233] GERSTNER'S INVESTIGATION. 417 



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



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



provided p = v /<r-g/(r 2 ................................. (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, 



jj "o y> dt~"^^* J ~ ifn ^ 



whence x -f/3sin o-tf, y^(\. coso-) (xi), 



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 <7/o- 2 , and the distance of the tracing point from the centre 

 is . The wave-length of the curve is X = 27r<7/<r 2 . 



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



(T 



where 6 is a function of , and conversely. It is evident that o- 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 



8/3 sin at and 8b dpcoso-t. 



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 



dyfdt and -dxjdt, 



respectively, and adding ; viz. we find 



/ \ 



(xiii). 



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



/>, 



27 



