52 Mr Buchanan, On a solar Calorimeter used in Egypt 



Abscissae are to be measured on a line at right angles to OP, 

 and they are positive when measured to the right. 



Join BB 1 ; and through B L draw a line OjPj parallel to OP, and 

 on it lay off B.A, = BA. 



Join AAj and produce it to a point B 2 , so that A X B 2 — AB. 



Join B^^; then B 1 B 2 is the line which represents in section 

 the innermost mirror. 



Through B* draw a straight line 0.,Po parallel to OP, and on it 

 lay off the length B 2 A 2 = BA. 



Join AA 2 and produce it to a point B 3 , so that A. 2 B 3 = AB. 



Join B»B S ; then B 2 B 3 is the line which represents in section the 

 second mirror of the series. 



Through B 3 draw 3 P 3 parallel to OP, and on it lay off 

 B 3 A 3 = BA. 



Join AA 3 and produce it to a point B 4 , so that A 3 B± = AB. 



Join B 3 B 4 ; then B 3 B 4 is the line which represents in section 

 the third mirror in the series ; and so on. 



It is evident from the properties of parallel lines that the angle 

 which the incident ray makes with the outer extremity of any one 

 of these lines is equal to the angle made with it by the line 

 connecting that point with the upper extremity of the focal line. 

 Therefore all the rays parallel to the axis which strike the outer 

 extremity of a line of section are reflected upon A, the upper 

 extremity of the focal line. In the same way all the rays parallel 

 to the axis which strike the inner extremity of a line of section 

 are reflected upon B the lower extremity of the focal line. Con- 

 sequently all the rays parallel to the axis which fall upon inter- 

 mediate points in the line of section are reflected upon the corre- 

 sponding points between A and B on the focal line. Therefore 

 all the rays parallel to the axis which strike the reflector are 

 reflected and condensed on the focal line AB. 



If the graphic construction is effected on the natural scale, all 

 the measurements, both linear and angular, can be taken from it 

 directly with sufficient exactness to enable the reflector to be con- 

 structed. On the other hand, the geometrical construction is 

 so simple that there is no difficulty in arriving at all the values 

 by calculation. It will be apparent from the diagram (Fig. 6) 

 that most of the elements of each section are contained in an 

 isosceles triangle A n B n B n+1 . In it the angle A at the apex of 

 one triangle is derived from the data of the previous triangle. 

 The angle of inclination to the axis of the mirror is 



1 = 1(180° -A), 



and the length of the base or width of the mirror is 

 m = 2AB cos i. 



