1889.] and the Vena Contracta. 9 



Again, by Fourier's theorem, suppose we have from y = to 

 tt/2 



2^8111 2/ + 2a 3 sin Sy = 1q 2n sin2ny (17). 



This will be true at both limits if 



«t-«. = (18). 



Then the second equation of (10) will be identical with the second 

 equation of (13) if 



%>+&*» + &„ = <> (19). 



Then if the constants satisfy equations (15), (16), (18), (19) the 

 motions on the left and right of OB will be continuous. 

 From (11) the equation to AFB is 



b^ x sin y + i b 3 6~ 3x sin Sy + X -^ e -2 "* sin 2ny 



+ %=& 1 -| + ^...(20). 



5. We will now suppose that 



y-|-£ < 21 > 



When (21) is fulfilled, the stream-line AFB will consist of an 

 infinite straight line AF, whose ordinate is it, and a curved portion 

 FB. The peculiarity at F will be presently explained. It must 

 be carefully observed that AF is not an asymptote. 

 It will be found that we get the following relations 



o 



a^^ + A,, ^ = -^-(17^ + 16^), 



a 3 = 8^+ SA V a 4 = - JL. (61a, + 644,), 

 a, n 2 L ( 4rc 4rc ± 12\ / ± 2 _ 18 \] 



&! = .«! -A, 6, = ^^ + 16^), 



ft. — e^-S^, 6 4 = I p^(29« 1 + 644 1 ), 



. J 6a, „ 8a, , s 



^=^- x , B = ^ (22). 



It will be noticed that for moderately large values of n, a 

 and b tn ultimately vary as l/?i* so that all the series employed 



