1889.] and the Vena Contracta. 11 



From equations (6) or (11) we shall get for the direction of 

 the fluid motion at B, 



t'_37T/ A 



u ~ 1 V + ^ 



For the first solution this becomes — '06, and for the second — 2"84. 



If we wished to carry the approximation to the sixth order, we 

 should have to introduce terms with the sine and cosine of 5y in 

 (6) and (11) and make the coefficient of z* in (26) vanish. This 

 solution would involve two arbitrary constants. 



By making in addition the coefficient of z® in (26) vanish we 

 could carry the approximation to the 8th order, and the solution 

 would involve one arbitrary constant. 



7. We proceed now to discuss the solutions obtained. And 

 first for the point F, where there is a peculiarity which was 

 pointed out to me by Sir George Stokes. 



From (20) the equation to AFB can be written (putting for 

 shortness z for e - *), 



B (y — ir) — h x z sin {y — ir) + \b z z z sin 3 (y — ir) 



-$ b f~sm2n(y-Tr) (28). 



Dividing out by (y — ir) and then putting y = it, we get for deter- 

 mining the abscissa of F, 



B^b.z + bZ-lb^ (29). 



For the first solution in (27) this gives us z = *58, from which we 

 get x — 23/43 nearly. 



Further it will be seen from (28) that for points near F, z 

 does not vary, for on account of the values of z and b (see 

 end of Art. 4) the series converges pretty rapidly. Therefore 

 the stream-line turns sharply at right angles at F. This curious 

 result may be further corroborated simply by comparing the first 

 equation of (11) and (29) from which we at once get u = at F. 

 It may be noticed that we are not compelled to make the ultimate 

 ordinate of the stream-line on the right of Oy equal to ir. For 

 instance, in the present case, we arbitrarily assumed (21) to hold. 

 If we did not assume this, the outer stream-line on the right of 

 Oy would have no sharp corner. The ultimate ordinate however, 

 whatever it is, must not exceed it, otherwise the motion would be 

 discontinuous. 



8. Kirchhoff has shewn (Lamb's Motion of Fluids, Art. 96) 

 that at such a point as H (where the motion of the fluid having 



