276 Prof. M. J. M. Hill. Motion of Fluid, part of [Feb, 7, 



February 7, 1884. 

 THE PRESIDENT in the Chair. 

 Dr. David Gill was admitted into the Society. 



The Presents received were laid on the table, and thanks ordered 

 for them. 



The following Papers were read : — 



I. " On the Motion of Fluid, part of which is moving Rota- 

 tionally and part Irrotationally." By M. J. M. Hill, M.A., 

 Professor of Mathematics at the Mason Science College, 

 Birmingham. Communicated by Professor G. G. STOKES. 

 Sec. U.S. Received December 29, 1883. 



(Abstract.) 



Clebsch has shown that the components of the velocity of a fluid 

 u, v, iv, parallel to rectangular axes x, y, z, may always be expressed 

 thus — 



u=^ + X d ±, v= r ^ + \ d -±, K = c k + X d ±; 

 dx dx dy dy dz dz 



where X, are systems of surfaces whose intersections determine the 

 vortex lines ; and the pressure satisfies an equation which is* equiva- 

 lent to the following — 



^--|-*{(l)' + ©* + ©'} + ^1©' + + (2-)'} 



where p is the pressure, p the density, and Y the potential of the 

 forces acting on the liquid. 



It is shown in this paper that an equation in A only can be obtained 

 in the following cases (that is to say, as in cases of irrotational motion, 

 the determination of the motion depends on the solution of a single 

 equation only) : — 



(1.) Plane motion, referred to rectangular co-ordinates x, y. 



X being a function of x, y, t, let — be expressed as a function of 



dy 



X, x, t by substituting in it for y its value in terms of X, x, t found 

 # British Association Report for 1881, p. 62. 



