152 W. D. Lamhert — Mechanical Curiosities 



also have physical interpretations.^^ Thns 





where R and N are respectively the radii of curvature of 

 the earth in the meridian and prime vertical. The quan- 

 tity c is connected with a and h by the relation 



2 (a + & + c)=-4 7r;b8 + 2(o2 (3) 



In this equation /c denotes the gravitation constant and w 

 the angular velocity of the earth about its axis ; 8 denotes 

 the density of matter at the point considered, i. e., 8 = 

 for points above the geoid and 8 = p for points in the 

 liquid. Equation (3) is really a modified form of Lap- 

 lace ^s equation or of Poisson's equation, according as 

 8 = or 8 = p, the modification being the term in w^, which 

 arises from the difference between gravity and gravita- 

 tion already mentioned (p. 129). The quantity h serves 

 to measure the rate of increase in gravity in going from 

 the equator towards the poles. Evidently by Taylor's 

 theorem for three variables 



where the subscript zero indicates the value at the origin. 

 If g denotes the intensity of gravity at any point in the 

 field. 



= (£)■ + m + (S)' 



9yJ 

 or by differentiating with respect to x, 



9^ _9_Y9'Y 9Y 9^ 5V 9^ ^ 

 ^ 9x - 9x 9''x ~^ 9y 9y9x ~*~ '9l 9y9z ' 



^° Helmert, Hohere Geodasie, Vol. II, Chap. I. Eotvos bases the theory 

 of his balance on an expression for the potential similar to (1) but contain- 

 ing terms here omitted because of sjnnmetry. The values of a and h do not 

 depend on any property of the gravity potential as such ; similar expressions 

 hold good approximately for any field of force having a potential and sym- 

 metrical in the way here supposed. Equation (1) would be exact for the 

 field of force due to gravity in the interior of a homogeneous rotating ellip- 

 soid; even in cases where additional terms would be needed to give an 

 adequate expression for V, the effect of these terms for the case here treated 

 would be nearl)' evanescent for reasons of symmetry. 



