PAPER BY PROF, HELMHOLTZ. 105 



Hence for this Itouiulaiy Hue result the equations 



e«^ cos {nii)=<i(co'^ ih cos ^— cos e) ') 



e'« sin (/<//) = — %in (//O sin ^ ..... (la) 



By the elimhiation of H this gives an equation between x and y as the 

 equation of the boundary liue. Beside the coustaut a which deter- 

 mines the initial point of the £P coordinate and thew which determines the 

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

 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 



by which f^cp^i becomes simultaneously a function of {x-\-i/i). When 

 /i=;/, then '/'i=0, so the boundary line on the lower side coincides with 

 the stream liue. When ?;=+(x then 



n{x-\-iji) = }}—id=^ [il:i+ cp{i] + n 



or 



ip^ = nbiX, 

 (pi=nbiy, 



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

 locity nbi. 



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

 value, 1 put 



Hence for ?/=A there results 



^ ,l-^=-nx-\-\ogf ij W/t-2 > T-"''.e-'"'. cos (ea) cos a6~\. 



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

 for i/=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 j?=— co we have 



cos ^.cos 7;i=cos£ 



sin f^ . sin ?/i=0 



