of certain Stream- Lines. 285 



At the point where the curves b = l cross the axis of x, both 

 the component velocities are null. The unit of velocity in each 

 of those expressions is the velocity of a particle at an infinite 

 distance in the positive direction from the axis of x } for which 

 particle we have w = l, v = 0. 



13. Suppose the plane of x and y to be vertical, and y to be 

 positive downwards; let the absolute value of the unit of mea- 

 sure (that is, the radius of the circle whose circumference is a 

 wave-length) be denoted by II; and let the heaviness (or weight 

 of a unit of volume) of the liquid be W. Then the stream- 

 lines for which b is not less than 1 may represent the profiles of 

 a series of forced waves, capable of travelling with the absolute 

 velocity 



c=Vg&, (IV.) 



being the same with that of free waves of the same length ; and 

 the absolute values of the velocities of any particle relatively to 

 still water will be 



horizontal component, c(u — \) = ce~y cosx; "" 



vertical component, cv= — ce'^sinx; L i ry\ 



resultant velocity, cV Uu — \) 2 -\-v 2 l =ce~v. 



14. Those forced waves differ from free waves in the following- 

 respects. First, in free or trochoidal waves, each wave- surface 

 is a surface of constant pressure, so that the upper surface of the 

 liquid needs no pressure to be applied to it to compel the waves 

 to travel ; whereas in the waves now in question the pressure at 

 each wave-surface is not constant, being expressed by the fol- 

 lowing formula, 



p= constant +Wb--^ — , . . . (VI.) 



of which the last term is variable ; and the upper surface requires 

 a pressure varying according to this law to be applied to it, in 

 order to compel the waves to travel. 



Secondly, free or trochoidal waves begin to break as they 

 reach the cycloidal form, in which the surface near the crest is 

 vertical, and the crest forms a cusp; whereas in the waves now 

 in question the steepest possible form, which cannot be passed 

 without breaking, is that of the stream-line 6=1, whose crest is 

 formed by two surfaces meeting each other at right angles, and 

 sloping in opposite directions at 45°. Thirdly, the particles of 

 water in free waves revolve in circles, and do not permanently 

 advance ; whereas the orbit of each particle in the waves now in 

 question is an endless coiled or looped curve, in which each revo- 

 lution is accompanied by an advance. The figure of that orbit 



