1890.] 



Mr Sharpe, On Liquid Jets. 



113 



Since Ox' = nrjp we must have 



. 7T , . Sw H - 07T . 



a, sin — I- Icl sin h Aol sin — = 



1 p 6 3 p ° 5 p 



.(3). 



It will be found convenient to replace a t , a 3 , a 5 by other 

 quantities such that 



a^^ + A^ a 3 = a 3 + A 3 , a 5 = a b +A t 



When x = 0, we have along OB, on the left of it, 



- u = (a, + Aj) cos y + (a 3 + A 3 ) cos 3y + (a 6 + ^4 5 ) cos 5;^ 



+ Xc n cos pny + A 

 v = (a 1 +A 1 ) sin y + (a 3 4- -4 8 ) sin 3y + (a 6 + -4 B ) sin by 

 + Xc n sin pny 



Let % be the stream function on the right of Oy, and on the 

 right of Oy let 



— t^ = - u = ^e"* cos y + 6 3 e" 3a; cos 3y + b 5 e~ 5x cos 5y ^ 



.(4). 



(5). 



+ 2,c n 'e~ p " x cos pny + B 



— -^ = v = — b t e x siny — 6 3 e _3a! sin Sy — b h e~ bx sin 5y 



(6). 



dx 



— 2c„'( 



sin pny 



.(7). 



(8). 



It will be found convenient to put 



When x = 0, we have along OB, on the right of it, 



- m = (a, - A t ) cos y + (a 8 - .4 3 ) cos 3# + (a s - 4 6 ) cos 5y 

 + 2c/ cos pny + B 



»=-(«,- A) sin y-i%- A 3 ) sin %/ - ( a s - A) sin 5 2/ 



-2c B 'sinpM/ 



By Fourier's Theorem, suppose we have from y = to 7r/p 

 2A t cosy+ 2J. 3 cos Sy + 2.4 5 cos 5y = Q + 2q n cos pny... (9), 

 which will be true at both limits. 



Identifying the first of (5) and the first of (8) we have 



C n ~C n ' + q n = (10), 



A-B + Q = (11). 



Again, suppose we have from y = to ir/p 



2^ sin y + 2a 3 sin Sy + 2a 5 sin 5y = %r n sin pny . . . (12), 



