Here, for real t and \t\ > 1, (r - 1) has the sign of t^ and - 7r/2 -g sin"^ — ^ 77/2. 

 For the particle velocity, Equation [113f] still holds. 

 On the lamina itself, t is real and |i!| > 1, also q ^- u, hence, from Equation [113f], 



i = + 



_ 1 + ?^ 



1/2 _ 1 - r 



2q 



{t'-l) = + 



2q 



[114e,f] 



the sign is negative where x > x , positive where x < x^, where x^ is the coordinate of C. 

 These equations and Equation [114d], in which now 2 = x, connect q with x. The distance of 

 C from the center is the value of s at 1^1 = 00 or 



I 



4 + 77 sin a 



2cosa(l + sina) + ( — -ajsinct 



[114g] 



where a is in radians. 



The treatment of the free streamlines is nearly the same as that in Section 113. Taking 

 w from Equation [114a] instead of Equation [113c], 



dz = dx + idy -2c (cos d + i sin 



sin G 



(cos 6 + cos a ) 



After separating real and imaginary parts, x and y are found by integrating and adjusting the 

 constant of integration. The formulas are most conveniently written in terms of distances 

 and angles measured positively with the stream, or y' = - y, d' = - 6, 6" = 6 + n = n - d'. 

 Right-hand streamline, Q <d' < n -a : 



I I sin'* 



— + 



2 



iJ cos y + cos a 



2 + cos a 



4 + '^ S'"" \ (cos 0'+ COSd )^ (l+COSd)^ 



[114h] 



/ sin I 



y = 



4 + 77 sina 



sina sin 6' (1 + cos a cos 6') 1 + cos (a - 6') 

 - In : 



(cos 6' + cos a Y 



cos a + cos a 



[114i] 



Left-hand streamline, ^ 6" < a 



1 I sin"* a / 2 cos 9" - cos a 2 - cos a 



2 4 + '^ sina I (cos (9"- cos a )2 (1 - cosa )' 



[114j] 



287 



