224 Wisconsin Acadetny of Sciences^ Arts and Letters. 



u + v 

 ^ , ^^ tan CPG + tan — -- — 



(u+v\ 

 CPG + — ^— 1 



tan (a±z) 



u + v 

 1— tan CPG tan 



sin {a ± z) = 



i/l+tan'^(a±z) 



As the chord elements of attraction from wedge to wedge vary in 

 mass as m cos^ a the above expression for sin(a±z) must be multiplied by 

 m cos- a. Thus modified and expanded, 



sin (a±z)=3m cos^ a sin <x(sin^ 3 + -E"- sin- 3-— „). 



sin (a-±z)=3m sin a(sin- 5+^^ sin^ ^—E- sin* ^—E^ sin* 3-4-J5J* sin^ 



3 4 23 



5— -g- sin^ 5 sin^ a+~q~ sin'' S- sin^ a— -^sin'3 sin- a-\-E^ sin^S^— „) 



The letters in the above expression are for the plane of the principal axes. 

 For any plane substitute a^, E,, m, and y, for z. As or, and E, are constant 

 quantities for any two opposite wedges, summation of chord elements for 

 opposite wedges can be made per expedient of Art. 13, by putting; 



sin^ 5 = An2. 

 5 



2 4 

 sin'*3= . — '- — n^, etc., for higher powers. 



5 . 7 



sin (a, ± y,) = A. m sin a, n^ (1 + E;- - — E,- n'- - — E* n^ 



+lA.E^n*-A. sin-^ a, + J^- n^ sin"- a, — ^- n^ sin^ a, + ^1, E,« — „) 

 ^ 7.9 ' 2 '21 '63 ' '^ 7.9 ' 



Substitute in the above expression the values given for sin a, and eccen- 

 tricity (£■„) in Art. 19. 



,, . • / , \ 6 m n- E- sin O cos O sin 5 



(b.) sm (., ± j,)=- ^^^ _.^.^ ^.^^ ^^.^^ .^ ^^ _ ^.^ ^ — ^ 



[1 + E-' COS'-* O ( 1 — f n^) — E'^ sin'-* O sin'^ 1(1 — f n-) + E^ cos-^ O 

 (1 - f n^ 4- -gSj n*) — W sin '' O cos'^ O siii'^ ^ (f - f f n^ + %^ n*) — f n'- 

 E* sin* O sin* 1(1 — f n-^) 4- J. 



21. To find the resultant direction of attraction for the whole Ellipsoid. 



For any two opposite wedges there are two corresponding opposite 

 wedges in the adjacent quarters of the Ellipsoid. Tlie resultant for these 

 four wedges is found by multiplying the expression for the resultant of 

 either of the two opposite wedges by the cosine of the angle [90° minus 

 the alpha elliptic angle (I— b)], or the sine of the alpha elliptic angle (?— b). 

 Let (a-, ± y„ be the angle for the resultant of the four wedges. 



.. rS - b^ = A, sin losing (1-E^sin^ O)^ 

 " ■ ^ a, (l-E-sin^Osin'^#' 



