56 Sir William Thomson on Stationary 



where 



0, denotes J A/D . -^ — ^ — =-3— ; 



[(t-|)ff-«i]a _fyi 



/ £ denotes e" D or e » ; 

 oci denotes (i— i)7r— 0,-; or the numeric between 



zero and 7r/2 which satisfies the equation 



[(i-i)7r-a l -]tan« i ~D/6 = 0; 

 D denotes the depth ; 



b denotes TJ 2 /g ; } (7), 



U denotes the velocity of the flow ; 

 a denotes the distance from ridge to ridge ; 

 A denotes the profile-sectional area of one of 



the ridges ; 

 S denotes, for the horizontal coordinate as, the 



height of the water above the mean level 



of places infinitely distant either upstream 



or downstream from the ridges. 



Take first the case of b>T>. In this case, as we have 

 already remarked in Part III., a 1} « 2 , • • • > a i are &H real ; and 

 therefore f ly f 2 , ... , /* are each real and less than unity. 

 Hence in this case the j series and the f series, of which the 

 sums appear in (6), are each convergent, and if we take ^'=00 

 and / = <*>, (6) becomes 



We have now the same expression for S whichever of the 

 ridges be chosen for the origin of as ; and the value for x=a 

 is equal to the value for as = 0. The water-disturbance is 

 therefore equal and similar in all the spaces from ridge to 

 ridge, and the solution (8), from #=0 to x = a, expresses 

 within the period the height of the water above a certain 

 level; not now, as in (2), the mean level throughout the 



period, but a level at a height I S . das/a above the mean level. 



Now, by integration of (8) , we find 



a Jo 



»=oo 



2C 



, B *-Ii^fe5 (9) ' 



To evaluate the series forming the second member of this 



