On the surface of B, r equals b. Furthermore, to include only terms of order 1/r^, it is 

 sufficiently accurate to write, by projection of the radius of B, 



•< "1^ 6 cos ^2 ^ sin 0. 

 T-j = r + b cos 02, — = , sin 6^ = , cos ^^ = 1; 



and the second term of <;A"may be omitted. 



The result of carrying out the integration, after adding and evaluating the corresponding 

 integral over A, is the following simple expression for the total kinetic energy of the fluid per 

 unit length of the cylinders: 



^ 2 



9 o o 4«^ i>^ 

 a^V^+b^W^ VW cos (a + 13) 



[95c] 



The next step is to express the energy in terms of coordinates representing the positions 

 of the cylinders. Let the Cartesian coordinates of the axes of A and B be a;,, y^, and x^, yj' 

 and let the corresponding components of velocity be denoted by x^, y., a;,, y," ^®*' ^^^ ^\Tie 

 of axes PQ make an angle 6 with the positive a;-axis. Then, by projection, 



V cos a ^ x^ cos 6 + y sm 6, W cos jB = x^ cos 6 + y^ sin 6, 



V sin a = - i, sin 6 + y. cos 6, W sin /3 = - x^ sin 6 + y„ cos 6. 

 Substitution for V and W in Equation [95c] gives 



4a^6^ r • . . . .... ^ 



— [{x^ x^ - y^ ^2) cos 2 + (ar^ y^ + x^ y^) sin 2 d] V 



r } 



[95d] 



Here r and Q are to be considered as functions of a;,, y,, aj., y,' 



Now let ^jg, y^g denote components of the force exerted by the fluid on unit length 

 of cylinder B. Then the reactive force on the fluid is --^i o> - ^ig) ^n<^ the Lagrange equation 

 for x^ is 



dx^ ' ' 



where t denotes the time. Hence 



237 



