IV. On a new Fundamental Equation in Hydrodynamics. By the Rev. James 

 Challis, ma., Plumian Professor of Astronomy and Experimental Philosophy in 

 the University of Cambridge. 



[Read March 6, 1843.] 



The object of this communication is to shew, that in addition to the two fundamental 

 equations of Hydrodynamics already recognised, a third is necessary to complete the analytical 

 principles of the science. 



For the purpose of reference I shall call the two known equations, the dynamical equation, and, 

 the equation of continuity of the Jtuid. The same notation will be made use of as in my last 

 paper : p is the pressure and p the density of a particle whose co-ordinates at the time t are x, y, x, 

 and the components of whose velocity V are u, v, w, in the directions of the axes of co-ordinates. 

 JC, Y, Z are the impressed forces in the same directions. A differetilial coefficient is put in 

 brackets to indicate that the differentiation refers both to the co-ordinates and the time : a 

 differential in brackets means that the co-ordinates alone are differentiated. All differential co- 

 efficients not in brackets are partial. 



1. It will be assumed that in any case of fluid motion an unlimited number of surfaces may 

 be drawn at each instant, cutting at right angles the directions of motion. In other words, it 

 is assumed that the directions of motion at every instant fulfil the condition of geometrical 

 continuity. In my last paper it was shewn that if 



u V w 

 d^ = — dx + —dy + — dz, (1). 



the factor — being such that the right-hand side of the above equality is an exact differential, 



the general differential equation of all these surfaces at all times is d\|^ = 0. It is not necessary 

 that the surfaces should be continuous : that is, it is not necessary that the equation of a given 

 surface should be the same function of the co-ordinates through its whole extent. But that the 

 condition of the geometrical continuity of the directions of the motion may be maintained, each 

 surface must be made up of parts, either finite or indefinitely small, which are surfaces of continuous 

 curvature. Hence the quantity N has a real value for every part of the fluid in motion; at least, 

 motions for which this is not the case, if there are such, do not come under our consideration. 



2. Let the integral of the equation d\p = be \l/(x, y, z, t) = 0, the arbitrary function 

 of the time being included in the function ^. The surfaces of which this is the general equation 

 I shall continue to call surfaces of displacement. Since the equation •v|/^(a!, y, x, t) = embraces 

 all the surfaces of displacement at all times, it will include the surfaces of displacement of a 

 given element of the fluid at two successive instants of its motion, if the pat/t of the element 

 in the interval he continuous. It is not necessary that the path of an element through its whole 

 extent should be determined by the same equations, but it is necessary for the continuity of 

 the motion that it should be made up of parts, either finite or indefinitely small, which are 

 geometrically continuous, and that the directions of motion at two successive instants should not 

 make a finite angle with each other. The condition of the continuity of the motion of each element 

 is therefore expressed analytically by the equation ^\^(ir, y, x, t) = 0, the symbol S having reference. 



