502 PROFESSOR CHRYSTAL 



Hence we shall now have 



<8 = £26,^ + 2e,<k (17), 



where b v has the same value as before ; and 



e, = 09p{Q.O*)-Q,(*)} (18). 



The equations for the motion of the lake from t = to t — T are now 



a y $„ + b v <f>„ + e„ = (v=l,2, ....) .... (19); 



and from t = T onwards 



a„<£„ + M>, = (v = l,2, . . . . ) . . . . (20), 



as before, with the condition that the values of <p„ and <p v must be continuous when 

 t = T. 



Since our equations of motion are all linear, and (15) are linear and homogeneous, 

 any admissible solution of (15) may be added to any solution of (19). It will therefore 

 be convenient first to find the integral equations of motion corresponding to our supposed 

 disturbance operating upon a lake initially wholly at rest. If we superpose upon this 

 motion that represented by the equations (6), (7), (8), we shall obtain the integral 

 equations of motion (after t = T) corresponding to our disturbance when it operates on a 

 lake in which the initial motion is given by 



Z = %k&Jw) cos n,r,, i 



1= - 2&„rc„Q' '„(w) sin n„T p ) 



The general solutions of (19) and (20) are 



<£„ = A'„ cos n v t + B'„ sin n y t - B„/b p (v=l, 2, . . . . ), 

 and 



<f>„ = A'', cos nj + B"„ sin nJL (y = 1, 2, . . . . ), 



where k! v , B^, A"„, B"„ are constants to be determined by the conditions that £ and t, 



shall vanish when £ = 0, and be continuous when t = T. 



We thus get 



t = 2/(1- cos n v t)Q' v (w) (22), 



W 



here 



• (24), 



^-=/=-i(2v+l)8p{Q,( M )-Q 1 ,(A)} . . . (23), 



when 0</<T ; and 



£ = %f v { - (1 - cos w„T) cos n„t + sin ra„T sin n v t}Q' v (w), 

 = 22/ sin (^ sin n v {t - JT)Q' F (w) 



when ^>T. 



From the second form of the equation (24) it follows (as is otherwise obvious) that, 

 ceteris paribus, the disturbing effect on the y-nodal seiche is greatest when T = ir/n v , i.e. 

 when T is half the period of the y-nodal seiche. 



3. General Case. — If now we suppose that initially the extreme amplitudes of the 



