## A History of the Progress of the Calculus of Variations During the Nineteenth Century |

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**PLANE CO - ORDINATE GEOMETRY AS APPLIED TO THE STRAIGHT LINE AND THE CONIC SECTIONS**; With Numerous Examples . By I. TODHUNTER , M.A. Second Edition . 316 pp . ( 1858 ) . Crown 8vo . cloth . 108. 6d . This Treatise exhibits the subject ...### Other editions - View all

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1+p² arbitrary constants axis Brunacci Calculus of Variations catenary chapter co-ordinates condition considered contains curvature curve Delaunay denote determined Differential Calculus differential equation double integral ds ds dx dx dx dy dz dx dz dx² dy dx dy dy dy dz dx dz dy dz dz Euler exact differential coefficient example expression formula geodesic gives Hence indefinitely small independent variable Integral Calculus integral sign investigation involves Jacobi's Jacobi's theorem Lacroix Lagrange maxima and minima maximum or minimum memoir method minima values notation obtain occupies pages occur ordinary Ostrogradsky partial differential equation plane Poisson problem quantities remarks respect result Sarrus second order shew solution Stegmann Strauch suppose surface theorem tion treatise triple integral vanish volume y₁ zero

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Page 140 - In fact this integral resembles the integral fVdydz ... and may be treated in the same way. We have merely indicated the transformations which must be applied to the portion JDUdxdydz ... of the variation SF; because since these transformations reduce to integration by parts they belong to the Integral Calculus rather than to the method of variations. It is true that one of the fundamental principles of this method consists in removing as much as possible the differential coefficients of the variations...

Page 37 - Sed quum calculus variationum integralium duplicium pro casu ubi etiam limites tanquam variabiles spectari debent, hactenus parum excultus sit, hanc disquisitionem subtilem paullo profundius petere oportet.