268 



Mr Sharpe, On Liquid Jets tinder Gravity. [Nov. 23, 



If we took a sufficient number of values of m to satisfy, in 

 addition to previous conditions, the condition S (w 2 a m ) = 0, we 

 see from (16) that the maximum error would be of the 3rd order 

 of smallness, and so on for higher orders. 



5. Suppose p instead of being large, were somewhat smaller, 

 we should then proceed thus. 



'■ F 



Fig. 2 



NA 



From F (fig. 2) draw FN perpendicular to Ax. 

 Let NF = irjp' where 



7T . ir 

 —: = sin — , 

 p p 



.(17). 



In equations (4) to (8) &c. put p' for p, and consider (4) to 

 apply to the right, and (6) to the left of NF. (8) also would have 

 to hold from y = to y = ir\p'. Of course from (17) p' could be 

 expressed as accurately as desired in terms of p. 



6. From (4) the equation to the outline FJ of the jet is 



Zd $ . o a ^ C n —pnx' • 



— r^sm^e + X— e * smpny- 



2a . 3tt 

 _ sm¥ ....(18). 



As the X term is of the order 1/p 3 , we see that in all solutions 

 obtained by the present method the shape of the jet is nearly 

 independent of the shape of the vessel, and is dependent only 

 on the angle which the orifice subtends at 0. 



Upon this point light may be thrown by the following 

 Article. 



7. Since writing the preceding, I have examined somewhat 

 carefully equation (14) which gives the outline of the vessel — 

 in the case where m has the values 8 and 9. In this case a m 

 and a n are determinate. I find that in (15) b is not perfectly 

 arbitrary, but appears to have limits in order that the curve 

 BEF may be continuous. I have tried to take it as large as 

 possible. I have actually taken it = 2tt/9 or about -6981, but 



