13G THE MECHANICS OF THE EAETH's ATMOSPHERE 



where the are whose sine is k is to be taken between zero and ^ 



/T 



For ip = + and qj > n_fc\2 



we have 



(to 

 dqj 



">+v?Z-^-(»hJ 



and for 



f/'= — and cp> (1 -,M 2 we have 



dx 



dqj 



1 dy /, /. 1 V 



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

 when we determine the constants of integration so that tbese lines 



start from the previously indi- 

 cated termini of the fixed walls, 

 are tbe free boundaries of the 

 moving liquid. The other 

 boundaries of the region of z lie 

 at infinite distances, as is seen 

 from the fact that when &?=oo 

 we have 

 dz 



JC 



doo 



= lc-iy/l-k 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 7c. Figure 7 illustrates the boundary of 

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

 line for which >/'=0, and <^<0. 

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



J v ' VI— er» 



and let ip vary between — n and +7r, but q> between — oo and +ao. 



From the point oo=Q draw a section for which ?/==0, and <p> 0, and 

 assume that for qj=-\-0, and j/<=+0, the real part of f(co) is positive. 

 The points of bifurcation of \Z/(&? )/(<*?) — 1 are the two points oo=0, and 

 G9=— log (1— W) both which are found upon the section that has been 

 drawn. The sign of the radical quantity \/ f(oo)f( go) — I is determined 

 by the rule that its real part shall be positive for ^= + 0, aud //<= + 0. 



