512 PROFESSOR CHRYSTAL 



added to p, as the integral of a constant pressure all over the surface of the lake must 

 obviously be zero. Hence we need only consider the disturbing pressure, which may 

 be expressed in centimetres of water as heretofore by dp =f(w, t). 



If, therefore, 3K,, denote the total increment of the energy of the y-nodal seiche by 

 the disturbing pressure dp =f(w, t) acting from t = to t = T, we have 



3K„ = ag \dwQ',,(iv) I dt k t ,n„ sin ??,,(/- t,.)/(w, t) . . . (43). 



The energy of the i-nodal seiche is equal to its potential energy in the configura- 

 tion of maximum potential and zero kinetic energy. Hence we have by (11) 



and therefore 



K '=2^V' ! ' ■ <">- ; 



m,=^!±w- (45). 



Strictly regarded, k v is a function of the time ; for the energy of the seiche is being 

 continually altered by the disturbing surface pressure, so that the extreme amplitude 

 k v of the seiche at each moment, which would be left if the disturbing pressure were 

 suddenly to cease, varies with the time. Inasmuch, however, as the variation of h v is 

 small, and f(w, t) is also small, if we neglect quantities of the order dkjdp, we may 

 regard k„ as constant in the integral on the right-hand side of the equation (43). 

 We thus get from (43) and (45) 



P 1 f 



dk v = J(2» + 1 ) dwq' v (w) dt n v sin n v (t - t v )/(w, t) (46), 



a formula which summarises our whole theory so far as disturbance of the extreme 

 amplitudes of the various seiches is concerned. It follows that 



o/i'„ = A' v cos n v r v + B'„ sin n v r v ; 

 where 



A', = h(2v + 1 ) dwQ'„(w) dt n„ sin n v t .f(io, t) , 

 B'„= - i(2i/+ 1) dw()'Aw) dt /,,. cos n v t ./(to, t) 



It will be found that the formulae (47) lead to the same results, so far as amplitude 

 is concerned, as we have already found in the special cases discussed above. We add 

 some important examples of its application. 



14. Example 1. — Let us consider the effect of a uniform time-change in a pressure 

 gradient which has a uniform space variation along the lake. This will be represented 

 by taking 



f(W, t) = \atw , 



