MECHANICS OF THE INNER EAR 73 



and the two constants a and iv. The former of these con- 

 stants is the area described by the initial point of the partition 

 in moving from one limit to the other, of whatever form this 

 area may actually be found to be. The latter is the width of 

 the partition at the initial point. 



The mathematical reader immediately sees that that quan- 

 tity F of displaced fluid for which room is made by a move- 

 The quantity of ment of any given section of the partition 

 fluid for which is determined by the following equation, 

 room is made which can be easily integrated. 



-/: 



1^:= I adx 



J X, 



In order to integrate this equation we have to express « 

 as a function of x. This has been done above under the tem- 

 porary assumption of a uniform increase of width. The re- 

 sult is stated in the equation just preceding the last. We 

 then have 



-/:; 



F= (iow+xydx= 



J X, I00w°^ 



= l(iow-f;t;,V — (lozv-'rx.y], 



where x, is the farther, ^ the nearer of the two points enclosing 

 whatever section of the partition is in question. 



If the section in question is an initial section of the par- 

 tition, then x^ is equal to zero, and the quantity of displaced 

 fluid is 



F= -^— [(lOK'-t-.r.V— (lozcyl . 



Let us regard the partition as consisting of sections each of 

 the length of w. We can find, then, the quantities of dis- 

 placed fluid for which room is made by the first section, the 

 first two, the first three, the first four, and so forth, sections 



