66-68] GENERAL FORMULA. 85 



68. As an example let us take the case of an infinitely long 

 circular cylinder of radius a moving with velocity u perpendicular 

 to its length, in an infinite mass of liquid which is at rest at infinity. 



Let the origin be taken in the axis of the cylinder, and the 

 axes of x, y in a plane perpendicular to its length. Further let 

 the axis of x be in the direction of the velocity u. The motion, 

 having originated from rest, will necessarily be irrotational, and 

 </> will be single- valued. Also, since fd^/dn . ds, taken round the 

 section of the cylinder, is zero, i/r is also single-valued (Art. 59), 

 so that the formulae (2) apply. Moreover, since d<f>/dn is given at 

 every point of the internal boundary of the fluid, viz. 



f- = u cos 6, for r = a ..................... (3), 



dr 



and since the fluid is at rest at infinity, the problem is determinate, 

 by Art. 41. These conditions give P n = 0, Q n = 0, and 



u cos 6 = 2" nar n ~ l (R n cos nO + S n sin n6), 



which can be satisfied only by making R = ua 2 , and all the other 

 coefficients zero. The complete solution is therefore 



ua 2 a ua 2 . a 



6= -- cos 0, ilr = -- sin 6 ............... (4). 



r r 



The stream-lines i/r = const, are circles, as shewn on the next page. 



The kinetic energy of the liquid is given by the formula (2) of 

 Art. 61, viz. 



2T = p ^d^ = puW T^cos 2 dd = m'u 2 ......... (5), 



if m', = 7ra 2 /o, be the mass of fluid displaced by unit length of the 

 cylinder. This result shews that the whole effect of the presence 

 of the fluid may be represented by an addition m' to the inertia 

 of the cylinder. Thus, in the case of rectilinear motion, if we have 

 an extraneous force X acting on the cylinder, the equation of 

 energy gives 



or (m + m')-2 ..................... (6), 



where m represents the mass of the cylinder. 



