132 MR JOHN SCOTT ON THE BURNING MIRRORS OF ARCHIMEDES. 



M 

 Also o s i n ft is th e intensity at F due to the light reflected from the part of the 



mirror intercepted between the small plane produced, and the co-ordinate plane 

 of xy. 



In the same manner it can be shown, that the intensity of the light concen- 

 trated at F, on the opposite side of the small surface, and reflected from the 

 remaining portion of the spheroid, is also equal to M. 



These results are independent of a, the radius of the luminous sphere, and are 

 equally true for all spheroids which have F for one of their foci, wherever the 

 other may be situated. 



It appears then, in conclusion, that the light emanating from a small luminous 

 sphere, with its centre in one of the foci of a prolate elliptic mirror, produces at 

 the other focus a nucleus of radiant light and heat, equal in intensity to the 

 radiation at the luminous surface diminished by the quantity lost by reflection. 



Again, putting /3 = (in Equation 2), we obtain intensity 



= — 1 1 sin0 cos 6 dd d<p , . . . (4). 



Integrating between <p = 0, and 2«r; 



M r 



intensity = 2«r x — / sin & cos 6 dd , 



sin 6 cos d dd ; 



= 2H /si 



and between 6% and zero, 



= hi sin H , . . . . (5). 



This expresses the intensity at the focus of the light reflected from a segment of 

 the spheroid intercepted between the vertex and a plane perpendicular to the 

 axis ; and the intensity produced by a zone intercepted between two planes, per- 

 endicular to the axis of revolution, is 



hi (sin 2 6 - sin 2 6) ... (6). 



Article 5. — The preceding proposition is true, independently of the size and 

 form of the luminous body in the focus /(figs. 3 and 4). 



Since radiant light and heat diminish in the inverse ratio of the square of the 

 distance, it follows that the quantities received from circular areas of equal angular 

 magnitudes are equal, whatever their absolute magnitudes, when the intensities 

 at the radiating surfaces are equal. Taking this principle in connection with the 

 fact, that a luminous surface appears equally bright when viewed at any angle, 

 the light emanating from CD, part of the surface of DCE, will therefore have at F 

 the same intensity as if it had proceeded from the small sphere AB (fig. 4). But 

 the light reflected from P to F can only emanate from some part of the surface 

 DC, which lies within the cone described by PA, revolving about P/. Hence the 



