122 Mr. W. H. Walenn on Unitation. 



U999 . . . N, and in which 8 = any number which has only nines 

 for its digits, there being n digits, includes the particular case 

 U 9 N. U 9 N may be styled the normal set of unitates, just as 

 the method of expressing an angle in terms of the angle sub- 

 tended by the arc equal to the radius may be called the normal 

 method, or the Napierian system of logarithms the normal 

 system. 



In the general unitation-formula (art. 31), if r = 10 n , n being 

 taken equal to the number of nines in 8 ( = 999 . . . ), the for- 

 mula has all its coefficients (of the form (r— S) w ) = l, and it 

 resolves itself into adding every set of n digits in N, and 

 repeating the operation until a number less than 8 is ob- 

 tained. The examples at the end of this article will make 

 this short method quite plain. Any reader wishing to verify 

 examples of U 999 ...N by actual division will be assisted by 

 knowing that all the multiples of 999 .. . are of the form 

 a n a n -i . . . a z a 2 a v writing the digits a h a 2 , a 3 , &c. as the number 

 is expressed decimally (for instance in 1760=a 4 a 3 a 2 « 1 j %=1, 

 a 3 =7, <2 2 =6, ax=0), and in which a n + a x =9j and 



a n —i ct n —2 • • . <^3 1?2= yyy . . . , 



there being (n — 1) nines. For instance, 9999 . 7 = 69993 ; 

 999999 . 6 = 5999994 ; and so on. Moreover the unit's digit 

 of the product = 10 — m, m being the multiplier. The whole 

 form of these multiples is therefore (m— 1)999 . . . (10— m). 

 Examples:—!, Find U 99 n, when N= 31415926536. 



U 99 n=U 99 (36 + 65 + 92 + 15 + 14 + 3) = U 99 225 



= U 99 (25 + 2) = 27. 



II. Find U 999 N, n being 10004. 



U 999 (4 + 10) = 14. 



III. Find U 9999 n, when n=31415926536. 



U 9999 n = U 9999 (6536 + 1592 + 314) = 8442. 



IV. Find U 999 n, when n = 34415926536. 

 U 999 n=U 999 (536 + 926 + 415 + 31) = U 999 1908 = 909. 



39. The form U im ...N, includes UnN, which is a very 

 useful system for practical work (11 being a prime number) ; 

 it is treated of in No. III., also in articles 4, 6, 8, 14, 16, and 

 27. If r=10 w , n being equal to the number of ones in 8, all 

 the coefficients are equal to unity, and the process of unitation 

 resolves itself into an exactly similar process to that used in 

 U 999 . . . n, excepting at the last step, when the next less product 

 of an integer and 8 must be subtracted from the amount to 

 obtain the required unitate. 



