MR JOHN SCOTT ON THE BURNING MIRRORS OF ARCHIMEDES. 129 



Jc T ?/ Jc T 11 



Therefore the intensity of the solar light at from area u x = circIe A p B p = — t > 



= kju 2 



"2 



3 a-r 2 ' 



W„ 



_ &r« n 



kl' 



Whole intensity at = — a Oi + ^ 2 + u % + O i 



VTT 



kl' x areaACBPA 



(1), 



= — 2 (2 x sector AFC - 2 triangle AOF + 2 x sector AOP) , 



= ^{s 2 t + rX«-JP)-§zs™*} ■ • (2), 



p sin 6 



because smp = , 



also z = gcostf— rcosp, 



= p cos 6— >J(r 2 — g 2 sin 2 0) . 



When p y r in the preceding expressions, and = 0, then (p = 0, z = p — r, and 



kT 

 intensity at = ^r x t 2 = M' ■ . . . . . (5). 



If, therefore, the circle ADPB falls wholly within the circle ABG, the in- 

 tensity of the illumination on the circular space, whose centre is F and radius 

 p — r, is constant and equal to that of the beam when it leaves the mirror, or the 

 intensity is the same as if all the rays from the sun's disc were parallel to one 

 another. 



When p = ?*, then, z — 0, and F is the only point whose intensity = kT . 



kl' 

 Next when p <^r and 9 = tt, then cp = ic, z — r — p, and intensity = —$ x pir' 1 



= ^f • • • (6). 



This formula expresses the intensity throughout the circle, whose radius 



is r — p- 



When z — r + p, the intensity vanishes. 



