The Variation in Attraction Due to tJie Attracting Bodies. 333 



allel to that plane are circles. The body thus c^n be conceived, made up 

 in an infinite nutnber of lamina? or layers or shells so that tlie mass of the 

 body within any layer may vary as the cube of the equatorial radius of 

 the layer. The body can also be conceived, made up of an infinite num- 

 ber of eccentric cones or cones with eccentdc bases, having; their bases in 

 the surface of the body and their vertices at the center of the equatorial 

 section or centt'r of bod}^. These c-ines may be taken with such areas of 

 base that tlieir volumes shall vary as dist ince from base to center of body. 

 In case the bodj'^ in equilibrium is an ellipsoid, it cia be conceived, made 

 up of an infinite number of elliptic cones or cones wiih elliptic bas s. If 

 each elliptic cone has an el iptic base with one principal axis, of length 

 due it per the system of dividiog the ellipse extending from either pole of 

 the ellipsoid to the eqaator per law of alpha or beta elliptic angles (see 

 Art. 14 and Diagram 5), and the other principal axis of length due it from 

 proportional distance from center of the ellipsoid, then the cones have 

 equivolent bases and their volumes vary as distances from bases to center 

 of body. Per the expedient of the layers and the cones, any layer by the 

 cones is conceivably cut up into infinitesimal paits, so that each part has 

 mass proportional to its distance fron the center of the body. 



Fluid equilibrium, or a stata of rest for any and all of the component 

 particles requires that the pressure from any infinitesimal part of a layer 

 on the fluid interior shall just equal that of any other part of same layer 

 having the sime layer surface area. 



Such ultimate parts of a layer may be called layer elements of mass. 

 As all Sections of the rotating fluid body, parallel to the plane of the 

 equator must be circles, and as the equatorial plane must divide the body 

 into two equal jjarts whatever the figure of a section in a plane of the axis 

 of rotation, a line of layer elements of mass in either polar radius must 

 balance a line of elements in an equatorial radius. It is safe the i to take 

 as granted that gravity at either pole to gravity at the equator is inversely 

 as polar to equatorial radii. As the sphei'e is the figure of equilibrium for 

 a fluid body not rotating, and as a sphere rotated on a center axis gener- 

 ates a centiefugal or repulsivd force in lines of ordicates of axis, propor- 

 tional to lengths of ordinate-", and as the oblate ellipsoid is the only figure 

 that has the required relations to the sphere to continue a fluid mass in 

 equilibrium in passing from a state of rest to a condition of rotaiioa, it is 

 safe to take as granted ihat such is the figure of equilibrium for a rot iting 

 ing mass, providing it be demonstrated that every element of mass of a 

 layer pi-esses, ea'di to each, equally on interior mass. 



The general expression for attraction on a particle at the surface of an 

 ellipsoid, is: 



Att. = -ir 1+^ li-- li sin-OT-^-\-— -^h-' siu- O— ilv sin-- 0-\-\\' ( )+"l 

 A-'L '55 35 350 50 i v ^ . j 



To satisfy the conditions the elliptic argle (O) in the above must be meas- 



