ON THE SPECIAL PROBLEMS OF DYNAMICS. 235 
tact with the tangent plane, and is a species of undulating curve, viz., the radius 
vector as it moves through the angles 6 to 0,+2II, 0,+ 20 to 6,+4I1, &c. as- 
sumes continually the same series of values. This is in fact evident from the 
mode of generation ; and it is moreoyer.clear that the serpoloid is an algebraical 
or else a transcendental curve according as II is or is not commensurable with 7. 
[Treating the poloid and serpoloid as cones instead of curves, the motion 
of the body is the rolling motion of the former upon the latter cone, which 
agrees with a previous remark. | 
There is a very interesting special case where the perpendicular distance 
from the tangent plane is equal to the mean axis of the ellipse. 
183. Poinsot remarks that the motion is such that [considering the plane 
of the couple of impulsion as drawn through the centre of the ellipsoid] the 
section of the ellipsoid is an ellipse variable in form but of constant magni- 
tude, and that this leads to a new representation of the motion, viz., that it 
may be regarded as the motion of an elliptic cone which rolls on the plane of 
the couple [the invariable plane] with a variable velocity, and which slides on 
this plane with a uniform velocity. 
184. The theory of the last-mentioned cone, say the “rolling and sliding 
cone,” is developed in the memoir, Liouville, t. xvi. p. 303, in the chapter 
entitled “ Nouvelle Image de la Rotation des Corps.” If a, 6, c signify as 
before (viz., A, B, C=Ma’, Md’, Mc’), and if h be the distance of the centre 
from Poinsot’s fixed tangent plane (h<a>c), then the invariable axis 
describes in the body a cone the equation whereof is 
(a? —h’) a +. (0? —h’) 7? +(C—h’?) 2=0 ; 
the cone reciprocal to this, viz. the cone the equation whereof is 
a? y 2 
oe Bet eae 
is the “rolling and sliding cone.” The generating line OT of this cone is 
perpendicular to the plane of the instantaneous axis OI, and of the invariable 
axis OG ; and the analytical expressions for the rolling and sliding velocities 
follow from the geometrical consideration that the motion at any instant is a 
rotation round the instantaneous axis OI: that for the sliding velocity is the 
instantaneous angular velocity into the cosine of the angle LOG, which is in 
fact constant ; that for the rolling velocity is given, but a further explanation 
of the geometrical signification is perhaps desirable. 
185. I may in this place again refer to Cohen’s memoir “On the Differential 
Coefficients and Determinants of Lines &c.” (1862), the latter part of which 
contains an application of the method to finding Euler’s equations for the 
motion of a rotating body. 
186. Rueb in his memoir (1834) first applied the elliptic and Jacobian func- 
tions to the present problem. Starting from the equations 
Ap* + Bq? +Cr*? =h, 
A’p? + BP? +Cr’=P*, 
and 
— —Bdq 
(A—C) rp" 
it is easy to perceive that by assuming y=a proper multiple of sin £, the ex- 
* 1 is Poisson’s &, the constant of the principal area ; it is the letter afterwards used by 
Jacobi ; Rueb’s letter is gy. In quoting (infra) the expressions for p, g, 7, I have given 
them with Rueb’s signs, but it would be too long to explain how the signs of the radicals 
are determined. { 
dt 
