Chessin — On the True Potential of the Force of Gravity 7 



9U 

 ( 13X0 = ^ + 4>fc^+ B^ + 6)2 8 cos 6> sin \ 



+ 0)2 sin X (^ sin X — ^ cos X), 



PC/" 

 5?/ 



(13), 0=3;; +^^, + i? +0,2^, 



( 13 )3 =^^ + 4)^^ + i?^ — ft)28 cos ^ cos X 



+ (i)2 COS X (^ COS X — I sin X), 



lie., R , Ro. being the components of the reaction of the 



constraints to which the particle is subjected, and (o the angular 

 velocity of rotation of the earth. 



At the point O, where |^'7=? = 0, the equations (13) 

 Teduce to 



^ a + jRj, + <o^S cos sin X, 



= R , 



V 

 = G -\- Re, — &)2S cos 6 cos X, 



if we assume, at the same time, that ^ = 0, that is, <l> >. = 

 <!>= <I) vjj = 0. But, then, the axis O^ would coincide with 



the direction of the mean * force of gravity at 0. If, there- 

 fore, we denote by g the value of the mean acceleration of the 

 force of gravity at that point we shall have 



Ri. ^ R =0 and Ri. = — q. 



Hence, 



a = — (o^B cos 6 sin X, 

 ^ ^ c =^ g -\- 0)^8 cos 6 cos X. 



8. Introducing these expressions of a and c into equations 

 ( 13) and putting for the sake of brevity 



B = h — &)2 sin X cos X, 

 (15) 2Z) = 2cZ + ft)2sin2X, 



2E = 2e -\- co\ 

 2F=2f + 0)2 cos 2\, 



See foot-note, § 3. 



