in tlie experiments of Fraunhofer. 115 



" There are, indeed, only single places marked by the pre- 

 cise middle of the largest common fringes, where the 100 sup- 

 posed systems of undulations are capable of mutually supporting 

 themselves, and of co-operating together. In these places the 

 elementary oscillations are exactly contemporaneous, in virtue of 

 the equality of their paths, like those of the middle : They fol- 

 low at intervals equal to an entire undulation, for it is evident 

 that in the second brilliant band, where the undulations of the 

 second system have lost an oscillation relatively to those of the 

 first, the undulations of the third system will have lost two of 

 them, since the third line is distant from the first double that 

 of the second, and the undulations of the 101st line will have 

 lost 100 undulations; the third band will be still one great- 

 er, but the effect will be the same throughout, and each band 

 will have the united force of all the 100 centres of diffraction ; 

 whilst, at a very small distance from the middle line, and such 

 that the light coming from the most remote points has lost 

 only an undulation by the difference of paths, the oscillations 

 will follow at distances almost equal throughout the circum- 

 ference of the circle which represents them, and the respective 

 velocities which are proportional to the positive or negative 

 cosines will mutually destroy one other, so that their sum will 

 be zeo'o. This distance is the 1 OOdth of the tenth of the ra- 

 dius, or the lOOOdth of the vvhole distance of the card ; and if 

 this whole distance is 100 inches, the width of the brilliant band 

 will be one-tenth of an inch on each side, or one-fifth altogether, 

 including the space imperceptibly illuminated by the enfeebled 

 light; the precise brilliant band being probably much narrower. 

 " Without knowing the law according to which the primi- 

 tive impulsions ought to diffuse themselves in all directions, 

 it is impossible to calculate exactly the illumination of the dif- 

 ferent points of the space on the card ; but we may suppose 

 these elementary impulsions equal in all directions, as Huyghens . 

 has done, or at least in all directions near the rectilineal one, 

 as M. Fresnel has done. It is also more convenient to presume 

 that they follow the law of the sines and cosines, which seems 

 to be that of the greater number of small natural vibrations. 



Then calling x the distance of any point of the card from 

 the middle of the nearest brilliant band, and supposing that x 



I 



