OF WHICH IS MOVING ROTATION ALLY AND PART IRROTATIONALLY. 367 
Moreover, since F(X, xfj) is an arbitrary function of A, t//; it can be so chosen that X' 
may be any required function of X, \Jj ; i.e., any vortex sheet. 
Therefore the X in ClebsciTs expressions for u, v, iv may be considered as the 
surface of any vortex sheet; and, consequently, as the surface separating the 
rotationally moving fluid from that which is moving irrotationally. 
Therefore, if <ft be the velocity potential of the irrotationally moving fluid, all over 
the surface X = 0, supposing the motion continuous there ; 
<7y d(f) d<p c?y dcf) 
das das’ dy dy ’ dz dz ’ 
Since also 
P 
+V= 
ci x \ 2 , 
d i-m)+$) >+** 
( Id’^r\^ ( (dy\r 
dx ) “H dy) ) 
in the rotationally moving fluid, and 
in the irrotationally moving fluid, it follows that the condition for the continuity of 
the pressure at the surface separating the rotationally moving fluid from that moving 
irrotationally is that < ^= < ~ all over the surface X=0. 
Now suppose that there exists a solution of the three equations to which y, X, i[/ are 
subject as given by Clebsch ; then to find <f>, it is necessary to find it so as to satisfy 
the above surface conditions. 
In any case in which y is the potential of a distribution of matter inside the surface 
X=0, together with many valued terms satisfying Laplace’s equation, then <fj is the 
potential of this distribution calculated for a point outside X=0, together with the 
same many valued terms, provided that it give zero values for the components of the 
velocity at infinity. 
With regard to the supposed distribution of matter, its total mass must be zero, in 
the case of an incompressible fluid. 
For total mass of supposed distribution of matter = — g jjq~dS, the integration 
being extended over the surface X= 0 
But this = —(total flux outwards across the surface) = 0. 
Therefore total mass of supposed distribution of matter ^=0. 
2. Plane Motion. Rectangular Coordinates. 
In the ordinary notation the equations are 
