the summation being for every composition of a given number a1 + a + ... + as into s or fewer parts a, a, ... a, such that a, i, for all values of s.

The Superior Index as defined is obtained by adding several numbers together. This is the simplest way of obtaining the index, but the numbers so added are not the most interesting that come up for consideration. If v >u the letter a, adds a number to the index if it precedes one or more letters au. Denote by p'u the number added to the index due to the positions of the letters a, a,. Moreover a, may precede 1, 2, ... or i, letters au. Denote by p'vu, o the number of letters a, which precede exactly a letters au. Every time an a, precedes exactly o letters au the number σ is added to the index. Hence

vu

P'vu = P'vu, 1+ 2P'vu, 2 + 3p' vu, 3+...+iuP'vu, in

Also if p" denotes, in regard to the whole of the permutations, the sum of the numbers added to the indices by reason of the relative positions of the letters a, a, and p’vu, o the number of times in the whole of the permutations that a letter a, precedes exactly σ letters au,

p",

vu

Now we know the value of from the following consideration. In any permutation consider merely the letters au, a,. If r'ue denotes the number added to the Inferior Index by the relative positions of these letters we see that

uv

P'vu + r'uv = iuiv,

for any one letter a, contributes to the sum of the Superior and Inferior indices the number i and therefore the total of i, letters a, contributes the number ii. Hence the average value of p'ou in a permutation is ii, and thence the number contributed to the Superior Indices of all of the permutations by the relative positions of au and a, is

which is derived from the original assemblage by adding an a, and subtracting an α. In any permutation fix the attention upon the +1 letters au. Call the one on the extreme right the last au, the one nearest to it the last but one au, the next one again the last but two au and so on. Now delete the last au but o from the permutation and substitute for it the letter an. We have thus an a, followed by a letters au and the assemblage is the original assemblage of letters. We thus construct a case of an a, followed by exactly σ letters a from every one of the

(i)!

i!ig!... iu! (iu + 1)! (i, − 1)! ip+1!... ig!

To illustrate these results take the assemblage aaßy wherein 1 = 2, ¿, = 1, i3 = 1.

and we verify that in the first column the numbers 0, 1 and 2 each occur 4 times.

and we verify that the sum of the numbers in the first column is 12.

and we verify that in the second column the numbers 0, 1 and 2 each occur 4 times and that the sum of the numbers is 12.

and we verify that in the third column the numbers 0 and 1 each occur 6 times and that the sum of the numbers is 6.

THE DOMAINS OF STEADY MOTION FOR A LIQUID ELLIPSOID, AND THE OSCILLATIONS OF THE JACOBIAN FIGURE

BY

R. HARGREAVES, M.A.

CAMBRIDGE:

AT THE UNIVERSITY PRESS

M. DCCCC. XIV.

ADVERTISEMENT

THE Society as a body is not to be considered responsible for any facts and opinions advanced in the several Papers, which must rest entirely on the credit of their respective Authors.

THE SOCIETY takes this opportunity of expressing its grateful acknowledgments to the SYNDICS of the University Press for their liberality in taking upon themselves the expense of printing this Volume of the Transactions.

V.

The Domains of Steady Motion for a Liquid Ellipsoid, and the

Oscillations of the Jacobian Figure.

By R. HARGREAVES, M.A.

[Received 8 February 1914.]

ONE of the oscillations of ellipsoidal type for Maclaurin's figure of equilibrium has, at the junction with Jacobi's series, a period exactly one-half that of rotation; i.e. if a day means a period of rotation, the natural equatorial tide is here semi-daily.

This isolated result was reached some twenty years ago, and seemed of sufficient interest to stimulate enquiry into the course of the periods of oscillation of the Jacobian figure.

As the present work is of recent date the stimulus has been tardy in operation. The scope of the investigation has been extended to cover other matters which, like the question of periods of oscillation, require for their complete discussion much laborious calculation with transcendental equations. The results are made accessible by the use of diagrams to represent the domains of steady motion for a homogeneous liquid ellipsoid under its own gravitation, and an inspection of these is sufficient to shew what kinds of steady motion are possible for an ellipsoid of given shape.

Special attention is given to the Jacobian form where a full series is treated with reference to shape, angular velocity and momentum, and kinetic energy; while the periods of the ellipsoidal oscillations are added for a smaller number of cases, sufficient to make the course clear through the entire range. In respect to the Jacobian an interesting feature is the connexion of the movement in values of angular velocity and momentum along the series, with the quantities on which secular stability depends.

With respect to motion about two axes the most interesting point is that the conditions laid down by Riemann for his Case II are entirely superseded by the condition of positive pressure.

It is proposed to describe the main results in general terms before proceeding to the analysis on which they are based.

For the spheroids the oscillations may be called polar and equatorial; in the former the equator remains a circle but its radius and the polar axis are subject to periodic change, in the latter the polar axis is unaltered, the equator suffers a periodic elliptical deformation.

The oscillations of the Jacobian near the opening of the series differ little from those of the spheroid; as the form moves away from the spheroidal the terms polar and equatorial fail to describe them, but in the ultimate position the oscillations become respectively equatorial and VOL. XXII. No. V.

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