246 SIR WILLIAM THOMSON ON VORTEX MOTION. 



where § denotes the radius of curvature, and -^ , -j- rates of variation of £ and -q 



from point to point along the curve at one time. 



59. (b). Now, instead of a single particle of unit mass, let an infinitesimal 

 portion, [x, of a liquid, filling the supposed endless tube, be considered. Let ®- be 

 the area of the normal section of the tube in the place where /jl is, and Ss the length 

 along the tube of the space occupied by it, at any instant ; so that (as the density 

 of the fluid is called unity), 



Further, let yr denote the rate of variation of the fluid pressure along the tube,. 

 so that 



Thus we have, by (1), 



Hi = — zr — OS 



as 



dt~ T + ?ds +v ds~ Ts ■ (2 > 



(c). Now, because the two ends of the arc Ss move with the fluid, we have, by 

 the kinematics of a varying curve, 



dBs d£ * £ j, .„. 



* = **-«* • • ( 3 ) ; 



and, therefore, 



dt= E* + t U*~ t Bs ) ■ ■ W- 



7C* 



Substituting in this for -k its value by (2), we have 



3P-(«f+«3-**rD». 



or 



<?(g&?) 



= W"/>) (5). 



if q denote the resultant fluid velocity; and 8, differences for the two ends of the 

 arc Ss. Integrating this through the length of any finite arc P^ of the fluid, its 

 ends P r P 2 , moving with the fluid, we have 



^^= (k 2 -p\-(h 2 -A .... (6), 

 the suffixes denoting the values of the bracketed function, at the points P 2 and 



and 



dr) 



in the plane of r\ 

 ds 



Hence 



and the formula (1) of the text is proved. 



