230 Wisco)isi)i Academy of Sciences^ Arts and Letters. 



nect points f b, gc, lid, etc., to Cs. By the law of the expedient of this 

 demonstration the time required for particle a to revolve to s acted on by 

 the combined forces is that required for a constant force equal to the 

 attraction of the ellipsoid at distance a C, to move the particle over the 

 distance aC. Let distance a C be represented by (a) and af by (c). Per 

 law of iiltimate ratio tangent ab^, arc a b and chord a b are equal each 

 to each. 



Chord^^' = v- = 2a X c. 2a = — 



c 



Chord of a quadrant, a m- = 2 a- = — ^ • 



c 



Per law of falling bodies : 



a =:= t* c. t = — r^ 

 vc 



a m'^ =^ V- 1'% a m = V t. 



V t is the length of the arc a s, and a m is th^ chord of a quadrant. The 



time required for the particle to revolve over the ai'c a s is ]_^ y^2 ^^ 



l/c 

 the length of the chord a m, when 2 7f is the length of a circumference 

 The time (T) rt quired for a complete revolution of the particle or the rota- 

 tion of the ellipse is : 



T = ^ '^^^ 2 a TT-* 



In ease of a homogenious oblate ellipsoid, the mass inside of the layer 

 having semi-major axis (a), varies as (a) cubed. The attraction, then, of 

 the ellipsoid on particle (a) in the plrine of the equator varies directly as 

 distance (a). As the ellipsoid rotates as one mass the time of revolution of 

 all component particles is the same, and all particles in the plane of the 

 equator rotate without pressure. 



Case 2d. — "When the repulsion from rotation is not suflficient to counter- 

 balance attraction. 



It is evident from case 1st, without further investigation, that the loss 

 of p essure fi-r particle a varies inversely as the time squared of rotation 

 of the ellipsoid to the time of rotation required squared for repulsion to 

 counterbalance attraction. 



Case 3d.— When the repulsion from rotation is greater than attraction. 



In Dia. 11 let arc a b,, be the velocity of rotation for an infinitesimal unit 

 of time. The initial impulse of rotation actiag alone would throw par- 

 ticle (a) in direction of tangent to ditance a b,, equal to arc a b„. Let a f 

 be the attraction for the first unit of tirrie. D aw b„ f perpendicular to 

 a C, then a f ' represents the force required acting in direct on a C to make 

 the particle revolve in a circle. The versed sine f ' a in direction C a is the 



