The subject of this short paper is one which has received much attention, and many theories and reasons have been given therefore, the principal object being to account for the upheaval of the water on the side opposite to the attracting body as well as upon the side adjacent.
My attention was called to this subject some time since by an extract from an article on tides and their cause, published in the Popular Science Monthly which criticised the accepted theory very
severely, and, as I thought, very justly; but there was one thing contained therein which I could not accept, and of which I had never heard before, viz. "It has been proved experimentally that all bodies
on the surface of the earth are heavier at midnight than at any other hour of the twenty-four; and that when new moon occurred at midnight this weight was very much increased."
Now this fact may have been proven as alleged, but I do not believe it. In my opinion the fact that there is a tide on the side of the earth opposite the attracting body is the best refutation of the
assertion.
Observations upon the tides show conclusively that there are two tide waves, the lunar and solar, and that for certain relative positions of the sun and moon these waves conspire and at others they are opposed. The reasons and conditions necessary to cause these results, together with the intermediate stages of the tide, I do not propose to enter into. I will simply premise that the observed tide is the resultant of two separate and distinct tides, the solar and lunar, and that both of these are the result of universal gravitation, are entirely similar, but different in amplitudes and are the effects of identical causes.
In discussing this question I will take up the solar tide alone, and will endeavor to show ample reason for a tide on the side of the earth opposite the sun, and will also attempt to show that anybody on the earth's surface is, from the sun's attraction, lighter at midnight than at any other hour of the twenty-four, excepting midday, and also that at these hours a body on the surface of the earth is, in regard to its weight, unaffected by solar attraction. It is well known that the earth, under the action of the sun's attraction, moves about the sun in an elliptical orbit of small eccentricity; the centre of inertia of the sun and earth, taken together, being situated at one of the foci. This last is subject to the modification that such would be the fact without the perturbating influences of the other members of the solar system, but in considering this subject these influences can be ignored as having no appreciable effect upon the tide.
By Kepler's first law, the sun's attraction is the centripetal force which causes the earth to follow its orbital path, and its direction is that of the line joining the centres of inertia of the sun and earth, and a discussion of the equations representing the laws of central forces as applied to the earth shows that this attraction or centripetal force is the resultant of the reactions of two forces, one in the direction of the radius of curvature of that part of the orbit in which the earth is found, and the other at right angles thereto—the first is the reaction of the centrifugal force which acts from the centre of curvature, the second is the reaction of the inertia of the earth in the direction of the tangent to the orbit.
Now suppose the circle A, B, C, D (Fig. II) to be a great circle cut from the surface of the earth by a plane passing through the centre of the sun S, and suppose any unit of mass on this great circle,
as E, to be assumed : Let
d = distance from centre of sun to centre of earth.
z=. " " " to unit of mass -£.
& = angular distance of unit of mass from sun.
0. angle at sun subtended by radius to £.
r = radius of earth.
m=: mass of sun.
k =. attraction of a unit of mass at a unit distance.
Now the intensity of the attraction of gravitation varies directly as the attracting mass and inversely as the square of the distance, therefore the sun's action on the unit of mass E is represented by:
The sun's attraction for the earth, as a whole, is made up from the attraction for the units of mass separately considered, and if the attraction for any unit of mass, as E, be resolved into two components, one parallel to the line joining the centres of the sun and earth, and the other perpendicular thereto, the first will be the component of the centripetal force acting upon the earth due to this unit. As shown before, this component is the resultant of the reaction of the centrifugal
force and the inertia of the unit of mass, and is neutralized by them.
Its line of direction is always towards the centre of the earth, indirectly. Resolving this force into two components, one in the direction of radius, and the other perpendicular thereto, we have two components.
The component represented by the first of these expressions is the one perpendicular to the radius at E, or tangent to the earth, and simply tends to impress upon the unit of mass E a motion of translation in its line of direction and cannot in any way influence the weight or specific gravity of E.
The other component represented by the second expression has its line of direction towards the centre of the earth, and therefore impresses itself upon the unit E and thereby adds the amount of its intensity to the weight thereof. The intensity of this component is a function of the angles ^ and 0, and depends upon them for its value. is also a function of ^p. By assigning different values to 9", deducing the corresponding value of and substituting in the expression, the intensity of this component for different positions of the unit E can be found.
The values of this component being zero at A and C and positive at B and D (the negative sign not affecting the intensity, but simply showing its direction), and being connected by the law of continuity, it follows that the effect of the sun's action upon the unit of mass is to increase its weight from A in both directions to B and D, and to decrease this weight from B and D to C; and as this is true of all sections of the earth by planes through its centre and that of the sun, it follows that waters of the ocean on and near the circumference of a section of the earth by a plane through the earth's centre, and perpendiculars to these, will, by the principles of hydraulics, press up those about A and C until an equilibrium of the pressures due to increased weight and increased height is established, thereby producing the solar tidal wave.
The same course of reasoning is applicable to the moon's action, showing that the weights of bodies are increased from the point of the surface of the earth immediately adjacent to the moon, until the bodies are at an angular distance of 90° from the moon in all directions, and that then the weights decrease until we arrive at that point of the earth's surface immediately opposite the moon.
I think nothing further need be said to show the fallacy of the so called "experimental proof" referred to in the paragraph quoted from the Popular Science Monthly in regard to the weight of a body at midnight. I do not wish to be understood as asserting that a body could not weigh heavier at the civil midnight, for the relative positions of the sun and moon can be such as to produce this effect; but if we suppose a hypothetical body that would produce the combined effect of the sun and moon and call its upper transit tidal midday, and its lower tidal midnight, a body would weigh the least at those times and the most at tidal six o'clock morning and evening.