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Ar Sr/Ar A^'r+i . Vi = ^n-ifi'n- THE ELASTICA. 227] 47 on which the lines are drawn and the extensions of the method to cases where the extremities are built in, or the supports are not all on the same level, the reader is referred to M. Levy's Statique A paper by Professors Perry and Ayrton in Graphique, t. il. S. Lond. Nov. 1879 may also be consulted. Proc. For details in regard to the scale to represent forces, K The 227. We Elastica. now consider the problem presented by a thin rod which is held in the shape of a plane curve by forces and couples applied at its ends alone.

CQ (BQ + 2GQ). and the equation for B, C, D is in like . {4^1\ the weight 223. W being at Q between B and G. Equation of Three Moments generalised. The theorem of three moments may be generalised so as to include the cases of variable flexural rigidity and variable load\ For this we begin with the case where there is no load between A and oi B or between B and G, but the flexural rigidity X. Fig. 28. 1 Webb. Gamh. Phil. Soc. Proc. vi. 1886. 33 is a function BENDING OF RODS IN ONE PLANE. 40 AB wq In the span have, measuring [223 x towards B, since as in equation (42) of the last article we may express the shearing force to the right of by taking moments about B.

1379. Theory of Sound, vol. i. 2—2 ch. x. HISTORICAL INTRODUCTION. 20 frequency-equation for the lowest tones and the character of the nodal lines. He remarked that if there are any materials which when they are slightly extended longitudinally, rectangular plates of such materials can vibrate like bars without being supported at the edges. The theory of the vibra- are not laterally contracted tions of a free rectangular plate has not in these cases. The first been made out except attempt to solve the problem of thin curved plates by the aid of the general theory of Elasticity was made He expressed the geometry of the middle-surface Herr Aron\ by means of two parameters after the manner of Gauss, and he by used the method employed by Clebsch for plates to obtain the or shells equations of equilibrium and small motion.

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A TREATISE ON THE MATHEMATICAL THEORY OF ELASTICITY by A. E. H. LOVE


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