1 „ T 1/2 T T 1/2 



sin i/= — ![(cc+ 1)2 + 2/2] _[(^_i)2 + y2] j_ ^j^j] 



On the real axis of s, where |a;| < 1, f = and sin~ 3 = sin" x = v and has its ordinary 

 meaning; where |a;| > 1, cos i^ = 0, a; = - cosh ^, and ^ may have either sign, so that 

 if a; > 1, 



if a; < - 1, 



• -1 • -1 + -1 



sin 2 = sin X = — - i cosh x: 

 2 



i cosh (-a;), 



2 ^ ' 



where either sign may be chosen. 



The variables v and ^ are doubly many valued, hence so is sin~^ z; as with real angles, 

 if one value is sin" z = u, then others are u - 2nn where n is any integer, and also 

 (1 -2n) 77 - u. This complication can be removed only at the expense of introducing discontin- 



uities. A convenient range for v may be ^ ^ ^ t7/2. Using this range, sin z is dis- 

 continuous along the a;-axis where \x\ = cosh ^ > 1; there the sign of ^ is indeterminate, where- 

 as elsewhere (f has the sign of y. With this convention, Figure 127 may be used by assuming 

 that on the plot c = 1 and 77 = (77/2) - v. 



'n the present problem the values of \/t lie on or below the real axis. For such values 

 continuity can be preserved with use of the ranges ^ < 0, - 77/2 <v< 77/2, provided it is agreed 

 that, for a real number a;, in terms of the positive cosh"^ a;, 

 if a; > 1, 



if a; < - 1, 



— 1 —1 



sin X = — - i cosh x\ [lllr 



2 



= - — - i cosh X. [llln] 



At 6', or at <= 1, s = a; at 6, or < = - 1, s = - a. Inserting these values in Equation 

 [lllh] and evaluating c and k, it is found that 



2a la 



In t, [lllo,pl 



2 + 77 2 + 77 



2a 



2 + 77 



t^(1?- -\)'^ ^s.m~^ —\. [Ulq] 



Since to = ^ + zi/i, <^ and xlt are now fixed, but they cannot be expressed in terms of 2 by means 

 of ordinary functions. 



The velocity of the fluid at any point is given by 



dw 1 „ 1/2 "^ 



-u-viv= = - — = [i + (;;2 _ 1) ] . [lllr] 



dz 4 



275 



