144 Proceedings of the Royal Society 
If T be the instantaneous axis of the element of fluid, whose 
velocity is <r-, we have — 
<! cn = - 2 r . 
But 
S <J 2 cj- = 0, 
whence, 
^ <] ’ 2 cn = V <1 r 
2 
and 
- ^ = <1 0 + <J 
! V<r. 
This contains the solution of the problem, treated by Helmholtz, 
to determine the linear velocity of each fluid particle, when the 
angular velocity is given. 
4. Mathematical Notes. By Professor Tait. 
The following self-evident propositions were employed for the 
deduction of several curious consequences — 
(a.) 4a? = (x + l ) 2 - (x — l) 2 , 
or, x3 ( x(x + 1) y _ ( x(x - 1) J ; 
or, “ Every cube is the difference of two squares, one at least of 
which is divisible by 9.” 
(b.) If 
x 3 + y 3 = z 3 , 
then 
( 'x 3 + z 3 ) 3 y 3 -f- (a ? 3 - y 3 fz 3 = (z 3 + y 3 ) 3 x 3 . 
This furnishes an easy proof of the impossibility of finding two 
integers the sum of whose cubes is a cube. 
Monday , 4 th April 1870. 
The Hon. Lord NEAVES, Vice-President, in the Chair. 
At the request of the Council Professor Wyville Thomson, Bel- 
fast, delivered an address on “ The Condition of the Depths of the 
Sea.” 
