1908-9.] 
341 
On Lagrange’s Equations of Motion. 
usual axes Ox, Oy, are to O z) are therefore Osin 0 + 0 sin 0 cos 0 and 
— 0 cos 0+0 sin 0 sin 0 respectively, 
The expression for the kinetic energy can now be written down, and is 
T = J{A(dsin 0 + 0 s i n 0 cos 0) 2 + B( - 6 cos 0 + 0 sin 0 sin 0) 2 + C(1 - cos #) 2 0 2 } 
+ i{A'(d 2 + 0 2 sin 2 ^) + C'(0-0(l-cosO)) 2 } . . . (46). 
From this, by the ordinary process, since the co-ordinates of any particle 
of the body are connected with the angles here specified by finite rela- 
tions, we can obtain the equations of motion ; and this, of course, is the 
simplest mode of proceeding. All the data for forming the equations in 
this manner will be found worked out in Thomson and Tait, § 330. The 
foregoing specification is only given here for the sake of the following new 
discussion of the problem. 
22. We refer the motion of any particle of mass m in the body to fixed 
axes coinciding for the instant under consideration with the principal axes 
A, B, C, and denote the angular velocities about these axes by p, q, r. The 
corresponding angular velocities for the fly-wheel are p, q, r. If x, y, z, be 
the co-ordinates with reference to these axes of a particle of mass m in the 
pendulum apart from the fly-wheel, and x', y', z\ those of a particle of mass 
m' in the fly-wheel, we have 
x = qz — ry , y = rx - pz, z=py — qx, 
x = qz — r'y', y = r’x — pz , z = py' — qx. 
Hence 
2T = ~%[m{(qz - ry) 2 + (rx - pz) 2 + ( py - qx ) 2 }] 
+ ^\in {qz - r'y') 2 + (r'x - pz) 2 + (py - qx) 2 }'] . . (47); 
where the second line refers to the fly-wheel and the first to the rest of 
the pendulum. 
From this we obtain 
0T 
dp 
= - (rx - pz)z + (py - qx)y}\ + %\in { - (rx' -pz)z + (py' - qx)y'}\ 
Calculating the total time-rate of variation of this, and taking account of 
the fact that the axes of reference coincide with the principal axes of the 
body, we get 
1 
dt dp 
= - z(rx - pz) — z( — pz + rx -J>^)}] 
+ \™{y{py - qx) + y(py+py - qx))] 
+ similar expressions for fly-wheel .... (48) 
From this we have to subtract, according to (13) above, 
3[m{ - z(rx - pz) + y(py - qx )) J + S[m'{ 
z(r'x -pz')+y(py' 
