118 Mr. W. H. Walenn on Developments and Applications 



its digits and of the powers of 10, the formula 



(\0-8) n -\a + (10-8) n -*b+(10-8) n ~*c+ . . . +{\Q-8)*s 

 + (10-8)/+« 



has the same remainder to 8 as the number in question has (this 

 being the fundamental formula of unitation), the following in- 

 vestigations in this paper refer to proofs and developments of the 

 points advanced in the previous paper 5 in relation to the unitates 

 of powers and roots. 



2. If m and q be integers, ^/q irrational, and n any integer, 

 excepting unity, so great that the expression n m — q is positive, 

 then the imitate of the mth root of q to the base 8 (or U 5 %/q) 

 is finite and integral when 8 is of the form n m —q. For if q 

 be found in the recurring series that belongs to the unitates of 

 the mth powers of numbers, T/q will be found in the correspond- 

 ing place in the series of the unitates of natural numbers, and 

 this whether y/q be rational or irrational. But for q to be 

 found in such a series of unitates, it is necessary that 8 should 

 be of such a value that r8 (r = an integer) being subtracted 

 from any complete mth power gives q } or that 



n m — r8 = q. 

 Now V s r8 = 8; and this equation becomes 



8 = n m — q. 

 \{n m —q be less than q, the expression V^q will become 



3. The following examples show some of the forms which 

 n m — q may assume; at the same time they are proofs by induc- 

 tion of the general theorem. 



Example I. — Find some of the values of 8, or some of the 

 series of unitates, in which \/2 has an integral unitate. 



8 = n m -q^2 <2 -2 = 2, 

 or 



3*-2 = 7, 

 or 



42_2 = 14, 

 or 



5 2 -2=23, 

 or 



G 2 -2 = 34, 



&c. &c. 



Therefore the most useful system of imitation, for work in which 

 a/2 enters (8 being less than 10), is that which has its base 

 equal to 7. 



