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vortex motion. In treating of the motion of a perfect fluid 
it is usual to consider the cases of acyclic irrotational mo- 
tion and of vortex motion, considering cyclic irrotational 
motion as an adjunct of the second. This appears to me to 
be an improper division of the subject, which should rather 
be treated in the inverse manner, since the result is that 
the phenomena which are actually due to the cyclic nature 
of the motion are sometimes attributed to the vortex motion. 
In cyclic motion the circulation in any circuit is either 
zero or constant according to the nature of the space which 
the circuit encloses. If the circulation in a circuit is not 
zero, the circuit either encloses a portion of space not 
occupied by the fluid, or a portion of fluid, either termina- 
ting afc a boundary of the fluid or bounded by a closed sur- 
face, for which no velocity potential exists. Let u, v, w 
denote the components of the velocity at any point in 
the fluid, and f ds denote integration completely round 
a closed circuit. Let J dS denote integration over any 
portion of surface bounded by the circuit, of which a portion 
may consist of the bounding surface of the fluid, and 
let l , m, n be the direction cosines of any portion dS of 
this shell. Then by a Theorem due to Stokes, the circula- 
tion in the circuit may be written 
/(“£ + v t + w t) ds -/(«+*»+ 
whence if the surface S can be drawn without passing out of 
the space occupied by the fluid, and if the integral does not 
vanish, £, ij, £ must have values different from zero at some 
portions of the suface S. Also the integral has the same 
value for all surfaces S drawn; therefore the strength of 
the vortices on a]] surfaces bounded by the circuit is the 
same, and if the vortex lines are capable of being cut at 
right angles by a surface, is equal to the total strength of 
the vortices. 
If the space be polycyclic the circuit can be replaced by 
