308 EXPLORATION GEOPHYSICS 



Since, 



dg da , , —da da . 



-f- ='f- cosS and ^-^= -f- sni <i 



dy ds ^ dx ds 



da da „ 



, cos <i>=-r- cos^ <i 

 dy ^ ds 



— r^ sin d) = -T" sin^ d) 

 dx ds 



Therefore : 



da da . da . ^ , . „ . dg 



^ cos .^ - ^- sm <!, = ^(cos =* + sm V) =^ 



The gravity g is itself a gradient of a potential, as (7 = dU/ds; where -\ 2 

 is the downward vertical direction heretofore indicated as h. The gravity 

 gradient in the s direction is then the second derivative of the potential, 

 viz., 



dg_ d'U 



ds '^2 "^s 



where g = + —- , 

 OS 



and further: 



-T-= ^ ^ cos cj> - :r^-^- sm <f> (47) 



ds dyds d^ds 



Using the following notation for the above partial derivatives 



and 



dxdz ^^ 



The gradient moment Mi = mlh {Uyz cos ^ — Ua-z sin ^] (48) 



is part of the fundamental equation of the torsion balance. The torsion 

 balance (with a hanging weight) furnishes, therefore, in any given azimuth 

 (/), the North-South component of the gradient Uxz and the East- West com- 

 ponent of the gradient Uyg. 



Units of Measurement. — The unit of measurement of the gradient 

 of gravity is the Eotvos Unit, as previously defined. The value of gravity 

 force is about 1000 dynes, or gals, and it varies in total between the equator 

 and the poles from 978+ to 983, approximately. An Eotvos unit repre- 



