292 ME. N. BOHE ON THE DETERMINATION OF THE 



To a first approximation we get as the general form of the vibrations 



r 



cos 



cos 



where q n 3 = - 3 (n 3 n). 

 oct 



As to the higher approximations, it is not possible to find the form for the general 

 vibrations in a corresponding manner, the single types of vibration only being 

 independent of each other to a first approximation. 



We shall now determine the next approximations of the pure periodical type of 

 vibration, of which the first approximation is defined by 



" -1 



= Ar" cos n3 sin qt, = - a" -1 A cos n$ cos qt, q" = ^(n 3 n). (14) 



q pa 



Second Approximation. 



From (6) and (7) we get 



and 



+ r 



a a 2 a' S3' a 3 '2a 3 a 



Introducing the values of (j>, {, q, defined by (14), in the terms of second order, 

 we get 



^ + R>1 = _ n2 ( 2n ~ 1 ) a 2 -W cos 2nS sin 2qt+ ^ a 2 - 3 A 3 sin 2qt, (17) 

 3t \_Sr_]r= a 4q 4q 



and 



4n 2) 

 - 



8K-1) - 

 Eliminating from (17) and (18), we get 



j / 1Q \ 



. . . (18) 



= _^ 3gn(n-l)(2n+l) a3n - 2 



4 ^71 T~ 1^ 



sin 2g. ( 1 9) 



