GRAVITATIONAL METHODS 



313 



Along the x" axis there is no change in gr/ and along the / axis no change in 

 g,'. Hence, the value of gx' at some point x' along the x axis is : 



^ dx' 3^ 



and for g^' at some point y along the y axis : 



9y 



gy = 



^ 3y« ^ 



For these axes 



3=C7 



^'U 



= and drops out of Equation 51. 



3 x' dy' dy' dx" 

 Equations 52 and 53 apply only to their respective axes. 



To consider the special case of points on an axis : 



(52) 



(53) 



. r 



Fig. 175. — A section along the x' axis 

 direction of an equipotential surface for 

 which Pj = the radius of curvature. 



Figure 175 shows a section along the x' axis. AO \^ approximately equal to CO = 

 px' := the radius of curvature of the equipotential surface in the x' direction. 



Px X' 



By similar triangles 4S = 4£ = -^=^ ; AC - x" ; 

 AO AC Px' x 



gj = — ^-^, the minus sign indicating the negative s^ direction. 



Substituting gj from Equation 52 in the above : 



"dgj . 

 g _.9x _ ^x' 

 Px x' x' 



In like manner for the y' section 



^_9l -_- -^U 



dx' dx" 



£^ _=^ 



pj 3 X' ' 







