The Rev. W. Cook on the Theory of the Gyroscope. 895 
gential co-ordinates in Curves of the Second Degree; and urged that 
eliminants ought to be introduced into the general treatment of these 
curves also, if only in order to accustom the learner to their use and 
gain uniformity of method. Thus, if the general eq. be 
Az’+By’+C+ 2Er+2Fy+G=0, 
then V=0 is the test of degeneracy. 
March 26.—Major-General Sabine, R.A., Tr. and V.P., in the Chair. 
The following communication was read :— 
“On the Theory of the Gyroscope.” By the Rev. William 
Cook, M.A. 
The explanation of the movements of the Gyroscope (as well as 
its mathematical theory) is founded on the principle enunciated in 
the two following verbal formulz. 
I. When a particle is made to move rato a plane by any 
applied force, but in consequence of its connexion with some rigid 
body on the same side of the plane, loses some of its momentum in 
a direction perpendicular to the plane ; all the momentum so lost is 
imparted to the rigid body, which is consequently impelled 
towards . 
3 as \ the plane. 
II. When a particle is made to move aa a plane by any 
applied force, but in consequence of its connexion with some rigid 
body on the same side of the plane, receives an extra momentum in 
a direction perpendicular to the plane ; all the momentum so gained 
is taken from the rigid body, which is consequently impelled 
ue the plane 
towards anes 
The mass of the dise of the gyroscope is supposed to be com- 
pressed uniformly into the circumference of a circle of given radius 
(r), and to revolve round an axis with a given uniform angular ve- 
locity (w). To facilitate the arithmetical computation of the for- 
mulz, masses are represented by weights; so that any effective 
accelerating force f is supposed to be due to a pressure P acting on 
a mass W, and their relation expressed thus, ee 
The mass of any arc of the circle is denoted by ae ; 8 being the 
angle at the centre, and ¢ the mass of a given length / of the cir- 
cumference. The terms of all the formulze are thus made homogeneous. 
The centre of gravity of the disc, axle, and the ring which carries 
the pivots of the axle is fixed, and the whole is moveable about that 
centre in any manner, subject to the condition that the line of the 
pivots of the ring is always horizontal, unless when detached from 
the stand. Let this straight line of the pivots be denoted by AB, 
the common centre of the dise and ring by O, the extremities of the 
axle by N and S; and ON=a. 
Let M denote the place of a particle of the disc, its position bemg 
determined by the angle AOM (6), and let M’ be another point in 
