74 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 51 



in which G is the vertical component of the accelerating force of 

 attraction and R the radius of curvature of that normal section 

 which is tangent to the direction of the absolute motion ; R t and R 2 , 

 however, denote the radii of curvature of the principal normal 

 sections of the rotating body, respectively parallel to the meridian 

 and to the circle of latitude for which we have in fact: 



1 . do 



— = — sin ' 



R t dr 



1 _ COS ip 



R 2 ~ r 



For 8 must be introduced the azimuth of the relative motion; 

 from (14), (15) and (16) we have 



V cos 8 =v cos 



V sin 8 = v sin 6 + rco 



From all this we obtain 



. v 2 cos 2 6 , v 2 sin 2 6 \ r 2 co 2 2 rvco sin d 

 N =G - - + 



R t R 2 / R 2 R2 



or, by introducing in the last two terms the preceding expression 



for R 2 : 



v 2 



N = G — rco 2 cos <p — 2 vco cos ip sin d — (24) 



R' 



in which R' denotes the radius of curvature of the normal-section 

 parallel to the direction of the relative motion. The first two terms 

 represent the local acceleration g of the force of gravity 11 



G — rco 2 cos <p = g (25) 



If the motion of the body has a vertical component, then the 

 same equation (24) will apply if the v therein is made to denote the 

 velocity of the horizontal projection of the motion, and 6 its azi- 

 muth. The magnitude of the force N will be changed slightly by 

 the vertical component of the motion only when this latter motion 



d 2 h 

 is not uniform and in fact the change corresponds to — which is 



di 



11 The apparent force of gravity or the vertical component of the attrac- 

 tion of the earth minus the vertical component of the centrifugal force. 



C. A. 



