Mr. W. H. Walenu on Unitation. 523 



eight-times Table, shows that it is the multiplier 2 which is 

 wanted for the tens digit, to reduce the series 8, 16, 24-, 32, 

 &c. to the form 8, 8, 8, 8, &c. ; for (calling the result of the 

 operation U 8 #), U 8 8 = 8, U 8 16 = 2.1 +6 = 8, U 8 24=2. 2 + 4 = 8, 

 U 8 32 = 2 .3 + 2 = 8, &c. The foregoing points also tend to 

 prove that the operation for the S-times table (or V s %) is clearly 

 one of multiplication by (10 — S), in respect of the tens' digit; 

 accordingly 3 is the multiplier for the 7-times Table, 4 for the 

 6-times Table, and so on. On reducing the theorem to general 

 terms from these particular results, it appears that the remainder 

 to a given divisor (8) = (10 — S)t + u, if the dividend be lOt + u. 

 This expression is now in a condition to be further expanded 

 into the formula given in article 11. 



13. In this place it may be convenient to elucidate the me- 

 thods of unitation employed in practice, although this point was 

 alluded to in the first paper on unitation. Practically, to uni- 

 tate a given number, it is best to commence at the left hand, 

 using the multiplier appropriate to the base of unitation, and 

 adding in the next figure to the right hand, at each step of the 

 operation. As soon as any number higher than 9 arises, the 

 result should be reduced (by a continued repetition of the pro- 

 cess) to a single figure, and the operation itself continued until 

 only a single figure results therefrom. For example, U 8 313 

 has the multiplier 2, and the operation is 



2.3 + 1 = 7; 2.7 + 3 = 17; 2.1 + 7 = 9; 9-8 = 1; 

 .-. U 8 313 = l. 



In this case all figures higher than hundreds do not influence the 

 result. Again, U 7 515 has the multiplier 3, and the operation is 

 3.5 + 1 = 16; 3.1 + 6 = 9; 9-7 = 2; 3.2 + 5 = 11; 

 3.1 + 1=4; .-. U 7 515 = 4. 



In some cases it is simpler to obtain the imitate by dividing the 

 given number by the base of the system. 



14. In most arithmetical works, the St. Andrew's cross, em- 

 ployed by Lucas de Burgo and others, is still used to formulate 

 the casting out of the nines. The methods which unitation 

 discloses, all point to operating with the unitates in the equa- 

 tional form. The following examples show this practical point. 



Example I. Given 



.27 = 2349 x 876 = 205772*; 

 then 



U, 1 a? = U 11 (6x 7) =9; 

 also 



U n 2057724 =9, 



thus checking the multiplication above indicated. 



