8 Mr H. J. Sharpe, On Liquid Jets [Oct. 28, 



be similarly used, and each assumption would give a distinct case. 

 The same also must be understood for the stream function and 

 velocities presently to be assumed on the right of Oy. 

 Then the equation to BUG is 



a L e x sin y + ±a./ x sin 3^ + 2 -^ i nx sin 2ny 



, CL a A.7T /H _ N 



+ Ay = a 1 -^ + ~. ..{!). 



Since the ultimate breadth of the jet is supposed to be 7r/2, we 

 must have 



«-x-K=o («)• 



It will be convenient to replace a v a 3 each by two new quan- 

 tities, such that 



a^^ + A^ a 3 = cx 3 +A 3 (0). 



Then when x = 0, we have at every point of OB, on the left of Oy 



-u= (a, +A 1 )cosy+ (a 8 + A 3 ) cos Sy + ta 2n cos 2ny + A ) 

 v = (a l + AJ sin y + (a 3 + A 3 ) sin oy + %a 2n sin 2ny ) 



On the right of Oy assume for the velocities 



dy \ 



— -^ = —u = b x e~ x cos y + b 3 e Zx cos Sy + 2,b 2n e~ 2nx cos 2ny + B 



I (11) - 



_ _* = v = - b x e x sin y — b 3 e 3x sin Sy - %b 2n e 2nx sin 2ny 



It will be convenient to replace b v b 3 each by two new quan- 

 tities, such that 



b 1 ^cc 1 -A 1 , b 3 = cc 3 -A s (12). 



Then when x = 0, we have at every point of OB, on the right 

 oiOy, 



-u = (a 1 - AJ cos y + (cc 3 -A 3 ) cos Sy + tb 2n cos 2ny + B~\ 



v = -(a l -A 1 )smy-(cc 3 -A 3 )smSy-Zb 2n sm2ny J 



By Fourier's theorem, suppose we have, from y = to 7r/2, 



2A 1 cosy+ 2A 3 cosSy=p + 2p 2n cos 2ny (14). 



This will be true at both limits. Then the first equation of (10) 

 Avill be identical with the first equation of (13) if 



* 2n -K+p, n = o ( 15 )> 



A-B+p = (16). 



