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NOTICES AND ABSTRACTS 
OF 
MISCELLANEOUS COMMUNICATIONS TO THE SECTIONS. 
MATHEMATICS AND PHYSICS. 
On the Principle of Continuity in reference to certain Results of Analysis. 
By Professor Youne, Belfast. 
Tue principal object of this paper is to examine into the influence of the law of 
continuity, as it affects the extreme or limiting values of varying functions, and to 
exhibit some remarkable verifications of the mathematical axiom that “ what is true 
up to the limit is true a¢ the limit.” 
Much error and confusion exists in the writings of analysts, more especially in those 
of Cauchy and Poisson, in consequence of the imperfect views generally entertained 
in reference to the principle of continuity, whenever extreme or ultimate values of a 
"variable come under consideration. In certain infinite series where the condition 
of continuity avowedly prevails, it is the common practice to neglect this principle in 
_ the limiting cases, and to treat such cases as if they were entirely isolated and un- 
connected with the general forms to which they really belong; whilst in other classes 
of series, those namely that have been called periodic, as also in definite integrals 
involving periodic quantities, it is a practice equally common to introduce the condi- 
tion of continuity where in fact it has no legitimate existence. Many false conclusions 
have thus obtained currency in analytical writings, and it is the aim of the paper, of 
which this is a very brief abstract, to inquire into the sources of these errors, and to 
supply the requisite corrections, 
As respects series, for instance, it is shown that the limiting cases of 
2 
v1 +5 += +..... ad inf., corresponding to#= 1land«= —1, 
are very different from what Cauchy and other writers affirm them to be. Cauchy 
says, that when these limits are reached, the series will be divergent in the first case, 
and convergent in the second: but it is proved in the paper adverted to, that “if x 
ascend from an inferior numerical value (that is from a fractional value either positive 
‘or negative), up to =1 or x = — 1, the limiting cases will both be convergent, 
like all the preceding cases; but if the same limits be reached through descending 
values of the variable, the extreme cases will then, on the contrary, be divergent.” 
bs In like manner the limit of the series 
os lt+oet2a2?4+ 2.3234 2.3. 424+ &c. ad inf. 
when x arrives at zero, and which is said by Cauchy to be equal to 1, is proved to be 
reality equal to a quantity infinitely great. 
‘The errors of Cauchy arise from his neglecting the influence of continuity in these 
miting cases: the errors of Poisson, in his researches into the theory of periodic 
series and definite integrals in the ‘ Journal de |’Ecole Polytechnique,’ and in his 
Théorie de la Chaleur,’ are of a directly opposite character: they arise from his 
arbitrarily introducing continuity where no such principle exists. Poisson admits 
that periodic series and periodic integrals are in themselves indeterminate; but 
he considers himself at liberty to overrule this indeterminateness, by introducing 
into the series the ascending powers of a quantity infinitely little different from unity, 
and by introducing into the integral the arbitrary multiplier e—47. By means of this 
unwatrantable artifice the periodicity is in both cases destroyed: the series is ren- 
—~«(1846. B 
