130 MR JOHN SCOTT ON THE BURNING MIRRORS OF ARCHIMEDES. 



The preceding limiting cases are also evident from the geometrical considera- 

 tion of the problem taken in connection with equation 2 ; and the same principles 

 can be applied to find the intensity, whatever the shape of the perpendicular 

 section of the incident beam. 



Article 3. — When there are n mirrors similar to the preceding, the light 



from each making an angle of incidence i, with the perpendicular to the plane 



on which it is thrown, the intensity of the central spot in each of the preceding 



cases becomes — 



In 1st case, nhl' cos i, . (1). 



In 2d case, «**?»» \ .... (2). 



Article 4. — Prop. A Small Luminous Sphere has its centre in one of the foci of a Prolate 

 Elliptic Mirror, to find the Intensity on any Small Plane surface situated in the other 

 Focus. 



Let/ (fig. 3) represent the luminous sphere. 



/3 = the angle which the small plane at F makes with FX : the axis of 

 z coinciding with the axis of the mirror, and the plane of ocz being 

 perpendicular to the small plane passing through F ; the axis of 

 y, which is at right angles to the plane of xz, will therefore 

 coincide with the small plane which passes through F. 

 r = FP. 



6 = the angle PFZ. 



(p = the angle which the projection of r on the plane of xy make with FX. 

 a — radius of the luminous sphere at/ 



V = the angle which r makes with the normal to the small plane at F. 

 I = the intensity of the light at the surface of the small sphere. 

 Since the well-known differential of a volume r 2 sin 8 d6 d<p dr has for its 

 perpendicular section, at the surface of the spheroid, r sin 6d6 d(p; we may sup- 

 pose the whole surface of the elliptic mirror to be divided into small areas, each of 

 which receives from the sphere at / and reflects to F, a pencil of light, whose 

 perpendicular section, at any point P of the mirror, is r 2 sin 9 d6 d<p. 



a 2 I 

 Moreover, the intensity at P perpendicular to P/"= -pp (by Art. 1, Equa. 3), 



2 7.T 



and after reflection it becomes -p-^- . 



The cone of rays reflected from the increment of surface at P, in the direction 

 PF, will have expanded at F into a circle (perpendicular to the radius vector), 



(XT ItOb 7 



whose radius by similar triangles is tw. and area p. 2 • 



By Art. 1. Equa. 6, the intensity of the light on this circle at F = r 2 sin 6 dB 



d(p x p „ 2 -5- p-ry = — sin ad d<p. 



