The Tariation in Attraction Due to the Attracting Bodies. 227 



Expression (b) Art. 21 gives: 



3 3 4 23 



tan(a — w)= g n-E'sinOcosO (1+ 5 E-— 2E'sin-0— ~ n"E- + on ^-"''^'^ 



fein-04-J. 



(a, ±y,) is the angle that D* makes witli the resultant of any two opposite 

 wedges, and (i?— w) is the angle that D also makes with the resultant for 

 the whole ellipsoid. The tangents of the angles having the same radius 

 (D) are in the same place perpendicular to D. To determine tlie distance 

 of the resultant of any two opposite wedges from the resultant of the 

 whole ellipsoiij, these tangents may represent the adjacent side of a tri- 

 angle having the included argle [90°— (|— b]. Let the tangent of angle 

 (a, ±yj be (b) and that of {a—w) be (a), then the third side (d) is the dis- 

 tance of the resultant, in tangent of angle, of any two opposite wedges 

 from that of the whole ellipsoid. 



Tugonometry gives: 



d=i/a--b--2a,b cos[90°-(|-b)] = T ^a^ - b-— 2ab sin(q-b)T 



3 3 4 23 



d = g n-E-'sin Ocos 0(1+ ^ E--2 E'^ sm-'O- ^ n- E'^ + 5^ n-'E'^ sin-^04-2 E'^ 



81 8 



sin'-Osin-'l — ;^n'- E- sin- O sin'-|— 2 E- sin -O sin^l-t-s, 



n'-E'-sin'-'Osin^? + X 

 Let c-=l-|-tan'- (a—w). 

 f =l+tan- (a-, — j). 

 Let r be the angle that the resultant of any two opposite wedges makes 

 with the resultant of the whole ellipsoid. 

 Trigonometry gives: 



c'2+f2— d'^ 

 cos r = - 3^^. - 



Cos r=l— 5»un-* E-" sin'^ O cos- O (l-f-3 E'^— 4 E- sin- O— f n-' E'+fJ n- E- 

 sin- 0+12 E- sin- i+4 E- sin- O sin- = — f4 n- E- sin- O sin- ; — 4 E- sin- O 

 sin^ ?+"/' ^1' E- sin- O sin-* l+J. 



Multiply expression (a) this Art. by the above value for cos r and the ex- 

 pression for the attraction of any two opposite wedges in the direction of 

 T.he resultant for the whole ellipsoid becomes: 



M 



Att.=g7,(l— ? n'^ E^ cos- 0+i? n- E- sin-^ O sin- |-g n-' E-* cos^ 0-/„ n* E-* 



sin- O cos- 0+-^ n' E' cos' 0+? n' E' sin' O sin' 1—4? E' sin- O cos- O sin- 1 



+ V- n- E" sin- O ccs- O sm- ?-i;|| n^ E^ sin- O cos- O sin- =+„). 



For the whole ellipsoid per Art. 15 



Sin- ==i. 



3 

 Sin' ? =z:^r-. Etc., for higher powers, 



*Line D extends from attracted particle to center of ellipsoid. 



