226 Wisconsin Academy of Sciences, Arts and Letters. 



This method of correction gives an expression exactly true to the eighth 

 power of eccentricity. Another order of correction, similar to the one 

 treated, commences with the eighth power of ecentricity, but its effect is 

 small when eccentricity is large, owing to the fact that each group of 

 eight wedges to be combiDcd in the next order of resultants, is so com- 

 posed of wedges from different parts of the ellipsoid, that the attractions 

 of all groups of eight are very neatly equal each to each. 



M* M 



^~l-sin-a, "~ 1— i E^ ein^ O cos- 0--|- " 



Substitute this value for M in the above expression for sin (ct— W) and 

 we get for the whole ellipsoid the angle that the direction of attraction 

 makes with the direction from the attracted particle to the center of the 

 ellipsoid. 



3 r 3 4 23 



(b) sin (a— W) = g-Mn- E'^sin O cos 0| I+Ej-^— 2 sin- O —i^ ^'+20 



n'^sin^O)+ EMJ4-"1 



When the attracted particle is at the surface of the ellipsoid n becomes 

 uni y and the expression is: 



3 r ,^13 17 



(c) sin (a-W) = gM E'^ sin O cos Ol 1+ E'^f j^ — g^sin-O J+E* U+„ \ 

 sin (a— W) = g-M sin O cos Oj E-+ eY j^ — ^ sin= O j + E" (J + 1 



In the above, in order to get the term depending on E' exact, it is to be 

 observed that ili"can be used for M, angle c, for (?— b) and arc of angle u 

 for sine or tangent, also so far as the expression is dependent on attraction 

 (c) Art. 18 the terms E* and sin- a are not involved. 



22. To find the attraction of an oblate ellipsoid on any outside particle. 



By requisite substitution, expression (c) Art. 18 for two ojiposite wedges 



becomes: 



m 3 3 3 3 



(a) Att.=j^,(l- gU^E- cos -O + gU-E- sin'-'O sin '-?— gU^E^ cos -^O + ;^n-»E -^ cos ^O 



3 43 32 



+ ~n-^E-*sin-'Osin-'c— 2gE^sin'-Ocos-Osin-c + -?- n'-E-'sin^Ocos- 



198 



O sin'-c— Ts^ n ^E^in'-'O cos -O sin-c+„). 

 17o - I «/ - 



Expression (b) Art. 20 modified for tangent and reduced gives: 



6 3 4 1 



tan(tr, + y )= g n-E-sinOcos03in?(l + 5 E'-cos'-O— ^ n'-E'-cos-O— ^E-sin-'O 



4 ^ 



sin-= + sn-E*sm-0sin-?4-J- 



* The Italic M represents greater mass than the Roman in this investi- 

 gation. 



