NOTES ON HYDRODYNAMICS, CHIEFLY ON VORTEX-MOTION. 13 



ends perpendicular to vortex-lines, and PA, QB are the directions of els' at 

 P and Q. 



Now for the plane P'A'C we have, since P'A' and P'C are at right angles, 



%= - 2w ss <g, 



3s 



or 



-^dt = - 2w 8 sds. 

 ds 



Thus, for the surface integral of rotation over P'A'C we have 



J£dtds'=-±mrfdsds .... (32). 



ds 



But if w be the resultant angular velocity of rotation of the fluid about the vortex- 

 line at P', and (p the angle between P'C (ds) and P'Q (the vortex-line), 



oj sin <f> = o)ss'> 

 so that 



^^dtds' = - ±(i>ds ds. sin <f> . . . (33). 



OS 



But ^dsds'sin(f) is the area of the eud PAC of the portion of a vortex-tube. Calling 

 this o-, we have 



gtMdt=-<*T .... (34). 



OS 



We can prove that this is the same for both sections of the tube. For, consider the 

 circuit PABQP, made up of the vortex-lines PQ, AB, and the two intercepted lines ds', 

 namely PA, QB. The line-integral 



reduces to d\js/ds'. ds' for PA, and a similar expression, with the minus sign prefixed, for 

 BQ. But for any closed circuit moving with the fluid the integral vanishes, and, 

 since \^ does not vary along a vortex line, d^/ds'.ds' is the same for PA and AB. 



Thus for both ends of the vortex-tube we have the same value of wo-, since we have 

 to take ds at the two sections to correspond to the same dt. Along a vortex-tube, 

 therefore, wo- is constant ; and the tube must either be endless or have its ends on the 

 free surface of the fluid, if a free surface exist. In both cases it is really endless. 



21. [February 12, 1909.] The equations of acceleration for any two directions 

 drawn from the point P are by § 2 



ou du _ 3V _ 1 dp \ 



dt ds 3./: p 3a; 



\ (35), 



dv dv = __8V_ 1_ dp_ 

 dt % ~dj p" dy J 



where dx, dy are infinitesimal steps, and u, v the velocities, in these two directions. 



