10 



Mr H. J. Sharpe, On Liquid Jets 



[Oct. 28, 



are convergent even for points very near Oy. I believe this 

 property is common to all the solutions obtained by the present 

 method. It is interesting to notice that unless we assumed the 

 equation (18) to hold, we could not establish, the obvious relation 

 A=2B. 



6. We will now make the curve HG as far as possible 

 identical with a line of constant velocity. It will be convenient 

 to put for shortness e x = z, 



aJA = c t , aJA = c 2 , aJA = c 3 , &c. 

 Then (7) can be written 



y = 9 — c x z sin y — \c 2 z 2 sin 2y — &c. 



and from (6), 



— j = 1 + c x z cos y + c 2 z 2 cos 2y + &c. 

 A 



v 

 —r = c x z sin y + c 2 2 2 sin 2y + &c. 



In (23) expressing y in terms of z, we have 



V = I - G i z + (K 3 - c i c 2 + i c 8 ) z 3 + &c. 



.(23), 



.(24). 



(25). 



It will be found that this series consists only of odd powers of z, 

 the simplification arising from our having taken 7r/2 for the ulti- 

 mate breadth of the jet. If we now substitute the above value 

 of y in (24) and form the value of u 2 + v 2 , we shall get 



25(m- + 0=1 + (3c*-2c 2 )z* 



+ (- f c/ + 8 Cl 2 c 2 + c 2 2 - ^ Cl c 3 + 2c 4 ) ** + &c. . .(26), 



the series consisting only of even powers of z. 



If we want to carry the approximation to the fourth order, we 

 shall have to cause the coefficient of z l here to vanish. This will 

 give us for determining the ratio of A x to a t the equation 



Solving this equation we get 



4,/^ = -1-0635 or -4-0127 nearly (27). 



Either of these gives a solution. If we take the first, we see 

 from the equations (22) at the end of Art. 4, that all the quan- 

 tities a t> a 2 , a 3 , a 4 are small, therefore the curve BHG clings all 

 along very closely to its asymptote. 



