234 ; REPORT—1862. 
itself also analytically developed, but without the introduction of the elliptic 
and Jacobian functions: to form a complete theory, it would be necessary to 
incorporate the memoir with that of Jacobi. 
179. The following is an outline of the ‘ Extrait ’:— 
The instantaneous motion of a body about a fixed point is a motion of 
rotation about an axis (the instantaneous axis); and hence the finite motion 
is as if there were a cone fixed in the body which rolls (without sliding) upon 
another cone fixed in space. 
The instantaneous motion of a body in space is a motion of rotation about 
an axis combined with a translation in the direction of this axis: this remark 
is hardly required for Poinsot’s purpose, and he does not further develope the 
theory of the motion of a body in space. The effect of a couple in a plane 
perpendicular to a principal axis is to turn the body about this axis with an 
angular velocity proportional to the moment of the couple divided by the 
moment of inertia about the axis. 
' And hence by resolving any couple into couples perpendicular to the prin- 
cipal axes, the effect of such couple may be calculated ; but more simply by 
means of the central ellipsoid (momental ellipsoid a?a?+67/°+¢e2°=k", if 
A, B, C=Ma?, MB?, Mc’), viz., if the body is acted on by a couple in a tangent 
plane of the ellipsoid, the instantaneous axis passes through the point of con- 
tact; and reciprocally given the instantaneous axis, the couple must act in the 
tangent plane. 
180. Considering now a body rotating about a fixed point, and taking as 
the plane of the couple of impulsion a tangent plane of the ellipsoid, the 
instantaneous axis is initially the diameter through the point of contact; the 
centrifugal forces arising from the rotation produce however an accelerating 
couple, the effect whereof is continually to impress on the body a rotation 
which is compounded with that about the instantaneous axis, and thus to 
cause a variation in the position of this axis and in the angular velocity round 
it. The axis of the accelerating couple is always situate in the plane of the 
couple of impulsion. 
181. Hence also 
1°. Throughout the motion the angular velocity is proportional to the length 
of the instantaneous axis considered as a radius vector of the ellipsoid. 
2°. The distance of the tangent plane from the centre is constant ; that is, 
the tangent plane to the ellipsoid at the = of the instantaneous axis 
is a fixed plane in space. 
Or, what is the same thing, the motion is such that the ellipsoid remains 
always in contact with a fixed plane, viz., the body revolves round the radius 
vector through the point of contact, the angular velocity being always pro- 
portional to the length of this radius vector. 
It is right to remark that in Poinsot’s theory the distance of this plane 
‘from the centre depends on the arbitrarily assumed magnitude of the central 
ellipsoid; the parallel plane through the centre is the invariable plane of the 
motion. 
182. The motion is best understood by the consideration that it is implied 
in the theorem that the pole of the instantaneous axis describes on the ellip- 
soid a certain curve, ‘‘the Poloid,’ which is the locus of all the points for 
which the perpendicular on the tangent plane has a given constant value (the 
curve in question is easily seen to be the intersection of the ellipsoid by a 
concentric cone of the second order) ; and that the instantaneous axis describes 
on the fixed tangent plane a curve called the Serpoloid, which is the locus of 
the points with which the several points of the poloid come successively in con- 
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