( 434 ) 



The envelope of the lines connectlnj paifs of points, through lohich 

 a curve of each of the pencils is possible, is of class 



3 rst — 2 {st-{-tr-\-rs) + (r+s+-0 - a{r-l) — /?(s-l) — y (^-1) -f rf = 

 = 3 rst — 2 (st^tri-rs) + (r+s^-^ - «'(r-1) - ^'{s~l) - r'(«-l) - 



S. If the pencils have no common basepoints then the class of 

 the envelope is 3 rst — 2 (.s-^ + tr + rs) -f (r + .s^ + t). By a common 

 basepoint Ast of the pencils {Cs) and (G) that class is lowered with 

 r — 1. IViis is because point Ast has separated itself from the 

 envelope r—1 times. Tn fact, the curve (■',. passing through Ast has 

 separated itself from the locus of the points P and P' . If we take 

 P arbitrarily on this C,, the corresponding point P' coincides with 

 Ast' So an arbitrary line passing through Ast is to be regarded 

 (^. — 1) times as a line connecting P and P' , as any of the r — 1 

 points of intersection with t; differing from Ast may be chosen for I\ 



If the three pencils have a common basepoint Arst the total envelope 

 of PP' remains definite (in contrast to the total locus of P and 

 P'). It is true P can be taken quite arbitrarily, but then P' 

 coincides with a point .4,,.^, so that the line 7V^' passes through that 

 point Arst^ and therefore is not quite arbitrary. As the class of the 

 envelope proper is lowered by the point Arst with r -\- s -];- 1 — 4 it 

 follows, that Arst separates itself (r -\- s -\- 1 — ^) times from the 

 envelope. As one of the points of the pair becomes entirely indefinite, 

 that multiplicity is not easy to explain, as far as I can see. 



Physics. — ''On a nem empiric spectra] formula." By E. E. 

 MoaKNDORFF. (Commuuicated by Prof. P. Zeeman). 



By the fundamental investigations of Kaysek and Runge and those 

 of Rydberg the existence of spectral series was proved. The fornnilae 

 of these physicists, however, give in general too great de\'iations for 

 the first lines of a series. I have tried to improve the formula given 

 bv Rydberg: 



11=1 A . 



(m -1- ay 



Particularly noteworthy in Rydberg's formula is tlie universal 

 constant 7V„. From Balmer's formula, which is included as a special 

 case in RvDBEiKi's formula, follows for hydrogou for the observation 

 corrected to vacuo X. = 109675. 



