This is the vortex-atom theory of Sir W. Thomson, dimly foreshadowed in the writings of Hobbes, Malebranche, and others, but only made distinctly conceivable in very recent times by the hydrokinetic researches of Helmholtz. Helmholtz, in 1858, first successfully attacked the equations of motion of an incompressible frictionless fluid, without introducing the great simplification which had been adopted by his predecessors, and which consisted in supposing the motion to be non-rotational. He proved, among other valuable results, that those portions of the fluid which at any time possess rotation preserve it for ever, and are thus as it were marked off from the others; also that these portions must be arranged in filaments whose direction is at each point the axis of rotation, and that the filaments are either endless, _i.e._ form closed curves (whether knotted or not), or terminate in the free surface of the fluid. Hence Sir William Thomson’s idea that what we call matter may consist of the rotating portions of a perfect fluid, which continuously fills space. This definition involves the necessity of a creative act for the production or destruction of the smallest portion of matter, because rotation can only be produced or destroyed by us in a fluid in virtue of its viscosity (or internal friction), and in a perfect fluid there is nothing of the kind. 134. Of course it may be objected to this theory that it merely shifts the difficulty one step further back,—after all, explaining what we call matter by certain motions of something which, as it must have inertia, it would appear we are bound to call matter also. We have been careful to mention this (latest) speculation as to the nature of matter for three reasons: 1st, because we shall have to make considerable use of it in the course of our argument, for purposes of illustration; 2d, because it shows one way of at once thoroughly accounting for the conservation of tangible matter; 3d, because it shows the possibility of forming an idea of a true atom which shall not require, even for perfect elasticity, the inconceivable quality of perfect hardness necessary to the atom of Lucretius. In fact, the few words which we have given above about Helmholtz’s investigations show that, to cut a vortex-atom, it would be necessary to give a free surface to the perfect fluid which on this theory is supposed to fill space, _i.e._ virtually to sever space itself! This suggestion of Thomson’s promises to be very valuable from one point of view at least, viz., the extension and improvement of mathematical methods; for in the treatment of its very elements it requires the application of the most powerful of hitherto invented processes, and even with their aid, the mutual action of two ring-vortices (the simplest possible space-form) has not yet been investigated except in the special cases of symmetrical disposition about an axis. Hence we are at present altogether unable