Prof. A. Gray : Notes on Hydrodynamics. 7 



Equation (11) is Lord Kelvin's well known theorem. 

 But (12) puts the matter in a new way and brings into view 

 the different causes o£ variation of I. The latter equation, 

 (12), is an integral equation of motion corresponding to the 

 differential equation 





+ 2&> ss < j sin 6 = 





(14) 



which has been obtained as (6) above. From this the whole 

 motion of the fluid can be derived. 



8. As has been noticed, if we take ds' in the direction of 

 the stream-line we obtain, since # = 0, 





(15) 



which is the equation of motion for the stream-line direction, 

 and yields at once the so-called theorem of Bernoulli. 



If we take ds 1 in the direction of the axis of spin, that 

 is along what has been called a vortex-line, we obtain again 





(16) 



Thus Bernoulli's theorem is true also along a vortex-line. 



It is thus possible, in the case of steady motion, to draw 

 through each point of the fluid a surface on which lie inter- 

 secting stream-lines and vortex-lines, and for every point of 

 which ^ has the same value. 



Let a normal be drawn to such a surface at any point, 

 and dn denote a short step from the point along the normal. 

 Then clearly we have, by (14), 



~dn 



+ 2< 



■ = 0. 



(17) 



If (j> be the angle between the stream-line and the vortex- 

 line which intersect at the foot of the normal, and o> be 

 the resultant regular velocity at that point, we have 

 G>sin cj) = a) sn , and therefore 



OY 



^ +2(oq sin eb- 

 on * T 



(17') 



This equation is 



in Lamb's ' Hydrodynamics/ § 164. 



It is the particular case of (14) in which 6= ^7r, w sn — (o sin $, 

 and 'dq'lot = 0. The theorem of (14) is quite general. 



