( 117 ) 



pencii, when the hasepoinU f<ill'm<j m S move away from S along 

 the osculat:in<i curve of the pencil. 



It is clear tliat the theorems V and VI invariably hold good when 

 tiie pencil contains cnrves degenerated in coinciding parts if only 

 they do not j)ass through point iS. 



Theorem VI is like theoiem IV an extension of the above mentioned 

 theorem of Hat,piikn and Smith. If namely we substitute for the 

 pencil of cnrves the pencil of straight lines with ^S'as vertex, theorem VI 

 passes into: 



Let S be the origin of a branch T of a curve lohikt that branch 

 hi intersected in t points S by an arbitrary straight line through S. If 

 now a point P moves aimy from S along the tangent in S, that jw int 

 S gives to that branch t points of contact ivith tajigents through P. 



It is not difficult to see that this is correlative to the formulation 

 given above of the SMiTH-HALPHEX-theorem. However when the pencil 

 of curves is not a pencil of straight lines the theorems IV and VI are 

 not correlative and so we have to I'egard them as entirely different 

 theorems. 



Sneek, June 1904. 



Astronomy. — ''On the apparent distribution of the nebulae." By 

 C. Easton. (Communicated by Prof, H. G. van üe Sande 

 Bakhuyzen). 



It being admitted that the results of the visual observations must 

 be kept separated tVom those oblained by photography, the systematic 

 investigation of tiie distribution of the nebuLae by means of photo- 

 graphy begun l)y Max Wolf should not prevent us from carefully 

 examining tiie xavy extensive material regarding nebulae which has 

 been formei-Iy olilained by (Urect observation and laid down in 

 catalogues; and the less so l)ecause it is higldy improl)al)le tliat even 

 in future a visual "Durchmusterung" of this kind, which for the rest 

 is very desirable, \viJl be cai-i-icd out on account of the different 

 method vvhicli now is bcijig foHowed at Heidelberg. 



It is noways unimaginable that the dislriliulion of the nebulae as shown 

 by phot()grai)hy will differ greatly from thai of the visually observed 

 nebulae; yet it is certain that the latter distribution shows remarkable 

 features which call for an explanation. William Herschel has found 

 that in the main the nebulae are numerous where the stars are 

 sparse. In a certain sense we have here the reverse phenomenon 

 from that of the stars; New comb (The Stars, p. 187) expresses it thus : 



