V=- p 2(0 



504 Mr. J. Prescott on the 



taken as axis of z, and let the axis of x be the equatorial 

 radius in the plane of the axes of rotation and of figure. The 

 direction -cosines of the axis of rotation are sin 0, 0, and 

 cos 6 ; and those of the radius through any point (x, y, z) 



X II z 



inside the earth are - , -, — , r beino; the radius vector. 

 r r r 



Now the potential of centrifugal force at (x, y, z) is 



^co 2 p 2 where p is the length of the perpendicular from 



(x, y, z) on the axis of rotation. Bat if <f> denotes the angle 

 between the radius through (x, y, z) and the axis of rotation 



ti z 



cos <b = - sin 6 + - cos 0, 

 r r 



therefore p 2 = r 2 (l — cos 2 </>) 



_ r 2 _ ^ s j u q _^_ z cos gy m 



Hence the potential V of the centrifugal force is 



1 * 



= 2 r " 0, 



-r 2 co 2 — -x « 2 -j x 2 sin 2 6 + 2xz sin cos 6 -I- z 2 cos 2 6 > 



lr 2 co 2 -r } co 2 [*Vin 2 + 2 2 cos 2 + sm ^ cos ^{(^/ """ (^T}] 



The first term JrV is the potential of a radial force 

 which causes no tide. Each one of the other terms repre- 

 sents a simple tide. The term — J^Vsin 2 6 denotes a negli- 

 gible negative tide with its depressions on the ^-axis. The 

 term — ^ 2 &> 2 cos 2 # represents a negative tide with its 

 depressions on the 2-axis. This gives the mean bulge at the 

 equator and does not shift relative to the earth. The expres- 



sions —jk ar >d / 9 nre the perpendicular distances of 



(a?, y, z) from the planes x + z=0, and x — z = 0, and the 

 corresponding terms in V give a positive and a negative tide 

 in latitude 45°, the crests and the depressions all lying on 

 one meridian circle, namely, the circle through the axes of 

 rotation and of figure. Since the plane containing the axis 

 of rotation is supposed to rotate in the earth about the axis 

 of z in 430 days, it follows that the crest of one tide and the 

 depression of the other will pass any place in latitude 45° 

 once in every 430 days. 



If the earth were perfectly rigid and the mutual attraction 



