High Tensions in Moving Liquids. 115 



If T is proportional to the excess of the velocity u above a 

 lower limit U, so that T=K(w— Uj where K is a constant, 



v't/v 2 = l/{ 1 + hr/KkcA {u - U) }, 



and this expression corresponds to the general laws of " ducks 

 and drakes/' for since it holds only for values of u greater than 

 U there is a certain minimum horizontal velocity required 

 by a solid which is to rebound from a liquid ; for horizontal 

 velocities greater than this there is rebound, but the energy 

 of the vertical motion after impact is always less than that 

 before impact, although it becomes more and more nearly 

 equal to it the greater u becomes ; also it appears from the last 

 equation that with finite velocities the angle of incidence for 

 which rebound is possible has a limiting value, because 

 tan i — u/v ; and u having the lower limit U, and v the upper 

 limit V of experimental possibility, tan i and therefore 

 i has a lower limit. Again, as h increases with ra, and A 

 generally increases with the size of the face of the solid, it 

 follows from the equation that the smaller the mass of the 

 body and the larger the face the more nearly does v' 2 equal 

 v 2 , whence the advantage of thin flakes of stone and shell for 

 getting " ducks and drakes." It is to be remembered that in 

 the case of a flat face the area over which there is finite 

 curvature is only the small strip of transition from the flat 

 surface to the rest of the surface, and thus A becomes very 

 small, but so also does r become small, and the effect of the 

 shape of the face is best expressed by making A the product 

 of an average width a and an average length b in the direction 

 of motion, so that A/r becomes ab/r, which depends on a and 

 the angle b/r. Jt should be noticed that the velocities are not 

 great which are required to produce the phenomena of ducks 

 and drakes, for at the last rebound of a series of ten or twelve 

 the motions are very gentle, but the tension called forth must 

 be remarkably high seeing that the impact lasts so short a time. 

 We all know the wonderful manner in which Kelvin has 

 helped us to grasp the coexistence in the sether of apparently 

 irreconcilable properties by his homely instances of jelly and 

 pitch,, and it seems to me that u ducks and drakes " carry in 

 themselves a suggestiveness only communicated to jelly and 

 pitch by the sagacity and imagination of a master mind. 



William Sutherland. 

 Melbourne, March 1, 1896. 



