134 PEOCEEDINGS OP THE AMERICAN ACADEMY 



equations similar to the equations expressing a strain potential. 

 Helmholtz has shown that in the case of rotation of a fluid element, 

 Eqs. (8) become 



Eq. 9 



and therefore " the existence of a velocity potential is inconsistent with 

 the existence of rotation of the fluid element." We have seen from 

 the equations of strain that the existence of a strain potential is incon- 

 sistent with the rotation of a material particle. Let us now see if 

 vortex movement can arise in a liquid from variation of density and 

 pressure. Following Ilelmholtz's notation, we have, if t/; is a function 



5i =:. ^ -]- rci^. + t>^ + w'!± Eq. (10) 

 St at ' dx "^ du ^ dz A V >' 



Calling S, y, ^, the components of the angular velocity, we can obtain 

 their variations by substituting them in succession iu Eq. (10). If we 

 eliminate X, T, Z, from Ecjs. (.'J) by the help of P^qs. (G), supposing 

 that h and p are functions of x,y^ z, t, we obtain, introducing the values 

 of ^, 7, "Q, from Eqs. (9) : — 



_, /-,,\ 8f v/f'y r dw\ I dv I ^dw , 1 /dh dp dh dp\ 



Eq. (11) ^ = - S(- + -^-) + r- + C^, + 2T.U '^~d, -i) 



and similar expressions for the variations of / and ^. If the variation 

 of h is infinitely small, we obtain by the aid of Eq. (5) : — 



Sf ^.dn I do I ydw 



— = 5 \- y h t — 



9( 'dx dx ' dx 



If it is not infinitely small, we have the term 



1 /dh dp fM dp\ 



2h\dz 'dy dy di) 



which is independent of ^, y, t, and depends upon the variation of h and 

 p. This term enters into the expressions for the variations in the 

 angular velocities ; and shows, therefore, that a vortex movement can 

 arise iu a process of diffusion by a variation in density and pressure, 

 without the aid of initial angular velocities. This condition can be 



