50 EKMAN. ON DEAD-WATER. [norw. pol. EXP. 



along the axis of coordinates x, y, z, and X, Y, Z, the components along the 

 same axis, of the extraneous forces. In addition, u, v, w, must satisfy the 

 "equation of continuity" 



9u to 2w _ 



dx ~ ty ~ iz v 



as well as the initial- and boundary-conditions, which may be of different forms. 

 If the linear dimensions be multiplied by X, the velocities by v, and so on, 

 the four terms in (8) are multiplied by 



(5w 2 7C , j v 

 "T ' X ' 7 ' * ¥ 



respectively. To satisfy (8) for both cases, it is therefore necessary that these 

 four quantities be equal, and the quantities X, v, S, it, y, and x must then 

 satisfy the equations 



n = v 2 d , (a) 



yX=v\ (6) 



x = dvl , (c) 



Equation (9), does not involve any restriction in the choice of I, v, 6, it, y, x, 

 but the initial- and boundary-conditions must accord with the choice of A, v, 

 and n. The boundary-conditions, are always fullfilled, if the boundaries con- 

 sist of: (1) perfectly welted, rigid bodies; either immovable and satisfying the 

 ratio X, or moving so as to satisfy the ratios X and v; (2) free surfaces, in- 

 fluenced by extraneous pressures satisfying the ratio it. The initial conditions 

 in the fluid are in accordance with any choice of X, v, it, if the motion be 

 steady, or if the water be initially at rest, in both cases. 



If all the above conditions be satisfied, the motion will be geometrically 

 similar in the two cases, and the ratio of the resultants of pressure upon cor- 

 responding surfaces is then 



p= li n = Pv'-d (d) 



In the case specially interesting us, the boundary-conditions are then 

 practically satisfied if: (1) the linear dimensions of the two vessels are in the 

 constant ratio X; (2) the linear dimensions of the basins, in which they move, 

 are in the same ratio X; or the basins are so wide and deep that their 

 boundaries have no influence upon the motion of the vessel; (3) the veloci- 

 ties of the two vessels are in the ratio v to oneanother. It is not necessary, 



