140 MR JOHN SCOTT ON THE BURNING MIRRORS OF ARCHIMEDES. 



It can be shown, as in Case 1, Equa. 7, that the concentration at 0, due to the 

 light reflected from the space around C, is 



2 k ELY cos 2 2i ffj /li . . . . . , 



/ / dr ad, and taken between j x and f 2 , 



*6 2 



2 /g Br cos 2 2 

 ,r£ 2 



4&RI'acos 2 2 



«r6 2 



if(l-*sm*6)ldd, 



because (fig. 10) 



fl = FP = z cos d + a (1 - L sin 2 0)i , 

 g 2 = FP' = 2 cos 4 - a (1 - ~ sin 2 *)* , 



fi-g 2 = 2(l- p sin 2 ^. 



Taking the integral between the required limits sin = - and = we obtain 



4ftRI'a cos 2 2a, , -□-, , iri , 

 ^ (A-H), . . (12), 



(A — H) z denoting the difference between the asymptote and the infinite hyper- 



b 



bolic arc whose major axis is unity, and eccentricity j , a finite quantity, though 

 A and H are severally infinite. But, if the distance from the centre to the focus 

 be equal to unity, the transverse axis is r and (A — H) z = j (A — H), which 

 expressed in complete elliptic functions of the second order gives 



(A - H^ = —j~ E ei — -jE e , 



e being put for - , the relation between the moduli e and e v being 



2_v5 _ 2jFz . 

 6 i ~~ 1 + e ~ z + b ' 



and e and e\ denoting the corresponding quantities when b' is put for b. 



Substituting in Equation 12, we get for the light reflected from the space 

 around C, 



4&RI'a cos 2 2i f z + b -^ z t? \ 

 Vb 2 1 ~~b~ " ~ b e j ' 



