On the Mathematical Principles o/Laplace's Theory of Tides. 545 



nates, oot l -\-ot = 0. Hence x l and y l are functions of t that may 

 be calculated. 



It has thus been shown that the coordinates x' } ij, z' are all 

 expressible as functions of t. We may now, if we please, sup- 

 pose the earth to have no motion of rotation, provided an angular 

 motion about the axis of z equal to that of the earth be im- 

 pressed on the moon at each instant in the direction from east 

 to west ; for thus the relative motions, and the values of x', 

 y\ z } will remain the same as before, only these coordinates 

 will apparently be referred to axes fixed relatively to the earth 

 and fixed in space, as plainly may be the case considering that 

 the earth is apparently absolutely motionless. Let x, y, z be 

 the coordinates of any point of the ocean referred to the same 

 axes. Then, if m be the attraction of the moon at the unit of 

 distance, and R be put for 



m (xx 1 + yy' + zz') m 



(*/2 + /2 + 2 / 2) f " Q(X-X ! )*+ (y-y>)*+(z-z'f)*' 



we shall have, as in Physical Astronomy, for the attractions in 

 the directions of the axes of x, y, z, the values -^— , -=— , — re- 

 spectively. 



We have next to take account of the attraction of the earth 

 itself at the same point of the ocean. The amount and direction 

 of this force will depend on the composition and dimensions of 

 the earth, and on the form of its superficies as determined by 

 the mutual attractions of its parts, both solid and fluid, by the 

 moon's attraction, by centrifugal force, and by its superficial 

 irregularities. Now, as we are here concerned only with central 

 forces emanating according to the law of gravity at each instant 

 from all the elements of the earth's mass, if we call the resultant 

 forces in the directions of the axes of coordinates X 1; Y v Z lf we 

 shall have, as is known, X^x + Y^y-j-Z^z, a complete differ- 

 ential. The values of X u Y v Z x may contain t. 



It only remains to take into consideration the action of cen- 

 trifugal force. It is here to be observed that we have already in 

 the previous reasoning had regard to the hydrostatical effect of 

 this force in modifying the earth's figure, and thereby exerting 

 an indirect influence on the amount of tide by altering the earth's 

 attraction at a given point of the ocean. At present we have to 

 take account of the hydro dynamical effect of centrifugal force — 

 that is, the effect of its action on fluid in motion, the motion 

 being such as is peculiar to a fluid, causing the forms and rela- 

 tive positions of the elementary parts to be continually changing. 

 Tidal motion is plainly of this kind. The centrifugal force in 



Phil. Mag. S. 4. No. 334. Suppl. Vol. 50. 2 N 



