266 



HARDW1CKKS SCIENCE-GOSSIP. 



3. The correct position of the bull's-eye to render 

 parallel the diverging rays of the lamp. 



I. The correct distance of the mirror from the 

 object depends partly upon the curvature of the 

 mirror. Assuming this to be unknown, we must 

 ascertain the whereabouts of the "principal focus." 

 It can be done thus : Place the microscope at one 

 end of the table, and the lamp at the other end, 

 taking care that the lamp-flame and the mirror are at 

 the same height from the table. At that distance the 

 rays will be fairly parallel. Turn the concave mirror 

 so that it accurately faces the lamp. Take a narrow 

 strip of thin paper, and hold it directly between the 

 lamp and the mirror, at the distance of a few inches 

 from the latter. Move the paper gently to and fro, 

 towards mirror and lamp alternately, until a position 

 is found at which the sharpest image of the flame is 

 projected from the mirror upon the paper. That 

 point is the " principal focus " of the mirror. Measure 

 the distance between strip and mirror ; it is the 

 principal focal distance. Double it ; the product is 

 the radius of the sphere, of which the mirror is a 

 part. 



Knowing the principal focal distance, the correct 

 position for the mirror, when parallel rays are to be 

 brought to a focus on the object, may be found for 

 any angle of incidence thus : 



Multiply the natural cosine of the given angle of 

 incidence by the principal focal distance of the mirror ; 

 the product is the required distance between object and 

 mirror. 



A table of natural cosines is given in any book of 

 mathematical tables, but the following short list, 

 carried to three places of decimals only, will probably 

 be all that the reader will require. 



Angle of Incidence 30 35 40 45 50° 55° 6o° 70 

 Nat. Cos. of Angle '866 '819 "766 "707 -643 '574 '5 '342 



Example : The small mirror represented in Figs. 

 158 to 161 (see last Art.) has a radius of three inches, 

 so that its principal focal distance is 1*5. If we 

 select the angles 30 and 6o°, and multiply their 

 cosines by I'S, the two products are "866 X .1*5 = 

 1 "299, and '5 X l - 5 ='75. The correct distances for 

 such a mirror from the object would be, therefore, 

 1 \ inch and f inch, in the cases supposed. These are 

 about the actual distances from D of the points of 

 mean focus, F 1 and F 3 , as found by construction in 

 the Figs. 159 and 161 of the last Art. 



The concave mirror of my own microscope is part 

 of a sphere of eight inches' radius, its principal focal 

 length being four inches. The following Table applies 

 to it, and to all mirrors of like curvature. 



Table of Foci for Parallel Rays of Oblique Incidence, after 

 reflection from a Concave Mirror of 8 inched Radius. 



Angle of Incidence 

 Distance of Meani 

 Focus from Cen 

 tre of Mirror in ' 

 inches and deci- 

 mals of an inch 



30° 35 40° 45° 50° 55° 60° 



70- 



If vulgar fractions be preferred, these lengths will 

 be very nearly 3J, 3J, 3, 2§, 2§, i\, 2 and if inches 

 respectively. 



The reader should construct such a table for his 

 mirror, and paste it inside his microscope- case. 



Instead of bringing the rays to a focus on the 

 object, it is sometimes better, as we have seen, to 

 take advantage of the more oblique marginal rays by 

 bringing the mirror somewhat nearer the stage, so 

 that the object may be in the position occupied by/ 

 in Fig. 160, and dispensing with the diaphragm. In 

 such a case the mirror may be set at about the 

 tabular distance of the angle, which is lo° larger than 

 the one actually employed. 



In the present case oblique rays are, by supposition, 

 to be excluded, and the angle of incidence is to be 

 30 . The correct position of our mirror of 8 inches' 

 radius will, therefore, be at a distance of 3J inches 

 from the object, and in the axis of the instrument. 

 The reader will see the utility of filing small notches 



.3*46 3-28 3-06 2-83 2-57 2'3 2-0 r'4 



upon the stem which carries the mirror, to represent 

 inches and half-inches of distance between the centre 

 of the mirror and an object on the stage. 



2. Having determined the position of the mirror, 

 we have next to determine that of the lamp. This 

 would in any case be comparatively easy, if we used 

 our microscopes in a horizontal position ; or, failing 

 that, if we used no other angle of incidence than 45 . 

 Directly we begin to tilt the tube, there is, with any 

 other angle of incidence than 45 , an element of 

 difficulty added. To simplify matters we must select 

 some two angles of slope for the tube of the micro- 

 scope, one of them being that which is most conve- 

 nient for the microscopist himself, the other for 

 friends whose height may differ from his own ; and 

 after a careful selection must employ no others. It 

 will also save much time and trouble if we have some 

 kind of tray beneath the instrument and lamp, on 

 which lines may be drawn representing correct 

 positions of the lamp, as determined by calculations 

 now to be explained. Fig. 166 represents such a tray 



