PAPER BY PROP. HELMHOLTZ. 105 



Hence for this boundary line result the equations 



(■"■■ cos (, /(/! = « (cos ik cos 6— cos e) ) 



(■■■■ sin (iiy)=— ^sin (ih) sin 6 \ ' ' -' ' ( la ) 



By the elimination of 6 this gives an equation between x and y as the 

 equation of the boundary line. Beside the constant a which deter- 

 mines the initial point of thea? coordinate and the n which determines the 

 wave-length this equation contains two arbitary parameters h and £ 

 that determine the form of the curve. 



We take x vertical, increasing upwards, and then for the space oc- 

 cupied by the upper fluid, for which we use the subscript j put 



<,"i + 9Pii=6i(//— h— id) 



by which ?/'+</>> i becomes simultaneously a function of (x+yi). When 

 h = i/, then ^i=0, so the boundary line on the lower side coincides with 

 the stream line. When 7/=+<x> then 



n(x+yi)=v—i0= T [fi+ <pii]+h 



or 



tp x =nbiX } 



(px — nbsj, 



so that at great altitudes the motion is a re ctilinear flow with the ve- 

 locity nb\. 



For the lower space where //</». and x has generally a negative 

 value, I put 



-™-nyi+\og(j,)+h-2 > a=1 la' e ' cos^M) J' 



Hence for >/=h there results 



1 //•,= -nx+ log( " ) + //-2> a'^"' C0S (al) C0S a ^]' 



When we determine the value of x from the equation (1) it is seen that 

 for rf=h there results */- 2 =0, therefore it is seen that the boundary line 

 for the second medium is also a stream-line. 

 According to equation (1) for x=— co we have 



COS tf.COS 7/i=COS£ 



sin 6 . sin 7/i=0 



