204 Wisconsin Academy of Sciences, Arts and Letters, 



10. To find an expression for sine sqtmre of angle a. 

 In triangle c C 1, 



sin (3^ + y) cl 

 sin a = — X 



c 1 ^ A cos -S~ T" 



B^ cos 5=A E-2 cos 5. 

 A 



A sin 3- 



sin(5 + y) = g 



A^ E^ sin 2 5 cos' 5 E^ sin^ 3- cos^ 5 



sin^ <x := 



~A,^ B,2 ~ (1-E' sin^ 3) (l-E-'cos" 3) 



E^ sin° 3- cos- 5 



1 - E= + E-i sin^ 3 cos' ^ 



III. 



OUTSIDE ELLIPSOIDAL ATTRACTION IN LINE OF POLES. 



First. — An oblate Ellipsoid and an outside particle in line of the poles at- 

 tract each other directly as product of mass of xjarticle into mass 

 of Ellipsoid midtiplied 6?/(l-fn'E/+fn*E/— fn''E/+etc.)awd^n- 

 'yerse?7/ as sguare of distance fl^om center of ellipsoid to particle.^ 



Second. — A prolate ellipsoid and an outside particle in line of the poles at- 

 tract each other directly as product of mass of particle into 

 inass of ellipsoid multiplied by (1 + f n-E"+f n'E*+f n^E^+etc.) and 

 inversely as square of distance. 



In the first case n equals the minor axis of the ellipsoid divided by the 

 distance and in the second, the major axis divided by distance, E,'^ eqiialsE- 

 dividedby (l-E^). 



11. To find the attraction of an ellipsoid of rotation on an outside 

 particle in line of axis of rotation. 



In Diagram 4 let C be the center of the ellipsoid having major axis A A 

 and minor axis B B , and let P be the attracted particle. 



On P C as minor axis described a semi ellipse, similar to the one 

 having C for its center. Also with C as the center and n C and 1 C as 

 semi major axes describe two semi ellipses similar to ones already de- 



*When not otherwise mentioned the ellipsoid is homogeneous. The 

 word particle here used represents finite mass condensed to the dimen- 

 sions of a particle. 



