1891.] Mr Sharpe, On Liquid Jets under Gravity. 2G7 



3. From (G) the equation to BEF is 



S fa e mx ' sin my) + 2 ^ e pii * sin F?i y + /ty 



= £ -^sin ■ — ) + (14). 



\m p ) p 



If y = b is the equation to the asymptote to BEF 



ji a (Q> m ■ ??i77 "\ Air ,, _. 



J.6 = £ ^- ra sin — + — (15), 



so that if 6 be finite, .4 is a small quantity of the order 1/^. 



Looking now at (6) we see that if OE is to be the surface 

 of the liquid, u and v must, when x' = —1, be small quantities 

 at most of the order ljp. A and the X term already satisfy 

 that condition. In the S term m has several values, enough 

 to satisfy conditions (11), (12) and (15). Suppose the particular 

 m in (6) to be the smallest of these values, and suppose m = \ogp, 

 then when a/ = — 1, the 8 term also satisfies the surface condition, 

 and the more accurately the larger p is, since \ogp/p diminishes, 

 as p increases. 



If FJ is to be a jet we must have, since AF is small, at every 

 point of the jet, nearly 



w 2 + v* = 2gr. 



But we see at once from (1) and (4) and Art. 2 that this con- 

 dition is fulfilled, the error being of the order 1/p 2 . 



4. To get some idea of the maximum value of this error, we 

 see from (4), since at F we have nearly 



u = a — c/ 

 that — 2c//a is a fair measure of this maximum. 

 From (10) and (13) 



c; = l ^S{m*aJ (16). 



From (11) and (12) we have nearly, since A is a small 

 quantity, 



-a + S(aj = 0, and ^-S(ma m ) = 0. 



If for instance we take 8 and 9 for the two values of w, then 

 p will be about 2981 and the maximum error about 



+ -0000143. 



21 2 



