MR JOHN SCOTT ON THE BURNING MIRRORS OF ARCHIMEDES. 131 



M 

 and — sin dO d<f> cos V = its intensity at F on the small plane, whose equation 



is z cos /? — x sin /3 = 0, ... . . . (1), 



cos V = - cos j8 sin (S = cos (3 cos 9 — sin (3 sin cos <ft. 



Substituting for cos V its value we obtain for the whole intensity at F, on the 

 small plane, 



— / / (cos /3 cos 6 — sin /3 sin 6 cos p) sin d c?0 c?p, . . (2). 



Denoting this integral by u, 



%=.--! (cos jS sin 2 6 + sin /3 sin d cos cos p — d sin j3 cos p) d<p + C 



Now cot = tan /3 cos <£, when V = 90°, that is, when PF coincides with the plane 



denoted by Equation 1, 



.'. 6 = cot — *(tan jS cos p) 



1 



sin 6» = 



cos 6 = 



(1 + tan 2 /3 cos 2 p)* 

 tan /3 cos p 



(1 + tan 2 /S cos 2 p)* 



Taking the integral between the limits = 0, and = cot - 1 (tan £ cos <f>), 



M /* f cos |S (1 + tan 2 j8 cos 2 p) ' . 1 , 



U = ^J I ' I + tan 2 /3 cos 2 p COt ^ tan * C0S ^ S1D ^ C ° S ° ) CZ? " 



H f /» . . , ,,, _ . /'sin |8 tan |3 sin 2 p dp \ 



= 2* \ J C0S * *' ~ Sm * Sm P COt (taD P C ° S ?) + / l+tan^cos 2 ^" J 



= g^ f y^os ^ c?p — sin /3 sin p cot- 1 (tan /3 cos p) -y cos £ tfp +y ^ 2;3 + tan 2^ j 



= o- { tan -1 (cos |3 tan p) — sin /S sin p cot -1 (tan |S cos p) > + C. 

 and between the limits (p = and <£ = 90°, 



H / IT It . \ 



M =2A2-2 Sin/3 > 

 = — (1 — sin /3), which is the intensity at F of the light reflected from 



that portion of the spheroid bounded by the planes passing through the 



co-ordinate axes of + x and + z, + y and + z, and the small plane produced. 



M 

 Similarly ^-(1 + sin/3) gives the intensity of the light, from that portion 



enclosed by the planes passing through the axes — x and + z, + y and + z, and 

 the small plane at F produced. Hence the intensity of all the light which can 

 fall on the side of the small plane towards A (fig. 3). 



==2(^(l-sm/3) + ^(l+sm0) }, 



= M, (3). 



VOL. XXV. PART 1. 2 L 



