436 



PROFESSOR YOUNG, ON THE PRINCIPLE OF CONTINUITY, 



to them, but are, on the contrary, indeterminate — like those already noticed *. That they are not 

 indeterminate however will be obvious from again adverting to the notion of quadratures: the 

 ordinatcs of the curves are evidently determinate throughout the whole extent of the integration, — 

 tliat at the superior limit CD being zero. The first of these integrals has been proved — by what 

 appears to me to be perfectly valid reasoning, though it has recently been objected to — to be 



altogether independent of the value of the constant a, and to be equal to — , or - — , according as 



the sign of this constant is positive or negative. Poisson indeed, following Euler and others, says 



tliat the values of the integral are — , 0, or , according as the constant is positive, zero, or 



neoativef. But it should be remembered, in obedience to the law of continuity, that if a become 

 zero, by passing through neighbouring values, and vanish positively, the value of the integral is still 



— ; and if it vanish negatively, the value of the integral is still , as in all the continuous series 



of cases which these terminate. 



The integration of the second of tlie preceding forms has however been effected by methods 

 which are really objectionable, notwithstanding the accuracy of the results obtained by them : and it 

 may not be uninstructive briefly to direct attention to this circumstance. 



Legendre commences his process by at once destroying the generality of the proposed integral — 



Skvr 



takine for the limits, not .v = 0, and ,v = os , but ,r = 0, and ,v = , k being a whole number ; and 



° a 



then, at a convenient stage of the investigation, making k infinite. By means of this artifice, the 



indeterminateness, which the method employed would otherwise have introduced at the limit 



.V = CO, is overruled by an arbitrary condition :j:. The true result however necessarily comes out; 



because that result is independent of all condition as to how the limit co is reached. 



In the other method of integration, the indeterminateness adverted to is not evaded, but is 

 allowed to enter into the process : it is however wholly disregarded ; and thus, by a sort of com- 

 pensation of errors, the true result is again obtained. This, I presume, is the metliod to which 

 Sir. W. R. Hamilton alludes, at page l(i of his profound and remarkable paper on Fluctuating 

 Functions^, where an accurate investigation of this integral is given ||. 



It may be proper to add, that when by applying differentiation to a determinate form, whether 

 an infinite series or a definite integral, we are led to indeterminateness, the step must be regarded 

 as inadmissible, and unless corrected, as leading to a false result. It is not difficult to see the reason 

 of this. In each case a certain constant is considered to be infinite ; for which extreme value 

 a particular function of the variable, that for all other values of the constant would have entered the 

 original expression, disappears; but which function if preserved, instead of being obliterated as zero, 

 would reappear in an indeterminate form, after differentiation. The suppression however of the 

 evanescent function in the original, precludes this reappearance; and thus leads to a defective 

 result II. This, I think, is rather an interesting fact : it shows that the differentials of certain forms 

 of analysis require indeterminate corrections, in a manner somewhat analogous to that by which the 

 ordinary determinate corrections are introduced into integrals ; and the omission of which indeter- 

 minate corrections has led to so many erroneous summations of certain trigonometrical series. From 



• Tramuclions of the Society, Vol. vill. Part lli. Earn- 

 shaw's Paper on sin co and cos co. It may be remarked here, in 

 reference to the two integrals in the text, that the function under 

 the sign of integration becomes in each case zero at the superior 

 limit CO : and that therefore, as was before observed of periodic 

 series, the foreign factor, €'", which Poisson introduces merely to 

 destroy indeterminateness at this limit, is inoperative, and may 

 therefore be admitted without incurring error ; and the same remark 



applies whenever the subject of integration, in integrals of this 

 kind, becomes zero for j- = co . 



t C/ialeur, p. 288. 



X Legendre ; Exercises de Calcul Integral, Tome I. p. 357. 



^ Transactions of the Royal Irish Academy, Vol. xix. Pt. ll. 



II For the faulty process, see Gregory's Examples, p. 481. 



f See a Paper by the author in the Phil. Mag. Vol. xxviii. 

 p. 213. 



