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

 with respect to 8, and after differentiation put s = ^, we get 



fV'arHogtfda? = r'(|). 

 Putting WBtnX' in these equations we get 



r e - mX '\'-ld\' = irlm-l, f° e~ mX ' \'-Hog\' d\' = m-h {r'(|) - «ijof ■»)], 



"o 



where that value of m~l is to be taken which has its real part positive. Substituting in 

 (91) we get 



limit of F t ir) = {—)*- \C" ♦ {*-W\ - log?) 2>"}. 

 Comparing with (88) we get 



»-(£)V + (.-»rt-«,r)in <*) 



30. We are now enabled to find the relation between C and D arising from the condition 

 that the motion of the fluid shall not become infinitely great at an infinite distance from the 

 cylinder. The determination of the arbitrary constants A, B, C, D will present no further 

 difficulty. We must have B = 0, since otherwise the velocity would be finite at an infinite 

 distance, and then the two equations (83), combined with the relation above mentioned, will 

 serve to determine A, C, D. The motion of the fluid will thus be completely determined, the 

 functions F^r), F 3 {r) being given by (84) and (87). When the modulus of mr is large, the 

 series in (87), though ultimately hypergeometrically convergent, are at first rapidly divergent, 

 and in calculating the numerical value of F 3 (r) in such a case it would be far more convenient 

 to employ equation (88). The employment of this equation for the purpose would require the 

 previous determination of the constant C'. It will be found however that in calculating the 

 resultant pressure of the fluid on the cylinder, which it is the main object of the present 

 investigation to determine, a knowledge of the value of C will not be required, and that, even 

 though the equation (88) be employed. 



Putting U= in (92), and eliminating C" and D" between the resulting equation and the 

 two equations (90), we get 



Vth 



C=0og--7r-*r'l)Z>; (93) 



o 



and we get from (83) and (84), observing that F 2 (r) = F 3 '(r), and that B = 0, 



A A 



— + F 3 '(a) = ac, + aF 3 "(a) = ac, .... (94) 



a a 



whence 



a'c + A aF 3 "(a) 



(95) 



a*c- A F 3 '(a) 



This equation will determine A, because if F 3 (a) be expressed by (87) the second member of 



