510 



ON PHYSICAL LINES OF FORCE. 



The whole force acting upon a stratum whose thickness is dz and area 

 unity, is r- dz in the direction of x, and r- dz in direction of y. The mass 



of the stratum is pdz, so that we have as the equations of motion, 



<Px_dX_, d*x d 1 dB' 



^ dt* dz dz* dz 4;r dt i-\Ar\ 



d*y dY , d*y d 1 da 



Now the changes of velocity -3- and -y- are produced by the motion of 



Gtt ' ' ' 



the medium containing the vortices, which distorts and twists every element 

 of its mass; so that we must refer to Prop. X.* to determine these quantities 

 in terms of the motion. We find there at equation (68), 



d d , d 



da = a -j ox-}- B -= ox + y -5- ox (68). 



dx dy dz 



Since 8x and 8y are functions of z and t only, we may write this equation 



da _ d*x 



f~ 7 ^ (147), 



and in like manner, - = y- r - 



dt ' dzdt J 



tut* 



so that if we now put k, = a?p. k. = b*p, and - y = c", we may write the 



47T p ' 



equations of motion 



dt*- l dz*~ S dz'dt (14g) 



~dt*~ ~* ~dz*~ ' dz*dt . 



These equations may be satisfied by the values 



x = A cos (nt mz + a)l 

 y = B sin (nt mz + a)/' 

 provided ,, i \ ,< _ > w> 



. \ (150). 



and (n mb) a*-~* 



* Phil. Mag. May 1861 [p. 481 of this vol.]. 



