A Treatise on the Calculus of Variations: Arranged with the Purpose of Introducing, as Well as Illustrating, Its Principles to the Reader by Means of Problems, and Designed to Present in All Important Particulars a Complete View of the Present State of the Science
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abscissa angle appear arbitrary constants assume axis axis of x become infinite become zero calculus of variations catenary co-ordinates consider cusps cycloid denote derived curve differential coefficients differential equation discontinuous solution discussion double integral equa equal Euler's method evidently extremities fixed points function give given curves Hence indefinitely small independent variable infinitesimal invariable sign Jacobi's Theorem Jellett last equation length limiting curve limiting values lower limit maxima and minima maximize or minimize maximum or minimum merely minimize the expression Moreover negative obtain ordinate partial differential particle pass plane curve Prob Prof quantities radius range of integration reduce regard render required curve required surface required to determine restriction result right line second member second order similar equation Substituting suppose supposition tangent Taylor's Theorem throughout tion transformation values of Sy vary Whence
Page i - Carll. — A TREATISE ON THE CALCULUS OF VARIATIONS. Arranged with the purpose of Introducing, as well as Illustrating, its Principles to the Reader by means of Problems, and Designed to present in all Important Particulars a Complete View of the Present State of the Science.
Page 549 - ... under the integral sign ; but the calculus of variations only indicates this operation and refers the execution of it to the Integral Calculus. CHAPTER VI. DELAUNAY. 133. THE Academy of Sciences at Paris proposed the following as the subject of competition for their great mathematical prize in 1842 ; To find the limiting equations which must be combined with the indefinite equations in order to determine completely the maxima and minima of multiple integrals, the formulas to be applied to triple...
Page 542 - PROBLEMA NOVUM, Ad cujus solutionem Mathematici invitantur. Datis in piano verticali duobus punctis A et .23, assignare mobili M viam AMB per quam gravitate sua descendens, et moveri incipiens a puncto A, brevissimo tempore perveniat ad alterum punctum B...
Page 403 - ... by a force in the given direction equal to the sum of the components in that direction of the external forces acting on the particles of the system.
Page 3 - The equation of the semicubic parabola ...... ... (79) is generally adopted for the developed curve of the rifling. The twist is assumed at breech and muzzle and the curve between these points is obtained from the above equation. The tangent to the curve at any point makes with the axis of x an angle whose tangent is dy/dx. The value of the tangent of the angle at any point is ir/n, see Eq.
Page 240 - ... Prentiss led him to believe that the otocyst was a static organ solely. It is probable that in the Brachyura the hooked setae and grouped setae have lost most of their functional activity owing to the absence of otoliths. The thread setae are, undoubtedly, the most important sensory organs of the otocyst. (For a further discussion of this subject the reader is referred to the paper by Prentiss.) The sensory setae of Cancer are of two kinds. The tactile setae may be present in various parts of...
Page 42 - For if ai - a0 be not = 0 we must have an equation which, as is easily seen, implies that the integral of an arbitrary function may be expressed (without determining, or even restricting its general form) in terms of the limiting values of itself and a certain number of its differential coefficients. This is manifestly untrue.
Page 345 - If tan-i m is the angle which the normal makes with the axis of x, m = - p, and equation (4) becomes 251.
Page 554 - Werkes wenigstens einigerroaassen i den Stand zu setzen: I. Lagrange, Lacroix. II. Dirksen, Ohm. III. Gauss. IV. Poisson. V. Ostrogradsky. VI. Delaunay. VII. Sarrus. "VIII.