This may be verified by means of the Helmholtz equations (14), which are in fact the conditions of integrability referred to. The hydrodynamical equations are accordingly of the forms where P is the pressure, is the potential of the ellipsoidal mass itself, and ' is that of disturbing bodies at a distance. The disturbing potential ' can be expanded, for points in the neighbourhood of the origin, in a series of solid spherical harmonics of positive degree. The terms of the first order are without influence on the motion relative to the centre of mass, whilst terms of higher order than the second are usually negligible. We write, therefore, n' = .......... † (A'x2 + B′y2 + C'z2 + 2F'yz + 2G′zx + 2H'xy), .(53) the coefficients, which are known functions of the time, being subject to the relation A' + B′ + C′ = 0, in virtue of the equation v2′ = 0. In the equations (14), (44), (55), (56) we have a system of ten equations connecting the ten dependent variables a, b, c, p, q, r, P1, 91, 71, λ with the time. It is to be noticed, however, that the equations (56) are precisely the equations which would be derived from (51) and (54) by expressing that the rates of increase of the angular momenta with respect to fixed axes coincident with the instantaneous positions of the axes of the ellipsoid are equal to the respective moments of the external forces. They are therefore equivalent to the system (13), where L, M, N may now be taken to refer to the disturbing forces alone, since the pressure-distribution given by (54) has zero moments about the axes. The direct identification of (56) with (13) is also not difficult. Although it is not essential to our purpose, we may substitute the values of a, ß, y obtained from (48) in (55). Eliminating A, we get aä a2 (q2 + r2 + q12 + r12) - 2caqqı = bb - b2 (r2 + p2 + r12 + P12) — 2abrr1 - 2bсрр1 + 2πph2 В。 + В'b2 2 c2 (p2 + q2 + P12 + 912) - 2bcpp1 - 2ca¶¶1 + 2πрc2уo + C'c2. ....(57) These, together with (13), (14), and (44), may be taken to be our fundamental system of equations. So far there is no approximation, and the equations would be applicable, for instance, to the finite oscillations of a Jacobian ellipsoid under a disturbing potential of the type (53). In the case, however, of a slight disturbance from a state of steady rotation about the axis of z, the quantities p, q, P1, 1, 71 will be small, whilst r will be approximately constant. It follows that, if we neglect small quantities of the second order in the first two of equations (13) and the first two of (14), the coefficients may be treated as constants. The precession is therefore independent of the tidal deformation, and is the same as if the fluid had been enclosed by a rigid envelope of negligible mass. The tidal motions of 'semi-diurnal' and 'long-period' types, on the other hand, are determined by the equations (57), together with (44) and the third equations of the systems (13) and (14), respectively. These latter, it may be noted, take the forms When the undisturbed ellipsoid is one of revolution about the axis of z, the precessional equations reduce as before to the forms (20) and (21). Moreover, in the astronomical application, that part of the disturbing potential which is effective as regards precession consists of terms of the form Ω = kr2 sin cos e cos (σt + ), .(59) where σ is nearly equal to w; cf. Art. 219 (1) and p. 348. In Cartesian co-ordinates we have The argument, leading to the conclusion that the precession is, under a certain condition, the same as if the mass had been solid, then takes the same course as in the preceding Art. Having regard to the form of ′ in (59), our solution may be regarded from another point of view as a determination of the 'diurnal' tides. In the present problem these cannot be discriminated from 'precession,' which is merely the name for their secular aspect. Barnes, H. T., 592 Barnes and Coker, 652 Basset, A. B., 127, 150, 175, 226, 591, 617, 635, 678, 684, 687, 692, 694 Beltrami, E., 81, 85, 142 Blasius, H., 86, 653, 667 Boltzmann, L., 101, 192 Boussinesq, J., 415, 417, 580 Bromwich, T. J. I'a, 287, 445 Burnside, W., 368, 383 Byerly, W. E., 103 Darwin, Sir G. H., 141, 274, 323, 327, 328, 329, Delaunay, C., 562 De Morgan, A., 288, 393 Dinnick, A., 498 Dirichlet, P. L., 116, 304, 392, 593, 689, 691, 692 Earnshaw, S., 478 Edwardes, D., 595 Eiffel, G., 665 Ekman, V. 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