So far, then, our assumption of section 3, and Saint-Venant's constant-volume assumption, yield concordant results. But on a more searching comparison, it is seen that the two results are not the same. Having found in each case that with no added axial pull, there is never plastic longitudinal stretch, let us see how much added axial pull may be applied according to the respective assumptions. This pull, as found from section 3, is Saint-Venant's assumption of constant volume during plastic strain seems to be applied without any sense of danger. As far as I have been able to see, it is recorded without comment in the History of Elasticity* in the plain remark: "The material in the plastic state is treated as incompressible." The assumption is quite a common one, but the most specific reference to any experimental or other substantiation which I have noticed refers to the case of a specimen in the Testing Machine, and no figures are givent. If we imagine a material, otherwise maintained in the plastic condition, subjected also to a fluid pressure, we cannot but suppose that this pressure will produce a shrinkage of volume. So that although it may be established that in the case of simple tension, plastic deformation is accompanied by no change of volume, yet it would seem that further justification is wanted before we may confidently apply the principle to the present, or to Saint-Venant's, complicated problem. A third assumption has been put forward, enabling us to find R, in the all-plastic condition. It is that the longitudinal stretch C is the same function of the tensions R, R2, R3 when the material is plastic under the shear (RR)/2, as when it is elastic; i.e. always This hypothesis yields the condition for no plastic longitudinal stretch Thus these three hypotheses all give different results. Fortunately for our investigation, they agree that in a cylinder, whose b/a does not exceed something between 4 and 7, and which is rendered plastic throughout by an internal pressure alone, the longitudinal tension is nowhere a component of the maximum shear. Todhunter and Pearson, History of Elasticity, vol. II. art. 246. +"During the ductile elongation, the area of crosssection decreases in practically the same proportion that the length increases, or in other words, the volume of the material remains practically unchanged." A. Morley, Strength of Materials. By Prof. B. Hopkinson, to whom I am much indebted for valuable criticism and advice. OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY VOLUME XXI. No. XV. ON THE DIFFERENTIATION OF FUNCTIONS DEFINED BY W. H. YOUNG, Sc.D., F.R.S. CAMBRIDGE: M.DCCCC.XI. ADVERTISEMENT THE Society as a body is not to be considered responsible for any facts and opinions advanced in the several Papers, which must rest entirely on the credit of their respective Authors. THE SOCIETY takes this opportunity of expressing its grateful acknowledgments to the SYNDICS of the University Press for their liberality in taking upon themselves the expense of printing this Volume of the Transactions. or XV. On the Differentiation of Functions Defined by Integrals. By W. H. YOUNG, SC.D., F.R.S. [Received October 14, 1910. Read October 31, 1910.] § 1. IN a paper presented to the Society last year, and published recently in its Transactions*, I discussed the problem of determining when the process of reversing the order of integration in a repeated integral is allowable, in other words the problem of integration under the sign of integration. In the present paper I propose to give a set of rules for determining when differentiation under the sign of integration is allowable, more generally when it is allowable to make use of the usual formulae for the differential coefficient of a function defined by an integral, when the limits, as well as the integrand, involve a parameter. This matter has been treated, it is hardly necessary to say, with considerable care in the more recent English text-books; it is hoped, however, that the account here given will be found more complete, and more up to date, than any at present in existence. Several of the theorems to be found below are, it is believed, new in substance or in form, or are the extensions of known results. I have once more avoided the use of the e-machinery and I have always stated my conditions without reference to the uniformity, or non-uniformity, of the convergence of the integrals, when these are improper, and accordingly defined as the limits of proper integrals. In this way greater generality has been secured, and in certain cases increased facility of application. For the sake of clearness I have, here and there, given not only the enunciations but also the proofs of certain results recently obtained by myself and published elsewhere. I have, of course, wherever desirable, made use of the results of the companion paper, above referred to, on "Change of Order of Integration." Our present problem cannot indeed, with our present methods of research, be successfully attacked, when the integrals are improper, without in certain cases making use of the theory of change of order of integration. This is due to the circumstance that, in applying the Theorem of the Mean to a function of two variables, which is being differentiated with respect to one of them, the which occurs is a function of the remaining variable; this limits to some extent the use we are able to make of this theorem. The present account will be found, however, to * W. H. Young, "On Change of Order of Integration in an Improper Repeated Integral," Camb. Phil. Trans. 1909, vol. xxi. pp. 361-376. VOL. XXI. No. XV. 53 |