MR JOHN SCOTT ON THE BURNING MIRRORS OF ARCHIMEDES. 141 



and similarly the light reflected from the space around D, 



4&RTa'cos 2 2* ( z + &'-,-, z -p X 



Hence the total intensity at is 



4JcTlI'acos 2 2i ( z + b j? z -p 1 , 4&RTa'cos 2 2i ( z + &'-,-, z -,-, ) 



M = ^F 1 T E * - & B <J + ^ i "F - ^ ~ F^ J ' 



and substituting for <z, b, a', V, R, and R' 



_ 4:c(v + z cot 2«)&I'sin4i f v + (2c — tan*)z -^ 2cz p, ") 



«r(i; — z tan*) 1 v — ztant e ' v — z tarn e J 



4c(v — zcot2t)&I'sin4i f v+ (2c + tani)z-p, 2cz ^ "I ,-_. 



«r(y + ztan*') 1 y + ztani 6 ' 1 w + sstant ^ J ' '' 



the values of the moduli in terms of c, v, z, and * being 



2 x /2cz(y — ztan*) _v — ztant / 2>/2cz(v + z tan i) _, , _ v + ztanz 



1 " v + (2c — tant)z ' 2cz ' * v + (2c + tan i) z 2cz 



Since the value of u will not be altered by substituting for v and z any two 

 quantities having the same ratio, it follows that the intensity of the reflected 

 light is uniform along the line which joins A and (fig. 7). 



Article 9. — Corollary. The value of k, the fraction which expresses the 

 relation between the intensities of the reflected and incident rays, may be found 

 by means of a conical reflector, thus : — 



Let R = distance from the axis of a small zone described by AB (fig. 11). 

 r = distance from the axis at which the reflected light or heat becomes, 

 by convergence to its axis, equal in intensity to the incident. 



Then hT :!' :: r :~R, 



. • . 7cRi' = rV , and k = w • 



This result, calculated on the assumption that all the rays emanating from 

 the sun are parallel, will not deviate perceptibly from the truth, except when r 

 is small compared with R. 



The value of k may also be found by using the combination of n plane mirrors. 

 By Article 3, Equation 2, intensity 



_ »&rg 2 cos i 



