39* Mr airy, on the INTENSITY OF LIGHT 



length of the path of any ray to the illuminated point is ^ + - {uf — m.w): 



and, by (2), the first differential coefficient of this quantity with regard 

 to w will be zero for those rays which pass according to the ordi- 

 nar)' rules of reflection and refraction. Performing this differentiation, 



3 w' — w; = : w = ± V ^ : the lengths of path therefore of the two 



rays are E — - V and E ■\- - \/ - _ : and the difference of these 



4 27 4 27 



. \ , /4w 

 IS - V 



„ _ „_ . The destruction of light would therefore take place, on 



this imperfect theory, when V -^=- = li or = 3, or = 5, &c. ; that is, when 



27 



'/27 */27 9 



m= V— -, or= \/ / , &c. ; or when to = 1-89, 3'93, &c. : and there 

 4 4 



would be no light whatever for negative values of m. We have found 



above, on the complete theory, that there is sensible light for negative 



values of m, and that the destruction of light takes place when »m = 2-48, 



4'4 (nearly). According to the imperfect theory, the intensity would 



be infinite when ?« = 0, and the next maximum would be nearer to 1*89 



than to 3 "93: perhaps when »i = 2"7: we have found above that the 



intensity is nowhere infinite, that the first maximum takes place when 



OT = 1'08, and the second when w? = 3 47. 



21. In figure 4, I have represented the intensity of the light by the 

 ordinates of a curve, of which the abscissa represents different values of m. 

 The strong line corresponds to the determination of the complete theory : 

 the dotted line, to that of the old theory of emission (supposing the 

 intensity inversely as the square root of the distance from the caustic) : 

 and the faint line, to that of the imperfect theory of interference 

 mentioned above, giving to the maxima values in some degree pro- 

 portionate to the ordinates of the dotted line. The absolute values of 

 the ordinates in the faint and the dotted line are not to be understood 

 as necessarily referred to the same unit as those in the strong line ; 

 but the abscissas correspond exactly in all. 



