MR JOHN SCOTT ON THE BURNING MIRRORS OF ARCHIMEDES. 137 



Let i = angle of incidence on the mirror, 



R = perpendicular distance of B from the axis, 



r = perpendicular distance of any point in the curve similar to HPG (fig. 9), 



as represented by FQ, 

 v = FA = FB, because BFA is an isosceles triangle (fig. 7), 

 u = intensity at 0, 



F = intensity of the sun's light at the earth's surface ; 

 then 2i — angle BFA, 

 90°— i = angle BAF, 

 a = bsec2i, 

 Trab = -rrb 2 sec2i = area of ellipse, semi-axes a and b, 



z — FO, in this case less than b. 

 Since the rays reflected from C and D (fig. 7) fall upon the same point 0, the 

 circumference of the circle described by in its revolution about the axis is 

 illuminated by the light reflected from the two annuli described by C and D. 

 Besides the point is situated so near the axis, that the perpendicular distances 

 of C and D from the axis may be considered as equal to one another. Thus, to 

 find the concentration which results from the converging of the rays to the axis, 

 we have, 



/ 2KI'\ 

 2*r : 4tB : : T : intensity at distance r from F, ( = I , 



the intensities being estimated on planes perpendicular to the rays. Wherefore 

 the intensity on the plane at F of the reflected light 



, 2EI' n . 2*EI'cos2* 



= k x x cos 2% — , 



which would give the intensity on the increment at Q (figs. 9 and 10), if the 

 sun's rays were perfectly parallel. But instead of this light being confined to the 

 increment at Q, it is spread over an ellipse whose area = irb 2 sec 2i, and hence the 

 intensity at due to this increment 



2k RI' cos 2i r dr d6 2k RI' cos 2 2i 



= x — Tq rr- — T5 dr dd : 



r ir¥ sec 2z no 1 > 



and the same is true for a corresponding increment on the other side of the axis 

 GH (figs. 9 and 10), 



2k RI' cos 2% r dr dd 

 u =1 72 x x 



6 2 sec 2i 

 4^RI'cos 2 2* 



L cos- 1 1% rp , 



^ Jj drd6 ' W' 



