238 
ON THE ROTATION OF A SOLID BODY ROUND A FIXED POINT. 
[37 
coefficients as fractions, the numerators of which are very simple rational functions of 
the second order of A, p, v, and which have the common denominator (1 + A 2 + p- + v-). 
These quantities may conveniently be termed the “coordinates of the resultant rotation,” 
and the denominator or the square of the secant of the semi-angle of resultant rotation 
will be the “modulus” of the rotation. The elegance of these results led me to apply 
them to the mechanical question, and I gave in the Journal (vol. ill. p. 224), [6], the diffe 
rential equations of motion obtained in terms of A, p, v : which I integrated as in the 
common theory, by supposing one of the fixed axes to be perpendicular to the invariable 
plane. Though my attention was again called to the subject, by the connexion of some 
of these formulse with Sir William Hamilton’s theory of quaternions, no other way of 
performing the integration occurred to me. The grand discovery however of Jacobi, of 
the possibility of reducing to quadratures the two final differential equations of any 
mechanical problem, when the remaining integrals are known, induced me to resume the 
problem, and at least attempt to bring it so far as to obtain a differential equation of 
the first order between two variables only, the multiplier of which could be obtained 
theoretically by Jacobi’s discovery. The choice of two new variables to which the equa 
tions of the problem led me, enabled me to effect this with the greatest simplicity; 
and the differential equation which I finally obtained, turned out to be integrable per 
se, so that the laborious process of finding the multiplier became unnecessary. The new 
variables Q, v have the following geometrical interpretations, Í1 = k tan \ 6 cos I, where k 
is the principal moment, 0 as before the angle of resultant rotation, and I is the incli 
nation of the resultant axis to the perpendicular upon the invariable plane, and 
v = k? cos 3 £ J; where, if we imagine a line AQ having the same position relatively to 
the axes in fixed space that the perpendicular upon the invariable plane has to the 
principal axes of the rotating body, then J is the inclination of this line to the above 
perpendicular. To the choice of these variables I was led by the analysis only. It will 
be seen that p, q, r are functions of v only, while A, p, v contain besides the variable 
0. In obtaining these relations a singular equation Í2 2 — kv — k 2 occurs (equation 13), 
which may also be written 1 + tan 2 \ 6 cos 2 1 = sec 2 \ 6 cos 2 \ J, in which form the inter 
pretation of the quantities I, J has just been given. The equation (17), it may be 
remarked, is self-evident: it expresses that the inclination of the resultant axis to the 
normal of the invariable plane, is equal to the inclination of the same axis to the line 
AQ. Now the resultant axis having the same inclination to the axes fixed in space as 
it has to the principal axes, and the line AQ the same inclinations to these fixed axes 
that the normal to the invariable plane has to the principal axes, the truth of the 
proposition becomes manifest. The correspondence in form between the systems (10) 
and (14) is also worth remarking. The final results at which I arrive are, that the 
time and the arc whose tangent is Í1 — k, are each of them expressible as the integrals 
of certain algebraical functions of v. The notation throughout is the same as that made 
use of in the paper already quoted. 
The equations of rotatory motion are 
P Q R A M N 
(1),