875 



We have established tlie mean luminositj law and also the mean 

 density and velocity laws for the whole sky and moreover we have 

 determined these principal laws for the 5 galactic zones separately. 



The numbers A',,, were derived from Table V of Puhl. (Troningen 

 N'. 27. It was necessary to represent the numbers found by inter- 

 polation-formulae. For these the following formulae were found: 



Whole sky log. .V,„ = — 4.2395 -f 0.63812 m — O.OJ1677 m' 

 Zone 1 = — 4.1848 + 0.65736 —0.01 J 243 



II = — 4.1841 + 0.63322 - 0.01 1463 



III =—4.6159 + 0.71857 —0.017257 



IV = — 4.4638 + 0.67891 — 0.01 6289 



V = — 4.5120 + 0.69565 —0.017976 



The mean parallaxes jr„,, as we have already communicated, have 

 been lent to us for our investigation with great kindness by Prof. 

 Kaptkyn and Dr. Van Rhijn. These may be represented as follows: 



Whole sky log. .t„, = 8.943 — 0.142 m 



Zone I = 8.883 — 0.142 m 



II =8.904 — 0.142 m 



III =8.957 — 0.142 m 



IV ■ =9.024 — 0.142 m 

 V =9.066 — 0.142 m 



We have deduced the mean parallaxes .%„_„ from those which 

 Kaptp:yn published in Publ. Groningen N°. 8, by making the constant 

 a in the formula 



in agreement with the values of jt,„ just mentioned. Then we find 

 the following formulae: 



Whole sky log. -t,„,„ = — 0.717 — 0.062 m - 0.142,^ 



Zone I* ' = —0.777 — 0.062 m -0.142^ 



Zone II = - 0.756 — 0.062 m — 0.142 g 



Zone III = — 0.703 — 0.062 m — 0.142 'g 



Zone IV = — 0.636 — 0.062 ?;i - 0.142 ly 



Zone V = —0.594 — 0.062 m — 0.142^ 



We have derived the principal laws from these data according to 

 the method that Schwarzschild has proposed in his article in J.f^ron. 

 Nachr. N".4557. There the relations, which exist between the coef- 

 ficients of the different formulae, have been communicated in extenso. 

 The whole computation has been made by means of the formulae 



