PAPER BY PROF. HELMHOLTZ. 65 



to the abox'e expression of x-^yi still another term cr -|- t t, which is also 

 always a function of (p + ip i, we ha^e then 



x = Aq)-\- Ae cos ip + o" / 



( (3a) 



y = A tp-{- Ae sin ip -\- t ) 



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

 tinuous surface where ij-= :^ tt we shall have 



^A-Ae"- + |^^Y+ (^)'= constant. 



This condition is fulfilled if we make 



^=0 or o- = Constant (36) 



and 



^=±AV2e*-/* (3c) 



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

 with respect to (^, and change the integral inco a function of (p-\-ip i by 

 substituting every where instead of ^ the expression cp+i {ip-\-7r). Thus 

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

 tain 



c+T i=Ai^yl -2e —e +2 arc sin J^'^ > 



The cusp points of this expression lie where 



{3d) 



_o. 



— -I, 



that is to say where 



^i = i (2 a -t- 1) 7r [a being any whole number], 

 and 



<p= log 2. 



Thus neither one lies between the limits from ^'= + ;rto ip = —7C. 

 The function ff-\-Ti is here continuous. 

 Along the wall we have 



ff+Ti=±A i I V2e'i'—e-'i>—2 arc sin I -r2 ^^ | 

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

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



portion of the lines ip=^7r therefore corresponds to the free portion of 

 the jet. 



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

 i A t n-, which latter is to be added to the value of t t and y i re- 

 spectively. 



80 A 5 



