159 
1914 - 15 .] 
The Theory of the Gyroscope. 
the nearly vertical top. If x, y denote small rotations about transverse and 
longitudinal axes respectively, we have 
x= -p^x - fSy - kx, y== l3x + Y . . . . (28) 
where /3( = Cn/A) represents the gyroscopic effect, k is the coefficient of 
damping, and 2irlp is the period of oscillation of the frame in the absence 
of friction and rotation. The symbol Y is written for N/A, where N is the 
couple exerted on the frame as the ship rolls. If we are to write down the 
equation of rotation of the vessel itself about the longitudinal axis, Y could 
be eliminated, and we might proceed to the consideration of the free and 
forced oscillations. The discussion is, however, very complicated,^ the 
equation for the free periods, for example, being of the fourth degree. The 
subject can, however, be illustrated to a certain extent by examining the 
effect on the flywheel of a prescribed oscillation of the ship, and the 
consequent absorption of energy. This is a comparatively simple matter. 
Assuming^, then, 
.... (29) 
. . . . (30) 
we have 
whence 
In real form we have 
y = 
(o-^ — ik<j)x — if^cry d ^ 
- ijScTX - ahj = Y j 
Y = - cr‘^y + 
(J-2 — 
Y= ~ 
COS at + cos {at + e) 
if 
This corresponds to 
pcose= 1 psine = A:/o-. 
y = ?/q cos at . 
and the rate of dissipation of energy is therefore 
N?/ = Aa^y^- sin at cos at - ^^ . 0 - sin at cos {at + e) 
The mean value of this is 
^ Aji‘^ay^ sin e ^ Akpiahj^ 
2 ~p “ '^{a^-p‘^f + kV 
(31) 
(32) 
(33) 
(34) 
(35) 
(36) 
If I be the moment of inertia of the ship about the longitudinal axis 
through its mass-centre, its energy of rolling is The ratio which 
the energy dissipated by the brakes in a period ( 27 t/(t) bears to this is 
27tA 
I 
27tC%2 
ka 
( 0-2 _ ^2)2 + (o-2 _ ^2yi + 
. (37) 
* The matter is treated very fully by F. Noether in Klein and Sommerf eld’s Theorie des 
Kreisels, pp. 794 et seq. 
