Solid Bodies through Viscous Liquid. 707 



When m = 0, h and k! are infinite and continually diminish 

 as m increases, until when ??i = co, k = l f k f = 0. For small 

 values of m the limiting forms for k, h' are 



*=1 + * ,r .„ h'= J: ; . . (39) 



7jr(log my m" log m 



from which it appears that if we make n vanish in (35), 

 while V„ is given, F comes to zero. 



We now seek the limiting form when t is very great. 

 The integrand in (38; is then rapidly oscillatory, and ulti- 

 mately the integral comes to depend sensibly upon that part 

 of the range where n is very small. And for this part we 

 may use the approximate forms (39). 



Consider, for example, the first integral in (38), from 

 which we may omit the constant part of k. We have 



cos (4zva~ 2 t.x)dx 

 x (log x) 2 

 . . . (40) 



J*^ ,7 tt (^ cos ntcln Airv C° 

 k cos nt dn= T \ — =-: < 9 = — 5- 1 



Writing kvt\a? = t' } we have to consider 



cos t'x . dx 



J' 



Jo 



(41) 



X (log X) Z v y 



In this integral the integrand is positive from x— to x = 7r/2t / , 

 negative from 7r/2i' to Zir\2t\ and so on. For the first part of 

 the range if we omit the cosine, 



C*/®' dx C d logs 1_ 9 . 



Jo xi^gx) 2 ^) iiogxy-\o g (2t'M ; ' t-) 



and since the cosine is less than unity, this is an over 

 estimate. When i' is very great, log (2t , \ir) may be identified 

 with logt', and to this order of approximation it appears 

 that (41) may be represented by (42). Thus if quadratures 

 be applied to (41), dividing the first quadrant into three 

 parts, we have 



IT 



cos 



12 , 37r. r l 11 57rr 1 



—. + cos -— — -.- 7-7 4- cos— 



log — l loo- log • I J loo- loo- 



£> — iV €> _ I I 



L 7T 7T J ~ 7T " 7r_j 



of which the second and third terms may ultimately be 

 neglected in comparison with the first. For example, the 

 coefficient of cos (37r/12) is equal to 



1 0.1 3tf' . 6t' 

 loo- 2 — loo; — . loo- — . 



77 7T 



