204 
of the space considered. For the Jacobian of a higher num- 
ber would vanish. 
The position of a point in the fluid in motion may be 
represented in terms of the three independent sets of 
surfaces = = (l>s = i^s where /mi, fi^ and ^3 are parame- 
ters. 
I have avoided the use of theorems and terms such as 
those 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 
f^i, ^2? fts) /n + Sjiid ^3 + 3^3. 
We will investigate expressions for the parts of this element. 
First, V01, V02, V(f)3, represent vectors in the directions 
of the normals to the three surfaces, such that if hi, h^, h^, 
stand for their tensors, the thicknesses of the element will 
be given by — and respectively. 
The directions of the edges are given by Y V02 V</>3 = a say, 
and the similar quantities (3 and y, so that 
V/3y= -Sv<^iV(/) 2 V^s. say. 
The length of an edge will be given by the thickness -j- 
the cosine of the angle between the corresponding normal 
and the edge. Thus length of the edge a is 
Sjui hi^a ^jjiiYa 
hi' S V 0ict H 
and the vector edge is - (1) 
Whence the area of the face 0, is ^^^^TY/3y = |T^/i 2 ^/a 3 
and the vector of the faces V </n (2) 
From either (1) or (2) we may get volume of element 
hfXi^lXiClX^ T h^lhfX 2 ^fX 3 , . 
— ■" 
