NO. 15.] APPLICATION OF SMALL SCALE EXPERIENTS. 51 



that the condition concerning the pressure on the free surfaces be satisfied. 

 For, as the water is practically incompressible, the motion is, in both cases, 

 unaltered, if the pressure in the water be diminished by a constant quantity 

 (for instance, the atmospheric pressure), so that the pressure be nil in the free 

 surface. It is then obviously necessary to leave the air-pressure consistently 

 out of account. Equations (b) and (c) give 



23 — x 

 A ~ yd* • 



As the possible variations of the density, of the gravity, and of the coef- 

 ficient of friction, are very limited, the right hand side of this equation cannot 

 be made many times greater or smaller than unity; and it is consequently 

 impossible to reproduce exactly similar motions on very different scales, if 

 gravity and viscosity have both to be taken into account. The gravity is of 

 essential importance, in cases where there is wave-motion. We must therefore 

 at first neglect the viscosity, and afterwards consider its influence as best 

 we can; equation (c) is consequently left out of consideration for the present. 



Gravity is furthermore practically constant, and so y = 1. If the densi- 

 ties of the water-layers are also the same in the two cases, d = 1 ; and (a), 

 (6), and (d) take the form 



n = V i ; A = u 2 ; P= in* = A 3 . 

 That is: If the linear dimensions be increased in the proportion X, and 

 the velocity of the vessel be increased in the proportion |T, the motion of 

 the water, on the larger scale and on the smaller scale, will be geometrically 

 similar to oneanother, and the resultant of pressure against the vessel 

 (the wave making resistance) will increase in the proportion I 2 . This is 

 the well-known rule which was used by Froude for his experiments with 

 ship-models, and which still bears his name. 



By an artifice, the equations (a), (b), (d) may be made to allow an impor- 

 tant variation of the experiment, other than that which is indicated by Froude's 

 rule. At the slow speeds which come under consideration, the waves in the 

 surface are quite insignificant, and the surface-disturbances caused by boundary- 

 waves, are also very small, because the difference of density between the 

 different water-layers, is small. The motion in the water will therefore not 

 be appreciably altered, if, by some means, the water-surface be held rigidly 



