &c. &c. 

 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 sunple 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 



