122 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



motion, which remains invariable except for the influence of direct ac- 

 celerating forces, while the quantities of flow p, and ^thereby receive 

 the significance of velocities. Thus the two problems in variations, 

 here solved, are completely analogous to the propositions developed by 

 me in the theory of polycyclic systems, that 



S{<P-L)=-2[P a dp ] {Be) 



tfg n =0 



when the velocities q a of the cyclic motions are maintained constant. 

 In this equation p a , are the variable coordinates, and P a are the forces 

 tending to increase these coordinates. Stable equilibrium, as is easy 

 to see, corresponds to a minimum of the ($—L). 



On the other hand, when we assume the moment of motion — - to 



be constant we have 



d($+L)=-2[P a dv a ] 



ofi^V (*/•) 



Here, also, stable equilibrium demands that the quantity (& + L), that 

 is to say, the total energy of the body be a minimum. 



The equation (2g) corresponds throughout to the above equation (3e) 

 for polycyclic systems, only that in the former the number of variable 

 coordinates SN of the surface elements ds is infinitely large and the 

 force which in it corresponds to P a , namely, the fluid pressure, is a con- 

 tinuous function of y; hence the integral is used instead of the sign of 

 summation. 



That stable equilibrium, even in the theory of waves, also corresponds 

 to the minimum of energy for a constant value of f is established when 

 we think of the influence of friction which can restore a disturbed stable 

 equilibrium but not a disturbed unstable equilibrium. Friction always 

 diminishes the store of energy that may be present. It can, therefore, 

 restore a disturbed minimum of energy but not a departure from a maxi- 

 mum. 



III. THE THEOREM OP MINIMUM ENERGY APPLIED TO LAYERS OF 



INFINITE THICKNESS. 



In the fallowing we shall consider the two layers of fluid on whose 

 boundary surface the waves form, as very deep in the vertical direc- 

 tion, therefore the values H x and Hi as very large and as respectively 

 increasing beyond all limits to infinity, in order to free the theory of 

 waves from those complications which are brought about by the influ- 

 ence of the upper and lower horizontal bouudary surfaces. 



Under these circumstances the motion on these two far distant hor- 

 izontal boundary surfaces does not differ sensibly from rectilinear uni- 



