Vibration of a System in its Gravest Mode. 571 



The first approximation to p 2 is therefore from (5), (21) 



p2=Z 3(2-4**) f > ^ 22) 



P = 1-1690 ( 9 /c)*, (23) 



which is Prof. Lamb's result*. 



For the second approximation we require also the terms in 

 (16) which involve q 2 and q 2 q iy and they are calculated as 

 before. The term in q 2 is 



8cVy 1 / L_ + _A_ 1 \ 8flV/16 7T 



■ • • (24) 

 The term in q 2 q 4 is made up of two parts. Its complete 

 value is 



64c 2 . . , K 



-1^«^* (25) 



Thus 



rp 4c 2 / 7T 2 \. „ 61C 2 . . 8C 2 /16 7T 2 \ . 2 . , 0/M 



which with (5) gives materials for the second approximation. 

 In proceeding to this we may drop the symbols c and g, 

 which can at any moment be restored by consideration of 

 dimensions. Also the factor 8 may be omitted from the 

 expressions for T and V. On this understanding we have by 

 comparison with (1), 



A= 3' B =~5' C= 7' 



U ir 8' 9tt j 1N 9tt 4' 



or on introduction of the value of 7r, 



L='2439204 ; M= --2829420, N=-3463696. 

 The coefficients of the quadratic (3) are thence found to be 

 LN-M 2 = -00443040, AC-B* = -0304762, 

 2MB -LC-NA=- -0284860 ; 

 whence on restoration of the factor (g/c)l>, 



Pl =l-16U( 9 /c)i, p 2 = 2-2b2b( & /c)i, . . (27) 



the first of which constitutes the second approximation 

 to the value of p in cospf, corresponding to the gravest mode 



* ' Hydrodynamics/ § 238. 



