a novel Arithmetical Operation. 347 



at the left hand and reducing the result of the operation to a 

 single figure as soon as any number higher than 9 arises ; for 

 instance, to unitate 34567 to the base 8, or, as it might be ex- 

 pressed, to find U 8 34567, the work is 6 + 4 = 10, 4 + 5=9, 

 2 + 6 = 8, 7; .-.Tig 34567 = 7. 



Regarding unitation in connexion with the well-known fact that 

 all direct operations upon the remainders to divisors are respec- 

 tively analogous to the same operations upon the dividends, also 

 taking into account the periodicity of series of unitates — the ve- 

 rification of direct arithmetical operations and of formulae invol- 

 ving them (with a certain degree of accuracy) becomes very easy ; 

 also the unitate of any unknown number may be calculated from 

 a known number with which it is connected by a known direct 

 operation, or by some inverse operations. 



In the following examples, JJx = the unitate of x to the base 9. 



Examples:— I. x = 25 (2558 2 -187 3 ) =104025. 



U#=U{7(2 2 -U7 3 )}=U{7(4-l)}=U(7x3) = 3. 



II. ^ = 537-2x23 = 12355-6. 



U*=U(8x5)=4. 



log#=2-7301360 + 1-3617278=4-0918638. 

 U(log,2?)=U(4 + 8)=3. 

 m , = 36326599 =27Q69+ 1 



1342 ' 1342 



ILr = 7 = 6 + l; 

 or, proving by multiplication, 



tf y =36326599 = (27069xl342)+l. 

 U^=7=(6xl) + 1. 

 IV. # = 58 247 ; find TJx, V 8 x, and V 7 x. 



In this case Ua? y = U {Vx v ^}, for the series of the powers of 

 U58, =4, has 3 (a divisor of 9) as its repeating period; 



58 247 

 .*. ILr=U4 4 = 4; i. e. — - — gives 4 as a remainder. V 8 x y = 



U 8 {VxVy} being true for U 8 58 247 , U 8 ^ = U 8 2 7 = 8, that is, 

 58 247 is divisible by 8 without a remainder. V 7 x = 2 l = 2, for 

 the series of the powers of U 7 58 has 6 for its repeating period. 



V. #=47 85 ; find ILr, \J 8 z, and Xf 7 x. The series XJxf, 

 U# w+1 , XL?"" 1 " 2 , &c. repeats after 6 terms; except when Vx l =2, 

 5, or 8, the same series repeats after 9 terms. U47 = 2, it 

 therefore repeats after 6 terms, and U<2? = 2 1 = 2, U & # = U 8 7 5 =7, 

 U 7 ^=5 1 = 5. 



