32 PROFESSOR CHALLIS, ON A NEW FUNDAMENTAL EQUATION 



as in the Calculus of Variations, to the function xjy, while the co-ordinates and the time vary with 

 the varying position of a given element. Hence, 



dt dx dy dz 



But Sx = uSt, Sy=vSt, and Sx = wSt. Consequently, 



-y^+ -j-it+-r-v + --iw = o (2). 



dt d.v dy dss 



The main object of the arguments in this paper will be, to shew that the equation just 

 obtained is a necessary and fundamental equation of Hydrodynamics. I propose to call it, with 

 reference to the principle on which it was investigated, the equation of continuity of the motion, 

 to distinguish it from the equation of continuity of the fluid. 



It may here be remarked, that in the place of the actual surface of displacement we might 

 have reasoned in the same manner on a surface having with it a contact of the second order 

 at tlie point xyx; for instance, the surface whose equation is, 



{x - ay (y - (if {z - yf 



— T^ + r + ^^ — ^ -1=0, 



m w p 



the six parameters a, (i, y, m, n, p, being functions in general, both of the co-ordinates and 

 the time. Writing F = for this equation, it is clear, that when the co-ordinates and parameters 

 vary with the change of position of an element, we shall have SF = 0, provided there be no abrupt 

 change of the parameters, and consequently no abrupt change of the curvature of the surface 

 of displacement and of the directions of the lines of motion. This equation, therefore, to which 

 the equation S-^ («, y, «, t) = is equivalent, expresses the condition of continuity of the motion. 



3. Before entering on the consideration of equation (2), it will be shewn by an example 

 that the two recognised fundamental equations are insufficient for the general determination 

 of fluid motion. One instance of contradictory results legitimately deduced from those equations 

 will suffice for this purpose. The example I have chosen is as simple as possible. 



Let the fluid be incompressible, and the motion be parallel to the plane of xy. The equation 



ft ti fj 1 f 



of the continuity of the fluid for this case is 1- -— = 0. If u = mx and v = — my, that equation 



dx dy 



is satisfied. These values make uda; + vdy an exact differential. Hence the dynamical equation 



gives, p - C (»' + y'), the arbitrary quantity being either constant or a function of the time. 



2C 

 Bv putting p = 0, we obtain x^ + y- = — for the equation of the free surface of the fluid, which 



m 



is therefore at all times cylindrical, and hence the velocity is every instant the same at all points 



dv V V 



of the surface. But the differential equation of a line of motion is — • = - = . The lines 



^ dx u X 



of motion are therefore rectangular hyperbolas having the axes of co-ordinates for asymptotes, 



and the directions of motion are consequently different at different points of the cylindrical 



boundary. Hence it is impossible that the boundary can be constantly cylindrical. This 



contradiction proves that the equations on which the reasoning was founded are either erroneous or 



insufficient. We have no reason to suspect any error in the principles from which they were derived, 



and must therefore conclude that they are insufficient. It will appear afterwards that this instance 



does not satisfy the conditions of continuity. 



