Lord Rayleigh's Notes on Hydrodynamics* 445 



from which the following points are calculated: — 



x=z-l -y = -0313 



x = '2 -y = -0932 



x = S -y=-1815 



#=•4 -y = -2985 



= •5 -y = -4509 



,i ? 



x— '6 — y= "6494 



#= -7 -y= -9203 



0= -8 -y=l-3127 



0= -9 -y=l-9915 



#=1*0 — ?/ = GO 



By means of these points the curve, fig. 6, is constructed. From 



(1^)> s= — log cos #; 



so that the radius of curvature is tan 6. The curvature is 

 therefore infinite at the origin, and diminishes continually as 

 s increases. 



In discussions on the cause of the contraction of the jet 

 doubts have been expressed as to the reality of the deficiency 

 of velocity in the middle of the orifice ; and it may therefore 

 be worth while to examine this point more closely. For this 

 purpose it will be convenient to express z or x -f iy in terms of ?. 



From (8) we get 



whence dco _11 — J 2 ( . 



^~?TT? 2 ^ b) 



Thus 



^=j^o > =j?J^=J^g^=2tan-Vf-r+a 



In order to determine the value of the constant of integration, 

 we may observe that £ when real varies between -co and —1, 

 and between + 1 and + oo . The values ± 1 correspond to the 

 edges of the orifice when z = and^=7r + 2. Hence tan -1 f 



varies between -j and -j-, and C = l— ^-. Accordingly 



0=2tan- 1 ?-?+l-| (16) 



If v be the velocity of the stream at any point y of the line of 



symmetry x= ^ +1, f= ; and therefore by (16) the rela- 



tion between y and v is 



i % 

 iy= —2 tan -1 - H 7r: 



J v v 



or, if tan -1 - be replaced by its logarithmic equivalent, 



1 t 1 + v /17 . 



^- lo gi^ ( 17 ) 



