On a Property of the Curvature of Rhizic Curves at Multiple Points.
By William Walton

Note on the Calculus of Logic. By Prof. Cayley.

PAGE

274

On the Inversion of a Quadric Surface. By Prof. Cayley
On certain Definite Integrals. By William Walton

294

On the Theory of the Curve and Torse. By Prof. Cayley.
Deformation of an Elastic Sphere Pressed between two Parallel
Planes. By R. Hoppe

Note on sin ∞ and cos co.

By William Walton

On the Summation by Definite Integrals of Geometrical Series of the
Second and Higher Orders. By J. W. L. Glaisher

318

326

328

On a Theorem Relating to Eight Points on a Conic. By Prof. Cayley 344 On the Analogues in the Theory of Quadrics to some known Properties in the Theory of Conics. By R. Townsend

On Gauss' Theorem of the Measure of Curvature at any Point of a
Surface. By Benjamin Williamson .

Notes on the Circle which Cuts Three Given Circles at Given Angles.

By John Griffiths

On a Pair of Definite Integrals. By William Walton

Review

1

THE

QUARTERLY JOURNAL

OF

PURE AND APPLIED MATHEMATICS.

ON THE RELATION BETWEEN THE ANGULAR VELOCITY ABOUT AND THE ANGULAR VELOCITY OF THE INSTANTANEOUS AXIS OF A BODY REVOLVING SPONTANEOUSLY ABOUT A FIXED POINT, AND ON THE AXES OF GREATEST AND LEAST MOBILITY.

By WILLIAM WALTON, M.A., Fellow of Trinity Hall. LET λ, u, v, be the direction-cosines of the instantaneous axis at the end of any time t in reference to a system of fixed rectangular axes of x, y, z, passing through the fixed point. Let w be the angular velocity of the body about the instantaneous axis, and let w, w, w, be the components of the angular velocity about the axes of coordinates.

and consequently, differentiating with respect to t, we have

Let r be a given length measured along the instantaneous axis from the fixed point.

VOL. XI.

Then

x= r cosλ, y=r cosμ, z=r cosv,

B

Imagine the axes of coordinates to coincide with the principal axes of the body at the end of the time t, and let w1, w, 27 W37 the corresponding values of w, w, w: then, as is well known,

dw, dw, dw,

2

do, do do. Thus dw dw

are equal respectively to "

dt

dt dt

dt

3

Let a, b, c, be the moments of inertia about the principal axes through the fixed point: then, by Euler's formulæ,

Again, ƒ and g being arbitrary constants, we have, as is well known, as consequences from Euler's equations,

then, from the equations (2), (3), (4), we obtain

...

.(5):

(6).

From the equations (6), combined with Euler's equations, we obtain, by the usual method, the known relation

From the equations (1), (6), and (7), we see that