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



Putting now 



A, - C - A tt , A„ = Dl-\ 



substituting in the above equation, and then making % vanish, we get 

 *,(r)-<C+Dlogr)(l+ — +5^+«) 



-< 



i^' Sl + i^' y2+ ^^^' y3+ - ) - • * * (87) 



The series in this equation are evidently convergent for all values of r, however great ; but, 

 nevertheless, they give us no information as to what becomes of F 3 (r) when r becomes infinite, 

 and yet one relation between C and D has to be determined by the condition that F 3 (r) shall 

 not become infinite with r. 



The equation (85) may be integrated by means of descending series combined with expo- 

 nentials, by assuming F 3 (f) = e^ mr {Ar a + B^...). I have already given the integral in this 

 form in a paper, On the numerical calculation of a class of definite integrals and infinite 

 series*. The result is 



„, ' . „ ,, l 2 1 2 .3 2 l 2 .S 2 .5 2 



F ■>( r) = C?e~ mr r~l \l + ■ i- I 



w l 2.4mr 2.4(4mr) 2 2.4.6(4jrar) 3 ' 



l 2 l 2 3 2 I 2 S 2 5 2 



+ D' e mr r~l{l + + -+ -— +...}. . (88) 



2.4,mr 2.1(4mr)' 2.4.6(4»»r) 3 * v ' 



These series, although ultimately divergent in all cases, are very convenient for numerical 

 calculation when the modulus of mr is large. Moreover they give at once D'= for the con- 

 dition that F 3 (r) shall not become infinite with r, and therefore we shall be able to obtain the 

 required relation between C and D, provided we can express D' as a function of C and D. 



29. This may be effected by means of the integral of (85) expressed by definite integrals. 

 This form of the integral is already known. It becomes, by a slight transformation, 



F 3 (r) = f*{C" + D"log(rsm i a ,)}(e mrcoa »+€- mrcosa ')da>, . . (89) 



C", If' being the two arbitrary constants. If we expand the exponentials in (89), and integrate 

 the terms separately, we obtain, in fact, an expression of the same form as (87). This trans- 

 formation requires the reduction of the definite integral 



P { m f * cos" a) log sin wdo). 



If we integrate by parts, integrating cos w log sin wdw, and differentiating cos 2 ' -1 to, we shall 

 make P t depend on Pj_i. Assuming P = Q , P i = ^Q 1 ..„ and generally 



_ 1.3...(£»-1) 



• Camb. Phil. Trans. Vol. IX. p. 182. 



