Chessin — On the True Potential of the Force of Gravity. 3 



southward direction perpendicular to 0^; finally, the axis 

 Or] parallel to CY, eastward. Then, denoting by S the dis- 

 tance of from the center of the earth, X the latitude of the 

 point O, and 6 the angle of the radius vector CO with the 

 plane of the equator, the formulas of transformation will 

 be: 



(4) 



X = 8 cos ^ + I sin X — ^ cos X, 

 z =B sin — ^ cos X — ^ sin X. 



9U 9U 917 

 4. The expressions of the partial derivations ?p , 5 — , jp-, 



as obtained from (3), readily yield the derivatives of the 

 same function with regard to ^, ?;, ^. Namely, 



(5). 



9U 

 W 



M 3iV \b Nz^ 



(h sin 6 + ^) 



+ -^ 8 sin ^ cos X — -^5 ^ cos2 X — -^5 ^ s"i ^ cos X, 





M ^N IbNz' 



(5)= 



9K 



M 3iV 15 iV^ 



i?3^ i25 Ri 



(S cos e — ^) 



»6 " Sin 6/ sin X — — ^ | sm X cos X — -^s ^ sin ^X, 



R'> 



R' 



where 



(6) e = X — (9, 



(7) i2 = i/S2 + |o2+2S(|sine — rcose), 



(8) p=^/|2 + ^2^^2. 



