TRANSACTIONS. 



I. — Notes on Hydrodynamics, chiefly on Vortex Motion. 

 By Professor Andrew Gray, F.E.S. 



(MS. received November 6, 1908. Read November 16, 1908. Issued separately April 15, 1909.) 



1. In the teaching of hydrodynamics many points of difficulty arise, both for 

 teacher and for student. The subject abounds in subtleties even in its very elements, 

 and the advanced student frequently finds himself in a state of doubt as to 

 fundamental questions which crop up unexpectedly in connection with various 

 problems. The following notes contain a discussion of a few of these fundamental 

 matters : for example, they deal with some theorems of energy which have been 

 found difficult by students, perhaps mainly through want of perfectly explicit state- 

 ment of their scope and purpose. Finally will be demonstrated a theorem of 

 vortex-motion, particular cases of which have been given by various writers, but 

 which I have not seen stated elsewhere in the same generality. This will be found 

 to lead to Lord Kelvin's well-known and far-reaching theorem of circulation, and 

 to other theorems of the vortex-motion of a perfect fluid, some of which are already 

 known. 



2. First it may be recalled that if q be the resultant velocity of a particle of 

 the fluid at any point P at time t (or indeed any other quantity characteristic of 

 an element of the fluid in motion), and ds the element of path actually described 

 by the particle in the interval of time dt beginning at t, the acceleration of the 

 particle is 



dt A ds 



This, of course, is the usual expression from which the component accelerations 

 parallel to the axes are deduced. 



But if ds' be an element of a line drawn from P at any angle to ds, and q' the 

 component of velocity in the direction of ds', the acceleration in this direction is 



dt *ds 

 This is a very useful expression, and will be of great service in what follows. By 

 putting u, v, w for q' we obtain easily the usual component accelerations. That the 



acceleration along ds' has this value is at once evident from the fact that, at time t, 

 y TRANS. ROY. SOC. EDIN., VOL. XLVII. PART I. (NO. I). 1 



