1890.] Mr Sharpe, On Liquid Jets. 119 



constant accelerating force g were acting on the fluid parallel to 

 yO, for we see from (33) and (34) that in that case the equation 

 u 2 + 2gy = constant would be satisfied accurately to the 3rd order 

 of small quantities all along BUG, as the coefficient of p 2 in (33) 

 is of the order 1/p. We thus get a sort of millrace BUG, and if p 

 be large enough, AF might almost be regarded as a free surface. 



8. It was proved in Art. 5, in the solutions already obtained, 

 that EF is a function of p which increases slowly as p increases. 

 Suppose it were required to find a solution wherein EF is a con- 

 stant quantity independent of p, but not zero. We could do so by 

 introducing into equations (1), (6) &c. additional terms involving 

 (say) sin 7y and cos 7y with 2 additional arbitrary constants. The 

 whole of the above process would then have to be gone through 

 and it would be found quite possible to satisfy the new conditions. 

 p would then have to be >70. The order of the errors would 

 of course remain unchanged. In this case there is no point G, but 

 the outer stream-line, after touching the asymptote at B, goes up 

 to F. The figure on the right of Oy would then have a much 

 closer resemblance to a cistern with an orifice at the bottom or (as 

 we may suppose the figure symmetrical with regard to the axis of 

 x) in the middle. There would be found to be an infinite number 

 of such solutions. It must however be observed that we cannot 

 find a solution of this kind wherein EF is actually zero. To prove 

 this we will first consider the case treated in Arts. 1 — 6 above. 

 Since from (31) and (35) c n ' — 0, we have from (6) to determine the 

 abscissa of F, 



0=-b 1 z-b/-b 5 z 5 + B. 



If this is satisfied by z = 1 Ave must have 



= -b 1 -b a -b 5 + B (39). 



But from (16), for large values of p, 



b t + b t +b B = Bir (40). 



(39) and (40) being incompatible, EF cannot be zero. An exactly 

 similar proof would apply if we introduced into (1), (6) &c. ad- 

 ditional terms involving (say) sin 7y, cos ly. 



9. In equations (1) (6) &c. the first three terms on the right- 

 hand side involve the three odd numbers 1, 3, 5. We are not 

 obliged to use odd numbers. Any integers will give a distinct case, 

 provided that p is large compared with the largest of them. If we 

 want a case which is compatible with the lowest possible value of 

 p, we should choose the numbers 1, 2, 3, and then p would be as 

 small as 30. So in Art. 8 if we chose the numbers 1, 2, 3, 4 we 

 could have p as small as 40 and yet have EF to a considerable ex- 

 tent independent of p. 



