26 Dr. Rankine on Stream-lines, 



the actual motion of the particles of water to be combined with 

 a uniform translation equal and opposite to the velocity of pro- 

 pagation of the waves. 



Thus, let -c be the velocity of the waves propagated in a 

 negative direction, as indicated by the arrow W in PL I. fig. 4; 

 and, as in article 13 of the original paper, let c(u — 1) and cv be 

 the horizontal and vertical components of the actual velocity of 

 a particle of water in the waves, so that cu and cv shall be the 

 components of the velocity of the same particle in the undulating 

 stream into which the waves are converted, and which flows in a 

 positive direction, as indicated by the arrows S, S'. 



The tangent of the slope of a stream-line or wave-line at any 



point is obviously -• 



The crest of a wave-line is a point where v — Q and u is a 

 minimum. The latter property distinguishes it from the trough, 

 where v=0 also, but u is a maximum. 



The equations of a crest are as follows : — 



A wave begins to break so soon as its crest ceases to be rounded 

 and becomes angular, as at C in fig. 4 ; and at such a point it 

 is evident that u must vanish as well as v. The ratio of two 

 quantities which vanish simultaneously is equal to the ratio of 

 their differentials. Hence the equations of a sharp crest, in ad- 

 dition to the equations (I.) and (II.), are the following: — 



dv 



dt v 

 u — } T m = — 

 du u 



Jt 



By putting for -=- the equivalent operation cu - — \~cv -=-, the 

 at asc ay 



preceding equations are transformed into the following. 

 At every crest 



v=0, (I.) 



du du TT . 



U ^ V Ty = °' ' W 



and at every sharp crest 



i« = 0, (III.) 



dv dv 



dx dy _ _ tfj Iy 



du du u 



doc dy 



