7. When we consider the moon as causing the tides, and the change from high to low as depending on the rotation of the earth, it becomes obvious that if there is resistance to the motion of the water that constitutes the tides, that resistance must directly affect the earth, and must react on those bodies, the moon and the sun, whose attractions cause the tides. The theory of energy declares, in perfectly general terms, that as there is frictional resistance, there must be loss of energy somewhere. We are not now merely content to say there is loss of energy by resistance, but the modern theory must account for what becomes of that energy. It is particularly to Joule that the full establishment of the true explanation as to what becomes of energy that is lost in friction is due. I suppose every one here present knows Joule's explanation, namely, that heat is generated. The friction of the waters against the bottom of the sea and against one another, in rubbing, so to speak, as they must to move about, to rise in one place and fall in another-the friction of waters especially

in the channels where there are tide races, gives rise to the generation of heat. Well, now, the end, where it altogether leaves our earth to be dissipated through space, is heat. The beginning to which we can at present trace the first source of that energy is in the motions of the moon and the earth. A little consideration shows us, by a very general kind of reasoning, that that particular component of the motion which at zero would give rise to no tides must tend to become zero. This we see as included in a very general proposition applicable to every possible case of action in nature. Now, if the motion of the earth in its rotation, relative to the moon in its revolution round the earth, were zero, there would be no rise and fall of water in lunar tides; the earth would always turn the same face to the moon, and then it would be always high water towards the moon, low water in the intermediate circle, and high water from the moon, but there would be no motion of the waters relatively to the earth and so no friction. The tendency of friction must then, according to the general principle, be to

It is easy to see that

reduce the relative motions of the earth and moon to that condition. However, it is satisfactory to know that we do not need to base a conclusion on so excessively general terms of the theory of energy as those. the mutual action between the moon and the earth must tend, in virtue of the tides, to diminish the rapidity of the earth's rotation, and increase the moment of the moon's motion round the earth.

8. "The tidal spheroid," you must understand, is not a reality, because the waters do not cover the whole carth, as we are here on terra firma to know. But there is a perfectly definite surface, being an elliptic spheroid calculated by mathematical rule, which is such that if it were the outer boundary of a distribution of water over a globe perfectly covered with water, this mass of water would exercise to an extremely close approximation the same force upon any distant particle of matter, and experience the same reacting force, as our tidally disturbed waters really do. That is what

is properly called the tidal spheroid.

It averages,

as it were, for the whole globe, the tidal effect of the disturbing body considered. The tidal spheroid averaging the moon's effect alone, is called the luni-tidal spheroid; and that for the sun is called the soli-tidal spheroid. The resultant tidal spheroid, representing on the same principle the average displacement of the water produced by the combined influence of the two bodies, is found by simply adding the displacements from the undisturbed figure, represented respectively by the luni-tidal and soli-tidal spheroids.

9. If there were no frictional resistance against the tides, each separate tidal spheroid would have its longest diameter perpendicular to the line joining the centre of the earth with that of the disturbing body, whether moon or sun.1 When the joint

1 This assertion is founded not on observation, but on dynamical principles. It depends on the truth that, if the tide-generating influence of either sun or moon were suddenly to cease, the period of the chief oscillation that would result would be greater than either twelve solar or twelve lunar hours. The period of this oscillation would be less than either twelve lunar or twelve solar hours, if the sea were very much deeper than it is, or if it were considerably deeper, and also less obstructed by land. (See § 11.) If this were the case, the greatest axes of the luni-tidal and solitidal spheroids would be in line with the moon and sun respectively,

influence of the sun and moon is analyzed by mathematical reasoning, it is found that there would be for either separately, a tidal spheroid fulfilling the condition just defined. Thus the dynamical result of the tendency of either body would be low water at the time of the high water of the imaginary equilibrium tide, and vice versâ, on the average of the whole earth. By the lunar tide, for instance, there would be low water when

and there would be average high water of either component tide when the body to which it is due crosses the meridian; also, the average times of greatest tide would still be those of new and full moon. But if the depth of the sea and the configuration of the land were such that the chief period of oscillation could be intermediate between twelve solar and twelve lunar hours, the greatest axis of the luni-tidal spheroid would be in line with the moon ; but that of the soli-tidal spheroid would be perpendicular to the line joining the earth's and moon's centres. In this case, the times of spring tides would be those of quarter moons. In the first of these two unreal cases, the effect of tidal friction would be to make the time of average high water somewhat later in each component tide, than the time when the body producing it crosses the meridian; and this deviation would be greater for the sun than for the moon. Thus, the time of spring tides would be, as it is, somewhat later than the times of new moon and full moon. But, in the second of the imagined cases, the effect of friction would be to advance the time of solar high water and to retard the time of lunar high water; and thus the time of spring tides would be somewhat before the times of the quarter moons.