GRAVITATIONAL METHODS 



305 



surface) times the distance ds, where a uniform change is assumed. In 

 equation form : 



g' = g ■\- dg/ds ' ds (36) 



With angles measured in radians, as illustrated in Figure 166, dh = 

 r de (in general), li de has a definite value, then dh = re. Applying this to 

 Figure 165 : 



ds'e = dh-dh' (37) /- "^ ^^^ ^ 



In general, as seen from Figure 165, for / 

 any pair of equipotential surfaces c • r = dh, 

 where r is the radius of curvature of the lines 

 of force of gravity as indicated. 



The three previous equations (35, 36 and \ 

 37) establish the following: 



/ 



\ 



g dh = g' dh' = (g '^ dg/ds • ds) (dh — e ds) 



(38) 



It can also be demonstrated that 



dg/ds = g/r 



\ 



_ Fig. 166. — Showing the rela- 

 tion of the length of arc dh, 

 angle de and radius r of a circle 

 in radians. 



(39) 



Equation 39 above sets forth that the gradient of gravity dg/ds equals 

 the force of gravity g (considered in general) times the curvature of the 

 lines of force of gravity (curvature is \/r, where r equals the radius of 

 curvature). 



The intermediate steps in this derivation are as follows. The gradient of gravity 

 along ds, assuming ds in a horizontal direction, is by definition dg/ds. 



g' = g + ^ds; e ds = dh - dh' ; ds= ^^~ ^^' 



ds e 



9" — 9 _ dg 

 ds ds 



JL 



dh — dh' 



^ (9' - n) 



dh — dh' 



Therefore, 



Since ^ = — — ; dh : 

 g dh' 



g' dh' 

 9 



e(9'-.Q) 



g' dh' — dh' 



9 



ds 



ML 



^ dh' — gdh' 

 9 



is. -.±9. 

 ds dh'. 



Using the more general dh for dh' ; 



dg __ eg 

 ds dh 



e jg' — 9) 9 

 {g' - g) dh' 



dg dh 



ds g 



(40) 



