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April 24, 1856.
The LORD WROTTESLEY, President, in the Chair.
The following communications were read:
I. "Elements of a Mathematical Theory of Elasticity." By Professor WILLIAM THOMSON, F.R.S. Received April 16, 1856.
This paper consists of two parts: Part I. on Stresses and Strains Part II. on the Mechanical Conditions of Relation between Stresses and Strains experienced by an Elastic Solid.
Part I.-The terms Stress and Strain are used in accordance with the valuable definitions by which they were first distinctively introduced into the Theory of Elasticity by Mr. Rankine*; with only this deviation; that instead of defining a stress as the reactive force exerted by an elastic body when in a condition of strain, the author of the present paper defines stress as "a definite external application of force to a body."
Various well-known theorems regarding the geometrical relations of the displacements among the parts of a body in a state of strain, and the geometrical representation of stresses and strains are enunciated, and briefly demonstrated, for the sake of convenience. A mode of expressing in absolute measure the magnitude of a stress or a strain, which the author believes to be new, is laid down nearly in the following terms. The amount of work done by a stress applied to a body of unit volume, while acquiring a strain of the same type as the stress, is measured by the product of the magnitude of the stress into the magnitude of the strain.
* "On Axes of Elasticity and Crystalline Forms," Proceedings, June 21, 1855. VOL. VIII.
When a stress and a strain are of the same type, they are said to be concurrent; or, if directly opposed, they are said to be negatively concurrent. When a stress and a strain are of any different types, the degree of their concurrence, or simply "their concurrence," is measured by the work done by the stress applied to a body of unit volume acquiring the strain, divided by the product of the magnitude of the stress into the magnitude of the strain. The measure of perfect concurrence is therefore +1, and that of perfect opposition -1. When work is neither spent nor gained in the application of a certain stress to a body while acquiring a certain strain, that stress and that strain, or any stresses or strains of the same types respectively, are said to be orthogonal to one another. The measure of their concurrence is zero.
A system of stress or strain coordinates involving symmetrically six independent variables, perfectly analogous to the system of triple coordinates for specifying the position of a point in space, is laid down. The concurrence of a stress or strain with six orthogonal types of reference being denoted by l, m, n, λ, μ, v, it is demonstrated that
l2 + m2 + n2 + λ2 +μ2 + v2 = 1,
and it is proved that if cos 0 denote the mutual concurrence between two stress or strain types, whose concurrences with six orthogonal types of reference are respectively (l, m, n, λ, μ, v) and (l', m', n', X', μ', '), we have
cos 0=ll' +mm' +nn' +XX' +μμ' + vv'.
The treatment of the subject in the text of the paper is quite abstract, but along with it a series of examples are given, illustrating the statements by applications to familiar types of stresses and strains.
Part II. commences with an interpretation of the Differential Equation of the potential energy of Elasticity of a solid, in terms of the mode of specification of stresses and strains laid down in Part I. The Quadratic Function expressing the potential energy of an elastic solid when strained to an infinitely small amount, is next considered; and its simplest possible form, that of six squares with coefficients, is interpreted. Hence it is proved that an infinite number of systems of six types of strains or stresses exist in any given elastic solid,
such that if a strain of any one of those types be impressed on the body, the elastic reaction is balanced by a stress orthogonal to the five others of the same system.
It is next shown that there is necessarily one, and in general only one, such system of six types of strain for an elastic solid which are all mutually orthogonal; and the types belonging to this system are called the Six Principal Strain Types of the body.
The characteristic of a Principal Strain Type is, that the stress required to keep a body in a state of strain of such a type, is of the same type as the strain. The six Principal Elasticities of a body are the six coefficients by which strains of the six Principal Types must be multiplied to find the stress required to maintain them.
In conclusion, reasons are given for believing that natural crystals may exist for which there are six unequal Principal Elasticities, and consequently six different, and only six different, Principal Straintypes.
A corollary regarding the property which certain liquids and crystals possess of causing a rotation in the plane of polarization of light passing through them, and Faraday's optical property of transparent bodies under magnetic force, is inferred, and is more fully considered in a subsequent communication to the Royal Society.
II. "On the Construction of the Imperial Standard Pound, and its Copies of Platinum; and on the comparison of the Imperial Standard Pound with the Kilogramme des Archives." By W. H. MILLER, M.A., F.R.S., Professor of Mineralogy in the University of Cambridge.-Part I. Received April 16, 1856.
The Commissioners appointed in 1838 to consider the steps to be taken for the restoration of the standards of weight and measure, to replace those which were destroyed by the burning of the Houses of Parliament, found provisions for the restoration of the lost standards prescribed to them by Sections 3 and 5 of the Act 5th George IV., whereby it is directed that, in case of the loss of the standards, the yard shall be restored by taking the length which shall bear a certain
relation to the length of the pendulum, vibrating seconds of mean time, in a vacuum, at the level of the sea; and that the pound shall be restored by taking the weight which bears a certain proportion to the weight of a cubic inch of water weighed in a certain manner. The Commissioners, however, in their Report dated December 21, 1841, decline to recommend the adoption of these provisions for the following reasons: "Since the passing of the said Act it has been ascertained that several elements of reduction of the pendulum experiment therein referred to are doubtful or erroneous. It is evident, therefore, that the course prescribed by the Act would not necessarily reproduce the length of the original yard. It appears also that the determination of the weight of a cubic inch of water is yet doubtful (the greatest difference between the best English, French, Austrian, Swedish and Russian determinations being about of the whole weight, whereas the mere operation of weighing may be performed to the accuracy of of the whole weight). Several measures, however, exist, which were most carefully compared with the former standard yard; and several metallic weights exist which were most accurately compared with the former standard pound; and by the use of these the values of the original standards can be respectively restored without sensible error. And we are fully persuaded that, with reasonable precautions, it will always be possible to provide for the accurate restoration of standards by means of material copies which have been carefully compared with them, more securely than by experiments referring to natural constants."
At the end of the Travaux de la Commission pour fixer les Mesures et les Poids de l'Empire de Russie, Professor Kupffer has collected the results of observations made in France, England, Sweden, Austria and Russia for finding the weight of a given volume of water. The resulting values of the weight of an English cubic inch of water in a vacuum at 62° Fahr., expressed in doli, of which 22504.86 make a kilogramme, are as follows::