218 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI 



so that the quantities denoted by (2, 3), (3, 1), (1, 2) in Art. 137 (21) vanish 

 identically. The equations therefore reduce in the present case to 



, dW r dW , dW 



where W=7rpa?(u?+v 2 + w*} .............................. (v), 



and A', Y, Z are the components of extraneous force applied to the sphere. 



By an easy generalization it is seen that the equations (iv) must apply to 

 any case where the liquid is in steady (irrotational) motion except in so far as 

 it is disturbed by the motion of the small sphere. It is not difficult, moreover, 

 to establish the equations by direct calculation of the pressures exerted on the 

 sphere by the fluid. 



When X, F, Z=Q, the sphere tends to move towards places where the 

 undisturbed velocity of the fluid is greatest. 



For example, in the case of cyclic motion round a fixed circular cylinder 

 (Arts. 28, 64), the fluid velocity varies inversely as the distance from the axis. 

 The sphere will therefore move as if under the action of a force towards this 

 axis varying inversely as the cube of the distance. The projection of its path 

 on a plane perpendicular to the axis will therefore be a Cotes' spiral*. 



141. If in the equations (21) of Art. 137 we put qi=Q, 

 <j 2 = 0,..., we obtain the generalized components of force which are 

 required in order to maintain the solids at rest, viz. 



Q-** Q- dK m 



y ' %' y ' %' 



We are not dependent, of course, for this result, on the 

 somewhat intricate investigation which precedes. If the solids 

 be guided from rest in the configuration (q l} q. 2l ...) to rest in the 

 configuration (q 1 + A^, q 2 + Aq 2 ,...), the work done on them 

 is ultimately equal to 



which must therefore be equal to the increment AK of the kinetic 

 energy. This gives at once the equations (1). 



The forces representing the pressures of the fluid on the 

 solids (at rest) are obtained by reversing the sign in (1), viz. 

 they are 



dK dK 

 ~d qi ' - 



The solids tend therefore to move so that the kinetic energy 

 of the cyclic motion diminishes. 



* Sir W. Thomson, I.e. ante p. 211. 



