708 Lord Rayleigh on the Motion of 



Proceeding in this way we see that the cosine factor may 

 properly be identified with unity, and that the value of the 

 integral for the first quadrant maybe equated to l/\ogt'. 

 And for a similar reason the quadrants after the first con- 

 tribute nothing of this order of magnitude. Accordingly 

 we may take 



kcosntdn = —s-, -. ('43) 



1 



o a 2 lo g t' 



For the other part of (38), we get in like manner 



\ lf . . 7 Sv C °° sin t'x . dx 

 I k sm nt an = 9 1 = 



Jo a 'o ^'log^ 



_8v C™ smx'dx' {aa\ 



In the denominator of (44) it appears that ultimately we 

 may replace log (t'\x') by log t' simply. Thus 



I 



7 , . , Attv ,. e , 



k sm ntan= -^ , , .... (4o) 



cr log t 



so that the two integrals (43), (45) are equal. We conclude 

 that when t is great enough, 



8,M' _ %vW 



a 2 \ogt'~ a 2 log(±vt/a 2 y ' ' K J 



But a better discussion of these integrals is certainly a 

 desideratum. 



§ 7. Whatever interest the solution of the approximate 

 equations may possess, we must never forget that the con- 

 ditions under which they are applicable are very restricted, 

 and as far as possible from being observed in many practical 

 problems. Dynamical similarity in viscous motion requires 

 that Va/v be unchanged, a being the linear dimension. 

 Thus the general form for the resistance to the uniform 

 motion of a sphere will be 



F = /ovVa./(Va/v), (47) 



where / is an unknown function. In Stokes's solution (1) 

 /' is constant, and its validity requires that Yajv be small*. 

 When V is rather large, experiment shows that F is nearly 

 proportional to V 2 . In this case v disappears. " The second 



* Phil. Mag. xxxvi. p. 354 (1893) ; Scientific Papers, iv. p. 87. 



