Prof. A. Gray: Notes on Hydrodynamics. 



Fig. 1, 



This is generally called Bernoulli's theorem (Daniel Bernoulli, 

 Hydrodynamica, 1738). 



2. Now if instead of considering, 

 as in (1), the acceleration along 

 the stream -line, we take the acce- 

 leration along a step ds' drawn from 

 P (fig. 1) in a direction making an 

 angle 6 with the step ds drawn from 

 the same point along the stream- 

 line, we get as the equation of 

 motion in the new direction 



If we write 



we have, identically, 



v 



3 V 1 -dp 



+ 3s' + p ^s' 



a? 

 "37^37 



Subtracting from (3) we obtain 



3s 3s 



3t + 3s' 



3* ? V3s 3*7 3s" 



(1) 



(5) 



But if co S s' denote the component of elemental rotation of 

 the fluid about the normal (drawn outwards from the paper, 

 see figure) to the plane of ds and ds' at P 7 we have, as we 

 shall prove below, 



sin v = ^- — ^ L 7, 

 0* OS 



so that (5) may be written 



3?' 



a* 



+ 2w ss '} sin 8 = — ^ 



a% 



09 



3. By turning ds' without altering 0, we can change the 

 plane of ds and ds' from that for which co ss t is zero to that — 

 inclined to the former at an angle ^tt — for which co ss > sin 6 

 is a maximum. Equation (6) thus shows how we can pass, 

 in any direction, from the value of ^ at a point P on one 

 stream-line to the value of % at an adjacent point P' on 

 another. Thus along any surface about the normals to 

 which at every point there is zero rotation &v, the value of 



