114 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



On the other hand, for the boundary lins, for which 



x = E x ) 



we have Vi — fa ) 



and for the other boundary line, whose equation is 



x = —H,) 



I (!<*) 



■we have </■■> = fa ) 



The quantities fa and fa, as is well known, give respectively the vol- 

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

 between the wave surfaces for which fa = fa = 0, and through the 

 upper or lower boundary surface. 



These are the quantities which I have above designated as quantities 

 of flow. In taking the variations of these quantities, I shall, in this 

 paragraph, consider fa and fa as invariable. 



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

 boundary surface of the two quantities of fluid under consideration 

 would be at rest, which is expressed by the equation 



(2/o+A 

 x dy = (le) 

 y° 

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

 it as sinks below it. 



Finally the space 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 



jx t u? 

 therefore for the same values of x 



'l'r= H 'L (1/) 



and 



dy" dy W 



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



dX }X 



or 



cp r — <^=constant (ity 



^ow it is known that equations (1) are resolvable when {$+<pi) can be 



represented as a function of (x+yi) % which function must show no dis- 

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



question. 



