revolving at small distances from their Primaries. 415 



In estimating the effects of centrifugal force and the disturb- 

 ing action of the primary in diminishing this attraction, we may 

 use, with some modifications, the formulas already obtained for 

 the case of a spherical satellite. The entire diminution arising 

 from both disturbances may be found by adding formulas (2) 

 and (5), substituting A for r in the result, and retaining only 

 the term containing the first powers of A, the rest of the series 

 being small in comparison to it. There results the amount 



JL ~^ nearly; (18) 



and this being subtracted from (17), we obtain 



the approximate value of the force of gravity at the points near- 

 est to the primary and most distant from it. To find the weight 

 or the pressure of a uniform column of the fluid extending from 

 either locality to the central region of the satellite, denote by S 

 the distance of any part from the centre ; the force of gravity 



operating on it will be ~-r-, and dV is equal to — -r — ; whence 



P=|£. ...... (20) 



Taking this integral within the limits of S = and S = A, and 

 substituting for G its value from equation (19), there results 



in which P represents the pressure when the transverse section 

 is unity. 



To determine the pressure of a similar column of matter coin- 

 cident with the axis of rotation, let either extremity of this line 

 be taken as the origin of three coordinate planes, — one being 

 tangent to the spheroid at that point, the second vertical and 

 ranging with the primary, and the third perpendicular to the 

 line in which both planes intersect. Through this line of inter- 

 section let planes be conceived to pass, dividing the body into 

 an infinite number of sections which are subdivided into infinitely 

 small pyramids, whose vertices all terminate at the origin of the 

 coordinates. If 6 denote the inclination of any section to the 

 second plane, and </> the inclination of any of its pyramids to the 

 third, the length of the pyramid being /, the angles at its vertex 

 will be dcj> and dd cos </>, and the vertical component of its attrac- 

 tive force will be 



gkH cos 2 j> cos 6d<i>dd (22) 



