114 THE MECHANICS OF THE EARTHS ATMOSPHERE. 



On the other hand, for the bouudary Hue, for which 



[ (It-) 



we have Vi = Pi ) 



aud for the other boundary line, whose e<iuation is 



X = — 17.) 



[ (1^7) 



we have //•■> = p2 ) 



The quantities pi and p2, as is well knowu, give respectivelj' the vol- 

 umes of the fluid that flow in the unit of time through every section 

 between the wave surfaces for whicb <■/•] = i/^z = 0, and through the 

 upper or lower bouudary surface. 



These are the quautities which I have above designated as quantities 

 of Jioic. In taking the variations of these quaurities, I shall, in this 

 l)aragraph, consider pi and p2 as invariable. 



That altitude will be adopted as the initial point for x, at which the 

 boundary surface of the two quantities of fluid under consideration 

 would be at rest, which is expressed by the equation 



X (hj = {le) 



/.. 



that is to say, x = is a plane such that as much water is raised above 

 it as sinks below it. 



Finally the si)ace within which lie the quantities that are subject to 

 variation is also bounded by two vertical planes that are separated 

 from each other by one wave length. Since the movements are to be 

 periodical and consistent with the wave length A, the velocities at the 

 right vertical surface and at the left vertical surface must be equal or 



^x ^x 

 therefore for the same values of x 



tr=^'l (1/) 



and 



~^-~w (^^^ 



According to Eq. (1) this last equation can also be written 



;)(p,_d(pi 

 dx ^x 

 or 



<7>,. — ^,= constant (1/i) 



Now it is known that equations (1) are resolvable when {i/:-\-(pi) can be 

 represented as a function of (x+j//), which function must show no dis- 

 continuity and no infinite values within the region filled by the fluid in 

 question. 



