T](e l^ariatioii in Attraction Due to the Attracting Bodies- 235 



Multiply this value for sin (4 — b) by expressioa (b) Art. 20, then, the ex- 

 l^ression for the resultaat of t'le four wedges is, 



sin (a, ±z yj= -| m n- E- sin O cos O [(l+l E--2 E- sin- 0-f n- EHI H" 

 E- sin- O) sin- ? +-f H' E'-' sin- O sin-" c =t„]. 



If tiie attraction of the pairs of wedges were equal, each to each, then 

 per Art. 15, the resultant of attraction for the whole ellipsoid would be 

 found by putting, 



3 



sin'* S== — , etc. for higher powers. 

 2.4 o 1 



Let (d + w') be the approximate angle for the resultant so found. 

 (a) sin (a ± w') = -| Ma- E- sin cos ^^^[1 + E'-(| — 2 sin- O— f n- +n- sin- 

 0)-^EX") +.,]. 



To correct this result let (s) be the angle that the resultant of any four 

 wedges having the angle ? greater than 45', makes with the resultant of 

 the four wedges so taken that the square of the sine of an^le ? in first plus 

 the squai'e of the sine of angle ? in the second shall equal unity. The 

 cori'ection for any such group of eight wedges is the difference of a cer- 

 tain two angles the sum of which is a certain angle (s). Let p and r be 

 these angles. Expressions for the sum and difference of the resultant 

 attractions for the two four wedge groups are obtainable from expression 

 (c) Art. 18. By a well known trigone rnetrical method the expression for 

 the tangent of the half difference of the two angles p and r is: 



tan g-'^'sir ^^N-' E^ siu= O cos O [(1+E-(J (2 sin- c-iy ± J. 



(2 sin- ;— 1)- = 1—4 sin- = cos'-|. 



The expression [sin- a cos- a+sin- (45°— a)c6s'-(45' — Si]=i, from well- 

 known trigonometrical fromulte, proves that the value of sin-| cos-| 

 for any and every group of eight wedges or for the average of all groups 

 composing the ellepsoid is Jr, therefore^ for the whole ellipsoid the cor- 

 rection becomes: 



p— r 9 

 tan --;r= YOO ^^'^^ '^' ^^^'' ^ ^°^ ^ [(l+E-(„)4-„]. 



p— r 9 



sin ^ = 10^ -^-^11' E-" sin'= O cos O [14-E-'(J+]. 



Tliis correction united additionally, as the conditions require it, to ex- 

 press'on (a) of this Art. gives: 



sin (<i— w)= i? il/n- E- sin O cos O [1+E- (■;;-2 sin- 0-in--f Ign- sin- O) 



+Et,)4-„J. 



15 



