118 Mr Sharpe, On Liquid Jets. [Nov. 24, 



To consider more carefully the form of the small portion GB. 

 As in this portion y < wjp we may put in (38) y, 3y, hy for sin y, 

 sin 3y, sin oy. We can then readily shew that in passing from the 

 curve given by + p 2 to the curve given by — p 2 we alter the 

 ordinate of GB by a quantity at most of the 3rd order of smallness*. 

 It is interesting to compare with this the fact that the corresponding 

 alteration in the ordinate of the jet at all points (except in the 

 neighbourhood of B) is of the 2nd order of smallness. Thus a 

 very small alteration in the vessel produces a disproportionately 

 large alteration in the jet, in fact changing a maximum into a 

 minimum ordinate or vice versa. 



The Problem thus seems to suggest a point of contact with 

 Lord Rayleigh's article " On the Instability of Jets " — given in 

 Vol. x. of the Proceedings of the London Mathematical Society, 

 though it must be admitted that in that article the author is con- 

 sidering not the effect of the vessel on the form of the jet, but the 

 instability of the jet itself due to capillary force. 



6. Of course there is nothing whatever to prevent us putting 

 p 2 = in Art. 4, in which case we see from (34) that not only at B, 

 but all along the jet the error in the velocity would be at most of 

 the 3rd order of smallness. 



We may remark that no attempt has been made to draw the 

 figures to scale, as that would be difficult. 



7. Supposing p 2 not to be zero, and the linear unit of measure- 

 ment to be comparable with 1 foot (or perhaps better say 2 / or S f ), 

 it would approximately make no difference to the solution if a 



* From (38) we have in curved portion GB (putting a for a x for shortness), 



j^ (V - t) = - (2a +p 2 ) zy + (g* + 2 Pz J ^hj - (^ a +p 2 J z^y. 

 Now keeping z constant, suppose we put -p 2 for +p 2 and let y become y x then 

 jg^ (Vi ~ t) = ~ (2« -p 2 ) zy x + ( g a ; - 2pA zhj x -^a-pA z 5 y x ; 

 .'. subtracting, we have nearly 



is? (y - 2/1) = (y - 2/1) [ - 2az + 3 azS - 5 a * 5 J + 2 y [ -v& + 2 ^ z3 -iv 5 ]- 



As in GB z<-628, 



we have approximately 



{y-y 1 )[-2ax'628 + &c.]=-2yp a x -628; 



.-. as in GB y < — . 



y -y 1 is at most of the order -= . 



