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considerable. These are among the curves given in the frontispiece of Prof. Pearson's Elastical Researches of Barré de SaintVenant.

116. Flexure of Holotropic Rectangular Beam.

We shall suppose the sides of the rectangle parallel to the axes of x and y, and of lengths 2a and 26.

When the two principal rigidities of the beam in the planes (z, x) and (z, y) are unequal we have to satisfy the equation

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Then x must be determined to satisfy the equation.

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at all points of the section, and the boundary-conditions

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where a' is written for a/(1+ L/M), and b' for b√/(1+M/L). This form is taken because X is an even function, and the condition at

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the y' boundary is satisfied identically. To determine the coefficients we have

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when b'>y'>-b'. Now between these limits we may expand y

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For an isotropic beam this reduces to

$ = B1 { [(1 + σ) a2 — žob2] x − † (2 + σ) (x3 − 3xy2)

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Returning to the general case, it is easy to see by symmetry

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therefore the terms in B, contribute no couple about the axis z.

Now suppose the beam bent by a load X parallel to ≈ applied at the end z = l, we have, as in art. 107, to add the solutions for

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where is the function determined by (142).

The curves of equal distortion have been traced by SaintVenant for a square beam of isotropic matter obeying Poisson's condition.

Investigations by means of conjugate functions might be given of the distortion of beams of various forms of section by flexure, but the problem is less interesting than the corresponding one of torsion on account of the comparative smallness of p.

CHAPTER VII.

CURVILINEAR COORDINATES.

117. Orthogonal Surfaces.

For many problems it is convenient to use systems of curvilinear coordinates instead of the ordinary Cartesian rectangular coordinates. These may be defined as follows:

Let f (x, y, z) = a, some constant, be the equation of a family of surfaces. Fixing our attention upon any point (x, y, z), one surface of the family will in general pass through this point. If small variations be made in x, y, z, i.e. if we pass to a neighbouring point (x+dx, y + dy, z + dz), this point will in general lie on a surface of the family differing from the surface ƒ (x, y, z) = α, but near to it. The surface on which it lies is given by the equation f(x, y, z) = a + da, where

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Thus a knowledge of a gives the surface of the family on which the point (x, y, z) lies, and a is called a curvilinear coordinate of the point (x, y, z).

If now we take three independent families of surfaces

fi (x, y, z) = α,

ƒ2 (x, y, z) = ß,

ƒ3 (x, y, z) = Y,

and fix our attention on the point (x, y, z), we find one surface of each family passing through the point. If a neighbouring point be taken one surface of each family will pass through the neighbouring point. The two sets of surfaces are taken to be (a, ß, y) for the

point (x, y, z), and (a+da, ẞ+dB, y+dy) for the neighbouring point. The quantities (a, ß, y) are called curvilinear coordinates of the point. Now, conversely, as any point will lie on three particular surfaces these determine the point; and, the region of space considered being suitably limited, if we attach to one point of this region a set of corresponding values of (a, B, y), and proceed in all directions from this point, by giving to (x, y, z) as functions of (a, B, y) values continuous with those at the chosen starting point, any point within the region will be given by its (a, ß, y).

The most convenient systems to choose, in applications of the theory of elasticity, are systems of surfaces which cut each other everywhere at right angles. Such systems are called orthogonal surfaces. It is well known that there exists an infinite number of sets of such surfaces, and, according to a celebrated theorem of Dupin's, the intersection of two surfaces belonging to different families of the same set of orthogonal surfaces is a line of curvature on each. In what follows we shall suppose the surfaces to be a, B, y, and shall suppose that these cut each other everywhere at right angles, so that the three relations

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The theory of orthogonal curvilinear coordinates is due to Lamé, and was developed by him in his Léçons sur les coordonnées curvilignes. The method we shall employ is founded on the particular case treated by Mr Webb in the Messenger of Mathematics, 1882. The problems at the end of the chapter have been considered by various writers, including Poisson, Lamé, Clebsch, Saint-Venant, and Mr Chree.

118. The line-element.

Let dn, be the length cut off from the normal to a constant at any point (x, y, z) by the neighbouring surface a+da of the family, and write h2 for the quantity

(da/dx)2 + (da/dy)2 + (da/Əz)2.

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