208 Wisconsin Academy of Sciences, Arts and Letters. 



from right angled triangles that have a common hypothenuse and each 

 continues to have the same length of perpendicular as hypothenuse in- 

 creases, therefore sin^ ipi representing the average of all the sin^ 5 varies 

 as n^. It can be shown likewise for sin* ipi, etc., which represent the aver- 

 ages of all sin-* 3", etc., that they vary as n*, n"^, etc. 



The expression, then, for the attraction of an oblate ellipsoid on a parti- 

 cle outside the pole is: 



M >- 3 3 3 



-^ ( 1— 5 Ji' E/'+^ n* jE,*— g n° E/ + 



The expression, etc., for prolate ellipsoid is: 



M / 3 8 3 \ 



^ ( 1+ gn=E^+;^n*E* + 9n'' E^^ + etc.^ 



IV. 



OUTSIDE ELLIPSOIDAL ATTRACTION IN PLANE OF THE EQUATOR. 



First. — An oblate ellipsoid and an outside particle in plane of the equator 



attract each other directly as product of mass of panicle into 



13 3 3 



mass of ellipsoid midtiplied by{l + — y^-^ n- W + -^-g X -~ n"* E-* 



3 5 3 



+ oXft X Q n'^E^^- etc.) and inversely as square ot dis- 



tance from center of ellipsoid to particle. 



Second. — A prolate ellipsoid and an outside particle in the 2^lctne of the 

 equator attract each other directly as product of mass of parti- 



1 3 



cle into mass of ellipsoid midtiplied hy {\ — ^X-s" n^ E," + 



'3 3 3 5 3 



— -jX^h'E*— -r-^ X ~7r n^ E,^ + etc.). and inversely as 

 2.4 7 ' 2.4.6 9 ' 



sguare of distance. 



14. This article finds a method of dividing an ellipsoid into an infinite 

 number* of wedges equal each to each, where the edges of the wedges are 

 in an equatorial diameter. 



In Diagram 5 let AB AB be a section of ellipsoid passing through the 

 center of the ellipsoid and at rightangles to an equatorial diameter. Draw 

 lines C a, C a^ C a^ etc., so that angles A C a, A C a\ AC a'^ etc., can be 

 represented by the alpha elliptic angles ^ — z, 3-1 — z^, S-g — Zg, etc.; also 

 draw lines C b, C b^, C bg, etc., so that angles B C b, B O b^, B C bg, etc., 

 can be represented by the beta elliptic angles 5+y, 3i+yi, -3g+yg, etc. 

 When the elliptie angle becomes 45° then the z angle equals the y angle. 



