72 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



case of waves on the free surface of water, then, according to the 

 remarks already made the ratio ^ must remain unchanged, therefore 

 q must be put =n^. Then will 



Y=n^y T=nt. 

 Z=nH 



Therefore when the wave lengths increase in the ratio n^ the duration 

 of the oscillations will increase only in the ratio ??, which corresponds 

 to the well known law of the velocity of propagation for the surface 

 waves of water, which velocity increases as the square root of the wave 

 length. Thus this result is attained very simply and for all wave forms, 

 without the necessity of knowing a single integral of wave motion. 



The same principle is applicable to the relative resistances that ships 

 having n^ times the dimensions and n times the velocity, experience by 

 reason of the waves that they excite on the surface of the water. The 

 total resistance in this case increases as g^r, and since for the same 

 fluid r-=l therefore tlie resistance increases as n^ and the work needed 

 to overcome it as w", therefore in a rather larger ratio than the volume 

 of the ship, wliile the supply of fuel and the size of the boiler that must 

 do the work can increase only in the same ratio as the volume of the 

 ship, namely as it. Therefore so long as lighter machinery can not be 

 applied (including the supply of coal) the velocity of such an enlarged 

 ship can increase above a certain limit only by a ratio that is smaller 

 than that of the square root of the increase of the linear dimensions. 



A similar computation holds good for the model of the bird in the 

 air. When we increase the linear dimensions of a bird and would take 

 into consideration the viscosity, we must put (/ and r equal to unity be- 

 cause the medium, namely the air, remains unchanged. Let w be a 

 vulgar fraction, then will the velocity be reduced in the same propor- 

 tion as the volume of the bird increases and the i)ressure (of the air) 

 against the total surface of the larger bird will only attain the same 

 value as for the smaller bird, therefore will not be able to bear up the 

 weight of the larger bird. 



If we allow ourselves to neglect the friction, which according to the 



above remarks we can do so much the more readily the more we increase 



the dimensions, or for the same dimensions increase the velocities, then 



q is arbitrary and the change of dimensions and velocities must be so 



made that the total pressure against the surfaces shall increase as the 



(fi 

 weight of the body or we must have 5^= -' or q=n^. In order to ex- 



n 



ecute the corresponding motions, the work that will be necessary will 

 be 



