126 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



The well known proposition that the living force of any complex me- 

 chanical system is equal to the living force of the motions relative to its 

 center of gravity plus the living force of the motion of the center of gravity 

 at which we imagine the ichole mass of the system to be concentrated, can, 

 with only a small change in the method of expression, be applied to 

 our case. For since the total mass of the system multiplied by the 

 velocity b of the center of gravity, gives the amount of the total mo- 

 mentum of the system in the direction of this velocity, therefore we can 

 also put the living force 8 of the center of gravity 



!>> = £ M b = i m b 3 (0), 



where M is the momentum of the whole system in the direction of b 

 and SJt is the mass of the system. If we now compare with each other 

 two different conditions of motion and configuration of the system in 

 which L x and L 2 are the living forces of the motions relative to the 

 center of gravity, #i and & 2 are the potential energies, bi and b 2 are the 

 parallel velocities of the center of gravity, then the difference in the 

 total energy of the system in the two conditions is 



E x - E 2 = $! - $ 2 + L x — L 2 + I m. bi 2 - -i m. b 2 2 . 



If now, without changing the relative motions, I in both cases add 

 the quantity c to the velocity of the center of gravity, then the above 

 difference of energies changes into 



Ex'-E 2 '=Ei-L} 2 +c {M x -M 2 ). 



If {Mi — ilf 2 )=0, then the value of the difference in euergy is not 

 changed by the addition of the velocity c. This must be true even 

 when Hi and H 2 , and therefore the masses of the moving fluids, increase 

 to infinity, since for our undulating fluids the differences (E l —E 2 ) and 

 (Mi—M 2 ) are finite for each wave length. 



Therefore the difference of the energy for stationary waves and for 

 stationary deep water will be equally great only for waves that satisfy 

 the condition (5a). According to the propositions above deduced, sta- 

 tionary waves of this kind must have less euergy than smooth water, 

 which is therefore also true in this case for this kind of waves above 

 quiet water. 



For waves that have larger values of a 2 , the addition of a common 

 velocity (— a 2 ), which brings the deep water into rest, changes the dif- 

 ference of energy between the two states, that of a smooth surface and 

 that of a wave formation, by the quantity 



Ei 1 — E 2 '=Ei— E 2 -\-a 2 {s 2 a 2 r 2 — sia x ri\ 



The index 1 refers to the billowy surface, the index 2 to the plane 

 surface, the accented E' refers to quiet deep water, the non-accented E 

 refers to stationary waves. 



