136 MR JOHN SCOTT ON THE BURNING MIRRORS OF ARCHIMEDES. 



the locus of the centres of all the preceding ellipses whose transverse axes 

 intersect in F and their circumferences in 0, 



(a 2 cos 2 S + b 2 sm 2 6j^ 2 + (a 2 sm 2 6 + b 2 cos 2 6)a 2 -2(a 2 -b 2 )a^smdcosd = a 2 b 2 (1), 



. • . (a 2 + h 2 tan 2 6) /3 2 + (a 2 tan 2 6 + 1 2 ) a 2 - 2 (a 2 - h 2 ) «/3 tan 6 = a 2 h 2 (1 + tan 2 6) . 



Substituting for tan 6 its value, we obtain 



{a\z + uf + L 2 I3 2 }!3 2 + {a 2 (3 2 + b\z + a.) 2 } a 2 - 2(a 2 - b 2 ) (e + «) a$ 2 = a 2 b 2 {(z + a) 2 + ,S 2 } , 

 .-.a 2 z 2 l3 2 + h 2 {(3 2 + a(z + a)} 2 = a 2 b 2 [(z + a) 2 + (3 2 } .... (2). 



By substituting in this z + a = p cos 0, and (3 = p sin 0, we have the polar equation 

 to the curve, F being the origin, 



a 2 2 2 g 2 sin 2 6 + b 2 (g 2 sin 2 6 + g 2 cos 2 6 — zg cos d) 2 = a 2 b 2 f , 



.'. a 2 z 2 i 2 sin 2 6 + b 2 g 2 ( s -z cos 6) 2 = a 2 b 2 s 2 , 



z 2 

 and g =zcos 6 ±a(l— ~z sin 2 6y> .... (3). 



It is evident that the form of the curve represented by the Equations 2 and 3 

 will vary with the relative values of the constants a, b, and z; but in every case 

 it is symmetrical with respect to the axis FX (fig. 8). 



"When s=Q, the point is situated on the axis of the mirror, and the curve 

 becomes the circle p = a . 



And if z — b, the equation breaks up into two circles whose centres lie in the 

 line FO, and which touch one another externally at the point F, their diameters 

 being a + b and a — b respectively. 



When the point F falls without the curve, the radius vector becomes a tangent 



for the value sin = - . 



Putting = in Equation (1), we get for a plane mirror 



a 2 !?- -\- Voir — ccb" , 

 an ellipse with for its centre, and equivalent to 



Now, if PHG (figs. 9 and 10) represent the curve, considerably magnified, 

 expressed by equation 3, in deducing the intensity at the proposition divides 

 itself into cases depending on the relative positions of the points F and (as in 

 figs. 9 and 10). 



Case 1. When the distance FO from the axis (fig. 7) is so small compared 

 with FB that the distance of any point within the curve similar to PHG from 

 CB, that part of the mirror where the light which overspreads O is reflected, may 

 be considered equal to FB. 



