342 Proceedings of the Poyal Society of Edinburgh. [Sess. 
With the substitutions y =rx—pz, z = r py — qx, y — rx' —pz f , z =py r — qx', 
B q = ^Z{m(xz — zx)} , A'q = 2 { m(x'z — zx ) } , the result becomes 
(A + A ')p - (?> - C + A' - C')qr + C q(r' - r). 
This is to be equated to the moment L of external forces round the axis of 
A. Hence we get 
(A + A')p-(B + A' -C-C’)qr + C'q(r -r) = L . . (49). 
If now we suppose the angle <p to be 7 r/ 2 , and insert (see § 21) O — p 2 sin 0 
for p, (psinO for q, — <^>(1 — cos0) for r, \[s — <£(1 — cos 0) for f, we get the 
0-equation of motion. Similarly the other equations can be obtained. 
If we suppose the pendulum to be symmetrical about the axis of the 
fly-wheel, (49) becomes, with the above substitutions, 
AO - {(C + C , )(l - cos 0) sin 0 + A sin 0 cos 0}4> 2 + C' sin 6if/<p — L . (50); 
where A is now used in the sense assigned above to A + A 7 . 
It will be observed that the process indicated in (13) could not be 
applied to the terms in (46) of which the kinetic energy is there made 
up. But in that expression for the kinetic energy the functions of the 
co-ordinates 6, <p, are exactly the same as those which would appear in the 
expression to which (13) is applicable. 
23. If A 7 = 0, (T = 0, that is, if there be no fly-wheel, (49) becomes the 
first of the set of three equations 
Ap - (B - C)qr = L j 
- (C — A)rp = M j (51), 
Cr — (A - B)pt/ = N ' 
which can all be obtained by the same process. The angular velocities 
p, q, r, are about fixed axes with which, at the instant under consideration, 
the principal axes of moment of inertia coincide. They are, of course, 
Euler’s well-known equations of motion for a rigid body one point of 
which is fixed. 
24. As another example, we take the equations of motion of the 
pendulum and of the fly-wheel about the axis. It is clear, by applying the 
third of (51), remembering that N is zero and A = B in each case, or by 
making the calculation indicated in (13) and then substituting as in § 21 
that we have 
Cr = 0, or Cr = c, 
for the pendulum, and 
C'r =0, or CV = c, 
for the fly-wheel. 
Substituting c/C for r and c/C' for r in (49), we see that since the 
