36 Mr. W. H. Walenn on Negative and Fractional Unitates. 



may produce chloride of aluminium and silicates of sodium, while 

 the magnesium of the chloride of magnesium may be replaced by 

 calcium ; lastly, a portion of the potassium in the sea-water appears 

 to have been replaced by the lithium of the granite. 



The alteration which has taken place in slates in the vicinity 

 of the cross-course has evidently been attended with the loss 

 of a large percentage of its silica. In fact this appears to have 

 been the chief chemical change effected, since, if we calculate 

 what would be the composition of the clay-slate after the reduc- 

 tion of its percentage of silica to the amount contained in the 

 altered rock, the calculated results will be found to agree very 

 closely with the analysis of the latter. One per cent, of lime, 

 however, has been exchanged for alumina, and about the same 

 amount of magnesia has been substituted for an equivalent 

 quantity of potassa and soda. 



III. On Negative and Fractional Unitates. 

 ByW. H. Walenn*. 



THE definition of a unitate, as originally given in the year 

 1868f, is: — The remainder to any given divisor (usually 

 a digit), determined by a certain theorem, without the know- 

 ledge of any multiple of that divisor. The theorem was stated 

 to be : — " If t be the tens' and u the units' digit of a two-figure 

 number, and $ be any integer less than 10, then 



(10-8)/+!* 



has the same remainder to 8 as 10/ + u;" and numerous exam- 

 ples of its use were given. 



In a paper read to the British Association at Liverpool in 

 1870 J, the expression (10 — h)t-\-u was expanded to 



(10-S)»- 1 «+ (10-8) n - 2 b + (l0-S) n - 3 c+ . . . + (10-S) 2 s 

 + (10-8)*+^ 

 n being the number of digits in the given number — so as to adapt 

 it to other than two-figure numbers ; and further examples of its 

 use were given. 



The examples showed that this theorem is useful, especially 

 when 8 = 9, in checking calculations and verifying Tables ; when 

 other values of 8 are used, it is applicable to quickly ascertaining 

 the remainders to divisors. 



In his address to the Mathematical Section at the British 



* Communicated by the Author, 

 t Phil. Mag. S. 4. vol. xxxvi. p. 346. 



X Report of the British Association for the Advancement of Science for 

 1870, Transactions of the Sections, p. 16. 



