1891.] Mr Sharpe, On Liquid Jets under Gravity. 



265 



with centre in the surface of the liquid, or as a small straight 

 line perpendicular to Ox. For simplicity we shall take OA the 

 radius of the circle (or the depth of the liquid) as unity. 



If g be the acceleration of gravity referred to this unit it 

 will be convenient to put 



a 2 for 2g (1). 



We shall take as the origin of Cartesian and Polar Coordinates 

 x, y, r, 6 and we shall put 



x for 0-1) (2). 



Let % be the stream function to the right of AF ; u y v the 

 velocities parallel to Ox m Oy. 



Further let AF=ir/p (3), 



where p is a large number. 



On the right of AF we will take 



JL = u = ar l cos £0 + % Cn ' e P nx ' cos pny 



dy 

 ax 



CO 



•ismifl + SoV*"*' 



sm pny 



.(4), 



where c n ' is an arbitrary constant. X indicates summation with 

 regard to n for all integral values from 1 to oo . 



Therefore along AF, we have on the right of it at every point, 

 nearly 



u = a + $c n ' cos pny ) ^ 



v = — \ay + 2c„' sin pny) 



[Of course AF is really half the small segment of a circle. 

 The equations (4) and (5) are only approximate (the more so 

 the larger p be taken) but it will be pointed out afterwards 

 (Art. 5) how their exact forms can be found — forms which would 

 be suitable for all values of p — but as these are rather complicated 

 it is better to begin with the simplest case first.] 



Let y{r be the stream function on the left of AF, and 

 on the left of AF we will take 





cos my) + %c n € pnx ' cos pny + A 



— -r-= v = — S {a^*^ sin my) — %c n e pnxf sin pny 



•(6), 



where a m c n and A are arbitrary constants, and S indicates 

 summation with regard to m for a finite number of values of 

 m, the largest of which is supposed to be small compared with p. 



VOL. VII. pt. v. 21 



