Connected with the Earth's Field of Force. 149 



and if t be reckoned from the time when <^ = y, i. e., when 

 6 = 7r/2, we find 



"^i \^^{l^''''y)- ^i^^'^'^y)^ 



(9) 



where the F denotes an elliptic integral of the first kind 

 with modulus sin y ; or in expressing <j> in terms of t, 



^ . , /sin <f>\ 



e = sin-' I — — j = en {ct) mod. si?i y (10) 



For numerical values we have for the sphere r.-,^.= ^, 



^ o-\-k 7 



/5= 0.00529, ^' = 9.8 meters per sec. and a = 6,368,000 

 meters. For the sphere supposed in the text h = 1000 

 meters. These values give, for the second as unit of 

 time, 



c = 0.000001351. 



The linear velocity (v) is a^, or by (8) 



V = a c V si?i^ y — sin^ <^ 

 a c 



= -Jy" ^ cos2<I> — cos 2 y. (11) 



With the above numerical values 



V = 6.09 V cos 2 <^ — cos 2 y in meters per second. 



The table given in the text (pp. 135-6) is readily com- 

 puted from (9) and (11). 



Appendix B. 



The Effect of the Earth's Field of Force on a Floating Body. 



As a simple example let us consider a sphere of radius 

 r (fig. 7) and density o- floating in a liquid of density p, 

 the outer surface of which forms a portion of the geoid. 

 From the center of the sphere draw a normal to the geoid 

 (conceived as continued into the sphere) and take the 

 intersection of the normal with the geoid as the origin of 

 a system of rectangular coordinates, with s-axis coinci- 

 dent with the normal, the positive direction being upwards 

 and with the rr-axis tangent to the geoid in the plane of the 

 meridian, the positive direction of x being towards the 

 equator. The forces acting on the sphere are the fluid 

 pressure on its submerged surface and the pull of gravity. 



