190, 191] VORTEX MOTION. 509 



191. Vortex Motion. We will now consider the case in which 

 no velocity potential exists , that is, the case of vortex -motion, 

 according to the methods of Helmholtz. 



From the equations 27), whose right-hand members are the 



derivatives of ( V + P + ^ en , this quantity may be eliminated by 



differentiation. Differentiating the last equation by y, the second by #, 

 and subtracting, we obtain 



*Wis U dz ~*~ * dz W dz ' 

 or otherwise 



-I!-: 



On the right the coefficient of w vanishes identically by 47), and 

 that of | is by the e 

 equation 55) becomes 



that of | is by the equation of continuity 12) equal to - ~> thus 



Now we have 



M ^ *| _ I *i 



dt\Q/ dt Q dt 



and accordingly we may write our equation 56) and its two companions 



i ^ , 1 ^ . 1 ^ 

 9 d# 9 ay 9 2z' 



rrrx _ | ^V 7] ^V ^V 



^i-Si"^-??? e 27' 



I fiw r\ diu g Bw 



7 ^ 7 3y "^7 a ' 



Thus the time derivatives of > > for a given particle are homo- 



geneous linear functions of these quantities. By continued differentiation 

 with respect to t and substitution of the derivatives from these equa- 

 tions, we see that all the time derivatives are homogeneous linear 

 functions of the three quantities themselves. Consequently if at a 

 certain instant a particle does not rotate, it never acquires a rotation. 



This we find by developing t> as functions of t by Taylor's 

 theorem, for if the derivatives of every order vanish for a certain 



