170 MR. R. F. GWYTHER ON THE VELOCITY OF 



2. First let us consider the circumstances accompanying 

 the forms. If SaSr is integrable by a factor, the condition 



So'Vo"=Oj or Scrp = Oj — 

 that is, the axes of vortex-motion perpendicular to the 

 lines of flow, a case satisfying the conditions of parallel 

 cylindrical vortices and vortex rings. 



If 0-— v^ be integrable by a factor, the condition is 



S(cr — v0) V {a-—V(f>)=0, 

 or S(cr— v^)p=o. 



From this scalar equation <f) could be found ; and we 

 may consider a='^(f> + k^'\Jr as a general form of expres- 

 sion for the velocity at any point. 



3. The kinematical condition that a must satisfy is the 

 " equation of continuity/' which if the fluid is incompres- 

 sible takes the form 



or 



V'«^ + ^VV + Sv^V^ = o; .... (II.) 



and the angular velocity at any point in the fluid is 



given by 



2p = Yvkvir (III.) 



The form taken by the equation of motion, when this 

 expression is submitted for a, will now be found ; for 



may be written 



D,(v<^ + ^V>^)=-v(v+0 

 or 



Now Di V = y . Di — AD^, where A acts only on the a in 

 D< or dt— (So-y) ; therefore 



