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



make the trial of the possibility ; and a fourth we shall see 

 later. 



[It may not be out of place to notice that from the 

 equation 



which may be written 



\ m 1 J 

 we may by operating with SV get the equation 



7) <J 



where P stands for V + ~ + — , 

 m 2 



an equation I have never seen stated.] 



4. In order to interpret as far as possible the expres- 

 sions here introduced^ we take first the last two con- 

 ditions, which express that the surfaces k and '^ move 

 with the fluid so as always to contain the same fluid- 

 elements j and; referring to the expression for the angular 

 velocity, we see that they intersect in the vortex-lines. 



It would be well to determine these surfaces more fully. 

 We have as yet treated them as distinct. However, the 

 surfaces k and i/r coincide or do not exist where vortex- 

 motion does not exist ; for then V . V^V'>/^=o at all points, 

 and the normals to the surfaces at all points are parallel ; 

 whence the surfaces k= const, and ■^■= const, coincide 

 except at points where vortex-motion exists. In order, 

 therefore, to apply this method to the solution of rotational 

 motion of a fluid, we should consider ksj-^ an additive term, 

 and take for k and "^ values such as to make the surfaces 

 A:= const, and -^^ const, move with the rotational fluid, 

 and always intersect in vortex-lines, while the term A(/) 

 would be taken to satisfy the general irrotational motion. 



