ANALYSIS TO THE DYNAMICAL THEORY OE THE TIDES. 
203 
it allows of our including in our analysis the effects due to the gravitational attrac¬ 
tion of the water ; and secondly, the convergence of the series obtained will be much 
more rapid. The latter circumstance is of particular value, as it has enabled me to 
treat with considerable success the problem of the free oscillations of the ocean. 
The general problem of the small oscillations of a rotating system possessing a 
finite number of degrees of freedom has been discussed by Thomson and Tatt but 
the extension to meet the case where the number of degrees of freedom is infinite 
involves analytical considerations of some delicacy. As a rule, the transition from 
the case of a system with finite freedom to that of a system with infinite freedom is 
effected by the employment of “ normal coordinates,”! and the chief difficulty in the 
solution of problems relating to the vibrations of the latter class of system consists in 
the discovery of these coordinates. The researches of Thomson and Tait just men¬ 
tioned shew however that in a rotating system these normal coordinates do not 
exist, and hence that the methods ordinarily employed to deal with the oscillations 
of a system about a state of equilibrium will no longer suffice for the treatment of 
our problem. In most “ gyrostatic ” problems which have been solved hitherto,| the 
solution has been obtained by means of a system of quasi-normal coordinates. When 
such coordinates exist, only a finite number of oscillations of certain particular types 
are possible, and, by constraining the system to vibrate in one of these types, we may 
treat it in the same manner as a system rvith a finite number of degrees of freedom. 
The period-ecjuation for the free oscillations of an assumed type will then only possess 
a finite number of roots, and will consecjuently l)e an algebraic equation usually 
most readily obtained in a determinantal form. It is shewn at the end of § 5 that 
the coordinates we have used possess this property when the depth of the ocean 
follows certain restricted laws ; but in general no such quasi-normal coordinates exist, 
and whatever coordinates be employed, the displacements in any of the fundamental 
modes of vibration can only be expressed by means of an infinite number of such 
coordinates. The most advantageous choice of coordinates will then be that which 
lead.s to most rapidly converging series. 
As however the oscillations of an assumed type can only be expressed by an infinite 
series of coordinates, it follows that an infinite number of oscillations of any assumed 
type must he possible, and that consequently the period-equation for oscillations of 
this type will have an infinite number of roots and will therefore be transcendental 
instead of algebraic in character. It is possible that the transition from systems 
with finite freedom to systems with infinite freedom may be treated with advantage 
by the employment of determinants of infinite order (as a means of expressing the 
transcendental period-equation), after the manner introduced into analysis by 
* ‘ Natural Philosophy,’ Part I., § 345. 
t Rayleigh, ‘ Theory of Sound,’ vol. I, § 87. 
t Cf. PoiNCARi, ‘Acta Mathematica,’ vol. 7 ; Brian, ‘Phil. Trans.,’ 1889. 
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