114 Mr. W. Sutherland on 



equal to that of the solid, for the surface of the liquid near 

 the solid has been set in motion by it. Let two sides of the 

 square flat face be parallel to the direction of u, then we may 

 liken the surface of the depression in the liquid to a stretched 

 membrane, and the front and back edges to portions of 

 cylindrical surfaces of radius r, over which the stretched 

 membrane passes with the tension perpendicular to the axes 

 of the cylinders, and therefore exerting pressure on the 

 cylinders ; at the side edges the tension is parallel to the axis 

 and is therefore devoid of pressure effect. 



Let b be the width of the strip of curved edge in contact 

 with stretched liquid, and let the tension near the surface of 

 the liquid be T per unit width, then the lifting pressure on 

 each area ab is abT/?*, making an angle bj'lr with the vertical ; 

 the total lifting force due to tension is therefore 



2 cos {b/2r)abT/r. 

 This result can obviously be extended to the case where the 

 face of the solid is not flat with rounded edges but is any 

 curved surface ; let A be the area of contact between solid 

 and liquid, T the mean tension, and 1/r the mean curvature 

 of the face in the direction of the tension, then each unit of A 

 is subject to a normal pressure T/r, and the total vertical 

 lifting pressure is the sum of the vertical components of all 

 the normal pressures. Thus, then, we can include all cases 

 in the one general expression sufficient for our purposes if we 

 say that the lifting force is equal to cAT/r where c is a con- 

 stant, A is that part of the face of the solid in contact with 

 the liquid and having finite curvature, the average value of 

 which is 1/r in the direction of T; c, A, T, and r being also 

 average values for the duration of the impact. 



Now suppose for a moment that the solid has only the 

 vertical velocity v at the moment of impact, and let h be the 

 distance below the free surface of the liquid to which its face 

 penetrates before it is brought to rest, then we may assume 

 the energy given to the liquid to be proportional to A 2 , and 

 then h 2 = kmv 2 /2 where m is the mass of the solid and h is a 

 constant. Then when u is not zero we have to take account 

 of the fact that the force cAT/r opposes the descent of the 

 solid, doing work hcAT/r against it, and therefore 



h2=k(mvy2-hcAT/r). 



Then the uplifting force cAT/r will in most cases act on the solid 



through distance h, and discharge it from the surface of the 



liquid with a vertical velocity v' upwards given b} r the equation 



mv f2 /2 = JicAT/r, 



l il - 1 



■"' v 2 ~~ 1+hr/kcAT 



