CONTENTS 
Nationai. Academy op Sciences, Bache Fund 16 
Errata 232 
Index 369 
MATHEMATICS 
The Riemann Geometry and Its Generawzation 
By L. P. Eisenhart and 0. Veblen 19 
Ricci's Principai. Directions for a Riemann Space and the Einstein Theory 
By L. P. Eisenhart 24 
Note on the Definition of a Lineai. Functionai^ By C. A. Fischer 26 
On the Rei/ATion of a Continuous Curve to its Compi^ementary Domains in 
Space of Three Dimensions By Robert L. Moore 33 
A Solution of the Linear Matrix Equation by Double Multiplication .... 
By F.L. Hitchcock 78 
On The Location of the Roots of the Derivative of a Polynomial 
By J. L. Walsh 139 
Normal Coordinates for the Geometry of Paths By 0. Veblen 192 
Principal Directions in a RiEmannian Space By J. L. Synge 198 
Principal Directions in the Einstein Solar Field By J. L. Synge 204 
Fields of Parallel Vectors in the Geometry of Paths . . . .By L. P. Eisenhart 207 
Spaces with Corresponding Paths By L. P. Eisenhart 233 
Number of Substitutions Omitting at Least One Letter in a Transitive 
Group By G. A. Miller 238 
The Meaning of Rotation in the Special Theory of Relativity 
By Philip Franklin 265 
New Properties of All Real Functions By Henry Blumberg 283 
Generalized Limits in General Analysis By Charles N. Moore 288 
Proiective and Affine Geometry of Paths By 0. Veblen 347 
ASTRONOMY 
The Effective Temperature of 16 Stars as Estimated from the Energy 
Distribution in the Complete Spectrum By W. W. Coblentz 49 
Note on the Problem of Great Stellar Distances By H. Shapley 69 
Dark Nebulae By H. N. Russell 115 
On the Distribution of the Velocities of Stars of Late Types of 
Spectrum By G. Stromberg 141 
Invisible Sun-Spots By G. E. Hale 168 
Comparison of Theory with Observation for the Minor Planets 10 Hygiea 
AND 175 Andromache with Respect to Perturbations by Jupiter 
By A. 0. Leuschner 170 
A Method of Deriving the Distance of the A-Type Stars 
By W. S. Adams and A. H. Joy 173 
The Larger Results of 20 Years of Solar Radiation Observations 
By C. G. Abbot, F. E. Fowle and L. B. Aldrich 177 
STATISTICS 
The Vital Index of the Population of England and Wales, 1838-1920 
By R. Pearl and M. H. Burger 71 
Seasonal Fluctuations of the Vital Index of a Population . .By R. Pearl 76 
iii 
