408 



VIII. NEWTONIAN POTENTIAL FUNCTION. 



the moon affects the motion of the earth, the earth disturbs the 

 orbital motion of the moon, which gives a fourth method of obtaining 

 the ellipticity, from which Helmert gives the value obtained from 



the moon's motion as ^=-5- 



AU I .0 



ISO. Potential of Tide -generating Forces. In order to 

 study the theory of the tides, it is necessary to obtain the expression 

 for the potential of the attraction of a distant body, such as the sun 

 or moon, considered centrobaric, as a function of the geographical 

 coordinates of a point on the surface of the earth. It is convenient 

 to consider the earth's center of mass reduced to rest by the principle 

 of 102, according to which we impress upon every point of the 

 earth an acceleration a, (cc x , a yj cc^) equal and opposite to that im- 

 pressed upon the earth by the distant body. But this acceleration, 

 which is the same for all points of the earth, is accordingly derivable 

 from a potential 



151) a x x -f K y y + a z z = ar cos (ccr). 



But if m is the mass of the distant body, D its distance from the 

 earth's center we have, 102, 





Fig. 140. 



Accordingly, if d is the distance of 

 the distant body from the point P 

 on the earth's surface (Fig. 140), 

 Z the angle between the radii, or 

 the zenith distance of the distant 

 body at P, we have for the whole 

 potential at P, 



1 K<r\ -rr m ^ ir rr 



152) F=^-^cos^. 



Now developing -5- by 105) and 



neglecting all but the first three 

 terms, 



153) V=~ 1 + ^ 



mr 



at/ 1 rj 



cos Z 



m . mr 



The first term is the same for all points on the earth, and therefore 

 may be neglected, so that the tide -genera ting potential is simply 



154) 



