220 Wisconsin Academy of Sciences^ Arts and Letters. 



In the above expression (b), for the two opposite wedges, angles a and 

 (<x ± y) as well as eccentricity E are constant quantities. In accordance 

 then with the expedient of Art. 13 for summation, an expression for the 

 attraction of the two opposite infinitesimal wedges on particle P is found 

 by putting 



8 

 Sin^ 3=^ n^. 

 5 



2. 4 

 Sin^ '^=57yi3'- 



Sin" 3=^-V^"- Etc., for higher power . 



5. 7. 9 



For the two opposite wedges in the plana of the principal axes of the 

 ellipsoid on particle P, 



m cos- a r 3 3 12 6 



Att.^ — jQ^ — I 1— g n- ^ --^~ n-* £'■'4-3 n- sin- a 



[ 



] 



1 3 



sm a sm y — g sin- {a ± j)—f.n^H'^ +^, 



Id any two opposite wedges let E, represent the excentricity and angle 

 (a, + y,), the direction of the resultant, then for any two opposite wedges: 



... m cos-a: r . 3 „ _, „ "I 



m r 3 ^ 12 



(c) Att ==^| 1—- n'E;' + -n' E* - sin'-a, + 3n'2 sin^ a -- 

 i)'-L 5 7 7 



ft -i o 



n-* sm- a, ± -n- sin a, sin y, — m sin'- (tr ± y ) — -n" £'/ + „~\ 

 5 3 "J 



19, To find an expression for angle a, and for the eccentricity -E,'- of any 

 wedge. 



Let O be the elliptic angle for the ellipse having A and B for semi major 

 and semi minor axes, and let ? be the elliptic angle for the ellipse having 

 A and A, for semi major and semi minor axes; also let a, be any semi 

 diameter of the ellipse having A and A, for semi major and semi minor 

 axes, and let b be the semi minor axis of the ellipse having A for semi 

 major axis and a, and B, for semi conjugate diameters. 



For any infinitesimal wed^e of the ellipsoid having its edge in diameter 

 B, B, , 



„. , a;B,2-A2b'2 



a - B- 



B;^=B-2 + (A=-B2) sin'^ O. 



a; = A^ — (A- — B'^) sin-' O sin^ €. 



a;^ + B;^ = A'^ + B' + (A- - B-') sin^ O cos^i 



