430 Mr. J. C. McConnel on the 



be many times greater than that of the sphere. But the 

 intensity of diffracted light is proportional to the square of 

 the area of the diffracting body. So the single filament sends 

 more intense light than a number of spheres, which together 

 occupy the same apparent area.) The steps of the calculation 

 for the filaments may be indicated as follows : — 



The filament, by reflecting and refracting in different direc- 

 tions part of the light that falls upon it, and greatly retarding 

 the remainder, behaves in diffraction like an opaque body, 

 and, by Babinet's principle, may be replaced by a similarly 

 oriented slit in an opaque screen. If the axis of the slit lie 

 in the " reflecting plane," fight from every point of any line 

 parallel to the axis will reach the retina in the same phase. 

 The oblique slit will therefore diffract light in the same 

 manner as a slit at right angles to the sun's rays, except that 

 we have to take as effective area the projection of its area 

 perpendicular to the sun's rays. But filaments lying within 

 a small angle of the reflecting plane will also contribute light, 

 and it may be shown that we get a fair approximation to the 

 amount of the total light, by supposing all filaments, for which 

 the retardation of one end of the axis relative to the other 

 does not exceed tt, to contribute the maximum light, and 

 other filaments to contribute nothing*. Further it may be 

 shown, with the aid of spherical trigonometry, that such fila- 

 ments must lie within an angle %=X/4fr«in7 on either side 

 of the " reflecting plane,'"' where b is the length of the fila- 

 ment, and 7 is half the angular distance from the sun. So of 

 the whole number we may consider only the fraction X/46 sin 7 

 to send light, for we suppose the filaments to lie at random 

 in all directions. 



When the slit is perpendicular to the sun's rays, the 

 intensity of light at a distance r in the direction opposite the 



* Let x denote the retardation of phase of any point in the axis rela- 

 tive to the middle point for any given direction of diffraction, and R the 

 value of x for one end. Then the amplitude in that direction is 



f+R 



1 cos x dx/'2H = sin R/R, 



J-r 

 that f or R = "being taken as unity. If the whole of the light for positive 

 values of R be supposed compressed between the limits R=0, R = n> tne 

 average value over this range is 



«• °° sin 2 R 7R . 7T 

 ~« 

 It is easy to verify by a rough numerical calculation that this last expres- 

 sion is approximately equal to unity. 



