246 
im. S. S. HOUGH ON THE APPLICATION OF HAEMONIC 
annual tides will be quite inappreciable, and we may take the above series as giving 
a good representation of the solar long-period tides, unless the effects of friction 
become important for such tides. 
The fact that when the period of the disturbing force'is increased without limit 
the free surface does not tend to approach its equilibrium form appears at first 
sight to be at variance with the general laws of oscillating systems. The explanation 
of this apparent anomaly may perhaps be made clear by considering a simple form 
of “ gyrostatic” system which possesses only two degrees of freedom. In the absence 
of frictional forces, the general equations of motion of such a system may, by a proper 
choice of coordinates, be expressed in the form 
X — coy n^x = X, 
y + cox -b m-y = Y.'' 
Here x, y denote the generalized coordinates of the system. Of the terms on the 
left, the terms x, y are due to inertia, the terms coy, cox are described by Thomson 
and Tait as “motional” forces, and the terms n^x, m^y as “positional” forces ; X, Y 
are the generalized components of the external disturbing force. 
If now X, y, X, Y be supposed proportional to we find from the above equations 
— Y'x — coiky + n^x = X, 
— k^y coi'kx + mhj — Y, 
whence we may obtain x, y in terms of X, Y. When the period of vibration is 
indefinitely prolonged, k will approach zero as a limit, and the limiting form of the 
solution will in general be 
X = X/rr', y — Y lm~. 
This implies that the displacements will in general tend to acquire their equilibrium- 
values as the period of the disturbing force is lengthened. There will, however, be 
an exception to this law if one or both of the positional forces n^x, m^y vanish. 
Let us first examine the nature of the free oscillations in such cases ; omitting 
X, Y, and supposing that n = 0 while m remains finite, we have for the determina¬ 
tion of the free motions 
— Yx — oiky ~ 0 
— k~y + coikx -f- wry =. 0 
or, if we denote by u, v the generalized velocity-components so that u = x = ikx, 
v = y = iky, 
* Thomson and Tait, ‘Natural Philosophy,’ vol. 1, p. 896 (1886 edition). 
