256 EXERCISES. 



25. If (r lt } ), (r a , O a ), ... be the polar co-ordinates at time t of a 

 system of rectilineal- vortices whose strengths are m lt m a , ... respectively, 

 prove that 



= const., 



[Kirchhoff.] 



and 2ww 2 -.- = - %m m . 



at ir l 2 



26. A series of long waves is propagated along a uniform canal, 

 arising from a small disturbance at the mouth which varies as a given 

 function of the time ; assuming that the effect of friction may be repre 

 sented in a general way by a retarding force varying as the velocity, 

 determine the motion. [Stokes.] 



27. Prove that the formula 



&amp;lt; = a (e jlf + e~ jy ) (e jz + e~ jz ) cos k (x - ct), 



k 

 where j= r , represents a possible form of wave motion in a uniform 



V^ 

 canal whose section is a right-angled isosceles triangle having its sides 



equally inclined to the vertical. (The axis of x is the bottom line of the 

 canal, and that of y is vertical.) 



Find the velocity of propagation in terms of the wave-length ; and 

 examine the form of the free surface. [Kelland.] 



28. Prove the formula 



&amp;lt; = ae k (v sin-zcos# cos k(x- ct), 



giving the motion of a series of waves parallel to the edge of a shore 

 sloping at an angle j3. 



Find the velocity of propagation, and the form of the free surface. 



[Stokes.] 



29. Calculate the tidal motion of a heavy liquid contained in a 

 square vessel of uniform depth, due to a small horizontal disturbing 

 force acting uniformly throughout the mass, whose magnitude is con 

 stant, and whose direction revolves uniformly in .the horizontal 

 plane. 



How could the forces here imagined be realized experimentally 1 



[Lord Rayleigh, Math. Trip., 1870.] 



