214 Wisconsin Academy of Sciences^ Arts and Letters. 



17. To find the relation in law of attraction existing between two con- 

 focal ellipsoids having the same foci on a particle outside of both and iu 

 line of poles or plane of equator. 



Let the semi major and the semi minor axes in the one be represented 

 by a and b, and in the other by A and B. 



. a^-b^ 



*"' • ■^' • • b'^ " • B^ 



Since the ellipsoids are confocal. 

 a=-b^=A^— B'. 



1 1 ) 



(a) e/:E;:.^, : g-,. 



1 1 



(b) e/ :E,*::r-^ : — , etc. for higher powers. 



It is evident from construction of right angled triangle Pa in Dia. 4 

 that any sin 5 for ellipsoid having ecentricity e,^ is to corresponding sin 3- 

 for ellipsoid having eccentricity E.- asb to B. therefore 



(c) sin- ^-.sin^ 3, :: b^: B' 



(d) sin^ 3 :: sin'* 3', : b'': B,* etc. for higher powers. 

 Product of (a) and (c) is, 



b^ B' 



e;^ sin^ 3 : E; sin^ ^, :: - : ^^ :: 1 : 1. 



b'-^ B'' 



Product of (b) and (d) is, 



b* B* 

 e * sin^ 5 : E,^ sin^ ^ ,:: ^ • :f^ : 1 : 1. 

 ' ' ' b* B^ 



etc. for higher powers. 



e,^ sin^ 3 + e/ sin* 5 + etc: E,= sin*^ 5-, + E,* sin* 5', + etc. :: 1:1. 



This proves that confocal oblate ellipsoids attract an outside particle in 

 line of poles directly as their masses. 



Likewise it can be proven that confocal prolate ellipsoids in line of axis 

 of rotation attract in conformity w^ith law of masses. 



As an ellipsoid can be made up of infinitesimal ellipsoidal wedges with 

 edges in an equatorial diameter, confocal ellipsoids attract an outside par- 

 ticle in plane of equators as their masses. 



It is evident ihat this law holds true for confocal ellipsoids made up of 

 laminae varying in density from lamina to lamina providing correspond- 

 ing laminae of each ellipsoid are of same density. 



