176 GRAVITATIONAL METHODS 



and the "gradient" moment is 

 Di = mhl 



(d'U 



\dydz 



cos a 



-:r^ sm a ) , 

 dxdz ) ' 



[Chap. 7 



(7-496) 



where d^U/dxdz = Us, is the north gradient of gravity and d^U/dydz = 

 Uyz is the east gradient. 



For a derivation of the curvature effect, assume that the beam makes the 

 angle a with the north direction and the angle ^ with the direction of mini- 



Fig. 7-59. Horizontal forces on balance beam, clue to curvature conditions in niveau 



surface. 



mum curvature (Fig. 7-59) . Then the component F at right angles to the 

 beam is the projection of the resultant of —dU/dx' and —dU/dy' and its 

 moment D2 = 2mlF. Since F = (dU/dy') cos ^ -{dU/dx') sin /? = 

 {d^U/dy'^)y' cos 13 - (d^U/dx'^)x' sin 13, the moment D2 becomes, by 

 substitution oi x' = I cos ^, y' = I sin /3, and 2f'm — K, the moment of 

 inertia of the beam, 



D2 = ^K sin 2/3 



\dy'^ dxV ' 



(7-5O0) 



