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



9U 

 < 13 )j = ^ + ^fc^ + i?fc + 0)2 8 cos e sin \ 



+ ft)2 sin \ (I sin \ — | cos X), 



9U 

 { 13 ), = -po + <I> o.^ + B^~ 0)28 cos cos \ 



+ o>2 COS X (^ cos X — I sin X), 



Up, M , Rl. 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;=:f = 0, the equations (13) 

 reduce to 



= a + Ry -f 0)2^ cos d sin X, 



Q = R , 



= c + R^ — 0)28 COS 6 cos X, 



if we assume, at the same time, that <1> = 0, that is, <E» c. = 

 4>^Q= <t>v'^ = O. 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 



Rt. = R =0 and Ry = — a. 



Hence, 



a = — (o^B cos sin X, 

 ^ ' c = g -\- 0)28 cos cos X. 



8. Introducing these expressions of a and c into equations 

 < 13) and putting for the sake of brevity 



B = h — 0)2 sin X cos X, 

 (15) 2Z) = 2tZ + 0)2 sin2 X, 



2E = 2e + 0)2, 

 2i^=2/ + o)2cos'X, 



♦ See foot-note, § 3. 



