Mr. W. H. Walenn on Unitation. 217 



for on expanding the (r — 8) m portion of each term by the bi- 

 nomial theorem, it has 8 as a factor in every term of the ex- 

 pansion (of any one term in the latter formula) except the 

 first, which is the same power of r as occurs in the correspond- 

 ing term of the original value of N. 



32. In obtaining the remainder to 8 of N, the formula in 

 art. 31 may be extended, by means of negative suffixes, into 



fl n (r-8) n - 1 + a»-i(r-.S)»- a + . . . + a,(r-8) 2 + a 2 (r-8y 



+ a 1 (r-8y + a_ l (r-8)- 1 + a_ 2 (r-8)- 2 + ..., 

 thus making it available for other numbers than whole num- 

 bers. In the operation for obtaining the remainder, the number 

 resultin g from the first substitution of the digits in the formula 

 is again subjected to the operation ; then this last number is 

 again treated in the same way, and so on, each treatment giving 

 a number less than the previous one, and divisible by 8 with 

 the same remainder that N has. If this treatment be continued 

 until a number less than 8 is obtained, that number is the uni- 

 tate of N to the base 8. This is according to the definition of 

 a unitate given in the Philosophical Magazine for November 

 1868, p. 346. 



33. This method of obtaining the unitate of 1ST is general, 

 and is therefore valuable. It also affords a means of compa- 

 ring the properties of U$N with those of N in a direct and 

 satisfactory manner. 



The repetition of the process of reduction by the formula is 

 peculiar to unitation ; and it may be symbolized by U^ N, 

 (n) being the number of times the formula is applied to a 

 given determination of U 5 N in order that the ultimate value 

 of U$N may be less than 8. This repetition has no analogy 

 in the expression of a number by means of the formula N. 



The following examples illustrate the repetition of the pro- 

 cess of reduction : — 



I. If N = 1234567, 



U7 N=l + 2 + 3 + 4 + 5 + 6 + 7 = 28. 



U 9 "N=2 + 8=10. 



IV'N =1 + = 1. Here (n) = 3. 



II. In obtaining U 7 N, if the formula containing the unre- 

 duced powers of 3 be used, 



U/ N=3 6 .l + 3 5 .2 + 3 4 .3 + 3 3 .4 + 3 2 .5 + 3.6 + 7 = 1636. 



U 7 "ST=3 3 .l + 3 2 . 6 + 3. 3 + 6 = 96. 



U/"N= 3. 9 + 6 = 33. 



U 1 ; N=3. 3 + 3 = 12. 



TT 7 N=3. 1 + 2 = 5. Here(?i) = 5. 



