1923] Proceedings of the Academy of Science 19 



of two dimensions. Extending these notions to higher dimensions we 

 consider quadratic differential forms in any number of variables as 

 characterizing manifolds of the same number of dimensions. 



The problem of measuring the curvature of surfaces, or manifolds 

 of two dimensions, is a problem well known to mathematicians. The 

 corresponding problem of obtaining a satisfactory measure for the 

 manifolds of higher dimensions has received much less attention, and 

 what has been done upon the problem is not readily accessible. 



For the case of four variables the matter is of vital importance to 

 mathematicians and scientists generally who desire to comprehend 

 the hypotheses advanced by Professor Einstein as regards the nature 

 of the universe in which we live. 



The present paper sets out to study some of the means that have 

 been suggested for measuring the curvature of manifolds. It men- 

 tions the measures of curvature of Gauss, of Sophie Germain, and of 

 Casorati for the case of two variables. Presupposing that the classical 

 account of Gaussian curvature is familiar because it is so readily ac- 

 cessible, the presentation of Gaussian curvature is that of an absolute 

 invariant of a quadratic differential form under a functional trans- 

 formation. The details of this consideration lead to algebraic quad- 

 ratic forms, Christoffel symbols and tensors of rank two and four. 



For manifolds in any number of variables the measures of curva- 

 ture suggested by Riemann and Einstein are developed in detail and 

 their relation to each other and to the curvature of Gauss pointed 

 out. A number of examples of the various measures of curvature 

 are given. 



Secondary Electron Emission from Tungsten and Iron. Otto 

 Stuhlman, Jr. 



It was found that when thermions liberated from a tungsten fila- 

 ment were accelerated and allowed to impinge on a metal grid main- 

 tained at a variable positive potential that secondary electrons were 

 emitted from the grid. The number of such secondarj^ electrons 

 emitted was measured by means of a galvanometer in series with the 

 grid and a plate maintained at a constant positive saturation potential. 



On plotting the secondary current as a function of the accelerat- 

 ing voltage acting on the primary electrons, a sudden change in the 

 slope of the curves occurring at critical potentials was interpreted in 

 the usual way. The energy-quantum relation V (volts) L (A) 

 = 12320 was used to compute the equivalent wave-lengths. 



