Plans Plate under Thrusts in its own Plane. 67 A very good illustration of the buckling of a circular plate is frequently afforded by the lid of a circular canister, in which the thrust is due to the tension of the rim. The "dint" in such a lid can be readily pushed from one side to the other, but it is impossible to keep the surface flat, as that position is unstable. The same principle is also illustrated in the " castanets," in which a " clicking " sound is produced by pushing a disc of

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... from one side to the other of the unstable plane form. 13. In all the cases discussed in this paper, the stresses in the surface are proportional to /3; and, therefore, to the cube of tho thickness of the plate. Since these stresses are distributed over the thickness of the plate, the strains they produce are proportional to the square of the thickness. If, therefore, the plate be thin, these' strains will be small, and there will be no rupture of the material accompanying the buckling. This accords with the general results obtained in my paper " On the Stability of Elastic Systems."* In a future paper, I hope to deal with further applications of the variational method, with special reference to the stability of a rectangular plate or strip in certain cases when the shear M does not vanish, and when the boundary conditions are different to those assumed in the present communication. On the Application to Matrices of any Order of the Quaternion Symbols 8 and V. By HENRY TABER, Docent in Clark University, Worcester, Mass. U.S.A. [Head Dec. Uth, 1890.] Properties of the Symbols S and V. The conception of scalar and vector parts of a quaternion, or matrix of tho second order, may be extended to matrices of any order.f Regarded as.a matrix, the scalar of any quaternion is one half the sum of its latent roots; following this analogy, I shall define the scalar of any matrix m of order w as the u> t h part of the sum of •