380 PROCEEDINGS OF THE AMERICAN ACADEMY 



If we conceive the whole surface of the earth to be covered with 

 w:ater, and consider only one section of it perpendicular to the earth's 

 axis, as that of the equatoi*, the effect of tlie moon is to cause the out- 

 line of the fluid to assume the form of an ellipse ; and if the tidal wave 

 moved without friction the vertices of this ellipse would be either ex- 

 actly in the plane of the meridian on which the moon is vertical, or 

 90° from it, the position depending mainly upon the latitude of the 

 place and the depth of the ocean. In either of these cases the attrac- 

 tion of the moon could not affect the time of the earth's rotation. But 

 as the fluid cannot move over the surface of the nucleus without fric- 

 tion, the position of the vertices of the ellipse, or tidal wave, is a little 

 displaced from the position which they would have without friction, in 

 the former case a little to the east, and in the latter a little to the west. 

 If the resultant of the moon's attraction upon the earth and the coun- 

 teracting centrifugal force be resolved into radial and tangential forces 

 perpendicular to the planes of the meridians, the latter only can have 

 any tendency to affect the earth's rotation. In the first quadrant east 

 of the meridian upon whicji the moon is vertical, and its opposite, these 

 tangential forces are towards the west, but in the other two quadrants 

 toward the east. Neglecting quantities of a very small order, these 

 forces in the different quadrants are exactly equal, and hence when the 

 quantity of matter in these quadrants is the same, they can have no 

 tendency to affect the earth's rotation. But we have seen that friction 

 tends to throw the greater part of the tidal wave into the quadrants in 

 which the tangential forces are toward the west. This is also the case 

 when the tide is produced by an oscillatory or rocking motion of the 

 fluid east and west, as in the Atlantic Ocean. Hence there is a pre- 

 ponderance of force toward the west, which causes a slight westward 

 motion of the water, and this motion, by means of friction, transfers 

 this force to the nucleus. 



In the third volume of Gould's Astronomical Journal I have shown, 

 upon the hypothesis that the outline of the whole tidal wave over the 

 surface of the earth assumes the form of a prolate spheroid, having its 

 vertices displaced 30° by friction, and that the height of the wave 

 above low water is two feet at the equator, that the rotatory motion of 

 the earth at the equator would, by the joint effect of the moon and sun, 

 be retarded about forty-four miles in a century. As the moon moves 

 about 84" of arc in its orbit while the earth at the equator moves forty- 

 four miles, the moon's motion from that cause would be apparently 



