146 MR JOHN SCOTT ON THE BURNING MIRRORS OF ARCHIMEDES. 



But a plane mirror which is a tangent to the paraboloid at the same spot, and of 

 just sufficient area to reflect the circular beam of light whose section is equal to 

 the circle DE, will also produce at the focus an intensity equal to kV (Art. 2, 

 Equa. 5) ; that is, the concentration at the focus is the same, whether a circular 

 beam of the section mentioned be reflected from the surface oi\ the paraboloid or 

 from its tangent plane ; and the same will evidently apply to any polygonal beam 

 capable of being inscribed in the circular. 



Hence a burning mirror, scarcely inferior in its effects to a parabolic one, may 

 be formed of plane hexagonal reflectors, their sizes, of course, depending on the 

 distance of the focus. For example, as the sun's image overspreads an area of 

 1 inch in diameter at a distance of 9 feet, a burning mirror of that focal length 

 may be formed of plane hexagonal pieces, each side about half an inch ; whereas 

 at 108 feet distance, the sides of the plane hexagonal plates need not be less than 

 half a foot, and so on in proportion. 



Plates of the latter size being greater than those with which Buffon performed 

 his experiments, we infer that his combination, at distances exceeding 100 feet, 

 would be little inferior in power to a parabolic segment of equal focal length, and 

 capable of reflecting exactly the same sectional area of the solar beams. 



Again, what has been proved true of plane mirrors, tangents to a paraboloid 

 of revolution, must be equally true of a series of tangential circumscribing conical 

 frustums. In all these cases, however, it is probable that the advantage in 

 practice will remain with the parabolic figure, from the light at its focus having 

 a greater area of maximum intensity. 



From these results, as well as from independent calculations, we con- 

 clude that refracting burning-glasses may be constructed, by placing at some 

 distance from an axis a series of acute-angled conical zones, or wedge-shaped 

 pieces of glass (fig. 18), built up like the compound lens of Brewster, 

 which will produce combustion at as great distances as Buffon's combination 

 of reflectors. 



Article 17. That the practibility of the Archimedean mirror may be made 

 still more apparent, we shall now apply Equations 1 and 2, Article 3, to find the 

 numerical intensity of the light in the focus of Buffon's combination. This was 

 attempted by Peyrard, but his conclusions are vitiated by the false premises from 

 which he set out. He assumed that the intensity is uniform at every part of 

 the luminous image reflected by a plane mirror, — a supposition proved incorrect 

 by Art. 2. In our calculation we shall suppose that each of Buffon's mirrors, 

 which were 8 inches by 6, produced an effect equivalent to a circular beam of 6 

 inches diameter, when it leaves the mirror. 



Taking k — \, and cos i = 1, we obtain by substitution, in the equations of 

 Art. 3, the following results : — 



