442 M. J. Plateau on some new and curious applications 



angle comprised upon the transparent disc between the radii 

 7' and 7-', will evidently be equal to the difference of these 

 angles. Thus, two radii traced upon the drawing of the 

 regular figure, and comprising between them the angle «, will 

 have for correspondents, upon the drawing of the distorted 

 figure, two radii comprising the angle a — /3, an angle the value 

 of which may be positive or negative ; the relation of these 

 two angles will then be 



1 I -IT • 



« U \n 



If we designate it again by M, the construction of the distorted 

 figure will be given by the expression 



M = l-^. (3.) 



But the results of this construction will be of a different 

 nature, according as we have V^ < V„ or V^ > V^; that is to 

 say, according as the velocity of the transparent disc is greater 

 or less than that of the black disc. We shall now proceed 

 successively to a consideration of these two cases. 



First case : V^ <"• V„ . 



In this case the relation M will necessarily be less thaii 

 unity ; whence it follows that the distortion will always con- 

 sist of an angular contraction, a contraction which will be 

 stronger as the velocities approach nearer to one another. 



If we designate by y the angle comprised between the two 

 radii which pass respectively by the first and by the last point 

 of the regular figure, and by y' the angle comprised between 

 the two radii which pass in the same manner by the extreme 



points of the distorted figure, we shall evidently have - =M, 



whence 



y=My (4.) 



Now it is evident, that here, in one revolution of the slitj 

 only a single regular figure will be manifested. In fact, for 

 this figure to be multiple, it would be necessary for the slit to 

 recur several times in its revolution, in coincidence with the 

 first point of the distorted figure ; now, according to a similar 

 coincidence, the point in question, in virtue of the less velocity 

 of the transparent disc, will remain behind the slit; so that 

 this last will have made its revolution before it, and conse- 

 quently no other coincidence can be produced. Nothing, 

 then, limits the angular extent which the regular figure may 

 have ; and consequently, when this figure is constructed to de- 



