871 



In Asfron. Nadir. N°. 4422 the following problems were discussed: 



a. Let Nh and (f{i) be given. It is required to determine D{r) 

 by solving (he inlegral equation (1). 



b. Let N k and. nu be given. It is re(inired to determine both D[r) 

 and </(/) by solving simultaneously the integral equations (1) and (2). 



The integral equations have been solved by Sohwarzschild by 

 means of the known properties of Fouriek coeffK'ients ^). 



It is clear that we can tind the velocity law in an analogous way. 



If it is required to determine simultaneously the density law, the 

 luminosity law and the velocity law this can be done in iho fol- 

 lowing ways : 



1^^ From A'„j, iVy and .t„,, 



2nd N N Tt 



3i-d „ iV„„ n-,„ ,, jr,„ 



4t'' „ A';/, Jr„^ „ Tly. 



In case we suppose that there is no connection between lumino- 

 sity and velocity, x\^„,„ and ;^„,« would be sufficient data from which 

 to determine the three functions we seek. 



Practically we have to reckon with the accuracy with which the 

 required data can be deduced from the observations. Theoretically 

 we can find D{i'), <i{i) and \y{ V) in each of the cases mentioned 

 by resolving the integral equations in accordance with Schwarz- 

 schii.d's method. 



Serious objections, however, may be raised against the use of 

 such integral formulae. If Nm, -^m, etc. were known to us as conti- 

 nuous analytical functions of the variables Schwakzschild's method 

 would be very well adapted to the determining of the required 

 functions from these data. As a rule, however, we only know the 

 values of N,u and rr,„ for whole numbers ot vt. If now we represent 

 these numbers by a formula, these relations, indeed, adapt them- 

 selves to interpolation. If, however, the relations have no theoretical 

 foundations and the parameiers no physical significalion, we cannot 

 attach any other value to them than that of interpolation formulae 

 There is always the risk, then, that by using these relations, we 

 keep properties of the original data hidden from view. 



A second objection is that empirical relations are extrapolated out 

 of the interval within which observations are available. If a function 

 in a determined interval represents the observations with sufficient 

 accuracy, it need not for that reason be of force outside the interval. 

 And particularly with the integral method it can be determined 



») Vide also Eddington, Stellar Movements, Chapter X. 



