SIR WILLIAM THOMSON ON VORTEX MOTION. 245 



Instalment, received Nov . — Dec. 1869 [§ 59 - § 64 (5)]. 

 length, and normal section, in any way, and with any speed. The average value of 

 the component velocity of the fluid along the tube, reckoned all round the circuit 

 (irrespectively of the normal section), varies inversely as the length of the circuit. 

 59. (a). To prove this, consider first a single particle of unit mass, acted on by 

 any force, and moving along a smooth guiding curve, which is moved and bent 

 about quite arbitrarily. Let f be the radius of curvature, and £, 7) the component 

 velocities of the guiding curve, towards the centre of curvature, and perpen- 

 dicular to the plane of curvature, at the point P, through which the moving 

 particle is passing at any instant. Let £ be the component velocity of the particle 

 itself, along the instantaneous direction of the tangent through P. Thus £, 77, £ 

 are three rectangular components of the velocity of the particle itself. Let Z be 

 the component in the direction of £, of the whole force on P. We have, by 

 elementary kinetics, 



§-« + ?.+ e§+iS .... o),* 



* This theorem (not hitherto published 1) will be given in the second volume of Thomson and 

 Tait's " Natural Philosophy." It may be proved analytically from the general equations of the 

 motion of a particle along a varying guide-curve (Walton, " Cambridge Mathematical Journal," 

 1842, February); or more synthetically, thus — Let 7, m, n be the direction cosines of PT, the 

 tangent to the guide at the point through which the particle is passing at any instant ; (x, y, z) 

 the co-ordinates of this point, and (as, y, z) its component velocities parallel to fixed rectangular axes. 

 We have 



f = lx + my + nz ; and Z = lx + my + nz , 

 and from this 



-- = lx + my + nz + lx + my + nz = Z+lx + my + nz . 



But it is readily proved (Thomson and Tait's " Natural Philosophy, § 9, to be made more explicit 

 on this point in a second edition) that the angular velocity with which PT changes direction is equal 

 t0 J(j 2 + ™ 2 + « 2 ), and, if this be denoted by a, that 



1 m n 



G>' O)' CO 



are the direction cosines of the line PK, perpendicular to PT in the plane in which PT changes 

 direction, and on the side towards which it turns. Hence, 



at 

 if K denote the component velocity of P along PK. Now, if the curve were fixed we should have 

 <° = g> % tie kinematic definition of curvature (Thomson and Tait, § 5) • and the plane in which 

 PT changes direction would be the plane of curvature. But in the case actually supposed, there is 

 also in this plane an additional angular velocity equal to -? , and a component angular velocity 



in the plane of PT and tj, equal to -? ; due to the normal motion of the varying curve. Hence 



the whole angular velocity co is the resultant of two components, 



K dP . 



- : + -~ in the plane of £ , 



VOL. XXV. PART II. 3 R 



