92 Prof. Maxwell on the Theory of Molecular Vortices 

 varied motion is 



, r , du 1 dot 



2 dz 47T^ (It 



Similarly, 



x=a, 



dx 



1 dp 



(145) 



The whole force acting upon a stratum whose thickness is dz 



and area unity, is -^—dz in the direction of w. and -=- dz in di- 

 * dz dz 



rection of y. The mass of the stratum is pdz, so that we have 



as the equations of motion, 



d*x 



dX 

 dz 



h 



d 2 x d 1 



dz 



+ H7Zf ir 



d*y _dY d 2 y d \ da 



dt 2 dz 



dz 4?r r dt 3 

 , dp 



dz 47r r dt 



(146) 



Now the changes of velocity -j- and ■— are produced by the 



motion of 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), 



Since &x and $y are functions of z and t only, we may write 

 this equation 



da. d 2 x 



1 



and in like manner, 



dt y dzdt' 



dfi_ dhj ; 

 dt~ y dzdt ' 



(147) 



so that if we now put k x =:a 2 p, k^tfp, and 

 write the equations of motion 



&*_,*&* ■ ' d s y -J 



1 fir 



47T p 



-Tz-—a>*-nr + c 



[dt 2 

 d 2 y 



dz 2 

 dz 2 



dz 2 df 



, d s x 



y = c i , we may 



(148) 



° dz 2 dt J 



These equations may be satisfied by the values 

 # = Acos [nt — mz + a),"\ 

 y=Bsin [nt— mz-\- a), J 

 * Phil. Mag. May 1861. 



(149) 



