232 Wisconsin Academy of Sciences, Arts and Tjctlers. 



the attraction of the ellipsoid on the particle varied inversely as the. square 

 of the distance, the path of revolution would be an ellipse. la Dia. 11, let 

 kl be any part of the elliptic path of revolution described in a unite of 

 time. Take gk to fa inversely as ck- to Ca'-, and draw gl and complte the 

 parallelogram kglm. )t is evident that the diagonal kl is the resultaat 

 effect of attraction kg being equal to the repulsion gk or Im. If the attrac- 

 tion then varied inversely as the cube or any other higher power of dis- 

 tance kg would be shorter than in the case just consiiered and the 

 diagonal of equilibrium would be a line from k to some point between m 

 and 1. With a spherical central body, then, the path of rev lution of the 

 particle would be an ellipse, but witli an ellipsoidal central body it is other- 

 wise. 



In the expression for the atti-action of an oblate ellipsoid in the plane 

 of the equator (Art. 22) subtitute (A) divided by (n) for (D) and the expres- 

 sion becomes: 



M 3 



Att.=^o (n-'-fiXf n^E-'+g-^X? n''E' [-etc.) 



. ., , Mn-" 3 Mn^E-^ 



ihe sum of the elements "TT^'^o — ^j — etc., make up the attraction of 



the ellipsoid. The first element acting alone would cause the particle to 

 revolve in an ellipse; but the other elements ac ing conjointly would 

 cause it to move in a path continuously increasing the distance from the 

 center of the ellipsoid, or the path would be a spiral evolved by a radius 

 vector increasing continuously in length and also decreasing in angular 

 velocity so as to generate equal areas in equal time. All of these elements 

 combintd evolve a resultant path having a radius vector increasing and 

 decreasing during a revolution as the radius vector of an ellipse would in- 

 crease and decrease while additionally and contiauosly receiving incre- 

 ments of length. An appropriate appellation for the resultant path is 

 elliptic spiral. The radius vector of an elliptic spiral generates equ il areas 

 in equal time. As far as observation has been able to determine each of 

 the heavenly bodies rotates on an axis; it is, therefore, good common 

 sense to conclude that every body in the heavens revolves in an elliptic 

 spiral orbit continually w^ith decreasing iucrements departing from its 

 primary. When the particle is an interior one, then its repulsion or out- 

 ward pressure depends upon the law of the interior attraction of the body. 



25. It is evident without additional investigation that the repulsive 

 effect from rotation parallel to the equator, decreases from equator to the 

 poles as A cos O. Angle O is the elleptic angle measured from the plane 

 of the equator. 



36. To find the figure of a mass of homogeneous fluid due to rotation 

 and mutual attraction of component particles, also to find the gravity at 

 any point on the surface or within the rotating body. 



From the nature of the case the two parts of the rotating fluid body as 

 divided by the equatorial plane are similar and equal, and all sections par- 



