The Variation in Attraction Due to the Attracting Bodies. 219 



Hyp, B,,^=B;- COS" 5 + A/- sin- 3 sin" a+A,^ sin- 3 cos- a— 2 A, B, sin 5 cos 

 5 sin a. 



=B,' [^i + £:-i sin' 3 -2 i/lT^ sin ^ cos 5 sin a: J 



Likewise, 



B,,," =B/ [14--B;- sin- 5+2l/l4-J5^ sin ^ cos ^ sin or]. 



Let s = ^^ sin^ 3^, and t = 2l/i+£" sin 5 cos 5 sin a. By substitution 

 and reduction. 

 Att.etc.,ata„= !!L(l+n) cos^ or „ 



' " TV-' 



Att. etc., at a„, = 



D-[l +(s — t)J^ 



m (1— n) cos^ or^ 



D= [ 1 + (s+t)]^ 



TrTi£t)]t = (^+^)t^-* («-^)-o ^^-*)'-^*^^- 



= (l+n) (l-f s-fft +„). 

 For mass at a„ in direction pc on p. 

 Att. == ^n(l + n)cos^a (i_3 g^jt+j. 



For mass at a„, etc. 



Att. = "^g-nWn: (1_3 g_B t+ J. 



Multiply the first series by (1-fn) and the second by (1— n) and divide the 

 sum of the results by two and we get 



mcos-tr 3 3 3.5 3.5,^,, 3.5 ^ ^ 



Average Att.= d2~*^^~2 "'"2'^*"'' 2^4 ^^ +2A ~r ~'"'' 



1= cos (a ± z) = 1— i sin- {a ± z) — k~J sin* {a ± z)— etc. 



Replacing the values of 1, s, n and t, 

 m 



cos" aF-. 



~& L " 



/„ \ Atf 1 1— - -EJ-sin^^ +_-^-*sin* 5 + _- sin- 5 sin- a-— — _ sm* 



{a.) Att— jy, L 2 2.4 2 3 



^ .5J^'sin^3 +^'^£J*sir^ - ■ ^'^ "-'-^ - ---^ - ^"^ " 

 2.4 



3 sin- a ± 3 sin-. 



3- sin crsin z— - - sin-" (a: ± z)—^i — '-^E^ sin" 3 +„ 1. 

 2 ^ 2.5.7. "J 



Let (a + y) be the angle that line PC makes with the resultant direction 

 of attraction for the two opposite wedges, then any two corresponding 

 opposite chord elements contribute in the attraction of the two opposite 

 wedges. 



3.5 



~2" 



/T N . ^. 7?icos-crr, 3 „ . , 3.5 „ . ^ 3.5 . „„ . „ 



(6.)Att.= — — , — I 1- T -^■"' si^^ •^H ^ ^^^ ~+ — ^^^ ^ si^ "~ 



■L*- ^- 2 2.4 2 



1 3 5 7 ~\ 



sin-* 3 sin= tr ± 3 sin- 3 sina sin y— — sin- (a±y)— ^-V^ E^ sin* -^ +,, I 



