DEDUCTIONS FROM THE TIDES. 37 



Poincar^, after a mathematical treatment of the influence of the water- 

 tides on the earth's rotation in the endeavor to simpHfy the case, reached 

 the following conclusion: 



L'influence des marges ocdaniennes sur la dur^e du jour est done tout k fait minime 

 et n'est nullement comparable k I'effet des marges dues a la viscosity et k I'^lasticit^ de la 

 partie solide du globe, efifet sur lequel M. Darwin a insists dans une sdrie de M^moires du 

 plus haut int^ret.' 



A MORE RADICAL MODE OF TREATMENT. 



To the foregoing method of treating the rotational effects of the tides 

 on the basis of the positions of the tidal protuberances and depressions, 

 as such, there seem to be, as previously intimated, graver infehcities than 

 those of mere complexity. The method appears to be defective in neglect- 

 ng the cooperating effects of the changes of kinetic and potential energy 

 that are associated with these differences in the distribution of matter. 

 These protuberances are not fixed masses of matter, but rather aggregates 

 of variations in the paths of the molecules of water in their revolutions 

 about the earth's axis. In the production of these protuberances and 

 depressions there are reciprocal increases and diminutions of the potential 

 and kinetic energies of the water particles involved. In analyzing the 

 influences of these on rotation, it will be serviceable to separate the factors 

 of inertia and friction — including under the generic term friction all obstruc- 

 tive effects growing out of the relations of one particle to another — because 

 the functions of these factors are contrasted, since the inertia tends to 

 perpetuate any given state of motion, while the friction tends to reduce the 

 amount of motion. There are also certain advantages in considering each 

 particle separately as a body in revolution about the axis of rotation. 



Let therefore the lithosphere be regarded as a perfect spheroid sur- 

 rounded completely by an ocean of uniform depth, and let the matter of 

 each particle be regarded as concentrated into a point and separated from 

 its fellow particles by a complete vacuum, but let the collapse of the particles 

 be prevented by a hypothetical force taking the place of the resistance to 

 condensation which affects the water in nature. We shall then have an 

 ocean made up of mass-points which move in perfect freedom from fric- 

 tional and other obstructive relationships; in other words, these points 

 will constitute satellites of the Hthosphere which may here be regarded as 

 a rigid body acting as a massive point at its center. The behavior of the 

 mass-points may then be treated, qualifiedly, according to the principles 

 of celestial dynamics. In fig. 4, let E represent the earth, the circle L the 

 surface of the lithosphere, and the circle A BCD the ideal surface of the 

 hydrospheric satellites when revolving without perturbation by the moon. 

 Then, according to the principles of celestial mechanics, first appHed to 

 this class of cases by Newton,^ the orbit of a particle, p, will be a closed 

 curve, abed, closely resembling an ellipse, whose major axis is transverse 

 to a line joining the centers of E and M. The general configuration, it 

 will be noticed, is that of the "inverted tides." The particle p in passing 



' Bulletin Astronomique, vol. 20 (1903) " Sur un Th^or^me G^n^ral Relatif aux Maries " 

 par M. H. Poincar^, p. 223. 



^ Moulton's Celestial Mechanics, art. 156, p. 243. 



