174 



GRAVITATIONAL METHODS 



[Chap. 7 



Since in the x'y' system the coefficient of y' in the first equation and the 

 coefficient of x' in the second equation are zero, it follows that tan 2X = 

 — 2c/(b — a) or, in regular notation, 



tan 2X = 



dx dy 



_(dUj 

 \dy' 



dx'^j 



(7-48/) 



Hence, (7-48e) becomes 



X' = x' {a cos^ X + b sin' X + c sin 2X1 

 Y' = y' {b cos' X + a sin' X - c sin 2X1 . 



(7-48sr) 



In the x' plane of minimum curva- 

 ture (see Fig. 7-58), AG/AO = AF/AC 

 or 



^2 



d'U 



Fig. 7-58. Relations of gravity 

 components and curvature of niveau 

 surface. 



g__ _ dx'^'^ , 



Px' ^ 



hence, d'^U/dx''^ = —(g/px'). Simi- 

 larly, for the y' plane, d^U/dy'^ = 

 — {g/py'), so that the difference of the 

 curvatures or the deviation from the 

 spherical shape is 



\pi Pi/ dy'^ dx^ 



where the notations p2 and pi (principal radii of curvature) have been used 

 for Px' and py'. 



The torsion balance does not indicate the quantity on the right side of 

 this equation, since the directions x'y' and the angle X are unknown. How- 

 ever, it is seen by comparing eq. (7-48d) with eq. (7-48c) that the coeffi- 

 cients a' and b' defined in eq. (7-486) are identical with the coefficients 

 u and w, so that 



a' = a cos' X + b sin' X + c sin 2X 



and 



b' = b cos' X + a sin' X — c sin 2X. 



By forming the difference of these two equations and substituting eq. 

 (7-48/), we obtain (b - a)/(b' - a') = cos 2X or 



(i-l)cos2X = (i-i) 



\Pl 92/ \Pv Px/ 



