HARDWICKE'S SCIENCE-GOSSIP. 



75 



THE TWO MIRRORS. 



By W. J. N. 



No. VI. 



FROM the results obtained in the last art. we 

 deduced the following important rule with 

 respect to a simple divergent pencil : — 



In order to bring the reflected rays to a focus on the 

 object, the lamp-flame and object must occupy the 

 positions of conjugate foci of the mirror. 



How then are we to determine the positions of 

 these foci ? In other words : when a position has 

 been assigned either to the mirror or to the lamp, 

 how are we, from this assigned and therefore known 

 distance, to determine what is the conjugate position 

 for the lamp, or for the mirror respectively ? 



Let A be the distance of L from D ; that is, of the 

 lamp-flame from the centre of the mirror. Let 8 be 

 the distance from D to the point of mean focus ; that 

 is, of the object from the mirror. Let R be the 

 radius of the mirror, and cos a be the cosine of the 

 angle of incidence. 



Then, if the mirror be set at a known distance 

 from the object, 8 is known and we need to find A. 

 But if, instead, we assign a fixed distance to the 

 lamp, A will be known and we shall need to find 8. 



5 R cos a 



In the first case A 



and in the 

 2 S— R cos a 



Example.-^-\n the last 



, . A R cos a 



second, 8 = - 



2 A — R cos a 



figure, L is 4I inches from D, and A, therefore, is 4*5. 



The mirror has a radius of 3 inches (= R) ; and the 



angle of incidence is 45 , of which angle the natural 



cosine is "707. Substituting these values in the 



formula for 8, we have — 



s _ 4-5X3X-7Q7 _ 4'5X'7Q7 = 3 '1815 

 2(4'5)-(3X'7°7) " 2(1-5)- -707 " 2-293 

 = 1*4 inch nearly. In such a case, therefore, the 

 little mirror represented would be correctly set at about 

 1 § inch from the object. This is the distance, in the 

 figure, of the point f- from the centre of the mirror, 

 as found by projection of the rays. 



From one of the above formulae the student 

 should construct a table of [foci for his own mirror. 

 The following table is for a mirror of 8 inches radius. 



The table is instructive. If the mirror be 4J 

 inches from the object, and the selected angle of 

 incidence be 30 , the lamp should be placed 15 

 inches from the mirror. This will give a satisfactory 

 result, provided the other conditions are observed, 

 the mirror being in the axis of the instrument, the 

 lamp-wick turned down low, and having one of its 

 corners (not the flat side) turned towards the mirror. 

 It must ever be remembered that only a minute 

 flame, turned with its narrow edge towards the 

 mirror, can, at short distances especially, fitly re- 

 present the point L in the figures. If the mirror be 

 brought half an inch nearer the object, the lamp 



Table of Conjugate Foci for a simple Divergent Pencil 

 obliquely incident on a Concave Mirror of eight inches 



radius. 



must be raised a little and removed 10J inches 

 further from the mirror. A great loss of light will 

 result, the intensity of the light upon the surface of 

 the mirror varying, as we have seen (Fig. 21), in- 

 versely as the square of the distance. The difference 

 in this case will be represented nearly by the ratio 

 15 2 : 25 2 or 9 : 25, so that nearly two-thirds of the 

 light will be lost. If the mirror be brought half an 

 inch still nearer to the object, the theoretically true 

 position of the lamp would be more than 25 feet 

 distant ; which tells us that for that position of the 

 mirror we should have to give up our simple diver- 

 gent pencil, and use a parallel one instead. Were 

 the mirror again moved half an inch, its distance 

 from the object being reduced to 3 inches, neither 

 a divergent nor a parallel pencil could be focused 

 on the object. The incident rays would, in that 

 case, need to be convergent. Taking the table as a 

 whole, it will be noticed that when the lamp is 

 within a reasonable distance of the mirror, the 

 mirror will need to be much further from the object 

 than when a parallel pencil is employed. For 

 instance, if the angle of incidence be 45 , the mirror 

 of 8-inch radius should for a parallel pencil be 2| 

 inches from the object (see vol. for 1886, p. 266) ; 

 whereas 4J, 4, or 3 J inches would be suitable 

 distances for use with our simple divergent pencil, as 

 will appear from an inspection of the table. 



When a compound divergent pencil is employed, 

 the positions assigned to the lamp, in the table for 

 simple pencils, will not apply. A deduction must 

 be made, which involves some further calculation. 



Fig. 33 represents such a pencil emanating from the 

 point /, and intercepted by the bull's-eye at a 

 distance slightly less than that of its principal focus. 

 In passing through the bull's-eye the rays are 

 refracted, and their new lines of direction become 

 the same as those of a simple pencil emanating from 

 the point L. Having L D given in our table for 



t 2 



