136 THE MECHANICS OF THE EARTH's ATMOSPHERE. 



where the arc whose sine is A- is to be taken between zero and ^^^ 



For ^/•= -fO and f/>> rf^r.-^-z ^e have 



and for 



dgj 



dcp 



'^^v^Z--^'-('^v,j 



?/•=— Oand (p> 



1+A-; 



we have 



= k 



1 



1 



^-^/-C-v^; 



The lines that are represented by the integrals of these equations, 

 when we determine the constants of integration so that these lines 



start from the previously indi- 

 cated termini of the fixed walls, 

 are the free boundaries of the 

 3/ moving liquid. The other 



boundaries of the region of 3' lie 

 at infinite distances, as is seen 

 from the fact that when a;=oo 

 we have 

 dz 



a: 



W 



dco 



= A'-?Vl-A-2 



Fig. 7. 



this equation shows at once 

 that at an infinitely great dis- 

 tance from the origin of coor- 

 dinates the flow takes place 

 with the velocity 1 in a direc- 

 tion that forms an angle with 

 the axis of x whose cosine is A. Figure 7 illustrates the boundary of 

 the region oi z ; besides this boundary the figure also gives the stream 

 line for which y=0, and <^<0. 



(III.) Still one more example may be introduced. Let there be 



and let >f: vary between —tt and +;r, but cp between — x) and -f-c. 



From the point cl?=0 draw a section for which //'=0, and q>'> 0, and 

 assume that for ^= + 0, and V= + 0, the real part of /(cj) is positive. 

 The points of bifurcation of y/J\(a)J\G3)^-[ are the two points (y=0, and 

 cj=— log (1— A-) both which are found ui)0u the section that has been 

 drawn. The sign of the radical quantity Vf{Gj)f{co) — l is determined 

 by the rule that its real part shall be positive for ^= + 0, and ?/-=+0. 



