NOTES ON HYDRODYNAMICS, CHIEFLY ON VORTEX-MOTION. 



where ds is in the direction of the resultant velocity q, and p is as usual a function 

 of p. The integrals are supposed taken along the stream-line from any chosen 

 starting-point to the point considered. If now we differentiate in any direction 

 inclined at an angle to the direction of ds and q, we get 



But we know that 



ds dt ds' ds' p ds' 



d l +q d i + d I+l d p=o 



dt H ds ds' P ds' 



for the first two terms make up dq'/dt, 

 and this, by the equation of motion for 

 the direction ds', is — (dV/ds' + 1 jp . dp/ds'). 

 Thus by subtraction we get 



OS \os OS / 



But if u> gg , denote the component of ele- 

 mental rotation of the fluid about the 

 normal to the plane of ds and ds', at P 

 their point of meeting, as indicated in the 

 figure, we can easily prove that 



- 2o)„, sin 0- 



dq 



: 8? 



d£ 



ds 



so that 



d\j/ 



= - 2w ss , q sin i 



(15), 



(16). 



12. I have not seen this theorem stated before, though various particular cases 

 of it are known from which it might be inferred. It asserts that, at time t, \J/- is 

 constant along a stream-line, and likewise along any line ds' so drawn that « M . is 

 zero for the plane determined by ds' and ds. \|/- is in general a function of t, and 

 the theorem shows how its variation from point to point depends on the motion of 

 the fluid. 



By turning ds', without altering 6, we can change the plane of ds and ds' from that 

 for which <a gl , sin 6 is zero to that — inclined to the former at an angle of 90° — for which 

 w gg < sin 6 is a maximum. The normal to the latter plane may or may not be the 

 direction of the axis of resultant rotation at P (see figure). 



13. Equation (15) may be regarded as an equation of motion for the direction ds' 

 inclined at the angle 6 to ds, the direction of flow. If we take ds' parallel to the fixed 

 axes Ox, Oy, Oz in succession, we obtain three equations : — 



^ + 2a> sx qsm0 sx = O, 



~- 1- 2w sy q sin sy = 



dxh 



■^ + 2o} sZ q sin 6 SZ = 



(17). 



