226 VISCOSITY. [CHAP. ix. 



of terms of the form (25), provided the coefficients be so chosen as 

 to make 



=0, for = ........................... (28). 



Thus, writing ft = ki in (26), and solving with respect to a, we find 

 a = - f n tf kc i, 



where 



(29). 



Substituting in (25), and making use of (28), we obtain the solution 

 Z^Ze-^^PcosJcc t + Qsmkc tfsmkx ......... (30), 



where the summation embraces all values of 1c. The coefficients 

 P, Q may be determined by Fourier s method so as to make the 

 sum of (27) and (30) satisfy any arbitrary initial conditions. We 

 see that the effect of the initial circumstances gradually disappears, 

 until finally the only sensible part of the disturbance is that due 

 to the forced vibration maintained at x = 0. 



It appears from (30) that the effect of friction is to diminish 

 the velocity of propagation of a free wave. 



184. Example 3. A sphere moves with uniform velocity in 

 an incompressible viscous fluid ; to find the force which must be 

 applied to the sphere in order to maintain this motion*. 



Take the centre of the sphere as origin, the direction of motion 

 as axis of x. If we impress on the fluid and the solid a velocity V 

 equal and opposite to that of the sphere, the problem is reduced 

 to one of steady motion. Further, assuming the motion to be 

 symmetrical about the axis of #, we may write, with the same 

 notation as in Art. 103, 



1 dtyr 1 (Mr 



u =--T-&amp;gt; v = --- ~f-&amp;gt; 



& dty TV ax 



whence we have, for the angular velocity w of a fluid element, 



dv du_ l/d&amp;gt; d&amp;gt; l&amp;lt;fy\ 



^ft&amp;gt; = -j --- 7-= -- 7 o H r^ o -- i I 



dx dy tx \ dx d^a -cr dw/ 



If we suppose the motion to be so slow that the squares and pro- 

 * Stokes. Camb. Phil. Trans. Vol. ix. p. 48. 



