266 Mr Sharpe, On Liquid Jets under Gravity. [Nov. 23, 



Therefore along AF we have on the left of it at every point 

 11 = S (a m cos my) + Xc n cos pny + A } . ^ 



v = - 8 (a m sin my) — %c n sin pny ) 



But as the motion must be continuous through AF, the w's 

 and v'a in (5) and (7) must be the same. Therefore we get 

 8 (a m cos my) =a-A+% (c,/ - c n ) cos pny) ^ 

 - S (a M sin my) + ±ay = % (c.' + e.) am pny J 



These must hold from y = to y = ir/p. But if we expand 

 the left-hand sides of (8) by Fourier's Theorem, we can get c n and 

 c ' as functions of n. 



n 



2. Doing so, we shall get 



c +c=2S/a sin — , „ \ ....(9). 



" n J ra j> 2 2 mV J p « 



\ W7r """7"/ 

 c'-c — aSf/n .^sin-^. C ° SW7r a A (10). 



Also ji-a + iSffa,, x -^- sin — ) = (11). 



Also in order that the second of the equations (8) may hold 

 at the limit y = wjp, we must have 



-S^,^— ) + 2 - = (12). 



We shall now prove an important property of c„ and c„\ 



As 1 is the least value of n, and as by hypothesis m/p is always 

 a small fraction, we may always (if need be) safely expand the 

 fractions in (9) and (10) in ascending powers of m'/pW. 



Doing so in (9), we shall get 



mir cos nir I ' m 2, „ \) a cos mr 



— . l + -2-2 + &c. n- — . 



p wrr \ pn J) p n 



(13). 



But by (12) the first and last term here disappear, so that 

 (c^' + cj is always a small quantity at most of the order 1/p 3 . 

 Similarly from (10) we see that (c n ' — c n ) and therefore c n and c„' 

 are always small quantities at most of the order 1/p 2 . We say 

 ' always ' — even when n = 1 when they are largest. 



C ;+c„ = 2,Sf ja>n- 



