450 M. J. Plateau on some nem and curious applicatiojis 



the head to the right and the tail to the left, and in the other, 

 the head to the left and the tail to the right. 



We arrive, then, at a curious conclusion, namely, that the 

 same transparent discs may be employed with two systems of 

 entirely different velocities. Now it is clear that this conclu- 

 sion is not limited to the above example, and that for every 

 system of velocities in contrary directions, there may likewise 

 be substituted a system belonging to the partial case under 

 our consideration, and vice versa. We shall evidently obtain 

 the general relation between the proportions of the velocities 

 in these two equivalent systems, by equalling to the value of 

 M given by the formula (1.) the value of this same quantity 

 given by the formula (3.), and taken with contrary signs. \\\ 

 then, in this last we designate the velocities by V'^ and V'„, 

 we shall have 



-1 = ^+1, 



= 9 



(8.) 



whence we shall deduce 



v V 



and it is seen that, as we have said, if the proportion of the 

 velocities of one of the two systems is a whole number, that of 

 the velocities of the other system will be so likewise. 



Let us now pass to the second partial case, that is to say, 

 let us suppose V^ to be comprised between V„ and 2V„. Then 

 the absolute value of M will be less than unity, and conse- 

 quently the distorted figure will be angularly contracted with 

 relation to the regular figure. Following always the same 

 mode of reasoning, we shall see that if V^ only exceeds V„ by 

 one unit, we shall be able to repeat the distorted figure, as 

 in the case of the systems in which, on the contrary, V„ ex- 

 ceeds Vrf by one unit ; so that there will be again two different 

 systems which may be substituted one for the other. For 

 example, if we take V^ = 5 and V„ = 4, the absolute value 



of M will be -; so that, as in the drawing of the figure 2, the 



4 



angular dimensions of the distorted figure will be four times 

 less than those of the regular figure, and we shall ascertain 

 without difficulty that the distorted figure may likewise be re^- 

 peated four times ; also the drawing of the figure 2 will pro- 

 duce identically the same regular figure, whether it be em- 

 ployed with the system ¥^ = 3 and V„=4, or with the system 

 Vrf=5 and V„ = 4, provided that, in the second, the disc which 

 bears it be reversed. For the secopd system, however, it will 



