SIR WILLIAM THOMSON ON VORTEX MOTION. 249 



parts by lines drawn across it, the circulation in the boundary of the whole is 

 equal to the sum of the circulations in the boundaries of the parts. This is 

 obvious, as the latter sum consists of an equal positive and negative flow in each 

 portion of boundary common to two parts, added to the sum of the flows in all 

 the parts into which the single boundary of the whole is divided. 



60. (c). Hence the circulation round the boundaries of infinitesimal areas, 

 infinitely near one another in one plane, are simply proportional to these 

 areas. 



(d). Proposition. Let any part of the fluid rotate as a solid (that is, without 

 changing shape); or consider simply the rotation of a solid. The "circulation" 

 in the boundary of any plane figure moving with it is equal to twice the area 

 enclosed, multiplied by the component angular velocity in that plane (or round 

 an axis perpendicular to that plane). For, taking r, 6 to denote polar co-ordinates 

 of any point in the boundary, A the enclosed area, and to the component angular 

 velocity in the plane, and continuing the notation of § 59, we have 



and therefore 



rcW 



2$8s = o)Sr 2 °™ s Ss = coXr 2 S0 =»x2A 



(e). Definition. (For a fluid moving in any manner), the circulation round 

 the boundary of an infinitesimal plane area, divided by double the area, is called 

 the component rotation in that plane (or round an axis perpendicular to that 

 plane) of the neighbouring fluid. 



In this statement, the single word "rotation" is used for angular velocity of 

 rotation: and the definition is justified by (c) and {cl)\ also by § 13 (2) above, 

 applied to (p) below. It agrees, in virtue of (p), with the definition of rotation 

 in fluid motion given first of all, I believe, by Stokes, and used by Helmholtz 

 in his memorable "Vortex Motion," also in Thomson and Tait's "Natural 

 Philosophy," §§ 182 and 190 (/). 



(/). Proposition. If £, 77, £ be the components of rotation at any point, P, of 

 a fluid, round three axes at right angles to one another, and w the component 

 round an axis, making with them angles whose cosines are /, m, n, 



co = %l + rjm + %n . 



. To prove this, let a plane perpendicular to the last-mentioned axis cut the other 

 three in A, B, C. The circulation in the periphery of the triangle ABC is, by (b), 

 equal to the sum of the circulations in the peripheries PBC, PC A, and PAB. 

 Hence, calling A and a, /3, y the areas of these four triangles, we have, by (0), 



a)A = %a + V j3 + £y • 

 VOL. XXV. PART IT. 3 S 



