HAEMOISIG ANALYSIS AND PREDICTION OF TIDES. 5 



face would be neither increased nor diminished by changing the posi- 

 tion of that particle to any other point in the same surface. On a 

 surface of equilibrium the resultant of all the forces at each point 

 must be in the direction of the normal to the surface at that point. 



The equilibrium theory assumes that the solid part of the earth is 

 covered yviih. water of considerable depth; that the water has neither 

 inertia nor viscosity and may move without friction. These ideal 

 conditions, differing so greatly from the actual conditions, it is not 

 to be expected that the liquid surface of the earth will attain the 

 state^of equilibrium assumed under this theory. The presence of the 

 great continental barriers, together with the inertia of the water, 

 would make such a state of equilibrium impossible on the rotating 

 earth. Nevertheless, the theory is of much service in the discussion 

 of the harmonic analysis, because it affords a convenient and com- 

 plete way of specifying the forces which act upon the ocean at each 

 instant. 



The attraction of either the moon or the sun will tend to draw the 

 earth out in the shape of a prolate ellipsoid of revolution with the 

 longest axis in the direction of the attracting body. Figures 2, 3, 

 4, and 5 illustrate the forms of tides which may be expected under 

 the equilibrium theory when either the moon or the sun is acting 

 alone. In these figures the oblateness of the earth due to its centrif- 

 ugal force is ignored. Figures 2 and 4 may represent any section 

 made by a plane passing through the center of the earth and the 

 center of the moon, but here they are supposed to represent especially 

 the section containing the earth's axis. In Figure 2 the moon is 

 assumed to be in the plane of the earth's Equator, and in Figure 4 

 the declination is taken at about 28.5° N., which is approximately | 

 the maximum declination reacfied by tlie moon. The great circles | 

 show the mean undisturbed surface of the earth and the ellipses the | 

 surfaces as modified by the attraction of the moon. In these figures I 

 the ellipticity of the modified surface has been made about a million | 

 times greater than the theoretical ellipticity due to the attraction of | 

 the moon. If drawn to true scale, the disturbance due to the moon 

 could not have been detected with the eye. This magnification of 

 the ellipticity will introduce some discrepancies in the figures when 

 compared with the true theoretical form but which are unimportant 

 at this time. In Figures 3 and 5 are shown sections of the undis- 

 turbed and the modified surfaces made by planes perpendicular to 

 the earth's axis in latitudes 0°, 30° N., and 60° N. 



Let us now consider what tides may be expected under the equi- 

 librium theory. We will suppose that the liquid surface of the earth 

 retains its ellipsoidal form with the major axis always toward the 

 center of the moon, and that the solid portion of the earth rotates on 

 its axis. It is evident that every point of the solid part of the earth 

 will describe a circle parallel to the Equator and with its center in the 

 axis of the earth, and that in passing around this circle the water 

 surface at the point will fluctuate in height. Referring to Figure 3, 

 let us take a point P^ of the undisturbed surface on the Equator and 

 directly under the moon, the latter being in the plane of the Equator. 

 At Pj it will be high water. As the earth rotates the point Pj will 

 move to Pj, the height of the water gradually diminishing until at 

 Pj it is a minimum or low water. In passing on to P^ the water will 

 rise to a maximum and then fall to another minimum at P^ and then 



