of Edinburgh, Session 1869-70. 143 
equal parallel vortex-filaments rotating either in the same or in 
contrary directions. 
3. On the most general Motion of an Incompressible 
Perfect Fluid. By Professor Tait. 
This is a quaternion investigation into the circumstances of fluid 
motion, especially with reference to the case of vortices. The 
method employed is very similar to that which I gave to the 
Society in 1862 ( Proc . R.S.E. April 28). 
It is shown that if <n be the vector-velocity of a particle of fluid, 
so that 
cn = iu + jv + Jew , 
and if we introduce the operators IV and 8 ^ such that 
d . d d d 
A + 
dt dx dy dz dt 
IV = V + uf + v-~ + 
dx dy 
together with Hamilton’s operator — 
<1 = 
.d 
% dx 
x • d , 7 i 
+ j-j- -f h - 
dy ' ~ dz’ 
the equations of fluid motion and of continuity are- 
<1P - Up = D,«r) 
S<jcn = 0, ) 
where r is the density, and P the potential of the applied forces. 
The principal transformation is effected by means of the 
curious theorem in kinematics 
- D <r<<r = 
Thus, for instance, we have from the equation of motion 
= o, 
because <l 2 ( p -0 is obviously a scalar. The above theorem then 
D»-<1 = 8,^, 
gives 
which proves that if <1 cr is ever zero for any particle of the fluid 
it must remain so for that particle. 
As an additional instance of the simplicity of the method 
employed, the following may be given in this abstract:— 
