233] GERSTNER S INVESTIGATION. 417 



which is a formula for the horizontal velocity. Combined with (ii), this gives 



) ........ .... (vii), 



provided /3 = VQ/O- - gja* ................................. (viii). 



Hence, for the vertical velocity, we have 



(ix). 



If the coordinates x, y of any particle on the stream-line be regarded as 

 functions of t, we have, then, 



)} ..................... 00, 



whence x = -t+fB8in&amp;lt;rt, y = /3(l cos&amp;lt;r) ..................... (xi), 



(T 



if the time be reckoned from the instant at which the particle passes through the 

 origin of coordinates. The equations (xi) determine a trochoid ; the radius of 

 the rolling circle is g/tr 2 , and the distance of the tracing point from the centre 

 is $. The wave-length of the curve is X = 



It remains to shew that the trochoidal paths can be so adjusted that the 

 condition of constancy of volume is satisfied. For this purpose we must 

 take an origin of y which shall be independent of the particular path considered, 

 so that the paths are now given by 



x = 0- t + /3 sin at, y = b-$ cos at ..................... (xii), 



&amp;lt;7 



where b is a function of /3, and conversely. It is evident that a- must be an 

 absolute constant, since it determines the wave-length. Now consider two 

 particles P, P , on two consecutive stream-lines, which are in the same phase 

 of their motions. The projections of PP on the coordinate axes are 



8/3sincr and db 8ft cos at. 



The flux (Art. 59) across a line fixed in space which coincides with the 

 instantaneous position of PP is obtained by multiplying these projections by 



dyjdt and dxldt, 

 respectively, and adding ; viz. we find 



coso-Z ............... (xiii). 



27 



&amp;lt;r 

 In order that this may be independent of t, we must have 



