MR JOHN SCOTT ON THE BURNING MIRRORS OF ARCHIMEDES. 127 



Whatever may be the form of the infinitesimal mirror at B, the perpendicular 

 section of the reflected rays, at all finite distances from B, is a circle. 

 Let a = the angle ABC = DBE. 

 r — radius of the sphere AC. 



c — the distance from B, measured along the slant side of the cone of 

 rays, at which the diameter of the circle of reflected light is equal 

 to unity. 

 I = the intensity of the light radiating from the sphere at its surface. 

 I' = intensity of the light at the mirror, on a plane perpendicular to the 

 axis of the cone of rays falling on B. 

 .• . kl' = intensity of the reflected light at the mirror, k being a constant less 

 than unity. 

 d = distance of the centre of the sphere from B. 



d' = distance of DE from B, measured along the slant side of the cone 

 of reflected rays. 

 S A = sectional area of the light incident on small mirror at B. 



Area of the circle DE = -n-d' 2 sin 2 " . . . . (1), 



4c 2 



(2), 



because c : d' : : \ : radius of the circle DE. 



Since the intensity of the light, emanating from the surface of a luminous 

 sphere and falling on a concentric spherical surface, is inversely as the square of 

 the distance from its centre, 



d 2 T 

 r 2 :d 2 ::I':I .-. I = ^f . . . (3). 



But ^1=^=1 ••• I =i • • (4) 



2 d 2c 



= 4c 2 I' . . . (5). 



Intensity of the reflected light at DE = k }' b \ =-= . . (6), 



area of circle DE v ' 



M'5A 



* f r 2 sin 2 ^ 



4c 2 M'5A 



(7). 

 (8). 

 When the plane on which the reflected light falls is not perpendicular to the 



VOL. XXV. PART I. 2K 



