500 



PROFESSOR CHRYSTAL 



The following table gives the values of Q„(w) and Q' v (w) for the first five values 

 of v : — 



V 



Q„(w) 



Q'v(w) 



1 



2 

 3 

 4 



5 



|(?« 3 - w) 

 |(5w*-6w 2 +l) 



f(7w 5 -10!o 3 + 3w) 

 T V(21w 6 -35w 4 + 15m: 2 -1) 



|(3w 2 -l) 

 f(5w 8 -3w) 

 |(35w* - 30?t> 2 + 3) 

 J(63w 6 - 70«> 2 +15w) 



It will be observed that Q' v (w) is the zonal harmonic of the v th order ; so that 

 Q,(d=l) = 0, Q sp {0) = 0, Q 2i ,_i(0) = (-l)n-3 . . . . (2p-S)/2*p\) and Q'„(l) = l, 

 Q'„(-l) = (-l)", QV-i(0) = 0, Q / JSp (0)=l-3 • . . (2p-l)/2'4 .... 2p. 



If now we put 



<£„ = ah v cos re„(rf - t„) , 

 = A„ cos ?j„i + B,/ sin « 



M i 



(6), 



we may write the general equations which represent the motion of the lake in the case 

 where the atmospheric pressure is uniform 





t = +2*Q» ■ 



(7), 



(8); 



and <j>i,<t>2, • • • , <£,o ■ ■ • •, infinite in number, may be regarded as the normal 

 co-ordinates of the motion in Lord Rayleigh's sense of the phrase. 



If X be the kinetic and 93 the potential energy in the case just supposed, we have 



r+a 

 X = U dxh{\ -x 2 /a?)p, 



a I aw 

 2h H^2 



{^„Q»} 2 , 



that is, 

 where 



<lw 



^„ 2 QA») ; 



X = |-aa,^ 2 



air, 



lwQ v 2 (w) 

 1 - w 2 



• (9), 



Since the co-ordinates (p x , (f> 2 , . . . . are normal, the products <jf> r <£ s do not appear 

 in the expression for £. 



