86 Prof. Challls on investigating genn-ally the partial 



differential equation of a surface of displaccjnefit, that is, a 

 surface which cuts at right angles the directions of motion of 

 the particles through which it passes. By integrating, the 

 equation of every such surface becomes 

 <t> {x,y, 2, /) = 0. 

 It is unnecessary to add an arbitrary function of the time, 

 as it may be supposed included in $. Now the coordinates of 

 a given fluid particle, which were a.', 3/, 2; at the time t^ \iG,- 

 come x + n d t, i/ + vdt, z + wdt, at the time ^ + ^^, dt being 

 indefinitely small, and these must ultimately be coordinates 

 of a surface of displacement. Hence 



<p{x + udt, y + vdt, z + wdt, t + dt) = 0. 

 Consequently, putting 4) for ^ [x, 3/, z, t), we have 



^^^udt + p-vdt + ^t^dt+^dt = 0, 

 dx dy dz dt 



which, since ($1 = 0, gives 



-jj+u.-P- + v.-~+'w.-j^ = (3.) 



dt dx dy d z 



U V 'W 



Again, from the equation d (p = -y;^- d x + -^ dy+^ dz, 



it follows that 



i^j d 't> -KT d'P XT d(p 



dx dy dz 



Hence, substituting in (3.), we obtain 



fl-'^C^+f-^);" <-) 



This equation serves to find N when ($ is known as a func- 

 tion of X, y, s, and t. In general, when the preceding values 

 of ?/, uand w are substituted in the equations (1.) and (2,), by 

 means of these and the equation (4.) N and p may be elimi- 

 nated, and the resulting equation in <$, x,y, ;^, and t is the ge- 

 neral partial differential equation which it was required to find. 

 The eliminations are far too complicated to be introduced 

 here, and I must therefore content myself with thus indicating 

 the process. For incompressiblefluids the equation(l.) is simply 

 du dv dw 

 dx dy a^ 

 and by substituting the values of w, v, and iv, it becomes 



MS 



(/2^ d^<p d^<p \ rfN d<p </N d<p 



'^ dy'^ dz^ J dx ' dx dy ' dy 



f/N d^ 

 + -7—.-— = 0. 



ax dz 



