MR JOHN SCOTT ON THE BURNING MIRRORS OF ARCHIMEDES. 139 



of different inclinations, there are two which produce a maximum effect at the 

 axis, viz., when 4i = 90° and 270° (in Equa. 10), that is, when BAF (fig. 7) 

 = 67^° or 22i°. 

 When 



z = 6 (Equa. 9), E z _(*) = 2, 



b 



and 



8cH'sin4i 

 u= . . . (11). 



If k = ^, the maximum intensity at the axis expressed numerically is 



u = 4cM' sin 90 = 4 x 107 x|I' = 214 1' . 



When k = ^ and 4i = 2° 36', it can be shown by Equa. 10 that u = 101' nearly, 

 a heat sufficient to ignite wood and other combustibles. This can be effected at 

 a distance of 130 feet with a segment of the reflector 18 inches broad, and having 

 a mean diameter of 6 feet. 



Case 2. When the point O is situated at a considerable distance from the axis, 

 z being much greater than b, the distances CO and DO will now differ perceptibly 

 from one another and from FA = FB = v (fig. 7). 



Let R and R' represent the distances of the points C and D respectively from 

 the axis, then 



R =fC sin 2i = (v + z cot 2z) sin 2i , 

 and 



CO = /C -/O = FA + F/-/0 = v + z cot 2i - z cosec 2i = v - z tan • . 



Similarly for the point D, 



E' = (v — z cot 2i) sin 2i , 



and DO = v + z tan i. 



The ellipses of light which overspread O reflected from each increment of the 

 space around C have for their minor and major axes respectively, 



; _ v — z tan i v — z tan i 



b ~ 2T~ ' a= 2c cos 2i ; 



and the corresponding quantities with respect to the space around D are 



7 , v + z tan i , v + z tan i 



2c 2c cos 2 % 



VOL. XXV. PART I. 2 N 



