order, of which this evolute is a right section, meets the surface 1) 

 moreover in a curve in space of the order 24. 



6. Tlie found image 1) can be useful in different researches 

 about the system S of the circles of Joachimsthal. By determining 

 the number of points common to this surface and a rectangular 

 hyperbola, a parabola and a straight line, we find that the system 

 S has the characterizing numbers 4,8,16; in other words, it contains 

 four circles passing through two given points, eight circles passing 

 through a given point and touching a given line, sixteen circles 

 touching two given lines. In the same manner is proved that it 

 contains sixteen circles touching two given circles, etc. 



7. If we are given a parabola instead of an ellipse, all the 

 circles passing through three conorraal points pass also through the 

 vertex of the parabola. Here the found surface of the eighth order 

 is reduced to the right cone .<•'- + :!/^ = '^, of which the vertex of 

 the parabola y^ = 2 px is the vertex. And the case of the hyperbola 



— ^ 1, ^^0 leads to the surftice 



a-* h^ 



4 a2 ;t2 4 62 w2 



" = 1 



(«2 f «2)2 («2—^2) 



and is ([uite analogous to that of the ellipse. 



Physics. — „(hi nirixunn and minima of apparent hrkjhl ness resul- 

 ting from optical illusion^ By Dr. C. H. Wind. (Commu- 

 nicated by Prof. H. Haga). 



1. If we see on a surface two zones of different (real) brightness 

 united by a transition-zone whose brightness decreases continuously 

 from the brighter down to the darker zone, this transitionzone 

 seems to be separated from the brighter zone by a still brighter line 

 (maximum of brightness) and from the darker zone by a still darker 

 line (minimum of brightness). 



2. This phenomenon, which — as will be seen from what 

 follows — presents itself under very different kinds of conditions was 

 first observed by me in a drawing carefully and succesfully executed 

 by Mr. van Grieken, of the firm van de Weyer at Groningen, 

 by means of lithography. This drawing of which fig 1 is a pho- 

 tographic reproduction (reduced to V4 of its size), which unsatisfactory 

 as it is, yet enables us to observe the phenomenon, consists of a great 

 number of parallel lines of equal thickness drawn at intervals of 1 

 m.M. in two outer zones, at intervals of 0.4 ni.M. in a middlezone, 



