148 W. D. Lambert — Mechanical Curiosities 



of gravity coincides with the normal at the level surface ; 

 at the parallel surface the components of force along the 

 tangent and normal are 



X. = mg' sin a, 



Z = mg' cos a, 

 or since a is small, 



X. = mg^ a 



Z=mg' ^ ^ ^ (6) 



in which g' is the intensity of gravity in latitude </> and 

 elevation h. The latitude <^ in formula (4) is usually 

 taken as the geographic latitude, but to the same order of 

 accuracy as is implied in equation (4) we may use any 

 other latitude, as the geocentric or the reduced latitude, 



or we may put <^ = -, a being the mean radius of the 



CL 



meridian ; to the same order of accuracy a may be put for 

 r in (5), and g' may be taken as constant. 

 With these substitutions (4) becomes 



= ^ sin 2<j>, (7)' 



^being written for ^r^^^ ^ ^'- 



Multiplying both sides of (7) by -^^ and integrating and 



determining the constant so that-^ = when <A = yv 

 where y is the initial latitude, gives 



/d^V _ c^ 



\dt) 



{cos 2 cf> — COS 2 y) 



= c' {si?i'y — si7i^ cf)) (8)' 



Equation (8) is similar to the equation for the motion 

 of a pendulum where the vibration is not restricted to 

 infinitesimal arcs. To integrate (8) put 



sin </) = sin y sin 0, 

 whence (8) becomes 



. „ ^ - = c at. 



VI — sin y sin 



This may be integrated in terms of the elliptic integrals, 



