4 ti M. J. Plateau on some new and curious applications 



complish one revolution or an entire number of revolutions, 

 it will be necessary that the angle a' be equal to unity or to 

 any entire number. Now we cannot suppose a'=l ; for, ac- 

 cording to the above expression, there would result from it 

 Vrf =0; thus the second coincidence can only be produced 

 after several revolutions of the slit. Now, as the numbers V 

 and Vrf have no common factor, it is clear that the quantity 



V 



-^T — '—- , or «', can only be equal to an entire number, if we 



have V^j — V^=l. We arrive, then, lastly, at this conclusion, 

 that the velocities should be taken such, that the numbers 

 which represent them differ among themselves only by a unit. 

 We shall then have simply a's=V^; that is to say, from one 

 coincidence to another, the slit will accomplish a number of 

 revolutions equal to V^^. 



We remark, moreover, that in this case the value of M 

 given by the expression (3.) is simplified, and becomes 



M=^ (6.) 



The expression (5.) will also consequently become 



n 



This being settled, let us examine more closely what passes 

 in the successive revolutions of the slit. If we always start 

 from a coincidence, it is at once evident that, in the first of 

 these revolutions, a complete regular figure will have been 

 produced ; or, in other words, that the slit will have passed 

 before the whole of the distorted figure. In fact, according 

 to the manner in which we have arrived at the formula (3.), 

 if we suppose that after a coincidence with the first point of 

 the distorted figure the slit has described an angle a, the 

 quantity M will represent also the relation between the angular 

 distance which then separates the slit from the point in ques- 

 tion, and that angle a ; if, then, « constitutes an entire revo- 

 lution, and is thus measured by unity, the above angular di- 

 stance, or, in other words, the portion of the transparent disc 

 before which the slit will have passed from the coincidence, 

 will then be represented by M, and consequently, in the pre- 

 sent case, by vv— . But, according to the expression (7.), the 



fraction vv- is the measure of the largest angle which the dis- 



