136 MR. GWYTHER ON THE LAGRANGIAN FORM OF 



This form enables us to express in a general way tlie cri- 

 terion that the vortex motion may exist in the form which 

 it will be seen to take in a perfect fluid. Taking the two 

 other similar quantities, and equating each to zero, we get 

 VVf)<o" = o or D^o- of the form yP; and, conversely, if 

 YvDiar=o, then Df/=o 



This notation also enables us to give a verbal expression 

 for V V c- Thus : let P Q U lie on a line of intersection of 

 two surfaces (p^ and f^, and let them be distinguished by 



(p^^ tpj f — , and <p^— -. Then the vector difference ot 



the velocities at Q and R is -r-h(p^; and resolving parallel 



to the face fi of the element, having P for centre, these 

 differential velocities on the parallel faces <pj, we get 



S<P,V -r- UV^, X area = ~- x area V ^— V<p, 

 ^' d(p, ^' Ih dp, 



= volume X Y~j—V<P,, 

 d<p, 



and similarly for the other faces. Therefore volume 

 X Vvo" = resultant differential velocity and Vyo" = 

 mean resultant differential velocity per unit of volume, 

 as the result of three shears and rotations. And by a 

 similar method we may find the force due to viscosity, 

 arising in consequence of the motion noticed above, upon 



