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



proceed to study successively the systems in which the two 

 discs turn in contrary directions, and those in which they turn 

 in the same direction. 



I. Velocities in contrary directions. 



Let us imagine a slit in one of its positions, which for greater 

 simplicity we will suppose vertical, and the radius of the 

 transparent disc, which is seen at this moment through this 

 slit, let us call r; the impression left on the retina by the 

 series of the points of the distorted figure ranged on this ra- 

 dius r, will belong to the regular image. Let us then imagine 

 the slit in a subsequent position, making with the first any 

 angle which we will designate by a. In this new position 

 there will be another radius of the transparent disc, which we 

 will call i\ behind the slit, and the impression produced in 

 the eye by the succession of the points of the distorted figure 

 ranged on this second radius, will belong also to the regular 

 image. In this image the two series of points will comprise 

 therefore between them the angle «. But whilst the slit, 

 quitting the vertical, has advanced up to its meeting the radius 

 r' whilst describing this angle, the radius r, also quitting the 

 vertical, has described, in a contrary direction, another angle 

 which we shall designate by /3 ; whence it follows that at the 

 instant when the slit comes before the radius r', at an angular 

 distance « from the vertical, the radius r is, on the opposite 

 side, at an angular distance /3 from this same vertical, and that, 

 consequently, upon the transparent disc, the radii /• and r' 

 comprise between them an angle equal to /3 + «. 



Thus, two series of points respectively ranged upon two 

 radii traced upon the drawing of the regular figure, and com- 

 prising between them the angle «, will have for correspondents 

 upon the drawing of the distorted figure, two series of points 

 ranged in the same manner upon two radii comprising between 

 them the greater angle /3 + a ; and, consequently, the deforma- 

 tion will consist simply of an angular dilatation of the regular 

 figure. 



It is easy to find the general expression of the relation 



^-— . In fact, this relation may be stated under the form 



- + 1 ; now the angle /3 is evidently to the angle « as the ve- 

 locity of the disc which bears the deformed figure is to that of 

 the black disc j if, then, we designate the first by V,; and the 



second by V^, and if we represent by M the relation , 



