XXXII. On the Priticiple of Continuity , in reference to certain Results of Analysis. 

 By J. R. Young, Professor of Mathematics in Belfast College. 



[Read December 7, 1846.] 



The mathematical axiom that " what is true up to the limit is true at the limit," is necessarily 

 implied in the general principle of Continuity. The recognition of this truth is essential to the very 

 conception of continuity ; of which indeed a sufficiently clear idea may be conveyed by the simple 

 enunciation of the axiom itself. In Geometry the continuity here mentioned refers to magnitude 

 only, irrespective of shape : in Analysis it refers simply to value. And in both, the limit spoken of 

 is that, whatever it may be, at which the continuous series of individual cases terminates; or. if 

 the expression be preferred, at which it commences. 



It is plain that different continuous series may start from, or terminate in a common boundary : 

 or the terminal limit of one series may be the commencement of another ; each series being governed 

 throughout by its own independent law. But there is a liability' to suppose the limit unique when 

 it is in reality multiple, or ambiguous; and indeed to confound the true limits with some unique 

 isolated form, having no connexion whatever with either series. 



Thus: — the tangent of x, when ,r commences in the first quadrant and continuously increases, 

 arrives at its limit when ,v reaches 90". In like manner, the tangent of .r, when x commences in the 

 second quadrant and continuously diminishes, arrives at its limit when .v reaches 90". But the two 

 limits (which are very liable to be confounded) are perfectly distinct. In the former case the limit is. 

 tan 90" = + X : in the latter case, tan 90" = - 03 . And, viewing the tangent independently, — that 

 is, as altogether unconnected with a continuous series, and therefore as uncontrolled by any law of 

 continuity, — the tangent of 90" is ambiguously ± m : and we cannot select one of these values, to 

 the exclusion of the other, without destroying the independence here supposed, and subjecting the 

 tangent to the operation of a law binding it in connexion with a continuous series of tangents. 



Again : the limit or extreme case of the continuous series of values of the progression 



\ — sc + x^ - ."fi -^ !v* - x^ + &c. ad inf. (1), 



furnished by the continuous variation of x from some inferior value up to ,< = 1, or from some 

 ■superior value down to a? = I, has been supposed in each case to he properlv represented bv 



1 - 1 + 1 - 1 + 1 - 1 + &c. arf itif. (y). 



But it has already been shown by the writer of these remarks*, that so far from this being the com- 

 mon limit, the two limits are totally distinct : — the one having for value A, and the other infinittj : 

 whilst the series (2) is not comprehended at all among the continuous cases of (1), hut is entirely 

 unconnected with, and independent of, those cases: its value is ambiguously I or n. 



In order that the influence of the law of continuity, which connects together all the indivi(lu;il 

 cases of (1), may not be overlooked or evaded in the extreme one of those cases, it will be desirable 



to change the notation: writing 1 for a;, when the limit 1 is to l)e reached through continu- 

 ous ascending values of ./•, and 1 + when it is to be reached through ccniiiuious dencending 

 values of X. 



• fhUonophicut Magazine for November ami December ni4.'i. 



