1062 Proceedings of Poyal Society of Edinburgh. [sess. 
ultimately into more and more nearly steady motion through 
greater and greater distances from O. The investigation of 
§§ 1-10 above (Feb. 1904), and particularly the results described 
in §§ 5, 6, and illustrated in figs. 2, 3, show that in our present 
case the commotion, however violent, even if including splashes ,* 
divides itself into two parts which travel away in the two direc- 
tions from 0, ultimately at wave-speed increasing in proportion 
to square root of distance (according to the law of falling bodies) ; 
and leaving in their rears, through ever broadening spaces, what 
would be more and more nearly absolute quiescence if the forcive 
were suddenly to cease after having acted for any time, long or 
short. 
§ 68. But if the forcive continues acting, and travelling right- 
wards with constant speed, ?>, according to § 67, the travelling 
away of the two parts of the initial commotion in the two 
directions from 0 (itself merely a point of reference, moving 
uniformly rightwards), leaves the water, as shown by fig. 26, in a 
state of more and more nearly quite steady motion through an 
ever broadening space on the rear side of 0, and through a small 
space in advance of O ; provided certain moderating conditions 
are fulfilled in respect to k, b, v. 
§ 69. To illustrate and prove § 68 ; first suppose v infinitely small. 
The water will be infinitely little disturbed from the static 
forcive-curve shown in fig. 25, and described in § 66. Small 
enough velocities will make very small disturbance with any finite 
value of k/b. 
§ 70. But now go to the other extreme and let v be very great. 
It is clear, on dynamical principles without calculation, that v 
may be great enough to make but very little disturbance of the 
water-surface, however steep be the static forcive curve. A 
“skipping stone” and a ricochetting cannon shot, illustrate the 
application of the same dynamical principle in three-dimensional 
hydrokinetics. By mathematical calculation (§ 79 below) we 
shall see that, when v is great enough, we have 
(97), 
A 
* However sudden and great the commotion is, the motion of the liquid is, 
and continues to be, irrotational throughout. 
