70 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 51 



Hence the velocity of the relative motion due to inertia is con- 

 stant at any latitude whatever on the earth's surface, just as found 

 before for the region of the pole. 



By a general theorem applicable to all rotating bodies, the radius 

 p of the geodetic curvature of a curve running in any direction what- 

 ever on the surface of a rotating body can be expressed by 



r cos 6 ds 



P = 



d (r sin 6) 



If in 



d (r sin 6) = r cos 6 d + sin d dr 



we substitute the two values derived from the first of equations (16), 

 having regard to (17) : 



cos ddO = --(~ + CL 



v \r 2 



and 



then we have 



sin = - ( — — ru> 



v cos 6 ds 



2cudr 



or, since 



cos 6 ds 1 



— dr sin <p 

 v 



2 io sin (p 



(18) 



This value of the radius of curvature p of the relative path due 



1 v 



to inertia corresponds perfectly to the value p = - . — found above 



2 co 



(compare equation (11') for the region of the pole, and shows that 

 the path is less slightly curved the more nearly the equator is 

 approached. For the equator itself (<p = o) the path becomes the 

 geodetic line or great circle itself. In the southern hemisphere <p 

 is negative and therefore the radius of curvature has the opposite 

 sign from that in the northern hemisphere. The center of curva- 

 ture in the one case lies on the right side and in the other on the 

 left side of the "inertia path." The proof of this statement is 



