CUM TENTS 
List of Publications of the National Academy of Sciences and of the 
National Research Council 339 
Index. .' . : . . 363 
MATHEMATICS 
On the Roots of Bessel's Functions By J. H. McDonald 66 
Normalized Geometric Systems By A. A. Bennett 84 
Problems of Potential Theory By G. C. Evans 89 
Quaternians and Their Generalizations By L. E. Dickson 109 
The Mathematical Theory of the Apportionment of Representatives 
By E. V. Huntington 123 
An Overlooked Infinite System of Groups of Order pq 2 . . . .By G. A. Miller 146 
A Formula for the Viscosity of Liquids By H. B. Phillips 172 
Geometric Aspects of the Abelian Modular Functions of Genus Four 
By Arthur B. Coble 245 
The Average of an Analytic Functional By N. Wiener 253 
Semi-covariants of a General System of Linear Homogeneous Differential 
Equations By E. B. Stouffer 273 
An Algorism for Differential Invariant Theory By 0. E. Glenn 276 
The Average of an Analytic Functional and the Brownian Movement 
By N. Wiener 294 
An Integral Equality and Its Applications By E. Hille 303 
On the Approximate Solutions in Integers of a Set of Linear Equations .... 317 
Geometric Aspects of the Abelian Modular Functions of Genus Four 
(II) By Arthur Coble 245 
Group of Isomorphisms of a Transitive Substitution Group 
By G. A. Miller 325 
Einstein Static Fields Admitting a Group G 2 of Continuous Trans- 
formations into Themselves By L. P. Eisenhart 328 
ASTRONOMY 
Internal Motion in the Spiral Nebula Messier 33 By A. van Maanen 1 
Measurement of the Diameter of Alpha Orionis by the Interferometer 
By A. A. Michelson and F. G. Pease 143 
Studies of Magnitudes in Star Clusters XIII. Variable Stars in N. G. C. 
7006 By Harlow Shapley and Beatrice W. Mayberry 152 
The Angular Diameter of Alpha Bootis by the Interferometer 
By F. G. Pease 177 
Changes Observed in the Crab Nebula in Taurus By John C. Duncan 179 
PHYSICS 
The Attachment of Electrons to NeutralMolecules in Air. By L. B. Loeb 5 
On the Reflection and Re-emission of Electrons from Metal Surfaces: 
and a Method of Measuring the Ionizing Potential of Such Surfaces 
By R. A. Millikan and I. G. Barber 13 
