MR JOHN SCOTT ON THE BURNING MIRRORS OF ARCHIMEDES. 135 



Article 7. — The intensity at the focus of a parabolic reflector is independent 

 both of the form and distance of the luminous body. 



Let BAC (fig. 6) represent a section of the parabolic reflector, and GH that of 

 a luminous surface of uniform intensity; it can be shown, as in the case of the 

 spheroid (Art. 5), that the concentration at F produced by the only rays which can 

 fall on it, namely, those emanating from GH parallel to the axis AF, is equal to 

 the intensity at the luminous surface GH, minus the quantity lost by reflection. 

 It is evident that the section of the luminous body must not be less than CB. 



Article 8. — Prop. When the Axis of a Mirror in the form of a Right Cone is directed to the 

 centre of the Sun, to find the Intensity of the reflected Light on any point in a Plane placed 

 perpendicular to its Axis. 



Let CAD (fig. 7) represent a section of the mirror, 



the point on which we wish to determine the intensity of the 

 reflected light. 

 Every small increment of the mirror gives rise to a cone of rays which casts 

 an ellipse of light on the plane at F, the major axis of which passes through the 

 point F. The light of all these ellipses, whose centres fall within a certain 

 distance of the point 0, will overspread it and increase the intensity at that 

 point. 



If P (fig. 8) be the centre of one of these ellipses NOM, considerably magnified, 

 whose circumference passes through 0, then P is a point in the curve within 

 which must fall the centres of all the ellipses whose light can overspread 0. 



To find the equation to this curve, 

 let FP = p 



F0 = 2 

 a and j8 = the co-ordinates of P, referred to rectangular axes whose origin 

 is at 0, 

 B = the angle PFO. 

 Now cfy 1 + b 2 x 2 = a 2 b 2 is the equation to the ellipse NOM, the centre being the 

 origin. 



When referred to the axes OX and OY; by substituting, 



y = (f — /3) cosd — (a/ — a) siu.6 , 

 x = (y'~ j8) sin0 + (x' — a) cos0 , 



tan 6 = -t— , we obtain 



(a 2 cos 2 U + b 2 sm 2 d)(y'~P) 2 + (a 2 sin 2 *) + b 2 cos 2 d)(x'-af - 2(a 2 -b 2 )(y'-P)(x'-a)sin6 cosS = a 2 b 2 . 

 Putting oc'= , yf= , there results the equation to the required curve, which is 



VOL. XXV. PART I. 2 M 



