116 
where is independent of the position and may be 
included in the (j). 
Putting for (p - (S<7v)</> and substituting for a we get 
or 2^ - (AfY + l\v<py = ^'y + ^ • • ■ IV. 
In support of the substitutions D ^ = 0 and DfXp - 0 I 
should state 
1 . That such surfaces can be found from the differential 
equation. 
2. That only three scalar equations have been used in 
determining a so as to satisfy the equation of motion. 
8. That as the intersections of such surfaces, if they 
exist, are to move with the fluid, it is not unnatural to 
make the trial of the possibility, and a fourth we shall see 
later. 
[It may not be out of place to notice that from the equa- 
tion — 
-(S,rvV = v(^V + £) 
which may be written 
; -2r<rp = Afv + ^' 
‘ \ m 2 / 
we may by operating with Sv get the equation 
ip^ - 2Syp<r = V^P 
where P stands for V + — + 77 
m 2 
an equation I have never seen stated.] 
4. In order to interpret as far as possible the expressions 
here introduced, we take first the last two conditions which 
express that the surfaces h and ;// move with the fluid so as 
always to contain the same fluid elements, and referring to 
the expression for the angular velocity we see that they 
intersect in the vortex lines. 
It would be well to determine these surfaces more fully. 
We have as yet treated them as distinct, however the sur- 
