9-10] SURFACE-CONDITION. 



P 



where R = 



dxj \dy 



1 d^ 

 this gives v = ~~Ti ~di ........................... * 



At every point of the surface we must have 



i&amp;gt; = lu+ mv + mu, 



which leads, on substitution of the above values of I, m, n, to the 

 equation (3). 



The partial differential equation (3) is also satisfied by any surface 

 moving with the fluid. This follows at once from the meaning of the operator 

 DjDt. A question arises as to whether the converse necessarily holds ; i. e. 

 whether a moving surface whose equation ^=0 satisfies (3) will always 

 consist of the same particles. Considering any such surface, let us fix our 

 attention on a particle P situate on it at time t. The equation (3) expresses 

 that the rate at which P is separating from the surface is at this instant zero ; 

 and it is easily seen that if the motion be continuous (according to the definition 

 of Art. 4), the normal velocity, relative to the moving surface F, of a particle 

 at an infinitesimal distance from it is of the order , viz. it is equal to G 

 where G is finite. Hence the equation of motion of the particle P relative to 

 the surface may be written 



This shews that log increases at a finite rate, and since it is negative infinite 

 to begin with (when =0), it remains so throughout, i.e. remains zero for 

 the particle P. 



The same result follows from the nature of the solution of 



dF dF dF dF , ,.. 



lfc + ^ + ^ + ^= ........................... (1) 



considered as a partial differential equation in F*. The subsidiary system 

 of ordinary differential equations is 



, dx dii dz ,.. 



dt = = ^- = .............................. (11), 



u v w 



in which a, y, z are regarded as functions of the independent variable t. 

 These are evidently the equations to find the paths of the particles, and their 

 integrals may be supposed put in the forms 



where the arbitrary constants a, 6, c are any three quantities serving to 

 identify a particle; for instance they may be the initial co-ordinates. The 

 general solution of (i) is then found by elimination of a, 6, c between (iii) and 



F=+(a,b,c) ................................. (iv), 



where \^ is an arbitrary function. This shews that a particle once in the 

 surface F=0 remains in it throughout the motion. 



* Lagrange, Oeuvres, t. iv., p. 706. 



