THE EQUATIONS OF FLUID-MOTION. 131 



If (p be a scalar function of the position of a point in 

 the fluid, its total differential after time 8/ is Dt<p^t; and if 

 I>tf = o, the property of the point of which <p is the analy- 

 tical expression is unchanged during the motion. Now 

 let f =/i be the equation to a surface all points on which 

 enjoy the same property; such a surface will, if 'Di<p = o, 

 move with the fluid. 



The number of independent surfaces of this kind which 

 can pass through any point, or the number of independent 

 integrals of the equation 'D(<p = o,is three, or the dimensions 

 of the space considered ; for the Jacobian of a higher 

 number would vanish. 



The position of a point in the fluid in motion may be 

 represented in terms of the three independent sets of 

 surfaces <p, = f^„ -p^^^,, <Pi=Fv where [x,„ fj,^, and /a^ are 

 parameters. 



I have avoided the use of theorems and terms such as 

 tliose of " circulation,^' as I think they are apt sometimes 

 to override and hide the more important facts which their 

 discoverers intended them to express; and I have en- 

 deavoured to bring the fundamental properties to the 

 surface. 



Imagine an element of the fluid separated from the rest 

 of the fluid by the surfaces which we shall denote by 



l-'-i, F-^., l^i, jw-.-f Sju-j, //., 4- V2;, ''^iid jtXj + S/Xj. 



We will investigate expressions for the parts of this ele- 

 ment. 



First, vf „ Vfz, VPi represent vectors in the directions 

 of the normals to the three surfaces, such that if h„ h^, h^ 

 stand for tlieir tensors, the thicknesses of the element will 



be given by -^j^, ^'■, and -p respectively. 



The directions of the edges are given by Vv'piV<Pj=a, 



