128 MR JOHN SCOTT ON THE BURNING MIRRORS OF ARCHIMEDES. 



axis of the cone of rays, as de (fig. 1), the intensity of the light at e will exceed 

 the intensity at d from the greater obliquity of the rays at the latter point. But 

 if the angle DBE be small, and the plane de cut the axis of the cone at a con- 

 siderable angle, the intensity will be nearly uniform over the whole ellipse de; 

 and the centre of the ellipse may be viewed as situated in the axis of the cone, 

 because the elongated cone approaches nearly to a cylinder. 



.-. intensity on de = r^-p ^-^ . . (9). 



J urea oi ellipse DE v ' 



In the case of the sun's light the above formulae will give pretty accurate 

 approximations, since a = 32' and c = 1074 nearly . . . (10). 



Article 2. — Prop. When a Cylindrical Beam of Solar Light is reflected from a Plane Mirror; 

 to find the Intensity on a Plane Surface perpendicular to the direction of the reflected Beam, 

 and at any distance from the Mirror. 



Since the rays which emanate from any single point in the sun's disc may be 

 considered as perfectly parallel, however large the mirror, it follows that those 

 from the centre of the disc will, after reflection from the mirror, form a perfectly 

 cylindrical beam of parallel rays, and will cast on the given plane a circle of light 

 of uniform intensity. 



Let ABG (fig. 2) represent this circle, each point in its area is the centre of a 

 circular image of the sun's disc reflected from a corresponding point, or infini- 

 tesimal area, of the mirror. 



If be one of these points, and the circle ADBP the image of the sun's disc 

 at the given distance, or, in other words, the base of a cone of rays whose apex is 

 at the mirror, as shown in fig. 1, Art. 1, the illumination or intensity of the light 

 at will be that due to the superposition of all the images of the sun's disc 

 whose centres fall within the area ACBPA ; for none of the images of the sun's 

 disc, whose centres fall without the above area, can extend so far as 0. The 

 point is therefore illuminated by a portion of the incident beam equal in area 

 to ACBPA, every increment of which, after reflection, gives rise to a conical 

 pencil of rays, a part of whose base overspreads the point 0. 



Let L = intensity of the solar light on a plane perpendicular to the direction 

 of the incident beam. 



Let AO - r, FA = p, FO = z, AFO = 0, AOC = <f>, and u x u 2 u 3 m 4 u 5 , &c, 

 the sectional areas, at the mirror, of the respective pencils whose light over- 

 spreads the point 0. Whatever may be the form of these small increments, the 

 base of the cone of light to which they give rise will be a circle at all finite 

 distances from the mirror, as shown Art. 1. 



