84 86.] MOTION OF A CIRCULAR CYLINDER IN LIQUID. 85 



a is moving with velocity V perpendicular to its length, in an 

 infinite mass of liquid which is at rest at infinity; to find the 

 motion of the fluid supposing it to have been started from 

 rest. 



The motion will evidently be in two dimensions. Let the 

 origin be taken in the axis of the cylinder, and the axes of z, y in 

 a plane perpendicular to its length. Further let the axis of x be 

 in the direction of the velocity V. The motion having originated 

 from rest will necessarily be irrotational, and &amp;lt;f&amp;gt; will be single - 



valued. Also, since I -~ ds, taken round the section of the cylinder 



is zero, -^ is also single-valued (see Art. 69), so that the formula? 



dd&amp;gt; 

 (32) apply. Moreover, since -~- is given at every point of the 



cylinder, viz. 



-?= VcosO, when r a ............... (33), 



the problem is determinate, by Art. 71. Since the region occupied 

 by the fluid extends to infinity we must in (32) omit the coeffi 

 cients P n , Q n . The condition (33) then gives 



V cos 6 = - 2* na~ n ~ l (E n cos n6 -f 8 n sin nd}\ 



which can be satisfied only by making R^ = Fa 2 , and all the 

 other coefficients zero. The complete solution is therefore given by 



TV Vn z 



&amp;lt; = - cos0, ^ sin0 ............. (34). 



These formulae coincide with those of Art. 80 (e). 



As this case is one which is readily comparable with experi 

 ment, we will calculate the effect of the pressure of the fluid 

 on the surface of the cylinder. The formula (4) of Art. 25 gives 



where we have omitted the term due to the external impressed 

 forces, the effect of which can be calculated by the ordinary rules 



of Hydrostatics. The term -J* in (35) expresses the rate at which 

 &amp;lt;f) is increasing at a fixed point of space, whereas the value of (f&amp;gt; 



