i^tS M. J. Plateau on some new and curious applications 



>•' must have advanced towards the slit ; whence it follows that, 

 upon the transparent disc, this radius ;•' is to the left of the 

 radius r. Hence it results, that all the points of the distorted 

 figure situated on the same side of the vertical will have their 

 correspondents, in the regular image, situated on the other 

 side of this same vertical, and that consequently the angular 

 distances will be measured in opposite directions in the dis- 

 torted figure and in the regular figure. It is clear, then, that 

 one of these figures will be inverted with reference to the 

 other ; in other words, all that in the one is found to the right 

 of the vertical, is in the other to the left of this same vertical, 

 and vice versa. It is clear now on what depends the negative 

 sign of M ; for this quantity designating the relation between 

 the corresponding angular distances in the two figures, if we 

 take as positive those which belong to one of these figures, 

 we must, on account of the opposition of direction, consider 

 as negative those which belong to the other, and consequently 

 the relation will take the sign minus. 



The absolute value of the relation in question is 



V 



Now, as the quantity tt— is only limited to the condition of 



being superior to unity, it is clear that three circumstances 

 may present themselves ; namely, 



or 



^-1=1- 



V * — • » 



which come back to those : 



V. > 2 V„, V. > V„ and < 2V , 

 or lastly, 



V,= 2V,. 



Let us begin by examining the first of these partial cases; 

 in other words, let us suppose that the velocity of the trans- 

 parent disc exceeds the double of that of the black disc. Then 

 the absolute value of M exceeding unity, it follows that the 

 distorted figure will be, as for the systems of opposed velocities, 

 angularly dilated with relation to the regular figure; and by 

 reasonings analogous to those we have employe(l with regard 

 to these systems, we shall recognize without difficulty that the 

 regular image will also be multiple; lastly, we shall ascertain 



