12 Mr H. J. Sharpe, On Liquid Jets, &c. [Oct. 28, 



been restricted suddenly becomes free) the radius of curvature 

 should be zero in the true solution. In the present approximate 

 solutions we cannot of course expect this to hold. But I am 

 inclined to believe that in all solutions obtained by the present 

 method there will be found on the line BHC (near where we 

 might expect the orifice to be) a point where the radius of curva- 

 ture is a minimum, and here I propose to place the point H. At 

 any rate there is such a point in the case of the first solution in 

 (27), as we proceed to shew. 



From (25) putting in it c 2 =|c 1 2 and from (8) c 3 = 3a a , the 

 equation to BHC is 



2 / = !-c 1 * + (-c 1 3 + c > 3 + &c (30). 



From (27), &c. 



_a A = Vj-A_ _?7r 0635 



It is evident therefore that c l is small and that (30) may be ap- 

 proximately written 



y=\ -c 1 (^-/)+&c. 



As dyjdx is always small all along BHC, we may get sufficiently 

 near the point required by finding the point where d 2 y/dx 2 is a 

 maximum. We see at once that this is got from the equation 

 z 2 is equal to 1/27 which gives x=— 71/43 nearly for the abscissa 

 of H. We can readily see that at H the curve is convex to the 

 axis and that a maximum value of d 2 y/dx 2 has been obtained, 

 also that the coefficient of contraction is "99545. 



9. It will now be interesting to calculate the limits of error 

 in the velocity at the point H. From (26) it can be shewn that 

 the coefficient of z 4, reduces to 



-\f-c l i -26c*+2c i . 



This is equal to '1654 nearly. Therefore at the point H only 

 •000011 of the velocity is variable. Of course as we pass to the 

 left of H this small proportion rapidly diminishes. 



10. If we examine equation (20) we shall find that the curve 

 FGB cuts the line y — tt/2 in a point G whose abscissa is about 

 3/43. Also if we imagine two points on the same curve whose 

 ordinates are 47r/6 and 57r/6 we shall find that their respective 

 abscissae are about 16/43 and 36/43. 



