343 
1908 — 9 .] On Lagrange’s Equations of Motion. 
terms involving r and r go out on account of the vanishing of the products 
of inertia, no change in the process which gives (49) is effected. 
Equations (49) and (50) have their notation changed. The latter 
becomes 
A(0 - sin 0 cos OR 1 ) + (c +- c) sin 0 . <p = L . . • (52).. 
25. An example of two distinct cases of motion in which the expressions 
for the kinetic and potential energy are the same, has been given by Appell 
( Mecanique Rationnelle, tome ii., art. 469). The first is a disk or hoop of 
mass m, which is symmetrical about an axis perpendicular to its plane, 
and rolls without sliding along a horizontal plane, on a sharp edge which 
forms the terminating circle of the plane drawn through the centroid per- 
pendicular to the axis. In the other case the disk or hoop rests with the 
edge on a horizontal plane without friction, while its centroid is constrained 
to move along a vertical circle of which the radius is the same as that of 
the circular edge. By properly choosing the moments of inertia in the 
second case, the kinetic energy can be made the same as in the former 
case, and since the mass of the body is taken the same in both cases, the 
potential energy is identical also with that in the former. 
In the latter case, Lagrange’s equations are directly applicable in their 
ordinary form ; in the former case, they are not. 
I shall here sketch the modified Lagrangian solution of the problem of 
the hoop or disk, in order to compare it with an elementary mode of 
solving gyrostatic problems which I have found readily applicable even 
in cases of very considerable complication. The Lagrangian solution does 
not differ, except in notation, procedure, and arrangement, from that con- 
tained in the paper by Ferrers referred to above. 
26. We refer the motion of the centroid to rectangular axes Ox, 0 y, Oz , 
in and perpendicular to the plane in which the hoop rolls, with origin O 
at the point of contact, and denote the co-ordinates of the centroid by 
x, y, z. Let be the angle which the vertical plane through the axis 
of the hoop makes with the fixed vertical plane containing Ox, and \js be 
the angular velocity of the hoop relatively to the former plane. The 
angular velocity of the hoop about its axis of figure is thus ^ + cos 0. 
A consideration of the geometry of the problem shows that the follow- 
ing equations hold: — (1) If the inclination, 0 say, of the axis of the hoop 
to the vertical remain unaltered, and the hoop roll through an angle Sx> 
Sx 1 = - yS x , Sy x = y cot . S x . 
(2) Due to the alteration of 0 we have 
Sx 2 — - a sin 0 cos . SO, Sy 2 = - a sin 0 sin cf> . SO, 
Sz — a cos 6S6 . 
