( 745 ) 
Weme Bn: B, are, as the form of F tells us, linear functions 
ORNE ABE, . . ay are. Constant and’-as a, can be 
expressed in a, we can write 8,,8,,.... 8, as linear functions of a 
So the coordinates g,,q7,.-+-g« feel the influence of the relation, 
however only in the phase, not in the amplitude. As «, (just as 
a,, B, and p,) was determined before already as function of f, we 
can also determine §,, ?,,.... 8 So the problem has been reduced 
to quadratures. 
§ 5. S=2(n,=n,-+ 0). All disturbing terms which we have 
to regard are again of order 4%. For this case the peculiarity appears 
that all disturbing terms of the second kind must be regarded at 
the same time as disturbing of the first kind. So a term q,g,” in 
the first equation gives as disturbing terms a term with cos (nt + 279.) 
and one with cos (nt + 4n8,—2n8,). 
Just as was done above for the case S= 4 we can prove easily 
also for this case that, apart from a modification of their frequency 
of vibration, the coordinates g, and g, behave as if we had to do 
with a mechanism with two degrees of freedom, whilst the remaining 
coordinates feel the influence of relation in their phase, but not in 
their amplitude. 
RELATION BETWEEN THREE OF THE FREQUENCIES OF VIBRATION 
FOR WHICH S= 3. 
$ 6. The only relation which for S= 8 remains to be discussed 
runs: 
n, + Nn, — NN, =O. 
Just as was done in $ 3 for the case of a relation between two 
of the frequencies of vibration for which S==8 we can also show 
here, that at first approximation only the coordinates g,,q,, and g, 
feel the influence of the relation and that q,,g,, and q, behave as if 
they were the only coordinates. So we can restrict ourselves to a 
mechanism with but three degrees of freedom; g,,q,, and g, are 
the principal coordinates. 
As in the equations of motion terms of order /? appear already 
among the disturbing terms we need not take into account any terms 
of a higher order than /’ in the expressions for the kinetic energy 
and the potential function. Hence 
ie TdS : ah: 
— gr’ + 4 Pape Wr i + 2 PreQr Js): 
nl Tl gl 
fp 
— = 
tol 
M 
