376 PROFESSOR CHRYSTAL AND MR JAMES MURRAY 



From (4) we get 



x = <7 + Le ~ x e - xM n cos n t ~ X s ^ n w 0/( w2 + X 2 )) 



where L is an arbitrary constant ; or, if we determine t by the equation 



tan nr = n/ x 

 x = d + Le ~ *' + A cos m sin n(t — t) 



(5), 

 (6). 



If a considerable time has elapsed from the beginning of the observation, so that 

 the water-level in the well has lost its initial disturbance, the term Le~ x< may be 



neglected ; and we have : — 



x- d = A cos nr sin v(f -t) .... (7). 



That is to say, the harmonic oscillation of the lake level as registered by the limno- 

 graph is retarded by the time interval ("lag"), 



T = (l/»)tan -\n/ x ) ■ .(8), 



and " damped" in the ratio 



cos m : 1 . . . . . . (9). 



If we put z = n/x, then t = (1/^) tan ~ 1 z/z. Hence, since tan ' 1 z/z diminishes con- 

 tinuously from 1 to as 2 varies from to + <x> , it follows that the lag diminishes 

 as the nodality of the seiche increases, and increases as x diminishes. 



It is obvious from (5) and (8) that the amount of damping increases (that is, the 

 ratio of damping decreases) as the nodality increases, or as x decreases. 



On the other hand, since 



r/T = (1/2tt) tan -\n/ x ) .... (10), 



the fraction r/T, which we may call the " rektive lag," increases with increase of 

 nodality or with decrease of x> its maximum possible value being \. So that the 

 greatest possible lag for any seiche is a quarter of its period. 



The following numerical tables will help to give a clear picture of the effect of 

 the relative sizes of well and access tube on seiches of different periods, chiefly such as 

 occur or might occur on Loch Earn : — 



(A) 



ForeVs Limnograph at Morges. 

 1= 840 cm., a= 160 cm., b = 6 cm. 

 X='1697, log x= 1*22961, log (2tt/ x ) = T56857. 



T 



r/T 



T 



cos nr 



sec 





sec 





4380 



•00135 



5-88 



•9999 



2100 



•00281 



5-88 



•9998 



600 



•00979 



5-87 



•9981 



300 



•0195 



5-85 



•9925 



60 



•0879 



5-27 



•8511 



