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OF THE

CAMBRIDGE

PHILOSOPHICAL SOCIETY

VOLUME XXII.
(1912-1923)

CAMBRIDGE

AT THE UNIVERSITY PRESS

AND SOLD BY

DEIGHTON, BELL AND CO. LTD. AND BOWES AND BOWES, CAMBRIDGE
CAMBRIDGE UNIVERSITY PRESS, LONDON.

M.DCCCC.XXIII

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 contributing towards the expense of printing this Volume of the Transactions.

PRINTED IN GREAT BRITAIN

CONTENTS OF VOLUME XXII,

I. On Compound Denumeration. By Major P. A. MACMAHON, R.A., Sc. D., LL.D., F.R.S.,

Hon. Mem. Camb. Phil. Soc.

II. A Class of Integral Functions defined by Taylor's Series. By G. N. WATSON, M.A.

III. The Hydrodynamical Theory of Lubrication with special reference to Air as a

Lubricant. By W. J. HARRISON, M.A., Fellow of Clare College, Cambridge

IV. The Superior and Inferior Indices of Permutations. By Major P. A. MACMAHON,

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

Jacobian Figure. By R. HARGREAVES, M.A.

VI. On the Fifth Book of Euclid's Elements (Third Paper). By M. J. M. HILL, M.A.,

LL.D., Sc.D., F.R.S., Astor Professor of Mathematics in the University of London

VII. The Invariants of the Halphenian Homographic Substitution to which is appended

some investigations concerning the Transformation of Differential Operators which

present themselves in Invariant Theories. By Major P. A. MACMAHON, F.R.S.

VIII. Vector Integral Equations and Gibbs' Dyadics. By C. E. WEATHERBURN, Ormond

College, University of Melbourne

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IX. On certain Arithmetical Functions. By S. RAMANUJAN

X. On the Fifth Book of Euclid's Elements (Fourth Paper). By M. J. M. HILL, M.A.,

LL.D., Sc.D., F.R.S.

XI. The Character of the Kinetic Potential in Electromagnetics. By R. HARGREAVES,

XII. The Field and the Cordon of a Plane Set of Points. An Essay in Proving the

Obvious. By ERIC H. NEVILLE

XIII. On certain Trigonometrical Sums and their applications in the Theory of Numbers.

By S. RAMANUJAN, B.A., F.R.S., Trinity College, Cambridge

XIV. Asymptotic Expansions of Hypergeometric Functions. By G. N. WATSON, Sc.D.,

Trinity College, Cambridge

XV. Asymptotic Satellites near the Equilateral-Triangle Equilibrium Points in the

Problem of Three Bodies. By DANIEL BUCHANAN, M.A., Ph.D., Queen's

University, Kingston, Canada

XVI. Terrestrial Magnetic Variations and their Connection with Solar Emissions which

are Absorbed in the Earth's Outer Atmosphere. By S. CHAPMAN, M.A., Sc.D.,

F.R.S., Trinity College, Cambridge

XVII. On the Representations of a Number as a Sum of an Odd Number of Squares. By

L. J. MORDELL, M.A. (Cantab.), Birkbeck College, London

XVIII. The Hydrodynamical Theory of the Lubrication of a Cylindrical Bearing under

Variable Load, and of a Pivot Bearing. By W. J. HARRISON, M.A., Fellow of

Clare College, Cambridge

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XIX. On Integers which satisfy the Equation t±3± y3 ± 23 = 0. By H. W. RICHMOND,

M.A., King's College, Cambridge

XX. On Cyclical Octosection. By W. BURNSIDE, M.A., F.R.S., Hon. Fellow of Pembroke

College, Cambridge

XXI. Congruences with respect to Composite Moduli. By Major P. A. MACMAHON,

XXII. On the Stability of the Steady Motion of Viscous Liquid contained between two

Rotating Coaxal Circular Cylinders. By K. TAMAKI, Assistant Professor of

Mathematics, Kyoto Imperial University, and W. J. HARRISON, M.A., Fellow

of Clare College, Cambridge

XXIII. On a General Infinitesimal Geometry, in reference to the Theory of Relativity.

By WILHELM WIRTINGER (Vienna)

XXIV. On the Fifth Book of Euclid's Elements (Fifth Paper). By M. J. M. HILL, Sc.D.,

LL.D., F.R.S., Astor Professor of Mathematics in the University of London

XXV. The Influence of Electrically Conducting Material within the Earth on Various

Phenomena of Terrestrial Magnetism. By S. CHAPMAN, M.A., Trinity College,

and T. T. WHITEHEAD

XXVI. The Escape of Molecules from an Atmosphere, with special reference to the

Boundary of a Gaseous Star. By E. A. MILNE, M.A., Trinity College, Cambridge 483

XXVII. Some Problems of Diophantine Approximation: The Analytic Properties of

Certain Dirichlet's Series associated with the Distribution of Numbers to

Modulus Unity. By G. H. HARDY, M.A., Savilian Professor, University of Oxford,

and J. E. LITTLEWOOD, M.A., Trinity College, Cambridge

XXVIII. Free Paths in a non-uniform Rarefied Gas with an Application to the Escape of

Molecules from Isothermal Atmospheres. By J. E. JONES, 1851 Exhibition

Senior Research Student, Trinity College, Cambridge

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OF THE

CAMBRIDGE

PHILOSOPHICAL SOCIETY

VOLUME XXII. No. I. pp. 1-13.

ON COMPOUND DENUMERATION

BY

MAJOR P. A. MACMAHON, R.A., Sc.D., LL.D., F.R.S.

HONORARY MEMBER CAMBRIDGE PHILOSOPHICAL SOCIETY

CAMBRIDGE:
AT THE UNIVERSITY PRESS

M.DCCCC.XII.

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