433 
of Edinburgh, Session 1868-69. 
differential equations containing the complete solution of the 
problem when no external forces act : 
dt 
2 
div 
W 
dx 
X 
dy 
Y 
where 
W= - x% - 
dz 
z; 
■r.e 
and 
X = w%l + y(& - z'B 
Y = wB + zM, — x® 
Z = w<& + 
^ = X ( a ( w2 “ : 
13 = ^ (^b{w 2 — : 
<sr = a 
y 2 — z 2 ) -p2z(ax-p%4-cz)-j-2w;(&2 — 
y 2 — z 2 ) -p2y(az-p6y-j-cz)-f-2w (cx—az)^j 
y 2 — z 2 ) -|-2z (az-p &y-|-cz) -}-2w(ay — &e) ^ 
Here A, B, C are the principal moments of inertia, and 
y ~ ia + jb + kc 
is the constant vector of moment of momentum. 
Thus we see that W, X, Y, Z are homogeneous functions of 
tv, x, y, z, of the third degree. Equations of this nature, but not so 
symmetrical, have been given by Cayley, and completely integrated 
(in the sense of being reduced to quadratures) by assuming the 
previous integration of Euler’s equations. 
Other modes of integration are employed ; and the problem is 
also solved by seeking the homogeneous strain which will bring the 
body from any initial position to its position at time t. 
This part of the paper concludes with the complete determina- 
tion of g for the case of no forces and two equal moments of inertia- 
The remainder of the paper deals with some simple cases of applied 
forces, when two moments of inertia are equal. If a denote a unit 
vector in the direction of the unequal axis of inertia, and if the 
motion be that of a heavy solid of revolution (such as a top) about 
a point in its axis, it is shown that 
B Y aa — Alia, = Vay 
where y is a constant vertical vector, and 
O = Sae = constant . 
This is the equation of motion of a simple pendulum disturbed by 
