258 SIR WILLIAM THOMSON ON VORTEX MOTION. 



The particular case of this example for which the curve is a circle, presents 

 us with the simplest specimen of cyclic irrotational motion not confined [as that 

 of Example (1) is] to a set of parallel planes. The velocity potential being the 

 apparent area of a circular disc (or the area of a spherical ellipse) is readily found, 

 and shown to be expressible readily in terms of a complete elliptic integral of the 

 third class, and therefore in terms of incomplete elliptic functions of the first 

 and second classes. The equi-potential surfaces are therefore traceable by aid of 

 Legendre's tables. But it is to Helmholtz that we owe the remarkable and 

 useful discovery, that the equations of the stream lines (or lines perpendicular to 

 the equi-potential surfaces) are expressible in terms of complete integrals of the 

 first and second classes. They are therefore easily traceable by aid of Legendre's 

 tables. The annexed diagram, of which we shall make much use later, show 

 these curves as calculated and drawn by Mr Macfarlane from Helmholtz's 

 formula, expressed in terms of rectangular co-ordinates. An improved method 

 of tracing them is described in a note by Mr Clerk Maxwell, which he has 

 kindly allowed me to append to this paper. 



Example 3. The motion described in Example 2 will remain unchanged out- 

 side any solid ring formed by solidifying and reducing to rest a portion of the 

 fluid bounded by stream lines surrounding the infinitely thin wire. Thus we 

 have a solid thick endless wire or bar forming a ring, or an endless knot as 

 illustrated in the first three diagrams of § 59, of peculiar sectional figure depend- 

 ing on the stream lines round the arbitrary curve of Example 2 ; and the cyclic 

 irrotational motion which, if placed in an infinite liquid it permits, is that whose 

 velocity potential is proportional to the solid angle defined geometrically in the 

 general solution given under Example 2. 



64. Kinetic energy of compounded acyclic and polycyclic irrotational motion — 

 kinetico-statics. The work done in the operation described in § 62 is calculated 

 directly by summing the products of the pressure into an infinitesimal area of 

 the surface, into the space through which the fluid contiguous with this area 

 moves in the direction of the normal, for all parts of the surface, whether 

 boundary or internal barrier, where the genetic pressure is applied, and for all 

 infinitesimal divisions of the whole time from the commencement of the motion. 



(a). Let TV denote the work done, and fdt time-integration, from the beginning 

 of motion up to any instant. At any previous instant let p be the pressure, 

 q the velocity, and <f> the velocity potential, of the fluid contiguous to any 

 element da of the bounding surface, k the difference of fluid pressures on the two 

 sides of any element, d% of one of the internal barriers, and N the normal com- 

 ponent of the fluid velocity contiguous to either da or ds. The preceding state- 

 ment expressed in symbols is 



W^fdtl-jrpmv + Z/jfmdi] .... (6), 



