PAPER BY PROF. HELMHOLTZ. 65 



to the above expression ofx + y i still another term g + t% which is also 

 always a function of q> + //> «, we have then 



x = Acp + Ae cos ip + a I 



4 f • (3a) 



y = A ?/' + A e sin //< + r ) 



and must determine a + ri so that along the free portion of the discon- 

 tinuous surface where ip= ± n we shall have 



This condition is fulfilled if we make 



- — =0 or a = Constant ....... (3&) 



d<p v ' 



and 



j£=±Ws.*-«* (3c) 



Since ip is constant along the wall we can integrate the last equation 

 with respect to cp, and change the integral inuo a function of q)-\-ip ihy 

 substituting everywhere instead of cp the expression cp+i (if:-\-7r). Thus 

 by an appropriate determination of the constants of integration we ob- 

 tain 



ff+ri=4i V-2e -e +2 arcsinl ^« ( ** + * 'J \ ■ ' ^ 



The cusp points of this expression lie where 



x (4+4 i) 



p — — 2- 



that is to say where 



ip=± (2 a x 1) rr and cp = log 2. 



Thus neither one lies between the limits from ip— -It- n to if: = — n. 

 The function a -f- ri is here continuous. 

 Along the wall we have 



& + ri = ± J. i | V2e*- e 2 * -2 arc sin [^4] } 



If ^ > log 2, then all these values become purely imaginary, there- 

 fore 6 = 0, while 7- has the value given above in equation (3c). This 

 portion of the lines tp=±7t therefore corresponds to the free portion of 



the jet. 



If <p< log 2 the whole expression is real up to the additive quantity 

 iiiff, which latter is to be added to the value of r i and y i re- 

 spectively. 



80 A 5 



