-A, 



^''^ "^It 



b x^ - {x^ cos 26 + y^ sin 2 i 



[{x^ x^ - yj y2) cos 2 + (a;^ y^ + x^ y^) sin 2 ^J — - 



4a^6 



2 i2 



.... . -5^ 1 



[(ij OTj - ^1 y^) sin 26 - {x^ y^ + x^ y^) cos 2 6\ — V 



A similar equation is obtained for Y ^g. 



For simplicity, after differentiating with respect to t, let the axes be rotated so that at 

 the instant under consideration 6 = 0. Then it is easily seen that at this instant 



dt 



^^ ^^1' dt 



y2~yi dr dr_ de_ 66 1 



dx„ 



dy^. 



dx„ 



dyr 



Hence, with axes chosen so that 6* = 0, the force upon B has components 



'IS = "P 



2a'b^ 



ia^ b' 



i^l - yl) 



[95e] 



^iB = '^P 



„ 2a^ b"^ .. %a^b^ 



^ vi ; — ^1 - — T- 



[95f] 



where two dots denote two differentiations with respect to the time, or a component of 

 acceleration. Here ij = F cos ct , y. = V sin a. 



Similar expressions are obtainable for the force on A, but it is also possible to use 

 the equations just found by interchanging notation between the cylinders. 



A stream at infinity can be introduced by changing to a uniformly moving frame of 

 reference. This change does not affect the forces or the accelerations; the equations for 

 X.g and y, D as they stand can, therefore, be used in all cases, with the understanding that 

 ip yj, V and a all refer to the velocity of cylinder A taken relative to the fluid at infinity. 



The first term in X^g or 7, „ represents the usual effective increase in the inertia of 

 B due to the presence of the fluid. The remaining terms represent the effect of A upon B, 

 and are valid only if a/r and b/r are small. It is readily seen that the errors due to the 

 approximations are of an order in 1/r that is higher than l/r^ in terms that involve the 

 acceleration and higher than 1/r^ in those that do not. 



238 



