It remains only to consider why the above assumption is legitimate. As far as the integral applies to the particles of d dx the rigid bodies, since Ex=8 for each particle, the trans dt dt formation is obviously justified. For the fluid, on the other hand, it is evident that, for an individual particle, Sx is not 8= dx dt d dt (This may be seen at once by considering the case of the motion of a flat piece of cardboard in a fluid, and supposing the time displacement perpendicular to its plane and the arbitrary displacement in its plane.) d To examine the meaning of the differences (8-8 (a 8.x - 8 da) dt, dt dt &c., let us suppose that the generalized coordinates are so taken that one only of the coordinates, say 92, alters with the time, so that the actual time displacement may be treated in the same way as a possible displacement 8q2. E C D The above may accordingly be written 88-86, where 81, 82 correspond to the variations 891, 892. Consider any point of the fluid A. Suppose the dispacement Sq followed by the displacement 8q2, and let the answering positions of A be B and C. Again, suppose the displacement 892 followed by 8q1, and let the answering positions of A be D and E. E will not coincide with C, but the displacement is that A which has for its projections the expressions above. B For, projection of EC=projection of DE-projection of AB ―(projection of BC-projection of AD) The bodies, however, after these two compound displacements are in identical positions; and consequently the displacements of the fluid (81-8182)x, &c., correspond to irrotational displacements of the fluid compatible with a position of the bodies momentarily fixed. We have then to show that, for such a displacement, Σdm(x8x+ÿdy+żdz)=0. This is evident; for the work done by the momenta of the particles of the system must for any possible displacement be equal to that done by the impulse; and the latter in this case vanishes, as the bodies remain fixed. The same thing may be shown analytically thus:— The first term vanishes, since the motion of the fluid in contact with the bodies is tangential; the second, since the fluid is incompressible. Addendum. Sir William Thomson has shown, in his paper on VortexMotion (Trans. R. S. E. vol. xxv.), that if one or more solid bodies are moving in an infinitely extended frictionless incompressible fluid, the motion of the fluid being supposed at one instant and therefore always irrotational, the impulse (i. e. the system of forces which would at any instant, if applied to the solids, generate the motion of the system of solids and fluids) would, if applied to a rigid body, represent a constant motive. It may be interesting to show that this conclusion follows directly from the Lagrangian equations. First, let there be only one rigid body. Take two systems of coordinate axes-the first (OX, OY, OZ) fixed in space, the second (O'X', O'Y', O'Z′) attached to the body. Let u, v, w be the components of the velocity of O' estimated along the moving axes; P, q, r the rotations of the body round these axes; x, y, z the coordinates of O' with respect to the fixed axes. Then, taking as generalized coordinates x, y, z, 0, 4, 0, we should have, were continuous forces applied to the body, O'Z, being a parallel through O' to OZ. It follows, integrating through the short interval of time during which the instantaneous impulse would act, that and similarly for the y-component and the z-component. Let us now proceed to find the physical meaning of and interpret the equation dT dy' It is clear from Green's theorem that T can be expressed as a quadratic function of p, q, r, u, v, w with constant coefficients; ᏧᎢ dT dp dT dq dT dr = + dy dp dy dq dự + + dr dy When p, q, r are expressed in terms of p, 4, 0, their coefficients do not involve the precessional angle . Therefore the first three terms vanish. are the force-components of the impulse with respect to O'X', O'Y', O'Z'; du dv dw dy' dy dy correspond to the displacement of a vector u, v, w fixed in space with respect to the moving axes, owing to the motion d of the latter; moment of a force R applied at end of vector u, v, w round axis O'Z; + moment of force of impulse applied at end of vector u, v, w round the same line =0. The left-hand member is obviously .. this last moment is constant, and the moments round OX and OY are also constant. We Secondly, let there be more rigid bodies than one. can now assume as the generalized coordinates the same as we have just taken which have reference to one of the bodies, together with other coordinates depending entirely upon the relative position of the rigid bodies amongst themselves; then dT dT da and are evidently the force-components along the axis dự OX and the moment round the axis O'Z, of the whole impulse, and the reasoning runs as before. XLVIII. Researches on Unipolar Induction, Atmospheric Electricity, and the Aurora Borealis. By E. EDLUND, Professor of Physics at the Swedish Royal Academy of Sciences. IT [Continued from p. 306.] § 2. Atmospheric Electricity and the Aurora Borealis. T is known that the earth may be regarded as a relatively good conductor of electricity; while, on the other hand, the atmospheric air, in the dry state and under the pressure to which it is subject at the surface of the earth, is a very bad conductor. Its conductivity, which depends almost exclusively upon the relative quantity of humidity which it contains, is consequently subject to incessant variations from the double point of view of time and space. When the density of the air diminishes, its conductivity increases; consequently there must exist at a considerable altitude above the terrestrial surface a stratum of air the conducting-power of which is better, yet without being particularly good. The terrestrial surface, both solid and liquid, is therefore immediately surrounded by a stratum of air endowed with feeble conductivity and subject to incessant variations. To this stratum succeeds another, the conductivity of which is greater and, as far as we know, sensibly invariable. The upper limit of the atmosphere has been fixed by astronomic methods at an altitude of between 70 and 80 kilometres. Truly speaking, these determinations signify only that the atmosphere up to that limit possesses sufficient density for its presence to be indicated by the ordinary methods of determination. That the atmosphere, though excessively rarefied, extends to a still greater elevation is most evidently proved by the fact that shooting stars have been observed at nearly 900 kilometres above the surface of the earth. These small bodies evidently can only become bright in consequence of a portion of their vis viva, transformed into heat by the friction of the air, augmenting their temperature to such a degree that they begin to shine. Now we can only perceive the falling body from the moment when it becomes luminous; and it is clear that at that moment it will have already traversed a certain length of path in the rarefied atmosphere before attaining so high a temperature. Therefore the upper boundary of the atmosphere must be situated at a much greater distance from the earth than has hitherto been admitted. The magnetic action of the earth cannot be explained entirely and in detail by the assumption that its magnetic force is due to a magnet of iron or steel situated in the earth, and |