No abstract

one accepts, \\-it11 J . \\.ads\\-orth (Q.]., , aFourier representation of pressure fluctuations as jlrysically valid over the entire symmetry interval, the usefulness of symmetry points for forecastinx follows immediately. Hoivrwr, there exists strong evidence that such Fourier representationc art. only mathertititically valid, and that t h v principal components will shon. frequent changes in amplitude, phase, and frequency, \shich iiiust be c1asat.d as un~r"dictabIt.---jii*t as the oc'curencc' and duration of their symmetry points.The occurrence of synimetries over large arcas i\ nut s u r p r i h g \\hen the size of normal pressure fot-niations is taken into account. Yet if it could be confirnicd that individual presstire serie? sho\s only the accidental frequency tlistribution of symmetries, it is h;trtl to he^' ho\v a set of thtvii could do better.Thus some statistical trc:itment of rither thc. prt'ssurc fluctuations or the symnietrieb remai,nb iiecessary to establish their non-accidental chararter. 1 should like t o record here the fact that credit for the first attempt a t treating symmetries along such lines is due to C. Reinsberg (.-lstuomniis,.hP Sarliriiliferr, Voi. Note on a paper by G. C. McVittieI n a recent paper G. C . McVittie (1948) introduces a ne\\ systein of co-ordinates for tlic study of atniosplirric motions. The equations derived by McVittie for hi3 " great rirclta " co-ordinate* differ slightly from those obtained for cylindrical polar co-ordinates. 1 t is thc purposc of the prewnt not9 to interpret this ditl'erc.iicc*.MrVittie', equations of niotioci ;ire :. \s emphasized by McVittie the terms containing ria as a factor do not occur in the cylindrical-polar form of the equations. I ? + V Z ,T h e tetimin (5.13) i y the centrifugal force \\ hich is due to the fact that a particle which nioves horiLoiitally on the earth's surface describes a circle. This force can be introduced as an additional inertial force as done b> J . Jauniotte (1930) ; it is, tiowcver, generally negligible.l h e other terms containing I / a also coiitain the Coriolis parariieters k or 1. This fact indicates that they are due to the \;riation of the Coriolis parameter* \\ ith latitude, and this is also true.32 I a

The mass-energy of spherically symmetric distributions of material is studied according to general relativity. An arbitrary orthogonal coordinate system is used whenever invariant properties are discussed. The Bianchi identity is shown to imply that the Misner-Sharp-Hernandez mass function is an integral of two combinations of Einstein's equations for any energy-momentum tensor and that mass-energy flow is conservative. The two mass equations thus found and the mass function provide a technique for casting Einstein's field equations into alternative forms. This mass-function technique is applied to the general problem of the motion of a perfect fluid and especially to the examination of negative-mass shells and their relation to singular behavior. The technique is then specialized to the study of a known class of solutions of Einstein's equations for a perfect fluid and to a brief treatment of uniform model universes and the charged point-mass solution.

Spherically symmetric relativistic spheres of perfect fluid are defined to be isotropic by Walker's (1935) isotropy condition. This condition permits the use of noncomoving coordinate systems, which, it is argued, are preferable to comoving systems in certain situations. It is assumed that these systems are such that the metric is orthogonal and involves three unknown functions. These functions are obtained by solving the equation expressing the isotropy condition in a number of cases defined by ancillary mathematical assumptions. Formulas are given for the pressure, density, and velocity components of the fluid, but the detailed physical analysis of the various cases found is reserved for a subsequent paper.