Dr. Rankine on Stream-lines. 



27 



Equation (IV.) may be reduced to the following form :- 



dv v /dv du\ 



dx u\dy dx) 



The well-known equation of continuity is 



du 

 dy 



:0. (IV. a) 



du dv _ 

 dx dy ' 



(V.) 



which reduces (IV. a) to the following 



du 



du 



dy 



s=o, 



r dv 



du ~\ 



J dx dx 1 1 



du du 2 



-dy dy 2 



(IV. B) 



(VI.) 



dv 



dx dx u 



whose two roots are as follows : 



du 

 v _ dx f 



u~~ du~ y/ 



dy 



and those two roots are the tangents of the slopes that meet at the 

 crest. 



It is known that in a perfect fluid the quantity known as the mo- 

 lecular rotation, or -I -= -j- ), is either nothing or a constant 



for each stream-line. The present proposition refers to the case 

 in which . 



du __dv _ ~ 

 dy dx ' 



(VII.) 



and consequently the tangents of the two slopes are the two 

 values of the expression 



du r du 2 ^ 



dy L dif^ 



but the product of those two values is = — 1 ; therefore at every 

 sharp or breaking crest of a wave in which there is no molecular 

 rotation, the two slopes meet each other at right angles. (J. E. D. 



28. The preceding demonstration applies to all waves what- 

 soever in which molecular rotation is null. In waves which, 

 besides having that property, are symmetrical at either side of 

 the crest, each of the slopes which meet at the crest of a wave 

 on the point of breaking is inclined at 45° to the horizon. 



29. The figure (which is reduced from a diagram exhibited at 

 the late Meeting of the British Association) represents one-half 

 of a forced wave, such as has been described in articles 13 and 



