1890.] Mr Sharpe, On Liquid Jets. 117 



If p 2 is positive, we readily see that a maximum has been 

 obtained. 



From (1) and (28) v vanishes at B. From (33) we see that 

 BHG is always above its asymptote. When z 2 = i (36) furnishes a 

 minimum, which, judging from the equation of continuity, should 

 give us a maximum ordinate, which we indicate by H in the 

 figures. This idea is corroborated, as we can prove from (1) that 

 z 2 = i- causes v to vanish. 



If p 2 is negative, the words " maximum and minimum " must 

 be interchanged in the above sentences. BHG is then always 

 below its asymptote, and at H there is a minimum ordinate or 

 Yena Contracta (fig. 2). 



In both cases, as we are not so much concerned with the sign, 

 as with the actual magnitude of the error in the velocity derived 

 from (34) we see, since the coefficient of p 2 in (34) vanishes when 

 z — 1, that the error is of the 3rd order of smallness at B and of 

 the 2nd order at H, as far as p is concerned. 



We now proceed to trace the curve AFGB on the right of Oy. 



From (7), (30) and (35) we get 



.(37), 



also from (32), pB = i|a 1 



therefore from (15) the equation to AFGB becomes, putting z for 



16a 



^-i ( y - ,r) = - (2a x 4- p 2 ) z sm y 



+ (fa + ti> 8 ) z* s^ % - (Jte + ip 2 ) z* sin 5y. . .(38). 



As p 2 is small compared with a x we see that it makes little differ- 

 ence whether we put +p 2 or — p 2 in the last equation, as the 

 curves obtained will only differ slightly in shape and position. 

 In fact, to get a general idea of the form of the curve, we may of 

 course practically put p 2 = 0. 



We thus get for the abscissa of G where the curve cuts the 

 line y = irjp 



T5 ^ + r ?' 

 whence z = "628 if x is about £. 



At F u = 0, so we get from (6) for the abscissa of F 

 0— jr + f^-^jP+A, 



EF therefore increases slowly as p increases (see Art. 8). If for 

 instance p = 100, EF = 5^ nearly. 



