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XV.—A Chapter in the Calculus of Variations: Maxima and Minima, for Weak Variations, of Integrals involving Ordinary Derivatives of the Second Order

  • A. R. Forsyth


The present memoir is devoted to the determination of criteria for the possession of maxima and minima by an integral which involves one independent variable, one dependent variable, and the first and second derivatives of the latter. All the variations, leading to tests for the existence of maxima and minima, are of the class called “weak”; not only are the variations themselves small, but their derivatives of all orders are small also.



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page 152 note * An implicit assumption is justified in § 22.

page 156 note * These results were first obtained by Clebsch, , Crelle, Bd. lv (1858), p. 269; the preceding analysis differs completely from that used by Clebsch.

page 161 note * It is to be noted that, if they are simultaneous partial differential equations for G, the most general primitive is

where, as regards the two partial differential equations, F denotes the most general arbitrary function of its four arguments.

page 161 note † For another mode of derivation, see § 14, post.

page 163 note * See § 15 (vi).

page 165 note * As x and t are assumed to increase steadily through the range of integration selected, x1 can neither vanish nor become infinite within that range.

page 169 note * It would be that actual resolved part of the displacement of P, if t denoted the arc of the curve ; but, as already stated, it is convenient to select, as the independent variable, a quantity less intrinsic than the length of the arc of the curve which is the main aim of the investigation.

page 176 note * So far as the primitive is concerned, or Q could have a limited number of isolated zeroes when expressed as a function of x alone. But no one of these zeroes could be of an order higher than four, if the integral is to be regular ; and other conditions would have to be fulfilled which here are irrelevant, because Q is devoid of zeroes within the range.

page 179 note * Should C2 be zero, the appropriate selection is to hand, without further calculation.

page 184 note * As special as, e.g., a point of inflexion, or a point of six-point conic contact, on an ordinary plane curve.


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