202 Mr CHALLIS's RESEARCHES IN THE THEORY 



Hence 



J dt df • r 2? • df ' 



Substituting in equation (B), 



j\-. - d^a FC0S9 b'^ ir, 2/, -am da b* COS^ 9 da „.^^ 



« ^-^-P = df ■ —r- + ap (2cos=0-sm=0) ^ - -^^ . ^ -M. 

 When /• = infinity, /o = 1 : therefore f{t^ = 0. Hence when /• = A, 



„,T. , «?^a , . COS 20 <:?a^ 



Where p = 1, let j9 = n = a^ Hence when (O = 1 + o-, p = e' (1 4- a-) = n + aV. 

 But because a- is very small, «^N. l.jo = «V very nearly. Therefore, 



„ d^a , - cos 20 rfa^ 



^ = n+^.*cos0 + -^.^. 



The total pressure resolved in the direction NA is ffp¥ eos6sm9d9dfi, 

 from /3 = to /3 = 27r, and from = to = 7r. It will consequently 



be found to be equal to — — . -^ : and if A = the ratio of the specific 



gravity of the ball to that of air, the accelerative force produced by 



1 d'a 



this pressure is — . -7-7 . But the accelerative force of gravity in the 



same direction, if SA = A, is ^ ( 1 ~ t" ) » taking account of the weight 

 of air displaced. Hence 



_ cP_a _g^(-._}\ j_ d^ 



d'a ^ 



or 



«^__^ ±^_§^(l_l] nearly 



1 + K 



