SIR WILLIAM THOMSON ON VORTEX MOTION. 253 



constant for that particular circuit. The cyclic constant must clearly have the 

 same value for all circuits mutually reconcilable (§ 58), in space throughout 

 which the three differential coefficients remain all finite. 



60. (x). When the function is cyclic with reference to several different 

 mutually irreconcilable circuits, it is called polycyclic. When it is cyclic for only 

 one set of circuits, it is called monocyclic. 



Example. — The apparent area of a circle as seen from a point (x, y, z) 

 anywhere in space, is a monocyclic function of x, y, z, of which the cyclic con- 

 stant is 4 77. 



The apparent area of a plane curve of the (2n)th degree, consisting of n 

 detached closed (that is finite endless) branches (some of which might be enclosed 

 within others) is an w-cyclic function, of which the n cyclic constants are essen- 

 tially equal, being each iir. 



Algebraic equations among three variables (x, y, z), may easily be found to 

 represent tortuous curves, constituting one or more finite, isolated, endless 

 branches (which may be knotted, as shown in the first three diagrams of § 58, 

 or linked into one another, as in the fourth and fifth). The integral expressing 

 what, for brevity, we shall call the apparent area of such a curve, is a cyclic 

 function, which, if polycyclic, has essentially equal values for all its cyclic con- 

 stants. By the apparent area of a finite endless curve (tortuous or plane), I mean 

 the sum of the apparent areas of all barriers edged by it, which we can dram 

 without making a partition. 



It is worthy of notice that every polycyclic function may be reduced to a 

 sum of monocyclic functions. 



(y). Fluid motion is called cyclic unless the circulation is zero in every closed 

 path through the fluid, when it is called acyclic. Rotational motion is (e) essen- 

 tially cyclic. 



{z). Irrotational motion may [ § 59 (/)] be either acyclic or cyclic. If cyclic 

 it is monocyclic if there is only one distinct circuit, or polycyclic if there are several 

 distinct circuits, in which there is circulation. It is purely cyclic if the boundary 

 of the space occupied by irrotationally moving fluid is at rest. If the boundary 

 moves and the motion of the fluid is cyclic, it is acyclic compounded with cyclic. 



61. (a). We are now prepared to investigate the most general possible irrota- 

 tional motion of a single continuous fluid mass, occupying either simply or multiply 

 continuous space, with for every point of the boundary a normal component 

 velocity given arbitrarily, subject only to the condition that the whole volume 

 remains unaltered. 



(b)" Genesis of acyclic motion. Commencing, as in § 3, with a fluid mass at 

 rest throughout, let all multiplicity of the continuity of the space occupied by it 

 be done away with by temporary barrier surfaces, /3 X , yS 2 . . . stopping the circuits, 

 as described in § 57. The bounding surface of the fluid, which ordinarily consists 



VOL. XXV. PART II. 3 T 



