ON THE SEICHES OF LOCH EARN. 501 



Also, the zero configuration being the lake at rest, we have 



/—a 



since the co-ordinates are normal, 



= +*2W (11), 



where 



6 F = -|.ldwQ' F (w)« (12). 



By a well-known property ol the zonal harmonic, divQ' v (w) 2 = 2/(2^ + 1). 



Hence 



*>,= 2 - y (18). 



(2v+l)o V ; 



Since the hypothesis of long waves involves the neglect of the squares and products 

 of <£i, 02, • • • ■ ; ^i, $2, • • • • m the equations of motion, the Lagrangian equations 

 for the motion of the lake reduce to 



l(|>Ir° <^ 2 > • • • ■<"» 



that is to say 



a„^ + 6^ = (v=l, 2, . . . .) . . . . (15). 



Since (15) must be satisfied by cf> v = ak„ cos n„(t — t„), we must have 



a v = b v /n v 2 , 



2a 



7iv(v+l)(2v+l) 



(16). 



2. .Effect of a Uniform Excess of Pressure dp over a Part of the Lake. — Let us 

 now suppose that an excess of pressure of dp (measured in cm. of water) extends from 

 the point corresponding to w = \ to the point corresponding to w = tx, and that this 

 excess begins at t = and ends at t = T. 



In the analysis everything will be as before, except that there will be an addition 

 to the potential energy of 



r r 



c/dpa dwt, = gdp dw'2(f> l ,Q' v (w) . 



+i 



* It follows, of course, that I w ^" = _ , which may he readily verified independently. 



1 - w 2 v(v + \)$v+\) 



