1884.] which is moving Rotationally and part Irrotationally. 279 



For a vortex of invariable form which moves parallel to the axis 



of z with velocity — , this equation becomes — 



dt 



rAdr* r dr dzV St j'j Sk J^^U' 



and the current function is 



S* - 2 ST" 



Suppose that in any of these cases any particular integral of the 

 equation in X is taken. 



It is shown that the components of the velocity can be expressed in 

 terms of X and differential coefficients of X, and that the current 

 function is also known. 



In the case of a fluid, part of which is moving rotationally and 

 part irrotationally, the boundary surface separating the rotationally 

 moving fluid from that which is moving irrotationally contains the 

 same vortex lines, and may be taken as the surface X=0. 



Now, if the integral taken of the equation in X do actually corre- 

 spond to a case of fluid motion in which part of the fluid is moving 

 rotationally and part irrotationally, the most obvious way to find the 

 irrotational motion will be to find its current function from the con- 

 ditions supplied by the fact that the components of the velocity are 

 continuous at the surface X=0. If after taking any integral of the 

 equation in X it be found theoretically impossible to determine the 

 current function of an irrotational motion outside the surface X=0, 

 which shall be continuous with the rotational motion inside it, then 

 the integral in question does not correspond to such a case of fluid 

 motion. 



In this method no assumption is made as to the distribution of the 

 vortex lines (as in the method of Helmholtz) before commencing the 

 determination of the irrotational motion. 



On the other hand, it is not a particular case of a method applicable 

 to motion in space of three dimensions. 



But it can be shown that Clebsch's forms for the components of 

 the velocity do also lead to a method which is applicable to the deter- 

 mination of the irrotational motion when the rotational motion in 

 space of three dimensions is known. 



For the rotational motion being known, the components of the 

 velocity are known for this part of the fluid. Let the components 

 of the velocity be expressed in Clebsch's forms, so that x» \ are 

 known. 



Moreover, let the forms be so arranged that the surface separating 

 vol. xxxvr. u 



