238 REPORT—1862, 
Ultimate Multiplier 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 in a simple manner ; 
and the differential equation which I finally obtained turned out to be inte- 
grable per se, so that the laborious process of finding the multiplier became 
unnecessary. 
195. The new variables Q, v have the following geometrical significations : 
Q=1 tan 30 cos], where / is the principal moment (A*p*+ B*q’+C'r’=P), 
6 (as before) the angle of resultant rotation, and I is the inclination of the 
resultant axis to the invariable axis; and y=? cos? 3J, where if we imagine 
a line AQ having the same position relatively to the axes fixed in space that 
the invariable axis has to the principal axes of the body, then J is the incli- 
nation of this line to the invariable axis. It is found that p, g, 7 are func- 
tions of v only, whereas \, x, v contain besides the variable Q. In obtaining 
these relations, there occurs a singular relation Q?=xv—l, which may also 
be written 1+ tan’ 30 cos*, I=sec* 30 cos’ 3J, where the geometrical significa- 
tions of the quantities I, J have just been explained. The final results are 
that the time ¢, and the arc tan-1 are each of them expressible as the 
integrals of certain algebraical functions of v. There might be some interest 
in comparing the results with those of Euler’s memoir of 1758, where the 
principal axes are also referred to an arbitrary system of axes fixed in space ; 
but I was not then acquainted with Euler’s memoir. 
The concluding part of the paper relates to the determination of the varia- 
tions of the constants in the disturbed problem. 
196. Cayley, “ Note on the Rotation of a Solid of Revolution ” (1849), shows 
the simplification produced in the formule of the last-mentioned memoir in 
the case where two of the moments of inertia are equal, say A=B. 
197. Jacobi’s final solution of the problem of Rotation was given without 
demonstration in the letter to the Academy of Sciences at Paris; the demon- 
stration is added in the memoir, Crelle, t. xxxix. (1849). The fundamental 
idea consists in taking in the invariable plane, instead of the fixed axes vy, a 
set of axes xy revolving with uniform velocity, such that the angular distance 
of the axis of « from its initial position is precisely = —n'w ; and consequently 
if 6’ be the longitude of the node of the equator on the invariable plane, mea- 
sured from the moveable axis of # as the origin of longitude, we have 
PL lat log ee (i= v— =I); 
lis 
=5; into a loga- 
rithmic function) in passing to the trigonometrical functions sin 6’, cos 6’ the 
logarithm disappears altogether; and we have in a simple form the expres- 
sions for the actual functions sin 6’, cos 6’, through which 6’ enters into the 
formule, and thus, Jacobi remarks, the barrier is cleared which stands i in 
the way when the expression of an angle is reduced to an elliptic ‘integral 
of the third class. 
198. For the better expression of the results, Jacobi joins to the functions 
H, 0, considered in the “ Fundamenta Nova,” the functions 0,vw=0 (K—vw), 
H (uw) =H(K—zw) ; so. that 
and in consequence of this form of the expression for @’ 
