Mr. W. H. Walenn on the Unitates of Powers and Roots. 351 



determined by the expression 8 = n m — q, as above stated. The 

 following are some of the series that may be used : — 



«tV, fl £ a* } cfi, a 1 , a 2 , a 4 , a 8 ; a*?, a%, a*, a 1 , a 3 , a 9 , cF t a 8] ; 

 «* f a^, a$, a>, a\ a™, « 125 , a 625 . 



ThusU 7 ^/2 = 3or4. 



Taking the leading principle of the function U ]0 >i oc as true 

 when x=^/q (an irrational quantity) — an extension that must 

 not be made without realizing the extent of the step involved 

 in it (namely, that the extreme right-hand figure or figures is 

 or are given by this class of unitates) — it would seem possible to 

 assign a value to the last figures of some incommensurable quan- 

 tities ; it would appear, for instance, that U 10V /5 must be 5. 

 If this be true, the real terra incognita of incommensurable 

 quantities does not always lie at their extreme right-hand end, 

 but in the middle region of their interminable decimal. This 

 supposition would make good an analogy between the curves to 

 certain equations (namely those that have a curve of finite peri- 

 meter at an infinite distance from the origin) and certain incom- 

 mensurable quantities. Although the subject must be dealt 

 with very cautiously, there does not seem any incongruity in 

 the conception; and the range of thought thus opened to the 

 mind is new. 



The points brought forward in this paper have been verified 

 and tested by inductive reasoning, by the laws of their existence, 

 and by examples, according to the methods more particularly 

 set forth in the last paper — that upon negative and fractional 

 unitates. 



The foregoing remarks and elucidation show that unitation is 

 not identical with any known process. As an operation it is a 

 direct process, dependent upon repetition for its completion. A 

 unitate is a function which, in some of its results, gives a finite 

 and integral value when applied to quantities that are neither 

 finite nor integral. 



The further the investigation of unitation proceeds, the more 

 unitates manifest themselves as functions that may be found 

 throughout all the domain of quantity, and as, in some instances, 

 interpretable when the quantity from which they are derived 

 is but imperfectly known. 



74 Brecknock Road, N. 

 March 1875. 



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