[44] PROFESSOR STOKES, ON THE EFFECT OF THE INTERNAL FRICTION 

 we get for the general term of F 3 '(a) 



, v.- i ^ m „ t 1 - 3— (2*— 3 



(- l)*- 1 Ce- mi 



2.4...(2i - 2)(4raa 



and the expression within brackets is equivalent to 



(2i- l)(2i + 1) 



)P I (2i- ]) 2 ' 2i - 1 1 



i) i_1 aM 2i.4ma 2a /' 



whence 



8ia 



aJf,'(a) = Ce- m «mal {- 1 - -±^_ + -il^l^ _ ...}, 

 1 2.4«ia 2.4(4ma) !i J 



and we find by actual division 



Wltn\ 



= - ma — A + * (ma) -1 ... 





35. When many terms are required, the calculation of the coefficients may be facilitated 

 in the following manner. 



Assuming aF 3 '(a) = v(a) F 3 (a) f we have 



F 3 \a) = a" 1 v(a) F 3 (a), F 3 "(a) = {a" 1 ./(a) - a' 2 v(a) + a- 2 (im) 8 } _F 3 (a). 



Substituting in the differential equation (85) which F 3 has to satisfy, we get 



av'(a)+ {via)} 2 - m?a 2 =0 (107) 



Assuming 



v (a) = - ma + A + A i (ma)~ 1 + A„(ma)~ s +..., . . . (108) 



and substituting in the above equation, we get 



- ma - 1 A-Ama)' 1 - 2A 2 (ma)~ 2 - 3A 3 (ma)~ 3 ... 



+ {-2ma + A + A x {md)-* +...] {A + A^ma)- 1 +...} = 0, 



which gives on equating coefficients, A = — ^, and for i > 



2A i+l = -iA { + A A t + A 1 A i _ l ...j r A i A Q , 



or, assuming to avoid fractions, 



4=2- 2i -'5,, (109) 



B {+1 = -2iB i + B B i + B l B i _ l ...+ B i B m .... (110) 



a formula by means of which the coefficients B„ B 2 , B 3 ... may be readily calculated one 

 after another. We get 



B =-l, £,= +1, # 2 = -4, #,= +25, £ 4 =-208, # 5 = +2146, J 

 B s = - 26368, B 7 = + 375733, B a = - 6092032. J 



We get now from (100), (101), (108), and (109) 



k-y/ - 1&' = 1 +2e 4 TO" 1 -iB e 4 HI" 2 - - B.e * ttr 3 ..., (112) 



2 4 



