136 Mr Basset, On the Application of Lagrange’s __[Nov. 28, 
2. The calculation of L can most easily be effected by employ- 
ing the current function instead of the velocity potential, for the 
former function is always single valued unless any sources or sinks 
exist in the liquid. 
Let u,, v, be the component velocities of any cylinder S, along 
rectangular axes fixed in the cylinder, w, its angular velocity, 
x, the circulation round any closed circuit which embraces this 
cylinder once only. 
Let the centre O of the cross section of the outer cylinder be 
the origin, and let «,, y, be the co-ordinates of the centre of inertia 
of the cross section of S, referred to rectangular axes fied in space; 
z,’, y,, the co-ordinates of the same point referred to moving axes 
through O which are parallel to the directions of u,, v,. Also let 
x be the current function and © be the velocity potential of the 
cyclic motion when all the cylinders are at rest. 
0 x 
In the figure let CA, CB be the axes of any one of the cylinders 
along which w,, v, are measured. then 
18 Mi DOr aay 
z= e|[ 7% du'dy 
da’ dix’ 
=~ pfx qe as+e| [x Gas], 
where the first integral is to be taken once round the circum- 
ference of the cross section of the outer cylinder, and the square 
brackets denote that the second integral is to be taken once round 
the circumferences of the cross sections of each of the moving 
cylinders. 
At the surface of each of the moving cylinders y is constant, 
hence the second integral vanishes, therefore 
, 
a 
d 
¥=— ply ds. 
