116 Mr Sharpe, On Liquid Jets. [Nor. 24, 



It will then be found from equations (4), (22) and (29), that 

 so far we may regard a t and a 3 as arbitrary, and A x , A s> A b , a 5 as 

 determined in terms of them by means of the following equations: 



A = - a i-P Z ' A 3 = - a 3 + 2 P2> A 5 = i( a i + Sa s)-P2\ /CAN 



« 5 = -*K + 3« 3 ) }"^> 



We shall further get, omitting a small term multiplied by p 2 in 



the bracket following, 



, 4 cos nir /a . « \ /oi\ 



o n = -c n '= v (2« 1 + 3« 3 ) (31), 



A=pB=%cc 1 + j.ct 3 (32). 



4. We shall now make BHC as far as possible a line of con- 

 stant velocity. In (2), putting for shortness z for e x and modifying 

 by means of (29) and (31), we get for the equation to BHC 



-i(2 ai + 3« 3 )S^^sin W ...(33). 



It is obvious that for any point on BHC the coefficient of p 2 

 cannot exceed f and that the S cannot exceed % (l/^ 3 ) which is 

 about 118. All along BHC therefore y differs from ir/p only by 

 quantities at most of the 2nd order of smallness. (They are in 

 fact of the 3rd order.) In (1) therefore we may put unity for 

 cos y, cos Sy and cos 5y, and then modifying (1) by means of (29) 

 and (31), we have at every point on BHC 



-u=p 2 (-z+2z s -z 5 ) 



+ i (2a, + 3* 3 ) 2 ^-^ *- cos W + A . . .(34). 



Jj lb 



From (1) and (31) and reasoning like the preceding, it is evi- 

 dent that v is at most of the 2nd order of smallness, because c n is. 

 Therefore (w 2 + v 2 ) only differs from A 2 by quantities of the order 

 1/p 2 . Practically therefore at every point on BHC (34) gives the 

 whole velocity. This velocity consists of a constant part A and 

 a variable part. A portion of this variable part can be made, if 

 we like, to vanish by putting 



2a 1 +3a 8 = (35). 



The remaining portion will be a maximum or minimum at a 

 point determined by 



-z+6z 3 -5z 5 = (36). 



This is satisfied by z = 1 or the point B. 



