„„,yBenY OF CAIHORNIA UtRARY 
 
 UC-NRLF 
 
 B 3 aic^ 7i5 
 
Digitized by the Internet Archive 
 
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 IVIicrosoft Corporation 
 
 http://www.archive.org/details/elementaryopticsOOunitrich 
 
1065 
 
 1065 
 
 ELEMENTARY OPTICS 
 
 AND 
 
 APPLICATIONS TO FIRE CONTROL 
 INSTRUMENTS 
 
 PREPARED UNDER THE DIRECTION OF 
 THE CHIEF OF ORDNANCE 
 
 May, 1921 
 
 WASHINGTON 
 
 OOVERNMKNT PRINTING OFFICE 
 
 1922 
 
PHWCS OK"- 
 
 ilF ?is 
 
 fMYSICS t)EPT, 
 
WAR DEPARTMENT, 
 
 Washington, 3Iay 10, 1921. 
 
 The following publication, entitled ''Elementary Oj)tics and 
 Applications to Fire Control Instruments," is published for the 
 information and guidance of all concerned. 
 \mi.\, A. G. o.] 
 
 By order of the Secretary of War: 
 
 PEYTOX Q. MARCH, 
 
 Major General, Chief of Staff. 
 Official: 
 
 P. C. HARRIS, 
 
 The Adjutant General. '-■ : 
 (3) 
 
 810829 
 
:n. 
 
 PREFACE. 
 
 The aim in the jDreparation of this pamplilet lias been to provide 
 a handbook of apphed optics and optical instruments suitable for use 
 as a textbook in the Army training schools , and for the information 
 of the officers and men who use optical fire control apparatus. 
 
 In the selection of material no attempt has been made to produce 
 an exhaustive treatise. It has rather been the purpose to treat more 
 fully those phases of the subject which are not dealt with in the books 
 ordinarily available, or which are particularly characteristic of fire 
 control instruments, and to avoid the unnecessary duplication of 
 material which is already accessible in elementary textbooks so far 
 as is possible without the undue sacrifice of completeness. 
 
 For the reader who desires more complete information, the follow- 
 ing books, all of which are available in English, are recommended: 
 
 Crude: Theory of Optics. Translated by I\Iann and Millikan. Longmans, Green 
 & Co. 
 
 Edser: Light for Students. Macmillan & Co. (Ltd.), London. 
 
 Nutting: Outlines of Applied Optics. P. Blaldston's Son & Co., Philadelphia, Pa. 
 
 Taylor: A System of Applied Optics. Macmillan & Co. (Ltd.), London. 
 
 Steinheil and Voit: Applied Optics. Translated in two volumes by P'rench. 
 Blackie & Son (Ltd.), London. 
 
 Southall: The Piinciples and Methods of Geometrical Optics. The ]\Iacmil]an Co. 
 
 Gleichen: The Theory of Modern Optical Instruments. Translated by Emsley and 
 Swjdne. H. AL Stationery Office, London. 
 
 Von Rohr: The Formation of Images in Optical Instruments. Translated by R. 
 Kanthack. H. M. Stationery Office, London. 
 
 Hovestadt: Jena Glass. Translated by J. IJ. ct A. Everett. Macmillan & Co. 
 (Ltd.), London. 
 
 (4) 
 
TABLE OF CONTENTS. 
 
 Page. 
 
 Tlio reflertion and refraction of light 7 
 
 T>aw of reflection — Law of refraction — Absolute and relative index — Re- 
 flection fi-om metallic surface — Reflection from transparent surface — 
 Total reflectioji. 
 
 The dispersion of light 10 
 
 De\iation by a prism — Minimum deviation — Production of spectra — 
 Physical nature of light — Spectral lines — ^fethod of specif>dng optical 
 glass. 
 
 Atmospheric refraction ^ . . 13 
 
 Path of light in a nonhomogeneous meditim — Influence of refraction ttpon 
 the visibility of distant objects — Effects due to irregular refraction by 
 atmosphere. 
 
 Plajie mirrors IG 
 
 Image of a point — Image of an extended object — Reversion — Image pro- 
 duced by two successive reflections — Image prodiiced by two perpen- 
 dicular mirrors. 
 
 Prisms 20 
 
 Measiuing wedge — Achromatic prism^ — Right-angle prism — Porro prism, 
 first and second type — Right-angle prism with roof — Rotating prism — 
 Brashear-Hastlngs prism — -One-piece erecting prism with roof — Aiming- 
 circle erecting system — Triple mirror — Penta prism — Penta reflector — 
 Rhomboidal prism. 
 
 The spherical reflecting surface 27 
 
 Image of an axial point — Focal length — Searchlight mirror — Parabolic 
 reflector. 
 
 The spherical refracting surface 30 
 
 Image of an axial point. 
 
 The thin lens 31 
 
 Image of an axial point — Image of an extended object — Convergent and 
 di^•ergent lenses — Simple magnifying lenses — Di^'erging lens in the floor 
 of an airplane — <."ylindrical lenses. 
 
 Tliick lenses and combinations 37 
 
 Location of principal points— Equivalent, back and front focal lengths — 
 Principal planes — Two thin lenses in contact — Two separated lenses. 
 
 The aberrations of a lens -40 
 
 Effect of aberrations upon the image — Spherical aberration — Chromatic 
 aberration — .\stigmatism — Coma — Curvature — Distortion. 
 
 Resolving power 4() 
 
 The telescopic system 47 
 
 Image of an axial point — Image of an extra-axial point — Magnifying jiower— 
 Galilean type of telescope — Reflecting telescope. 
 
 The telescope and tlie eye 51 
 
 Parts of the eye — Myopia — Hypermetropia — Astigmatism — Field of A-iew — 
 Combined action of eye and telescope — Focusing of telescope for the 
 individual eye — Depth iji the image — Stereoscopic vision — Radius of 
 stereoscopic vision. 
 
 C5) 
 
6 
 
 Page. 
 
 The components of the telescope 5G 
 
 (Objectives — Eyepieces —Reticules Adjustments necessary for focusing— 
 Parallax — Fixed-focus telescopes. 
 
 The telescope with an erecting system 62 
 
 Lens erecting system — Different types of prism erecting systems. 
 
 The field of view and brightness of image f!7 
 
 True and apparent field Position of the eye — Exit pupil — ^Night glasses. 
 
 The selection and use of a telescope 70 
 
 Desirable optical properties for hand telescopes — Focusing of the telescope — 
 Tests of definition — Tests for film on reticule. 
 
 The optical characteristics of service fire-control instrumejits 73 
 
 Telescopic musket sight, model 1913— French aiming circle— Macliine-gun 
 panoramic sight— Azimuth instrument, model 1918 — Azimuth instru- 
 ment, model 1910 — Observation telescope, " Longue Aaie Monoculaire " — 
 2-inch telescopic sight — 3-inch telescopic sight — Lewis depression ])osi- 
 tion finder— Telescopic sight for tank gun— Telescopic musket sight- 
 Telescopic sight for 37-mm. infantry gun— Officer's trench periscope 
 No. 10; battery commander's periscope — Periscopic azimuth instru- 
 ment, model 1918 — Right-angle telescope — Panoramic sight, 4 power — 
 Panoramic telescope, 4 and 10 power — Binocular, type EE — Battery 
 commander's telescope— Aiming circle, model 1916. 
 
 The collimator 82 
 
 Principles of collimator— Tj^pes of construction— Use of Collimator- 
 Collimator on Michelin bomb sight. 
 
 The coincidence- type self-contained range finder 85 
 
 Fundamental triangle — Optical system of range finder — Types of field — 
 Measuring wedge — Adjusting wedge. 
 
 The construction of the range finder 89 
 
 Range scale— Rotating compensating prisms— Equal-magnification lens - 
 Ocular prism— Optical tube — Astigmatizer— Optical characteristics of 
 range finders used by the service — Transmission of a range finder. 
 
 The azimuth type of coincidence range finder 93 
 
 The stereoscopic range finder 94 
 
 The errors of the range finder 96 
 
 Accidental errors— Table showing effect on range readings— Systematic 
 errors. 
 
 The infinity adjustment of the range finder 
 
 By known range — By Tise ,of celestial object— Adjusting lath — Errors of 
 adjusting lath— Barr & Stroud type of internal adjustment— Internal 
 adjustment by triple mirrors— Zeiss absolute internal adjustment. 
 
 99 
 
ELEMENTARY OPTICS AND APPLICATIONS TO FIRE-CONTROL 
 INSTRUMENTS. 
 
 THE REFLECTION AND REFRACTION OF LIGHT. 
 
 The path of a ray of light traveling in a homogeneous medium is a 
 straight line. When, however, a ray is incident upon a surface sep- 
 arating two media, the path in general is no longer straight. The 
 ray is divided into two portions, one of which does not enter the 
 second medium but is reflected back into the first, while the other 
 continues into the second medium but is refracted and travels in a 
 different direction. 
 
 If a normal is erected to the surface at the point where it receives 
 
 the incident rav, the angles which the incident, 
 
 refracted and re- 
 respectivoly, the 
 
 fleeted rays make with the normal are termed. 
 
 angles of incidence, refraction, and 
 
 reflection. The laws of reflection 
 
 and refraction, which state the 
 
 relations existing between these 
 
 angles, are the fundamental laws 
 
 upon which instrument design is 
 
 based. 
 
 The law of reflection. — The angles 
 of incidence and reflection lie in 
 the same plane and are equal. 
 
 The law of refraction. — The 
 angles of incidence and refraction 
 lie in the same plane, and the sine 
 of the angle of incidence divided by 
 the sine of the angle of refraction 
 is equal to a constant n characteristic of the two media and termed 
 the index of refraction of the second medium with respect to the 
 first. These two laws are sufTicient to determine the path of a ray 
 when the initial path and boundary and indic(>s of refraction for each 
 medium are given. In mathematical form they may be expressed 
 ])y the two e([uations: 
 
 sin (' = /( sin i' (1) 
 
 i = i" (2) 
 
 where i is the angle of incidence, /"' the angle. of refraction, and i" 
 
 the angle of reflection. 
 
 (7) 
 
 «% 
 
 ^ 
 
8 
 
 The refraction and reflection of a ray of light is ilkistrated in figure 
 1. AB is the trace of the refracting surface which separates the first 
 and second medium. CN is the incident ray incident upon the re- 
 fracting surface at N. It is there divided into two portions, the 
 reflected portion which remains in the first medium traveling along 
 the line ND and the refracted portion which enters the second me- 
 dium, is deviated and travels along the line NE. The angle d, be- 
 tween the refracted ray and the incident ray produced, is termed the 
 "angle of deviation" and is the angle through which the ray is bent 
 from its original path. From the drawing it is evident that it is 
 equal to i — i'. It should be noted that if a ray is incident normally 
 upon a surface sin i is zero, and by the equation (1), sin i' is also zero. 
 Angles i and i' are equal and the ray is not deviated. 
 
 The constant n in equation (1) is the index of refraction of the sec- 
 ond medium with respect to the first and is a relative index. The 
 index of refraction of any substance with respect to a vacuum is 
 termed the ''absolute index." If n^ and lu are the absolute indices 
 of refraction of two substances, the value of ri, the index of the 
 second with respect to the first, is "'/„,. Equation (1) is therefore 
 often written in the more symmetrical form: 
 
 n^ sin 1 = 712 sin /' (3) 
 
 From this equation it is seen that the index of a first medium with 
 respect to a second is the reciprocal of the index of the second with 
 respect to the first. 
 
 The absolute index of air at atmospheric pressure is 1.00029. The 
 absolute index and index with respect to air are commonly consid- 
 ered identical, a relationship which would be strictly true if the index 
 of air were 1 instead of 1.00029. The indices of refraction of the 
 types of glass used in optical instruments lie between 1.5 and 1.7. 
 
 Of two substances, the one which has the greater absolute index of 
 refraction is said to be the more dense optically. When a ray passes 
 from a less dense to a more dense medium, it is bent toward the nor- 
 mal as shown in figure 1. If bent away from the normal, it is passing 
 from a more dense to a less dense medium. 
 
 In the above, it has been tacitly assumed that the surface con- 
 cerned is a polished surface. If the surface is rough, as for example 
 the ground surface of glass, and a beam of light falls upon it, rays 
 falling upon different portions of the surface after reflection or re- 
 fraction do not proceed in common directions. This is not because 
 the laws as stated above do not hold. Rather, the rough surface is 
 to be considered as composed of numerous elementary surfaces 
 turned in different directions, at each of which regular reflection and 
 refraction takes place. The net result of the many small surfaces is 
 the scattering of the beam. Such reflection and refraction is termed 
 
9 
 
 "irregular" or " diffuse, " in contradistinction to regular reflection 
 and refraction at ])olislied surfaces. 
 
 The magnitudes of the reflected and refracted i)ortions of the light 
 are dependent upon the characteristics of the surface, the material 
 of which it is made and the angle of incidence. In general a polished 
 metallic surface reflects the larger portion of the light. The refracted 
 part which is very small is entirely absorbed before it penetrates very 
 far into the metal. That some of the light does actually enter the 
 metal is shown by the fact that metal in very thin films is transparent. 
 A freshly polished silver surface may reflect as much as 98 per cent 
 of the incident light. 
 
 If we have a polished transparent surface, the amount of reflected 
 light varies with the index of refraction of the material and the angle 
 of incidence. For a surface sep- 
 arating glass and air from 4 to 6 per 
 cent is reflected when the angle of 
 incidence is small, that is, when 
 the light falls upon the surface 
 at nearly normal incidence. The 
 amount reflected increases with 
 the angle of incidence until at graz- 
 ing incidence all is reflected. 
 
 Equation (3) may be written in 
 the form: 
 
 sm t = sm I . 
 
 (4) 
 
 ■Angle of incidence less 
 angle 
 
 If the ray is traveling within a i 
 dense medium and falls upon a 
 surface bounding a less dense medium, the quotient "'/„, is greater 
 than 1. 
 
 It is seen that under such circumstances there may be values of 
 sin i, for which equation (4) indicates a value of sin /' greater than 
 1, for which obviously there is no corresponding angle. This wnll be 
 the case for any value of i such that sin i is greater than ""/„,. In 
 this case there is no refracted ray but all the light is reflected, the 
 ordinary law of reflection holding. Such reflection is termed '' total 
 internal reflection." For any angle i, the sine of which is less than 
 "Vn,. we have both reflection and refraction. The angle, the sine of 
 which is "V„„ is termed the ''critical angle." For this angle the 
 refracted ray makes an angle of 90 degrees with the normal f that is, 
 it proceeds in the second medium along the boundary surface. Fig- 
 ures 2, 3, and 4 show the course of a ray of light incident respectively 
 at angles less than, equal to, and greater than the critical angle. 
 The lettering is the same as in figure 1 . 
 
10 
 
 The light is travehng from a more dense to a less dense medium. 
 Accordingly, in figure 2 the ray after refraction is bent from the 
 normal. In figure 3 the ray is incident at the critical angle and the 
 refracted ray travels along NB. In figure 4 we have total internal 
 reflection and there is no refracted ray. All the light is reflected at 
 AB and travels in the direction ND. 
 
 For a surface separating glass and air, if the index of refraction 
 is 1.5, the critical angle is 42°, since 
 
 If, therefore, a ray of light traveling in glass incident on a surface sepa- 
 rating glass and air makes an angle with the normal greater than 
 42°, none of the light will pass into the air, but all of it will be reflected. 
 
 Fig. 3.— Angle of incidence equal to critical angle. 
 
 D B 
 
 Fig. 4.— Angle of incidence greater than critical 
 angle. 
 
 THE DISPERSION OF LIGHT. 
 
 Let FGH, figure 5, be the cross section of a prism of glass bounded 
 by the polished faces FG and FH. The angle a between these faces 
 is termed the refracting angle of the prism. Since the rays of light 
 to be dealt with are not incident upon the face GH, its location or the 
 condition of its surface is not essential. The ray AB incident at B 
 is deviated at the first surface through the angle i^ — i'i, and at the 
 second face is turned further through the angle i'^ — i^. If d is the 
 angle between the entrant ray AB and the emergent ray CD, then 
 d = i^ — i\-\-i\--i., (5) 
 
 We have the following relations existing between the four angles 
 of the right-hand member: 
 
 sin I, = 11 sin i\ 
 
11 
 
 The value of d is a function of a, n, and i^. For a given prism a 
 and n are constant and d then is a function of i^. It can be. shown by 
 differentiation that d is a minimum for the value of i^ which makes 
 ^l and i\ equal. The ray then travels through the prism in a sym- 
 metrical manner, as shown in figure 6. This minimum value of d 
 which we shall denote by do is termed the ''angle of least deviation." 
 Its value may b(» determined ])y the equation 
 
 sin 2 (a+do) 
 
 (0) 
 
 It is by means of this equation that the index of refraction of a prism 
 is commonly deter- 
 mined. The angles F 
 do and a are meas- 
 ured by means of a 
 spectrometer and 
 the value of n ob- 
 tained by the substi- 
 tution of the values 
 in the above equa- 
 tion. 
 
 We have thus far 
 spoken of a prism or 
 piece of glass as 
 though it has but a 
 single index of re- 
 fraction. The true 
 situation is not so 
 simple. The index 
 of refraction takes 
 on a different value 
 for each color. 
 
 This is illustrated in figure 7. White light is not to be considered 
 as a fundamental type of light, but is rather a composite mixture 
 of all colors in a certain definite proportion. Suppose AB represents 
 a ray of such light. When it falls upon the first face of the prism at 
 B, each color of which the ray is composed is bent through a differ- 
 ent angle as the indices of refraction for the different colors are 
 different. Consequently the beam is spread out fanwise, as indi- 
 cated with the red bent the least, the blue the most, and the 
 other colors in intermediate positions. At the second face of the 
 })rism the colors are still more widely separated and if the beam 
 is allowed to fall on a white surface we obtain a long colored strip, 
 red at one end and blue at the other. Such a colored .band is 
 termed a '"spectrum." Tl:e breaking up of the original beam into its 
 
 Deviation of ray by a prism. 
 
12 
 
 phenomena are propagated. 
 
 separate colors is said to be due to the dispersion of the prism, which 
 is anotlier term for the variation of index with color. 
 
 It has been shown that light is periodic in nature and is })ropagated 
 as waves in a manner analogous in some respects to the propagation 
 of sound. In the case of sound the waves are air waves and the 
 pitch of the sound depends upon the frequency of the waves. We 
 know that light travels through a vacuum and it is therefore evident 
 that the waves do not depend uj)on the air for propagation. It is 
 considered that the light waves are waves transmitted by a hypo- 
 thetical substance termed the " ether," which is assumed to occupy all 
 space including even that occupied by matter and which is the 
 medium by which optical, electrical, magnetic, and gravitational 
 
 Light of different colors differs only in 
 the frequency with 
 which the waves 
 succeed each other; 
 that is, light differ- 
 ing in color differs 
 in a manner analo- 
 gous to tha t in which 
 tones of different 
 pitch differ. For 
 the blue light, the 
 frequency is higher 
 and the wave length 
 shorter than for the 
 red. 
 
 An examination 
 of the spectrum pro- 
 duced by an incan- 
 descent body shows 
 that it is made up, 
 not of a definite 
 number of distinct colors, but that no two parts are the same and that 
 different portions blend into each other by passing through a large 
 number of intermediate colors. This indicates that the frequency 
 of the light varies continuously from one end of the spectrum to 
 the other. 
 
 The spectrum of the sun contains numerous dark lines superposed 
 on the continuous background. These dark lines correspond to fre- 
 quencies which are absorbed by the atmospheres of the sun and earth 
 and do not reach the observer. As the relationship between these 
 lines and the colors of the spectrum are invariant the lines are con- 
 venient "landmarks" for designating parts of the spectrum with 
 pre(;jJ9Jyn. These lines have been designated by letters and the 
 
 G' ^H 
 
 Fig. 6.— Path of minimum deviation through a prism. 
 
13 
 
 C, D, F, and G' linos are the ones commonly referred to in optical 
 computations. The D line lies in the yellow portion of the spectrum 
 and is near the region of greatest luminosity, i.e., for light of approxi- 
 mately this color the greatest reaction upon the eye is produced in 
 proportion to the energy present. 
 
 The C and F lines lie on either side of the D line in the red and blue- 
 green and sufficiently near the ends of the visible spectrum to be 
 suitable for use in computations carried on to determine whether an 
 optical system is satisfactorily corrected for the entire range of color. 
 If the instrument is for photographic purposes, the G' line, further 
 in the violet than the F line, is used in making the corrections, as the 
 photographic emulsion is more sensitive to the blue end of the 
 spectrum. 
 
 In selecting glass for use in a telescope it is important to know the 
 index of refraction for the C, D, and F lines. The color to which an 
 index refers is indicated by a subscript as Hd referring to the D line. 
 The quantity np-iic is termed the 
 mean dispersion, as this quantity is a 
 measure of the extent of the spectrum 
 produced by a prism. 
 
 The differences no—nc and np — no 
 are partial dispersions. The quotient 
 
 —^ is represented by the symbol ^ 
 
 V and is generally tabulated in cata- 
 logues of glass, as its value is of im- 
 portance in determining the desir- ^' ^. '. T""*^ . 
 
 r fv r ^ Fig. 7.— Dispersion by a prism. 
 
 ability of the different types of glass 
 
 for use in lenses. While it is universally represented by the symbol 
 V, no name has been proposed which has met with general accept- 
 ance. The demand of the instrument designers has led to the 
 development by the manufacturers of a large number of different 
 types of glass, the optical constants of which are tabulated in their 
 catalogues. 
 
 ATMOSPHERIC REFRACTION. 
 
 At a surface separating two diflerent media, the index of refrac- 
 tion and direction of a ray of light changes abruptly when passing 
 from one side of the surface to the other in a manner consistent with 
 equation 3. If we have a single medium in which the index of re- 
 fraction changes gradually as the ray proceeds from point to point, 
 the course of the ray will also change gradually and will be a curved 
 rather than a straight line. Accordingly .the commonly made state- 
 ment that light travels in a straight line is incorrect unless the path 
 is restricted to a homogeneous medium. The importance of this 
 
14 
 
 lies in the fact that the air is ridt homogeneous, but departs from 
 homogeneity to such an extent that the refraction arising therefrom 
 must be taken into account in all precise levelling or operations of a 
 similar nature. 
 
 The index of refraction of the air is influenced by its density, 
 which varies with height, temperature, and amount of water vapor 
 present. The result is that a ray of light traveling through the air 
 does not in general follow a straight line but is refracted and follows 
 a curved path. 
 
 For points near the horizon, the bending is so great that the setting 
 sun is seen after it is completely below the horizon, as determined 
 when atmospheric refraction is ignored. The most important effect 
 of atmospheric refraction is a deviation of the ray in a vertical plane 
 which under normal conditions is in such a direction that an object 
 appears farther above the horizon than it actually is. This is illus- 
 trated to an exaggerated degree in figure 8. The arc CD represents 
 < the circumference of the earth. A 
 
 P star in the direction AO sends 
 
 light to the observer along the path 
 EAO. The star actually appears 
 in the direction of EA'O', which is 
 the tangent drawn to the path of 
 the light at the point where it 
 enters the eye of the observer. 
 
 FIG. 8.-Refraction by the atmosphere. ^^ astrOUOmical WOrk, COrrCCtionS 
 
 are applied to the observed position of stars in order to obtain the 
 true position. 
 
 If there were no atmospheric refraction, the distance L at which 
 objects on the earth's surface could be seen would be given by the 
 equation : 
 
 L=VDH (7) 
 
 where D is the diameter of the earth and H the height of the observer 
 above the surface of the earth. If L is measured in yards and H in 
 feet, this equation becomes: 
 
 L = 2155VH (8) 
 
 Atmospheric refraction is based upon so many factors that its 
 influence is variable. As a first approximation it is commonly 
 assumed that when atmospheric refraction is taken into account, 
 the distance L' is given by the equation: 
 
 VI 
 
 DH (9) 
 
 It will be seen that this is equivalent to the result one would 
 obtain if there were no atmospheric refraction and the height of the 
 
15 
 
 observer were increased one-sixth. 
 H in feet, the equation becomes: 
 
 If L' is measured in yards and 
 
 L' = 2328VH 
 
 (10) 
 
 On the basis of these two formulre, atmospheric refraction increases 
 the range of vision approximately 8 per cent. The following table 
 shows the distance at which an object on the surface of the earth 
 can be seen from different elevations. If the object viewed is ele- 
 vated above the earth's surface, it can be seen at a much greater 
 distance. Accordingly, additional columns are given showing the 
 distance at which objects 25 and 50 feet high may be seen. These 
 values are based upon the approximate refraction as embodied in 
 equation 9 and are the limiting values for possible paths of light 
 computed on this basis. The absorption of the atmosphere is not 
 taken into consideration and may reduce the range of vision greatly. 
 In particular, the large ranges given at the latter end of the table 
 will be more often realized over land than over sea, due to the greater 
 freedom of the atmosphere from water vapor. Furthermore, it 
 should be noted that departures from the normal refraction on which 
 this table is based are not unusual and may cause the geometric 
 range to differ greatly from the values here tabulated. 
 
 observer, 
 in feet. 
 
 Extreme distance, in yards 
 
 at which object can be seen. 
 
 On surface of earth. 
 
 25 feet above surface 
 of earth. 
 
 50 feet above surface 
 of earth. 
 
 No 
 refraction. 
 
 With 
 refraction. 
 
 No 
 refraction. 
 
 With 
 refraction. 
 
 No 
 refraction. 
 
 With 
 refraction. 
 
 10 
 25 
 50 
 75 
 100 
 
 6,800 
 10,700 
 15,200 
 IS, 700 
 21,500 
 
 7,400 
 11,600 
 Ifi, .500 
 20,200 
 23,300 
 
 17, .500 
 21,400 
 25,900 
 29,400 
 32,200 
 
 19,000 
 23,200 
 2s, 100 
 31,S0O 
 .34, 900 
 
 22,000 
 25,900 
 30,400 
 33,900 
 36,700 
 
 23,900 
 28,100 
 33,000 
 36,700 
 39, SOO 
 
 ^12.5 
 150 
 175 
 200 
 
 24,100 
 2<>,400 
 2S,500 
 30,500 
 
 26,000 
 2S,500 
 30,800 
 32,900 
 
 34,800 
 37,100 
 39,200 
 41,200 
 
 37, 600 
 40, 100 
 42,400 
 44,500 
 
 39,300 
 41,600 
 43,700 
 45,700 
 
 42,500 
 45,000 
 47, 300 
 49,400 
 
 225 
 250 
 275 
 300 
 
 32,300 
 34, 100 
 35, 700 
 37,300 
 
 34,900 
 36,800 
 38,600 
 40,300 
 
 43, 000 
 44,,S00 
 46,400 
 48,000 
 
 46, ,500 
 4S, 400 
 .50,200 
 51,900 
 
 47,500 
 49,300 
 50,900 
 52,500 
 
 51,400 
 53,300 
 .55, 100 
 56,800 
 
 325 
 350 
 375 
 400 
 
 as, 800 
 40,300 
 41,700 
 43, 100 
 
 42,000 
 43, 600 
 45,100 
 46,600 
 
 49,500 
 51,000 
 52,400 
 .5.3, 800 
 
 5.3,600 
 5.5,200 
 56, 7(X) 
 5S, 200 
 
 54,000 
 5.5,500 
 .56,900 
 58,300 
 
 ,5S,500 
 60,100 
 61,600 
 63, 100 
 
 425 
 450 
 475 
 600 
 
 44,400 
 45, SOO 
 47,000 
 48,200 
 
 48,000 
 49, 400 
 50,700 
 52,100 
 
 .55, 100 
 .56, 500 
 57, 700 
 O.S900 
 
 59,600 
 61,000 
 
 62, 300 
 
 63, 700 
 
 .59,600 
 61,000 
 62,200 
 63,400 
 
 64, .500 
 6,5,900 
 67, 200 
 6.S, GOO 
 
 525 
 550 
 675 
 600 
 
 49,400 
 50,000 
 51,700 
 52,800 
 
 53,300 
 54,600 
 55,800 
 57,000 
 
 60,100 
 60, 700 
 62, 400 
 63, ,500 
 
 64,900 
 66,200 
 67,400 
 6,s, 600 
 
 64,600 
 6.5,200 
 66,900 
 6«,000 
 
 69, ,S00 
 71,100 
 72, ,300 
 7.i, ,500 
 
 625 
 650 
 675 
 700 
 
 53,900 
 54,900 
 56,000 
 57,000 
 
 58,200 
 59, 400 
 60,500 
 61,600 
 
 64,600 
 
 65, rm 
 
 66, 700 
 67,700 
 
 69, ,SOO 
 71,000 
 
 72, 100 
 
 73, 200 
 
 69,100 
 70,100 
 71,200 
 72,200 
 
 74,700 
 7.5, (KX) 
 77,000 
 7.S, 100 
 
 725 
 750 
 775 
 800 
 
 58,000 
 59,000 
 60,000 
 61,000 
 
 62,700 
 63,800 
 64,800 
 65,800 
 
 6,s, 700 
 69,700 
 70,700 
 71,700 
 
 71, 300 
 75,400 
 76,400 
 77,400 
 
 73,200 
 74,200 
 75,200 
 76,200 
 
 79, 200 
 
 80, .300 
 81,300 
 82,300 
 
IG 
 
 The depression position finder determines range by measuring the 
 angle of depression of the target. The atmospheric refraction is so 
 great as to make the results entirely useless unless some compen- 
 sating device is employed to correct for the refraction. Different 
 inventors have shown their ingenuity in effecting this correction by 
 various devices, and it is here that the fundamental differences 
 between the several types of depression finders are found. 
 
 The bending of the path of a ray of light in a horizontal plane 
 is not due to any systematic variation of the refractive index and is 
 in general so small that it can be neglected in readings taken with 
 fire-control instruments. In very careful triangulation its effect is 
 eliminated by repeating the observations on different days and under 
 different conditions. 
 
 In addition to the normal effects of atmospheric refraction of the 
 preceding paragraphs, anomalous effects often occur. Over large 
 areas of heated sand or over water, conditions are such as to produce 
 strata of air differing greatly in temperature and refractive index. 
 This condition is favorable to anomalous atmospheric refraction 
 and images, erect or inverted, and sometimes much distorted, are 
 formed and can be seen from a great distance. In this manner at- 
 mospheric refraction gives rise to the various forms of mirages. 
 
 On a hot day the columns of heated air rising from the earth are 
 optically different from the surrounding air and a ray of light is 
 irregularly refracted. The air is turbulent and the conditions are 
 changing all the time. Consequently, an object viewed through 
 such a layer of air appears to be in motion about a mean position. 
 In such a case the air is said to be ''boiling" or the image is ''dancing" 
 due to the presence of "heat waves." This condition is particularly 
 detrimental when a high-power telescope is employed. In fact, 
 this is such a source of trouble that it is usually impossible to use a 
 terrestrial telescope of more than twenty power with any real benefit. 
 
 With an astronomical telescope, higher power can in general be 
 employed than with the terrestrial telescope, as the rays do not pass 
 through the strata of dense air near the surface so obliquely and the 
 greater portion of the path of the light is further removed from the 
 surface of the earth. 
 
 PLANE MIRRORS. 
 
 A plane polished surface used to reflect light is termed a "plane 
 mirror." Looking into the mirror one sees objects which are actually 
 in front of the mirror as though they were located back of the re- 
 flecting surface. The course of the rays of light is showTi in figure 9. 
 Suppose O is a point sending forth rays of light. The ray ON in- 
 cident upon the mirror at N is reflected along NF, where NF is so 
 drawn that the angle of reflection is equal to the angle of incidence. 
 It may be easily shown that NF will satisfy this condition if it is 
 
17 
 
 Fig. 9. — Image of a point ii> a plane mirror. 
 
 drawn through N and the point O' located on the normal to the mirror 
 
 through O, and as far back of the mirror as O is in front of the mirror. 
 
 Similarly, the ray 
 
 OM is reflected 
 
 along the line ME, 
 
 where ME produced 
 
 also passes through 
 
 O'. In fact, all rays 
 
 from O which fall 
 
 upon the mirror will, 
 
 after reflection, 
 
 travel along lines 
 
 which intersect at 
 
 O'. Therefore, if an 
 
 observer in the 
 
 neighborhood of E 
 
 or F looks into the 
 
 mirror, the rays re- 
 ceived by the eye 
 
 apparently proceed 
 
 from O' and the 
 
 point O is seen in 
 
 the mirror appar- 
 ently at O'. Under these conditions O' is said to be the image of 
 
 the object O. Such 
 A an image is termed 
 
 ' ' virtual ' ' because the 
 rays do not actually 
 pass through O' but 
 only appear to do so. 
 With curved mirrors 
 or lenses we shall have 
 examples of images 
 termed "real" in 
 which the rays actu- 
 ally pass through the 
 point where the im- 
 age is located. 
 
 If the object before 
 the mirror is an ex- 
 tended one, instead 
 of a single luminous 
 point, each point will 
 have its image back 
 
 of the mirror and the ontii'e object will appear in true proportion. The 
 
 image will, however, be reversed in a manner which is termed " re- 
 ■18918°— 21 2 
 
 Fir,. 10.— Reversion by a plane mirror. 
 
18 
 
 version." If we imagine the drawing, figure 10, to represent a 
 plan view, the arrow viewed from a point P, (with the mirror re- 
 moved) will be seen with the point to the right of the observer. If 
 
 ABCDE 
 FGHI J 
 
 NORMAL IMAGE 
 
 3G08^ 
 L I HOI 
 
 REVERTED IMAGE 
 
 Fig. 11.— Normal and reverted images. 
 
 the image in the mirror is viewed from P,, the point is to the left. 
 A clear idea of reversion may be obtained by viewing the image of 
 printed matter in a mirror. This is illustrated in figure 11.^ With 
 a single reflection the letters are reverted. If a second mirror is 
 
1!) 
 
 employed so that the image foimed by the iirst mirror is viewed in 
 the second, the characters are reverted twice and are again normal. 
 In general, any even 
 number of reflec- 
 tions leaves the im- 
 age in the normal, 
 any odd number in 
 a reverted aspect. 
 
 In -figure 12, the 
 method of locating 
 an image formed 
 by two successive 
 reflections at two 
 mirrors is sho^vn. 
 The object is at 
 O and the image 
 formed by the first 
 reflection in the 
 mirror AB is at 
 O/. If then we 
 consider 0/ as an 
 object and locate 
 the image of it 
 formed by the mir- 
 ror CD, we obtain 
 O,". This is the 
 
 
 -B 
 
 ima<re of O 
 
 V 
 
 -Images formed by 
 
 formcd by 
 
 Fig. 13.— Image formed by two perpendicular mirrors. 
 
 wo successive reflections. 
 
 two successive reflec- 
 tions, the first of 
 which occurs at the 
 surface of AB. If 
 in a similar man- 
 ner we locate the' 
 image of O in the 
 mirror CD and then 
 the image of this 
 image formed in 
 mirror AB, we ob- 
 tain the point O./', 
 which is the image 
 of O, formed by 
 two successive re- 
 flections, the first 
 of which occurs in 
 tlie mirror CD. 
 the two mirrors, sees 
 
 The observer at E, therefore, on looking int 
 the object at O and the two images 0/ and O,.' formed ])y single reflec- 
 
20 
 
 tions. He will also see the two images which are formed by two 
 
 reflections, one at O/' and a second at O2". In addition there will 
 
 be other images visible, formed by more than two reflections and 
 
 which are not indicated on the drawing. The images formed by the 
 
 successively greater number of reflections become progressively 
 
 weaker as a result of the absorption at each reflection until they 
 
 finally become too faint to be seen. But for this limitation, the 
 
 number of images formed is unlimited, except in the special cases 
 
 when the angle between CD and AB is commensurable with 360°. 
 
 In figure 13 such a special case is represented in which the angle 
 
 between AB and CD is a right angle. An eye at Ej sees the image 
 
 formed by two reflections, the first in the mirror AB. An eye at 
 
 E2 sees the image in the same position formed by two reflections, the 
 
 first of which is in the mirror CD. The coincidence of these two 
 
 images is due to the fact that 90 degrees has been selected as the 
 
 angle between AB and CD. This special case is very important as 
 
 it is the fundamental basis for the design of all roof angle prisms. 
 
 (See Prisms.) 
 
 PRISMS. 
 
 In a fire-control instrument it is frequently desirable to bend the 
 rays of light through an angle in order to bring the eyepiece into a 
 more convenient position or to 
 permit of a periscopic arrangement 
 of parts. Plane mirrors might be 
 employed for this, but the silvered 
 surfaces are a source of trouble, as 
 they cause a loss of light which, as 
 a result of tarnishing, grows more 
 serious as the instrument becomes 
 older. Prisms are therefore em- 
 ployed for this purpose and form a 
 very important part of the optical 
 system of an instrument. Even 
 when the relative location of eye- 
 piece and objective does not necessitate the use of prisms, they are, in 
 many cases, used to erect an image which would otherwise be inverted. 
 
 The bending of the rays by a prism may be due to refraction, as 
 shown in figure 7. As shown in that drawing, however, it is evident 
 that the rays of different colors are bent through different angles. If 
 the object to be viewed is small and light colored, viewed through the 
 prism, it will appear as an elongated streak made up of spectral colors. 
 When the angle through which the rays are bent is very large, the 
 colors are so pronounced as to make the use of the prism impos- 
 sible, but if the angle is small a prism in which the deviation is 
 wholly produced by refraction may be used. The "measuring 
 wedge" in a range finder (fig. 14) is a prism of this sort. If the inci- 
 dent ray is normal to the first surface, there is no deviation until the 
 
 rf^ 
 
 -y 
 
 'r' 
 
 i / 
 
 
 
 
 ~' ^^~~ 
 
 [f:^A^ 
 
 
 U. ^Measuring wodgo of range finder. 
 
21 
 
 second surface is reached. If a is the rofracliu*,' auj>;le of tlie prism, 
 the angk's of incidence and refraction at the second surface are 
 respectively a and r where sin /• = ii sin a- Since the angles are small, 
 
 we can assume the angles and the sines identical. Then, 
 
 (11) 
 
 .Vchromalicpris 
 
 and the angle of deviation d is given by the equation: 
 
 d = na-a={n- \)a (12) 
 
 The- total angle through which the rays are bent by the measuring 
 wedge of a range finder is only a few minutes and the separation of 
 the different colors is so slight that 
 it is not taken into account in the 
 smaller range finders. 
 
 When the angle through which 
 the rays are to be bent is large, an 
 achromatic prism may be em- 
 ployed. The construction is illus- 
 trated in figure 15. Two prisms are 
 cemented together, made of two 
 different kinds of glass. The prism 
 having the smaller refracting angle 
 is made of heavy flint glass which 
 has a large mean dispersion. This prism is turned in the opposite 
 manner to the first prism which is of crown glass. The flint prism, 
 by reason of its large dispersion, neutralizes the dispersion due to 
 
 the crown prism without entirely 
 neutralizing the deviation. There- 
 fore, there is a net residual devia- 
 tion due to the two prisms which is 
 nearly constant for all colors and the 
 deviated image is almost entirely free 
 from color. In some range finde-. 
 an achromatic measuring wedge is 
 employed instead of the single prism 
 described above. 
 
 Even with the achromatic prism it 
 is impossible to obtain complete free- 
 dom from color. When reflection 
 takes place there is no breaking up of 
 the ray into its constituent colors as is the case with refraction. There- 
 fore, the prisms used in instruments in nearly all cases are of the type in 
 which a reflecting surface is used to produce the deviation. Figure 16 
 shows such a prism. The ray indicated by the full line falls upon the 
 first face perpendicularly and proceeds as a single ray without being 
 separated into its component colors. At B the angle of incidence is 
 such that we have total reflection and at th:? face HG the incidence is 
 again normal. Consetiuently the ray traverses the entire prism 
 
 M 
 
 ling prism. 
 
22 
 
 without being divided into its different colors. Tlie case is somewhat 
 different for a ray falling upon the first surface obliquely. The ray 
 shown by the dotted line incident at J is refracted and we should 
 expect to find the final image colored. The ray, however, is again 
 
 Fig. 17.— Right angle reflecting prism 
 
 Porro prism. 
 
 refracted at L on emergence and the two refractions are such that 
 the one neutralizes the other and all the parts of the beam, regardless 
 of color, are bent through the same angle and proceed in a common 
 direction after traversing the prism. The prism just described is 
 known as the right-angle reflecting prism and is very often used in 
 
 fire-control instrn- 
 ments when it is de- 
 sired to bend the 
 rays through 
 
 an 
 angle of 90°. Fig- 
 ure 17 shows the 
 same prism in per- 
 spective. As there 
 is only one reflect- 
 ing surface, the im- 
 age is reverted. 
 
 The Porro prism 
 shown in figure 18 
 is similar in shape 
 to the right-angle 
 prism, but the path 
 of the rays is different. The image is inverted in the plane in which 
 reflection takes place, but, as there are two reflections, there is no 
 reversion. Two Porro prisms placed as shown in figure 19 are used 
 in many telescopic systems to change an inverting to an erecting 
 telescope. It is this system which is commonly used in binoculars. 
 
 
 XT-I. 
 
 
 /^^ 
 
 .— ^— -^=== 
 
 -- 
 
 
 / ■ ' /^ 
 
 
 
 \^<L 1 -""""' 
 
 ~~~-~~I2*''"~-^ 
 
 1 
 
 
 
 
 F(G. 1 'J. —Porro prism system. 
 
23 
 
 In figure 20 there is shown a second type of Porro prism which is 
 often used for securing an erect image. This is substantially the 
 same in principle as the first type. In each case there are four reflec- 
 tions, two in each of two planes, but the reflections occur in diflFerent 
 order in the two types. 
 
 The right-angle prism with roof is showTi in figure 21. It may be 
 considered as made up of a right-angle reflecting prism w^th the 
 hypotenuse face replaced by two faces inclined to each other at an 
 angle of 90°. These two faces form the ''roof" of the prism. The 
 lower prism of the panoramic sight and the prism used in the right- 
 angle telescope of antiaircraft fire-control apparatus are of this type. 
 This prism when inserted in a telescopic system bends the axis of the 
 telescope through 90° and also erects the image. The course of a ray 
 through the prism is shown in the drawing. Two reflections take 
 place, one at each of the two in- 
 clined faces, and as a result an 
 object viewed through this prism 
 
 
 
 ^ 
 
 /,-'"' \;- 
 
 
 xy 
 
 — ^ \^ /y 
 
 
 Fig. 20.— Second type of Porro system. 
 
 Fig. 21.— Right angle prism with roof. 
 
 is not reverted. This prism is one o'f the most difficult to manu- 
 facture, as the angle between the two inclined faces can not differ 
 from 90° by more than a few seconds if the prism is to function satis- 
 factorily. The angle between two such faces in any prism is com- 
 monly spoken of as a "roof angle" and the same high degree of 
 accuracy in its construction is always demanded. If the angle is 
 not accurately made, the image in the center of the field will appear 
 doubled and the prism is entirely useless. This doubling arises from 
 the fact that a portion of the rays are incident upon the two faces 
 in one order, the remainder in the reverse order. The images by the 
 two portions do not coincide. If the angle is precisely 90°, the images 
 produced by the two reflections, regardless of the order in which 
 the reflections occur, coincide, as shown in figure 13. The doubling 
 of images then disappears. Angles having the requisite accuracy can 
 
24 
 
 not be produced directly, but must first be carefully approximated 
 by usual manufacturing methods after which the faces must be care 
 fully ground by a skilled operator and individually 
 tested until the angle is so nearly correct that the 
 image is not doubled. An error of a few seconds is 
 sufficient to make the prism worthless. This 
 tedious method of production limits the output, as 
 few men become sufficiently skilled to do this 
 local retouching. 
 
 In the rotating prism shown in figure 22 the light 
 undergoes a single reflection, after which it contin- 
 ues in its original direction. The image is reverted 
 by the single reflection. It is by means of this 
 prism that the image in the panoramic sight is 
 caused to remain erect as the upper head is rotated 
 in azimuth. 
 
 The Brashear-Hastings prism (figure 23) is a 
 type of erecting prism in which there is no dis- 
 placement of the ray. Four reflections take place, 
 two at the upper sloping faces and two at the lower 
 pair of faces, which are inclined to each other at 
 an angle of 90° and which form a roof. This prism 
 is subject to the same difficulty in production as 
 the roof angle prism, previously described, as the 
 angle between the two lower faces must be accu- 
 rately 90°. 
 
 In figure 24 there is shown another type of 
 
 erecting prism which is used more frequently by 
 
 Em-opean than by American manufacturers. The 
 
 Fig. 22.-Rotating prism, advantages 'of this prism are that it can be made 
 
 ^i. 
 
 t =. 
 
 p? 
 
 
 > -M 
 
 U' 
 
 V 
 
 ^ 
 
 
 — 1 
 
 
 
 \y , ••?-/ yi^ 
 
 
 Fig. 23.— Brashear-Hastings erecting prism. 
 
 in one piece and is of such shape that it can be held very rigidly in 
 an instrument. The last two reflecting surfaces form a roof angle. 
 This is designated as the Sprenger prism. 
 
25 
 
 The erecting prism used in (he Amei;ican aiming circle, model 1916, 
 is shown in figure 
 25. This design of 
 erecting prism is 
 only used in aiming 
 circles of this type 
 where itlends itself 
 to the design of a 
 compact instru- 
 ment. Inthedraw- 
 ing only three re- 
 flections are indi- 
 cated and theimage 
 will be reverted, as 
 an odd number of 
 reflections always 
 implies a reverted 
 image. This erect- 
 ing prism is used 
 with a right angle 
 reflecting prism 
 which is inserted in 
 the optical train at 
 a different point. 
 
 ^^^^^^"^-^ 
 
 
 M 
 
 ^+ 
 
 Vks) 
 
 
 Fig. 24.— Sprenger erecting prism. 
 
 The four reflection,^; 
 
 an erect image. The 
 second reflection 
 indicated in figure 
 25 occurs at an an- 
 gle of incidence less 
 than the critical an- 
 gle, and the surface 
 at which this reflec- 
 tion takes place is 
 therefore silvered. 
 
 The triple mirror 
 shown in figure 2G 
 has three reflecting 
 facts which are mu- 
 tually orthogonal. 
 It has the peculiar 
 property of deviat- 
 ing any ray which 
 enters it through 
 an angle of 180°— 
 ,.„ o- TT .■ . , • • ■ , that is, the ray is 
 
 Jir.. 2i.— Erecting system of aiming circle. ^ 
 
 returned along a 
 course parallel to its original path and in the opposite direction. 
 
26 
 
 If 11 
 
 angles are accurately made, this parallelism is true, no matter 
 how the prism is turned so long as 
 it receives the incident light. Use 
 is made of this prism in signalling 
 devices and in the internal adjust- 
 ment device of small range finders. 
 The penta prism illustrated in fig- 
 ure 27 is used in the range finder as 
 an end reflecting prism. The devi- 
 ation of the ray, which is 90°, is 
 independent of any small rotation 
 of the prism in the plane of reflec- 
 tion. The fact that the deviation 
 Fir.. 2o.-Tripie mirror. ^^ coustaut is the feature which 
 
 makes this prism 
 useful in range fin- 
 ders. The angles 
 measured by the 
 range finder are so 
 small that if a con- 
 stant deviation 
 prism were not 
 used the variation 
 in the angles of de- 
 viation at the end 
 reflectors brought 
 about by the flex- 
 ure of the tube and 
 consequent rota- 
 
 
 / 1 / 
 
 
 
 / J?"'^ 
 
 
 
 
 \ 
 
 ' 1 
 
 1 ^> 
 
 
 
 k 
 
 Fir.. 2S.— Rhomboidal prism. 
 
 fore substituted. Two silvered 
 
 Fin. 27. — Ppnla prism. 
 
 tion of the prisms would be suffi- 
 cient to impair the accuracy of the 
 range finder. 1 1 is necessary to sil- 
 ver the reflecting surfaces of the 
 penta prism, as the rays fall upon 
 the reflecting surfaces at angles less 
 than the critical angle. We there- 
 fore do not have total reflection. 
 
 In the very long base range 
 finder the penta prisms are so large 
 that it is diflicult to find a block of 
 glass sufficiently large and homo- 
 geneous for the construction of a 
 prism. Penta reflectors are there- 
 3-lass mirrors are held in a metal 
 
27 
 
 frame in such a way that they occupy positions analogous to the two 
 reflecting surfaces of the prism. It is necessary that the mirrors be 
 held rigidly and permanently in the desired positions and that temper- 
 ature changes, so far as possible, be eliminated. 
 
 Figure 28 illustrates the rhomboidal prism. This may be con- 
 sidered as made up of two right angle reflecting prisms built in one 
 piece. It does not invert or revert the image nor change the direc- 
 tion of the beam of light but displaces it parallel to itself. This 
 prism forms one of the components of the ocular prism of the range 
 finder and is also used in one of the systems for internal adjustment. 
 
 THE SPHERICAL REFLECTING SURFACE. 
 
 In figure 29 the arc passing through V with center at C represents 
 the trace of a concave spherical reflecting surface. The line OV is 
 
 . Fig. 29.— Reflection at a spherical surface. 
 
 termed the '' axis " of the mirror and the point V, the " vertex." The 
 location of the image of the point O is to be determined. A ray from 
 O incident upon the mirror at I is reflected in the direction 10', 
 where 10' is so drawn that angle OIC equals angle O'IC. 
 
 The following sign convention is to be adopted: The length of a 
 line generated by a point starting at V and moving in the direction 
 of the incident light is positive. Lengths extending in the opposite 
 sense are negative. 
 
28 
 
 In accordanco with this statement 
 
 VO, which will be represented by u, is negative. 
 VO', which will be represented by u' , is negative. 
 VC, which will be represented by r, is negative. 
 
 This convention which in the present case makes all lengths nega- 
 tive has been adopted in order that a generalization may be made, 
 including the equations for both reflecting and refracting surfaces. 
 
 In triandes OIC and O'lC 
 
 u — r sm ^ 
 r ~"sin I 
 
 (13) 
 
 r — u' sin i' 
 
 (14) 
 
 r sin V 
 
 Since the angles of incidence and reflection are equal, 
 
 
 sin i = sin i' 
 
 (15) 
 
 Combining these equations, then, 
 
 
 u~r sin Z' 
 r — u' sin Z 
 
 (16) 
 
 r — u sm 
 
 r 
 
 
 
 
 
 If I is near V and consequently angles 
 
 I and V 
 
 are 
 
 small, 
 
 we may 
 
 make the approximation: 
 
 sin / = — 
 u 
 
 h 
 
 
 
 
 
 (17) 
 
 sin?'=-^p 
 
 
 
 
 
 
 Equation 16 then becomes 
 
 
 
 
 
 u — r u 
 r-u~^u' 
 
 
 
 
 
 which may be written 
 
 
 
 
 
 1 1 _ 
 
 u u' 
 
 2 
 r 
 
 
 
 
 (18) 
 
 The focal length, which will be denoted by/, is equal to the recip- 
 rocal of the right hand member of equation 14. 
 
 It should be noted that in the case here selected /" is positive, 
 although the negative sign appears explicitly in the equation. This 
 arises from the fact that with the convention of signs which has 
 been adopted r is negative. Equation 14 may be written, 
 
 _1-Vi (20) 
 
 u u' f 
 
29 
 
 -Parallel rays projected by 
 mirror. 
 
 The approximations which were made in equation 17 have a simple 
 physical interpretation. A ray from O, incident upon the mirror at 
 I, cuts the axis OV at a point near 
 O'. If I is considered as a variable 
 point approaching V as a limit, then 
 this point is also variable and ap- 
 proaches as a limit the point O' lo- 
 cated by equation 19 or 20. The rays 
 from O, incident upon the mirror, af- 
 ter reflection, do not all pass through 
 a single point, but if IV is not too 
 large they do pass very near to the 
 point O'. Therefore O' is the image of O, although there is not the per- 
 fect concurrence of the rays after reflection as with the plane mirror. 
 
 The rays after reflection from 
 the plane mirror did not actually 
 pass through the image but only 
 appeared to proceed from a com- 
 mon point. The image was said to 
 be virtual. In the present case 
 the rays actually pass through O' 
 (to the degree of approximation 
 indicated above) and the image is 
 accordingly real. 
 
 In equation 20, if u increases 
 indefinitely, u' approaches —/as a 
 limit. As u increases, the rays fall- 
 ing upon the mirror become more 
 nearly parallel. If the rays are 
 strictly parallel, we have the lim- 
 iting condition and the image is at 
 O', figure 30, where VO' equals /. 
 The focal length may therefore be 
 defined as the distance from the 
 vertex of the mirror to the image 
 formed of an object at an infinite 
 distance. 
 
 Conversely, if the point O ap- 
 proaches the mirror, u' increases 
 and becomes infinite when u equals 
 -/. A source at O', figure 30, then 
 sends forth rays which, after re- 
 flection, are parallel. It should be 
 observed tliat tlie paraUelism of rays here referred to is only ap- 
 proximate and is conditioned upon the as.sumption introduced in the 
 derivation of equation 17, i. e., that IV is small. 
 
 Fig. 31.— Projection by a spherical and parabolic 
 mirror. 
 
30 
 
 If we wish to illuminate a distant object, it is advantageous to 
 place the source at the focus of a concave mirror, as sho^\^l in figure 
 30. All the rays which fall upon the mirror are then projected in 
 a common direction and fall upon the distant point. It is further- 
 more seen that it is advantageous to make the mirror as large as 
 possible in order that it will receive a greater proportion of the rays 
 from the source and hence concentrate more light upon the distant 
 object. But as the mirror is made larger, the assumptions of equa- 
 tion 1 7 become more and more inaccurate and the parallelism of .the 
 projected rays becomes less perfect. If, instead of the spherical 
 surface, the mirror is made a paraboloid of revolution and the source 
 placed at the focus, the geometrical properties of such a surface are 
 such that all the rays are projected in a parallel beam, no matter how 
 large a portion of the surface is utilized. Figure 31 shows on an 
 exaggerated scale the difference in the action of the spherical and 
 parabolic reflector. 
 
 In the drawing it appears as though the rays from the spherical 
 mirror converged and that this form might be the more favorable. 
 The converging rays, however, cross at a point relatively near the 
 searclilight, after which they diverge, so the net result in an actual 
 application is a divergence. The fundamental parts of the search- 
 light are a brilliant small source of light and a paraboloid reflector. 
 
 THE SPHERICAL REFRACTING SURFACE. 
 
 The arc AB (fig. 32) is a curved refracting surface with center at 
 C and vertex at V. The object is at O and the course of a ray is 
 represented which is incident upon the lens at I and, after refraction, 
 passes through the image point at O'. 
 
 If the same sign convention adopted for the reflecting surface is 
 used, 
 
 VO, which will be represented by u, is negative. 
 
 VO', which will be represented by u', is positive. 
 VC, which will be represented by r, is positive. 
 
 In the triangle OIC 
 
 OC —u + r sin i 
 
 IC r sin I 
 
 Similarly in triangle O'lC, 
 
 (21) 
 
 CO' -r+ri'^_^su^^ ^^2) 
 
 IC r ~ ^sin I 
 
 Combining these two equations 
 
 -u+r ^sini^' (23) 
 
 — r+u' sin i' sin I 
 
31 
 
 Since, in accordance witli tlio law of refraction ' • ^, = - this may be 
 ' sin in, -^ 
 
 written 
 
 ■ li + /• 
 
 ■r +u' 
 
 n, sin I 
 
 (24) 
 
 Now if, as in the case of tlie sj)lieri(»il reflectin<i^ surface, we assume 
 that I api)roaches V as a limit, "^. , ajjproaches as a limit the value 
 
 Substitutinj 
 
 u + r 
 r+u' 
 
 n.,u 
 
 (25) 
 
 Fig. 32.— Refraction at a spherical surface. 
 
 and this may be A\Titten : 
 
 77, na^ri;- 
 u u' r 
 
 (26) 
 
 As in the case of the curved mirror, the image at O' is ap{)roximate 
 and the approximation improves as IV is made smaller. 
 
 It will be noted that if n, = 1, 713= — 1, equations 26 and 18 become 
 identical. In general the equations for reflecting surfaces can be 
 obtained directly from the corresponding equations for refracting 
 surfaces by making: the above substitution. 
 
 THE THIN LENS. 
 
 In the elementary development of formuhe for the action of a lens 
 it is usual to assume that the thickness of the lens may be neglected. 
 This results in a great simplification of the formuhe and the approxi- 
 mation is sufficiently accurate to be useful, as in a large number of 
 lenses the thickness is very small in comparison with the other lengths 
 
32 
 
 which are involved. Formulae in which this assumption is made are 
 referred to as "thin lens formulae." 
 
 A lens consists essentially of two refracting surfaces which act 
 successively upon the light, and by applying formula 26 for the single 
 refracting surface twice we obtain the resultant effect of the lens. 
 
 In figure 33 the lens is bounded by the two spherical surfaces 
 having vertices at Vi and V2. The image formed by the first surface 
 is at 0\. Let VjO and ViO'i equal u and u\, respectively. Then 
 for r?! and % of formula 26 substitute 1, the index of air, and n,- the 
 index of the material of the lens. The equation becomes 
 
 _i+iL_'l^ (27) 
 
 _L=r!iiLi+ni (28) 
 
 The image formed by the first surface is to be considered as the 
 
 object for the second surface and the final image produced by the 
 lens located. The distance from the second surface to the object 
 is V20'i, but since the thickness is to be neglected this is the same 
 as VjO', which has already been determined. A second application 
 of formula 26 with iiy and % replaced, respectively, by n and 1 gives 
 
 _r!izii+n+L,=i^ (29) 
 
 Transposing 
 
 -! + 7' = (^-l)R-n (30) 
 
 In this formula the sign convention previously laid dov^m is to be 
 followed, i. e., w and u' are to be measured from the lens to the ob- 
 ject and image, respectively, and the length is positive if it extends in 
 the direction of the incident light. In figure 33, u is negative and u' 
 is positive. 
 
 Radii of curvature are positive when convex toward the incident 
 light. In figure 33, r^ is positive and r., is negative. 
 
33 
 
 If wc write 
 
 equation 30 l)ecomcs 
 
 f< 
 
 «-'{'^] 
 
 u u 
 
 (31) 
 
 (32) 
 
 The value of /has the same significance as with the reflecting sur- 
 face. If parallel rays fall upon the lens, u is infinite and u' equals/, 
 i. e.,.the rays pass through a point 
 ilistant/from the lens. If a source 
 of light is placed at this point, 
 the rays which are intercepted hy 
 the lens are projected as a par- 
 allel beam. This explains the use 
 of the "bull's-eye" lens to project 
 the beam of light from a flash 
 lamp. For a searclilight the lens 
 is not so adaptable as the mirror, 
 as the lens would require for its construction glass of a homogeneity 
 difficult to secure and the thickness at the center would be so great 
 that the lens would absorb a large part of the light. 
 
 Divergent lens. 
 
 DOUBLE CONVEX. PLANO CONVEX. MENISCUS CONVERGING. 
 
 CONVERGING. 
 
 DOUBLE CONCAVE . PLANO CONCAVE . MENISCUS DIVERGING. 
 
 DIVERGING. 
 
 Fig. :i5.— Types of len.^es. 
 
 A thin lens of the form shown in figure 33 renders a pencil of light 
 falHng upon it more convergent. A convergent lens, often called a 
 positive lens, is characterizecl by the fact that it is thicker in the ccn- 
 48918°— 21 ?, 
 
ter than at the edge and has a positive focal length as defined by 
 equation 31. Figure 34 illustrates a divergent or negative lens 
 which is always tliinner at the center than at the edge. A pencil of 
 light diverging from the point O is rendered more divergent by the 
 lens and apparently proceeds from O', which is a virtual image. The 
 
 classification of con-r 
 verging and diverg- 
 ing lenses based 
 upon their shape is 
 illustrated in figure 
 35. Figure 36 illus- 
 trates the action of a 
 converging and di- 
 verging lens, respec- 
 tively, upon a par- 
 allel beam of light. 
 In the first case a 
 real image is formed 
 distant / from the 
 lens. In the second 
 case the image is vir- 
 tual, distant /from 
 the lens and on the 
 side as the 
 
 same 
 
 3rgent and divergent leu.ses 
 
 source. 
 
 Thus far only the 
 images of points ly- 
 ing on the optical 
 axis of the lens have 
 been considered. In figure 37 let the arrow OOi represent an 
 extended object. The image of O is at O'. To locate the image 
 of 0„ consider the path of the ray OjC. If the thickness of the lens 
 is neglected this ray will be equally refracted at the two coincident 
 surfaces and will be 
 
 a ^ ^ ^ 
 
 undeviated. It may 
 therefore be ex- 
 tended as a straight 
 line. The ray OJ., 
 which is parallel to 
 the axis of the lens, 
 will after refraction 
 pass through F, the focus of the lens. Since the intersection 
 of any two rays serves to locate the image, 0/ is the image 
 of Oi. These two particular rays were selected because theu" 
 courses after refraction could be easily determined. The image 
 O'O/ is inverted and in general will differ in size from the object. 
 
 Fig. 37.— Image of an extended object by a thin leii 
 
35 
 
 (33) 
 
 If y aiul y' are the length of ima<;e and object, respectively, 
 
 y OiO, u 
 
 This equation gives the magnification of the object. In the case 
 represented in figure 37, '/ is negative and u' positive. The quotient 
 is negative and tliis is to be interpreted as denoting that the image is 
 inverted. , 
 
 Figure 38 illus- Q 
 trates the relation- '[^^"^-'^v-^ 
 ship of object and 1 ^^^:^--.^ 
 
 a 
 
 --Q---- 
 
 Fig. 3S.— Magnification prcxluced by a positive lens. 
 
 image, for a con- 
 verging lens when 
 the image is vir- 
 tual. The image 
 of the arrow is 
 much enlarged, 
 erect and oii the 
 same side of the lens as the object. This illustrates the action of the 
 simple magnifying lens or reading glass. 
 
 The image formed by a diverging lens is illustrated in figure 39. 
 For the case here selected the image is virtual erect and much smaller 
 than the object. Under these conditions, then, a diverging lens does 
 not magnify an object but makes it appear smaller. The lens placed in 
 the floor of an airplane for observation operates upon this principle. It 
 makes objects appear smaller than when viewed with the unaided eye 
 but has a compensating advantage in that the field of view which 
 
 may be observed through the rela- 
 tively small opening is much larger 
 than would be possible if the lens 
 were not used. 
 
 The lenses treated above have had 
 only spherical or plane refracting sur- 
 faces. Spectacle lenses having one 
 surface cylindrical instead of spher- 
 ical are commonly employed in order 
 to correct the eye for astigmatism. The action of such lenses is 
 illustrated in figure 40. In planes passing through the object point 
 O and parallel to the elements of the cylindrical surface, as indi- 
 cated by the shading of the drawing, there is no converging or 
 diverging effect. In planes perpendicular to these first mentioned 
 planes and passing through O, the rays are caused to converge or 
 diverge, accordingly as the lens is thicker or thinner in the center 
 than at the edge. 
 
 Fig. 39.— KediKtioii produced by a negative 
 
36 
 
 The upper drtu\dng illustrates a converging cylindrical lens. The 
 ray 01 which passes symmetrically through the cylindrical lens is 
 undeviated. A plane pencil of rays, as represented by rays OA, 
 OB, and OC remains in the original plane but is caused to converge 
 and is brought to a focus at the point I,. The symmetrically sit- 
 uated plane pencil is brought to a focus at I.,. Intermediate plane 
 pencils are brought to a focus at intermediate points and the final 
 result is that all rays proceeding from the point O which are re- 
 fracted by the lens pass through the line 1, h. This line is there- 
 fore a real image of the point O. 
 
 If the refracted rays are produced backward through the lens, 
 they are seen to pass through the line I3 I^. This line is therefore 
 a virtual image of the point O. 
 
 The cylindrical lens therefore forms two images of the point O, 
 each of which is a straight line. Furthermore, it is easily seen that 
 the two lines are perpendicular to each other and intersect the line 
 
 OX 
 
 Fig. 40.— Action of eylindrical lens. 
 
 01 at right angles. In general, a cylindrical lens forms two such 
 images of a point and as the position of the object or type of lens is 
 changed, they may be both real, both virtual, or one real and one 
 virtual, as in the illustrated case. In the last two drawings the 
 action of a diverging cylindrical lens is illustrated. In this case 
 both images are virtual. 
 
 When a point is imaged as two perpendicular straight lines, the 
 image is said to be astigmatic. A further treatment of such images 
 is given in connection with the aberration of a lens. 
 
 In the large horizontal base range finders two cylindrical lenses 
 are mounted so that they may be swung into the path of the light. 
 They are termed " astigmatizers " and are employed when ranges 
 are to be read and the only target offered is a searchlight. Without 
 the cylindrical lenses the searclilight appears as a small bright spot 
 and it is difficult to keep it upon the dividing line in the range finder. 
 When the cjdindrical lenses are in position the image of the search- 
 
37 
 
 liglit is drawn out into a luminous lino perpendicular to tlie dividing 
 line and the readin*!; of the ran<jje is niucii facilitated. 
 
 THICK LENSES AND COMBINATIONS. 
 
 For many purposes the approximate formulae already given are 
 sufficiently accurate. So long as the thickness of the lens is negligihle, 
 it is a matter of indifference from what portion of the lens, center or 
 surface, the distances u and v are measured. When the thickness 
 of the lens is too great to be ignored, equation 32 will still be true if u 
 and u' are measured from the two principal points and if for /' we 
 write the equivalent focal length. 
 
 The two principal points lie on the axis of the lens and are conjugate; 
 that is, one point is the image of the other. Therefore any incident 
 ray which passes through the first point has a corresponding emergent 
 ray which passes through the second point. Furthermore, two such 
 
 rays make equal angles with the optic axis. For this reason these 
 two points are sometimes called points of unit angular magnification. 
 The principal ])oints are shown in figure 41 at P, and P... 
 
 The locations of the two principal points are given by 
 tions : 
 
 V P ^^' 
 
 * 1-^ 1 
 
 n{r. 
 
 V..P, 
 
 
 t{n-\) 
 
 tlie e([ua- 
 (34) 
 
 (35) 
 
 n{j\ — To) — <(n— 1) 
 
 Tlie thickness of the lens is / and the distances, when measured in 
 the sense indicated, are j)()sitive if they extend in the direction of the 
 incident light. 
 
 For a double convex or concave lens the two principal points are in 
 the lens and, if the two surfaces are equally curved, are equidistant 
 
from the two surfaces. If one surface is more strongly curved than 
 the other, both points tend to move toward the more curved surface. 
 If one surface of the lens is plane, one principal point lies at the vertex 
 of the curved surface. If the lens is meniscus, one or both points will 
 lie outside the lens. 
 
 A tliick lens has three focal lengths, the equivalent focal length 
 (E. F. L.) the back focal length (B. F. L.), and the front focal length 
 (F. F. L.). The equivalent focal length is the focal length of a 
 "thin" lens which will form an image the same size as that formed 
 by the thick lens provided that the distance from object to thin lens 
 equals the distance from object to first principal point of thick lens. 
 The value of the equivalent focal length is given by the formula : 
 
 E. F. L. 
 
 in-l)[n{r,-r,)+t{n-l)] 
 
 (36) 
 
 Fig. 42.— Method of locating the image formed by a thick lens. 
 
 In figure 41 the principal points are shown at P^ and F^. Parallel 
 
 rays from the left 
 are brought to a 
 focus at Fj, similar 
 rays from the right 
 at ¥,. 
 
 The distance P^Fj 
 or PjFi is the equiv- 
 alent focal length. 
 The back focal 
 length is VaF,, i. e., 
 it is measured from 
 the vertex of the 
 lens instead of the principal point. The corresponding distance ViFi is 
 the front focal length. These last two focal lengths may be found 
 ■by equations 34, 35, and 36. The equivalent focal lengths on the 
 two sides of the lens are equal but the back focal length and front 
 focal length are equal only when ViPi equals P.Vj which is only true 
 for a symmetrical lens. 
 
 Two planes passing through the two principal points and perpen- 
 dicular to the axis of the lens are termed the two principal planes. 
 These are sometimes referred to as the planes of unit magnification, 
 as a ray incident upon the lens falls upon the first principal plane at 
 the same distance from the axis as the conjugate emergent ray inter- 
 sects the second. This fact, together with the unit angular magnifica- 
 tion at the principal points, enables one to easily determine graphically 
 the position of the image of any point not on the axis of the lens. Let 
 O in figure 42 be the point in question, APi and A'Pj the principal 
 planes, and Pj and P, the principal points. The lens itself is not 
 shown, as the principal planes and equivalent focal length P^F define 
 
39 
 
 all its properties. The ray OPi will emerge at P, and the entrant and 
 emergent rays make equal angles with the axis of the lens. This 
 enables P.O' to be drawn. The ray OA is parallel to the axis of the 
 lens. The conjugate ray must be drawn through A' (APi = A'P2) 
 and will pass through F. The intersection at O' is the image of O.- 
 The conjugate ray to any ray OB can now be easily dra\\Ti. It passes 
 through points B' and O'. 
 
 To determine the location of the image of an axial point, equation 
 32 mray be used. The distances u and v' are measured from the prin- 
 cipal points, and for/ the equivalent focal length is to be used. 
 
 If two thin lenses of focal lengths,/ and/, are placed in contact, 
 the focal length of the resulting system is given by the equation: 
 
 ^Vlien the lenses are thick lenses or are not in contact, the equivalent 
 focal length of the system is given by the equation: 
 
 In this equation /i and/, are the equivalent focal lengths of the two 
 component lenses and ,s is the distance from the second principal 
 point of the fu-st lens to the first principal point of the second lens. 
 The usual sign convention holds for s and the measurement is to be 
 made in tiie direction indicated. It should be noted that when the 
 two lenses are in actual contact, as the two components of a cemented 
 objective, -s in general is not zero, as the distance is measured between 
 the two principal points and not between the two surfaces. 
 
 The system of two lenses may be considered as equivalent to a thick 
 lens having an equivalent focal length given by equation 38 and two 
 principal points given by the equations : 
 
 P.f. = y7^|'r:, (39) 
 
 The distances p^Pi is measured from the first principal point of 
 the first component lens to the principal point of the equivalent thick 
 lens, the distance p.P, from the second principal point of the second 
 lens to the second principal point of the equivalent lens. 
 
 If the ef[uivalent single tliick lens is to be determined corresponding 
 to three or more component lenses, two may first be combined by the 
 preceding equations. The equivalent lens thus obtained may then 
 be combined with a third and the process continued successively 
 until all are combined. 
 

 Z-L ^^^^==5^ 
 
 
 / "~* ^~^^^ 
 
 
 /___,--— -;^ 
 
 40 
 
 THE ABERRATIONS OF A LENS. 
 
 In determining the location of the image formed by a lens, certain 
 approximations have been made in the mathematical equations. 
 It has been assumed that the angles dealt with were so small that 
 •angles, sines, and tangents could all be considered equal. This is 
 equivalent to assuming that the only rays which form the image are 
 those traveling in a direction nearly parallel to the axis of the lens 
 and not far removed from the center. Such a condition can be real- 
 ized if one places in front of the lens a diaphragm which excludes all 
 light except that passing through the center of the lens and if the field 
 of view is small. But such a diaphragm lets so little light through 
 that the image for most purposes is not sufficiently bright. If the 
 course of the rays through a lens is traced exactly and no approxi- 
 mations are made, it is found that in general there is no single point 
 through which all the rays pass after refraction by a lens. All the 
 rays, however, pass relatively near the image point which was located 
 
 by the approximate method. This 
 failure of the rays to pass through a 
 single point after refraction gives 
 rise to a lack of perfection in the 
 image which is said to be due to the 
 presence of aberrations. As a result 
 ^ ,., ^ , , , ^ of the aberrations, a point in the 
 
 Fig. 43.— Spherical aberration. _ ' f 
 
 object corresponds, not to a point 
 in the image, but to a small area. In this regard, the image is analo- 
 gous, in some respects, to a mosaic made up of small blocks. It is 
 desirable that the blocks of the mosaic be made as small as possible, 
 as details of the object which fall within a single block can not be 
 distinguished. It thus becomes the problem of the instrument 
 designer to so design the optical system of an instrument that the 
 aberrations shall not be sufficiently great to make vision through the 
 instrument unsatisfactory. 
 
 The aberrations of a lens arise from several different causes and 
 for convenience each is spoken of as producing a specific type of 
 aberration although the aberrations seldom exist independently. 
 The aberrations most commonly considered as distinct are spherical 
 and chromatic, coma, astigmatism, distortion, and curvature. 
 
 Spherical aberration. — Figure 43 illustrates the formation of an 
 image by a lens when there is a large amount of spherical aberration. 
 The rays which pass through the center of the lens and those near 
 the edge, after refraction, do not cut the axis at a single point. If the 
 lens is a single positive lens, the rays farther from the center cut the 
 axis nearer the lens than the center rays. For a negative lens the 
 variation is in the opposite sense. If we choose an appropriate non- 
 spherical surface of revolution, the rays may be caused to cut the 
 axis of the lens at one point. From this standpoint; then, the aber- 
 
41 
 
 ration may be said to be due to the fact that the lens is spherical and 
 it is therefore termed spherical aberration. The spherical aberration 
 can also be eliminated by combining a positive and negative lens to 
 make a single compound lens. As the aberrations of the lenses are 
 in opposite directions they may be made to neutralize each other in 
 this manner. A large astronomical telescope usually has a surface, 
 which by local polishing is caused to depart from the spherical form 
 in order to make the elimination of spherical aberrations more perfect 
 than can be secured bj the use of two lenses. The chief application 
 of the nonspherical surface in military equipment is in the search- 
 light mirror, already described. It is desired that the rays from the 
 arc, after reflection from the mirror, shall emerge in a parallel beam. 
 This requires freedom from spherical aberration and is secured by 
 making the reflecting surface of the mirror a paraboloid of revolution. 
 In fire-control instruments, the spherical aberration is commonly 
 eliminated by the use of a positive and negative lens combined. 
 
 CTiromatic aberration. — The focal length of a lens has been shown 
 to be 
 
 1 
 
 (n-1) 
 
 K4] 
 
 Since n, the refractive index of the glass of which the lens is con- 
 structed, is different for each color, it follows that a lens has a differ- 
 ent focal length for each color of the spectrum. Consequently even 
 though spherical aberration is eliminated, the rays of different color 
 will be brought to a focus at different points. For a simple positive 
 lens the focal length is shorter for blue than for red light. Like 
 spherical aberration, chromatic aberration is eliminated by making 
 a lens of two separate lenses, one positive and one negative, placed 
 together. The positive lens is made of a crowTi glass and the nega- 
 tive lens of flint. The fundamental characteristic of flint glass is 
 that the index of refraction varies much more for the different colors 
 than is the case with the crown. The negative flint lens is so designed 
 that its variation of focal length with color, which is in the opposite 
 sense to that of the crown, is sufficient to compensate for the varia- 
 tions in the crown lens. But since the flint glass is used the negative 
 lens does not neutralize the convergent effect of the crown component. 
 The two lenses, together, therefore constitute a convergent lens with 
 the chromatic aberration eliminated. It is the usual practice to 
 make the two adjacent surfaces of the components of the same 
 curvature so that they may be cemented together with Canada 
 balsam, a transparent cement. This reduces the loss of light by 
 reflection at the two surfaces and makes of the two lenses a single 
 unit, the parts of which are not subject to relative displacement. 
 In hot or humid climates the Canada balsam deteriorates and it is 
 
42 
 
 necessary to have the lenses taken apart and recemented from time 
 to time. 
 
 Astigmatism. — Figure 44 shows the manner in which an image of a 
 point is formed by a pencil of light proceeding from a converging lens 
 when affected by astigmatism. If no aberrations are present the 
 
 
 
 /f^ 
 
 ^ 
 
 
 ,— -1:=^^ 
 
 ^^ 
 
 
 
 1 •/ 
 
 s»— — — — 
 
 ^-.E^^^^^ 
 
 
 .--''^'^ 
 
 "Y 
 
 if 
 
 1^ 
 
 --^^^^"""^ 
 
 b^ 
 
 
 '^ 
 
 h 
 
 :i^ 
 
 
 ^^^^ ^^..--^ 
 
 
 
 
 ^^'^ 
 
 
 
 -■'^ -^ 
 
 \ 
 
 ^ 
 
 If-^^ 
 
 
 
 
 "-" 
 
 "^ 
 
 v^ 
 
 ' 
 
 
 
 
 Fig. 44.— Astigmatism. 
 
 course of the rays in the neighborhood of the image is as shown in 
 figure 45. Cross sections of the pencil of light at several intervals 
 are given and it is noticed that they are circular in form. The best 
 image will be found at A, where all the rays pass through a point. 
 Many times the rays are not so symmetrically arranged but approach 
 
 Ooo ooO 
 
 Fig. 45.— Image of a point. No astigmatism present. 
 
 the condition shown in figures 44 and 4G. To secure clearness, some 
 of the rays are omitted in the perspective dra^nng but reference to 
 the series of cross sections show that the rays, instead of passing 
 through a single point, pass through two short straight lines which 
 are perpendicular to each other, as shown at B and C. Midway 
 
43 
 
 between these two lines, at 1), the cross section is approximately cir- 
 cular. When the image is formed by a bundle of rays of tliis type, 
 we say that astigmatism is present, and the image is said to be astig- 
 matic. A lens which produces an image without this fault is said to 
 be anastigmatic, i. e., without astigmatism. 
 
 The effect of astigmatism upon the images of horizontal and ver- 
 tical lines is interesting. Consider the image of a horizontal line as 
 formed in the plane of B (fig. 46). If the line is as at E (fig. 47), the 
 image of each point of the line will be a short vertical line. There- 
 fore the image of the entire lines will be as showTi at F. The image 
 will be broad and with ill-defined edges. In the plane of C (fig. 46) 
 the image of each point will be a short horizontal line and the com- 
 posite image of the entire line built up of these partial images will be 
 the sharply-defined lines as sho\vn at G. With a vertical line, the 
 
 O o 
 
 D 
 
 C 
 
 Fig. 4f).— Image of a point. Astigmat i^m present. 
 
 opposite is true. The sharp well-defined image shown at I is in the 
 plane of B (fig. 44) , the hazy image in the plane of C. Therefore a 
 series of orthogonal lines form a convenient test object for astig- 
 matism. Either vertical or horizontal lines can be focused sharply, 
 but both can not be focused simultaneously. As for the other aber- 
 rations, astigmatism is lessened by the use of different kinds of glass 
 and of several spherical surfaces bearing the proper relation to each 
 other. The cross section at D in figure 44 which is nearly circular in 
 shape is termed the ''circle of least confusion," and it is in this plane 
 that the most satisfactory image is secured when the object is an 
 average object and does not consist of lines running in a prevailing 
 direction. A lens constructed with ordinary care will not show any 
 astigmatism for a point on the optical axis of the lens but only for 
 points off the axis. 
 
 Coma. — If the different portions of tlie cone of rays in the neigh- 
 borhood of the image come to a focus in approximately the same 
 
44 
 
 IM^^^^ r 
 
 H 
 
 plane, but fall at clifTerent points instead of being superposed, we have 
 coma. 
 
 Figure 48 shows such an image. The lens may be considered as 
 divided into circular concentric zones. The inner zone forms the well- 
 defined image at A. The image formed by the next zone is the small 
 
 circle to the right of A. The image 
 ' ~ L_ formed by each successive zone is a 
 
 slightly larger circle, displaced to 
 the right. The final image is as 
 shown at C. The resemblance to a 
 comet gives rise to the name of this 
 aberration. The symmetry of the 
 lens is such that this displacement 
 of the successive images is always 
 in a direction radiating from or 
 toward the center of the field of 
 the lens. Like astigmatism, coma 
 occurs at the edge of the field 
 rather than at the center. It should 
 be borne in mind that in dealing 
 with astigmatism and coma the 
 simplest possible cases have been 
 treated in which it was assumed 
 that all other aberrations were ab- 
 sent. The image of a point formed 
 by a lens when viewed by a micro- 
 scope is usually found to be affected 
 simultaneously by all types of aber- 
 rations and often has a very fantastic shape only remotely resembling 
 those discussed. 
 ' Curvature. — The two remaining aberrations to be discussed, curva- 
 ture and distortion, differ fundamentally from the first four. Spher- 
 ical and chromatic aberration, astigmatism and coma have to do with 
 the imperfection of the image of a 
 point. Curvature and distortion 
 are aberrations in which the rela- 
 tive location of the images of dif- 
 ferent points are incorrect while the 
 perfection of the separate image 
 point is not considered. The 
 images of the different points of a plane image should be in one 
 plane. However, oftentimes they lie on a curved surface with the 
 points at the edge of the field l3^ing nearer the lens than those at 
 the center. (See fig. 49.) This is particularly disadvantageous in 
 a photographic lens. The photographic plate is plane and the 
 
 ■ 
 
 I 
 
 Fig. 47.— Astigmatic images. 
 
45 
 
 image, if it is to be in focus on the whole phite at once, should 
 also be plane. A field without curvature is termed "flat." Flatness 
 of field is secured by 
 the use of particular 
 kinds of glass and 
 by means of dia- 
 phragms which 
 cause the rays 
 forming images at 
 the edge and cen- 
 ter of the field to traverse different portions of the lens. 
 
 Distortion. — Figure 50 shows the image of an object viewed through 
 a lens which produces distortion. Any straight line extending across 
 
 Fig. 49.— Curvature of field. 
 
 jO.— Dislurli 
 
 the field is curved and for different lenses the curvature may be from 
 or toward the center. The distortion is termed *' barrel-shaped " in 
 
46 
 
 the first case and "hourglass" or "pincushion" in the second. The 
 two lower diagrams show the appearance of a net of orthogonal lines 
 when imaged by a lens affected with either type of distortion. Such 
 an object is frequently used as a test object to determine the amount 
 and character of distortion present. 
 
 RESOLVING POWER. 
 
 A study of the physical nature of light shows that it is a wave 
 phenomenon. It is a commonly observed fact that the ocean waves 
 break around the end of a breakwater and invade the portion of the 
 surface which would be completely sheltered if the waves traveled 
 only in a straight line. In a similar manner, an opaque body may be 
 considered as a "breakwater" sheltering the portions which lie 
 within the shadow from the action of the light waves. And further- 
 more, a careful study of the boundary of the shadow will show that 
 to a very shght extent the light waves bend in around the edge of the 
 opaque object and invade the portion which we should expect to lie 
 
 within the shadow. 
 
 With waves on the water this action is relatively 
 large, as the successive waves do not follow each 
 other closely. Light waves succeed each other so 
 rapidly that the angular extent of the bending, 
 except in special test cases, is very slight. Phe- 
 nomena of this sort are said to be due to diffrac- 
 tion. It follows that some of the conclusions 
 which have been developed regarding the aber- 
 rations are very useful approximations, but are not rigorously 
 accurate, as the effects of diffraction enter as disturbing elements 
 and were not taken into account. 
 
 Referring to figure 45, the image at A is shown as a geometrical 
 point. Such will not be the case even though all the spherical 
 aberration is eliminated. As a result of diffraction, the image is a 
 small disk surrounded by a series of concentric rings which rapidly 
 fall off in intens-ity, as shown diagrammatically on a greatly enlarged 
 scale in figure 51. It follows that details in the image which are less 
 than the diameter of the central disk can not be distinguished. 
 After the aberrations have been brought beneath a certain limit, the 
 diameter of this disk can be further decreased only by enlarging the 
 size of the lens. When two near points viewed through a lens are 
 so close together that they can not be distingushed as two d' t 
 points, it is said that they are not resolved or that they are so close 
 together as to make separation impossible without the use of a lens 
 of greater resolving power. The angle subtended by two points 
 which are just far enough apart to permit resolution is commonly 
 used as the measure of the resolving power and is termed the limit- 
 ing angle of resolution. 
 
If the aberrations arc reduced to such an extent that they do not 
 affect the resolving power and if the rays are parallel and incident 
 upon a lens of diameter d (measured in inches), then the limiting 
 angle of resolution in seconds denoted by r will be given by the 
 equation: 
 
 -^ »" . 
 
 If the aberrations are not suitably corrected, the resolving power 
 may be less than that indicated. The larger the value of d, the 
 smaller is the limiting angle of resolution and the greater the resolving 
 power. Diffraction therefore sets an ultimate limit to the sharpness 
 of image formed by a lens of a given size beyond which it is impossible 
 to go. It also estabhshes a limit beyond which the further elimina- 
 tion of aberrations is unprofitable. In practice, the resolving power 
 is made so great that the resulting lack of sharpness is too slight to 
 be perceived by the eye. 
 
 THE TELESCOPIC SYSTEM. 
 
 The optical system illustrated in figure 52 consists of two con- 
 verging lenses of focal length /j and fn, respectively, separated by 
 the distance f^+f-^- Let it be assumed that the object is to the left 
 of the system and so far removed that the rays proceeding from any 
 one point may be considered as sensibly parallel. The rays AG, 
 BH, and CI are representative of a bundle of parallel rays originating 
 at the point of the object lying on the axis of the lens system. As 
 the rays are parallel, the image of this point, formed by the first 
 lens, is at L. After passing through L the rays are received by 
 the second lens, termed the eyepiece, and since the point L is dis- 
 tant /. from this lens, the rays are rendered parallel and finally 
 emerge along lines MS, NT, and OU. 
 
 The rays DG, EH, and FI are representative of a bundle of parallel 
 rays proceeding from a second point of the object lying at such a 
 distance from the axis that the two points, viewed from the center 
 of the objective, subtend the angle BHE. The image of this point 
 lies on the prolongation of EH, a line passing through the optical 
 center of the lens. Furthermore, since the object is at an infinite 
 distance, the image must lie in a plane through L perpendicular to 
 the optical axis. (It is, of course, assumed that the curvature of 
 field is so slight that it may l)e neglected.) The image is therefore 
 located at the point K and the path of rays DG and FI after refrac- 
 tion is along lines GK and IK. 
 
 Since K is in the focal plane of the eyepiece, the pencil of rays 
 passing through K, after the second refraction, will form a bundle 
 of parallel rays. To determine the direction of this l)un(lle of rays, 
 draw the fictitious ray KZ passing tlu'ough the optical center of the 
 
48 
 
 second lens. As it passes through the optical center it will not be 
 deviated. But all rays passing through K are parallel after refrac- 
 tion. Therefore rays PV, QW, and RX may be drawn parallel to KZ. 
 
 The two bundles of parallel rays selected or any other bundles 
 •originating at the distant object, after passing through the entire 
 system, emerge as bundles of parallel rays. The object was at an 
 infinite distance. But since the emergent rays are parallel, the 
 image is also at an infinite distance. It at first seems paradoxical 
 that such a system can have any use since both object and image 
 are at an infinite distance. 
 
 However, important modifications have been introduced, the most 
 important of which is the change of angle between any two bundles 
 before and after passage through the system. The angle between 
 the two incident bundles is BHE. The corresponding angle between 
 the two emergent bundles is TNZ. It is evident from the drawing 
 that the angle between the two bundles has been greatly increased 
 
 Fig. 52.— Telescopic system. 
 
 by passage through the system. Two points in the object which 
 subtend a relatively small angle when viewed by the naked eye will 
 subtend a much larger angle viewed through the system. As in- 
 creased angular size is associated with nearness of the object, the 
 use of such a system makes the object appear nearer to the observer. 
 It is by use of a combination of this type that the telescope serves to 
 magnify a distant object. The ratio of apparent size of an object 
 viewed through a telescope to the size as viewed by the unaided 
 eye is termed the magnification or the power of the telescope. This 
 ratio is identical with 
 
 tan TNZ 
 tan BHE 
 
 which equals 
 
 tan KN L 
 tan KHL' 
 
 Furthermore, it is easily seen that 
 
 tan KNL _ /, 
 tan KHL /. 
 
49 
 
 The negative sign is introduced, as one of the angles is negative 
 with respect to the other. If the power or magnification is denoted 
 by a, we have 
 
 (42) 
 
 Before incidence the bundle embracing rays DG, EH. and FI is 
 proceeding from below upward. After refraction by both lenses 
 this bundle is proceeding downward. Any point which appears 
 below the axial point when viewed by the naked eye appears above 
 when viewed through the telescope. Therefoi;-e, the image is an 
 inverted one. Points to the right and left are similarly interchanged, 
 and it is evident that this is true inversion and not reversion. This' 
 inversion may be considered as indicated by the negative value of 
 a yielded by eciuation 42. 
 
 The cross section of the bundle of rays before and after passage 
 through the system is greatly changed. The entering bundle has 
 the diameter GI. The diameter of the emergrent bundle is MO. If 
 
 (43) 
 
 Fig. 53.— Galilean telescopic system. 
 
 the two diameters arc denoted, respectively, by rf, and tZ,. it is easily 
 seen that 
 
 Equations 38, 39, and 40 give the focal length and location of 
 principal points for a system composed of two lenses. But s =/i +/,. 
 Therefore, in all these expressions the denominator vanishes for the 
 particular combination which has been discussed. The focal length 
 of the system is infinite, and the principal points lie at the infinitely 
 distant points of the axis. This is the fundamental characteristic 
 of the telescopic system. The telescopic system may therefore be 
 defined as any optical system of infinite focal length with the prin- 
 cipal points at an infinite distance. The particular case illustrated 
 in figure 52 is one of the simplest of many possible variations. 
 
 An important variant is the Galilean type of telescopic system 
 shown in figure 53. The positive eyepiece is replaced by a diverging 
 48918°— 21 4 
 
50 
 
 Ions. As/, is negative, the eyelens is placed between the objective 
 and its focal plane, thus making the distance between the two lenses 
 equal to the algebraic sum of / and f^- The image formed by the 
 objective is at KL. This image is only virtual as the rays, before 
 reaching KL, are received by the eyepiece. Since KL is in the focal 
 plane of the eyepiece, the pencil, after passing through both lenses, 
 proceeds as a bundle of parallel rays. 
 
 The axial bundle of rays is easily traced as it continues along the 
 axis of the system. As in the preceding case, the dh'oction of the 
 oblique bundle may be located by drawing a ray through K and the 
 
 Fig. 51.— Telescopic system. 
 
 optical center of the eye lens. The emergent bundle wdll be parallel 
 to this ray. 
 
 If figures 52 and 53 arc compared, it will be seen that in figure 52 
 the oblicjue bundle, after passing through the system, proceeds from 
 the left do\vnward, while in 53 the corresponding bundle proceeds 
 from the left upward. The telescope shown in figure 52 was an in- 
 verting telescope. In 53 an erecting telescope has been secured by 
 the use of the negative eye lens. 
 
 A telescopic system of the elementary type shown in figures 52 
 and 53 is not particularly useful, as the relatively small number of 
 surfaces makes it impossible to secure a very satisfactory adjustment 
 of the aberrations. In ordinary practice, each component, as above 
 
 Galilean telescopic system. 
 
 described, is replaced by a lens system, in which case the distance 
 from the second principal point of the first system to the first prin- 
 cipal point of the second system is made equal to the sum of the 
 ec[uivalent focal lengths of the two systems. The inverting telescope 
 and Galilean telescope, as actually constructed, are shown in figures 
 54 and 55. 
 
 For astronomical work, one of the two elements may be a spherical 
 mirror instead of a lens. Such a telescope is termed a "reflecting" 
 telescope when it is desired to distinguish it from the refracting tele- 
 scopes in which lenses are employed. The largest astronomical tele- 
 
51 
 
 scopes are of the reflecting type, as it has not yet been found possible 
 to produce the very hirge discs of ghiss sufficiently homogeneous for 
 the construction of lenses, although the less severe requirements to 
 be met by glass destined for use in the construction of mirrors may 
 be met. 
 
 THE TELESCOPE AND THE EYE. 
 
 A section of the eye is shown in figure 56. Light entering the eye 
 is first refracted by the cornea, which is the strongly curved trans- 
 parent portion of the eyeball, showTi at A. The space B, next trav- 
 ersed by the light, contains the aqueous humor. The cross section 
 of the beam is then limited by the iris CC. The iris is a diaphragm 
 and the aperture through which the light passes is termed the pupil. 
 The pupil is the black circular area of the eye which is surrounded 
 by the colored annular portion. After passing through the pupil, 
 the light traverses the crystalline lens D, the space E which contains 
 the vitreous humor, a jellyhke transparent substance, and the image 
 is finally received upon the retina. 
 
 The retina is built up of 
 microscopic rods and cones 
 which are arranged end-on 
 to the light and which 
 cover the back portion of 
 the inner concave surface 
 of the eyeball. It is par- 
 tially covered by a fluid 
 termed the visual purple. 
 The retma is the light sen- 
 sitive part of the eye and 
 the sensations received by it 
 optic nerve, showTi at H. 
 
 The function of the parts before the retina is to produce a clear 
 well-defined image of external objects upon the retina. For objects 
 at different distances, the focusing of the optical system must be 
 different if the images are to be equally well defined. Among optical 
 instruments, the method of focusing employed in the eye is quite 
 unique. Ordinarily, focusing is accomplished by altering the dis- 
 tance between the optical system and the screen receiving the image. 
 In the eye, focussing is performed by varying the curvature of the 
 crystalline lens. Muscles attached to the periphery pull upon the 
 lens until it becomes flatter or relax and permit it to become more 
 curved. This process of focusing is termed ''the accommodation of 
 the eye." The limits of accommodation are the two limiting dis- 
 tances between which objects must be placed if the image is to be 
 sharply focussed upon the retina. It is commonly assumed that 
 the normal eye can accommodate itself for any object lying between 
 
 Fig. 50.— Section of eye. 
 
 ire transmitted to the brain bv the 
 
52 • 
 
 infinity and a point about 12 inclies in front of the eye. Actually, 
 the limits are greater than this. Most eyes can accommodate for an 
 object beyond infinity, i. e., for a convergent pencil of light or for an 
 obiect 4 or 5 inches in front of the eye. The limits of accommodation 
 decrease with age. 
 
 The optical system of an eye in which some of the refracting sur- 
 faces are too strongly curved forms an image in front of the retina. 
 Under such conditions the object will not be seen clearly. If the object 
 is brought very close to the eye, the image is thrown back and may 
 be focused upon the retina. Such an eye is said to be "nearsighted" 
 or myopic. A diverging spectacle lens placed before the eye may be 
 used to compensate for the too strongly curved surfaces of the eye 
 and to permit clear vision at normal distances. 
 
 Conversely, if the surfaces are too slightly curved, the image is 
 formed back of the retina. If the lack of curvature is only slight, the 
 eye may be able to accommodate for far, but not for near objects. 
 Such an eye is said to be "farsighted" or hypermetropic. The 
 remedy is the use of a converging lens to supplement the too slight 
 curvature of the surfaces of the eye. 
 
 Sometimes the surface of the eye is not accurately spherical, but is 
 curved differently along different meridians. This defect is known as 
 astigmatism. The effect upon the image is similar to that previously 
 described under "Astigmatism." It should be noted that in dealing 
 with lenses, the astigmatism arose from oblique refraction, whereas 
 in the case of the eye it arises from a departure of a surface from a 
 truly spherical form. Astigmatism may be corrected by the use of a 
 cylindrical lens which is so turned that its unequal curvature along 
 the different meridians compensates for the nonuniform curvature 
 of the eye. 
 
 The iris by dilating or contracting prevents too much light from 
 entering the eye. It involuntarily contracts when the eye is exposed 
 to a bright light and relaxes again when the illumination is reduced. 
 For intense illumination, the diameter of the pupil is approximately 
 one-tenth of an inch. For very faint illumination, its diameter will 
 increase to 0.25 or 0.3 inch. The rubber eye shield on the eyepiece 
 of a fire-control instrument shuts out all stray light which would other- 
 wise enter from the sides and as a result, the pupil opens up wider than 
 otherwise. For dimly illuminated objects, this is a great help, as the 
 larger pupil permits more light to enter the eye. 
 
 The retina is cellular in construction. If two points of an object 
 are so close together that the images fall on a single cell of the retina, 
 it follows that they can not be distinguished. It is commonly con- 
 sidered that points subtending an angle of one minute are the closest 
 that^n bo separated and this is the limiting angle of resolution for 
 the JHSMMNwormal eye, the different surfaces and media are so 
 
53 
 
 combined that the aberrations of the S3'stem are too shght to inter- 
 fere with distinct vision. 
 
 The angidar extent of an object which may be seen sharply at one 
 time is termed the field of view. For the eye, this is exceedingly 
 small. An ordinary photographic objective gives a sharply defined 
 image corresponding to a field of view of 40° to 60° and objectives 
 constructed for special purposes have much greater fields. With the 
 eye, however, objects are clearly seen only when the image falls near 
 the center of the retina. The angle of view for distinct vision is not 
 
 Fig. 57.— The telescopic system and the eye. 
 
 more than half a degree. Portions of the image falling on the remain- 
 der of the retina are seen indistinctly. The outer portions serve as a 
 finder to locate objects, after which the eye is turned in its socket 
 until the image falls up(m the central portion and- distinct vision is 
 secured. The mobility of the eyeball is so great that the narrow field 
 is not a handicap. In fact, the process of viewing in rapid succession 
 the different portions of an extended object, is usually accomplished 
 without conscious effort. 
 
 Although distinct vision is only afforded by the center of the 
 retina, the peripheral portions are much more sensitive to faint illumi- 
 
54 
 
 nation. It is for this reason that faint objects are more distinctly 
 perceived when the eye is turned so that the image is near the bound- 
 ary of the retina. A faint star just at the limit of visibility may be 
 seen much better if viewed ''from the corner of the eye" than if the 
 eye is turned directly toward it. 
 
 The action of a telescope in connection with the eye is shown in 
 figure 57. At A, two bundles of parallel rays from two very distant 
 points are shown entering the naked eye. These two bundles are 
 brought to a focus at points E and F. At B the course of the rays is 
 shown when a telescope is placed before the eye. The rays entering 
 the eye from any one point are parallel at B, as well as at A, and conse- 
 quently the eye is accommodated for an infinitely distant object in 
 each case. It will be noted that at A, E is above F, while at B, F' ig 
 above. This illustrates the inversion produced by the telescope. 
 The points on the retina are farther apart at B than at A, showing 
 that the image on the retina has been magnified by the telescope. 
 
 C and D illustrate the focusing of the telescope for eyes not accom- 
 modated for infinity. A nearsighted eye focuses with less fatigue for 
 near points, i. e., for divergent rays. If the lens nearer the eye is 
 moved closer to the other lens than the normal separation, the rays 
 from the telescope will be divergent and, for the proper adjustment, 
 the image will be sharply defined upon the retina. If the proper 
 spectacle lens were worn before the eye, the eye and lens would be 
 accommodated for parallel rays without any discomfort and the ad- 
 justment of the telescope would be as shown at B. But it is prefera- 
 ble to remove the spectacle lens and secure the compensation by fo- 
 cusing the telescope as the use of spectacles prevents the eye from 
 coming properly up to the rubber shield and may cause the eye to be 
 so far from the eyepiece of the telescope that the field of view is 
 restricted. Furthermore, there is a loss of light by reflection at the two 
 surfaces of the spectacle lens which is eliminated when the lens is 
 removed and the corrections secured by focusing the instrument. 
 
 If the eye is such that it accommodates with less fatigue for con- 
 vergent rays, the eye lens is drawn back from the front lens as shown 
 at D. For an eye seriously affected by astigmatism, the spectacles 
 should not be removed, as no compensating adjustment can be 
 secured by focusing the telescope. 
 
 The final image of any object is formed upon the surface of the 
 retina and therefore has no genuine relief. The observer, however, 
 appreciates the fact that different objects lie in different planes and 
 is able to allocate to each object its appropriate distance with an 
 accuracy depending upon practice. At least three elements conspire 
 to give relief to an object. They are light and shade, perspective, 
 and stereovision. 
 
 A comparison of a shaded and an unshaded drawing shows at once 
 the added relief (hat results from proper shading. On the stage a 
 
55 
 
 silhouette of a round column with capital, if properly shaded, resem- 
 bles closely a true column. Painted panels correctly shaded are fre- 
 quently used in ceiling deco- 
 rations with an effect com- 
 parable to that of true 
 panels. For an example of 
 perspective, one again re- 
 verts to the stage. When it 
 is desired to present the ap- 
 pearance of a vast extent 
 back of the stage, a back 
 curtain, if painted by a man 
 who understands perspec- 
 tive, and properly illumi- 
 nated, gives a very perfect 
 illusion. This is achieved by 
 the combination of perspec- 
 tive and light and shade. 
 
 The true stereoscopic 
 effect is entirely different 
 and is based upon the fact 
 that the two eyes observe 
 an object from different 
 points. Assume that a cube 
 level with the eyes of the 
 observer is the object. The 
 plan is shown in figure 58 
 It will be seen that the right 
 eye sees the front face and 
 a portion of the right-hand 
 side, while the left eye sees 
 the front face and a much 
 foreshortened image of the 
 left side. The two images are shown in figure 59. The images pre- 
 sented to the two eyes are different, and entirely unconsciously, as a 
 
 result of experience, this is inter- 
 preted in terms of relief. The fact 
 that the two eyes receive different 
 impressions differentiates the true 
 stereoscopic relief from the relief 
 obtained by the use of perspective 
 or light and shade. Tlie illusion 
 produced by the last two would 
 persist if one eye were closed, 
 while it is absolutely necessary that both eyes be employed if stereo- 
 scopic relief is to be secured. 
 
 Fig. .W. — StPreospopip vision. 
 
 
 1 
 
 1 
 1 
 
 Fig. 59.— Stereoscopic picture. 
 
56 
 
 When the distance from an observer to an object exceeds a certain 
 distance, the angle subtended by the two eyes is so small that the im- 
 ages received by both eyes are sensibly alike. True stereoscopic efTect 
 then ceases to exist and any appearance of relief beyond this point is 
 due to perspective or light and shade. Some authorities consider 
 that the distance at which the eyes subtend an angle of 30 seconds is 
 the limiting distance or radius of stereoscopic vision, as it is termed. 
 The eyes are separated approximately 2.6 inches. On this basis the 
 radius of steroscopic vision is approximately 500 yards. 
 
 If, however, a pair of binoculars is employed, the radius of stereo- 
 scopic vision is much increased. In the first place, if the binocular 
 is 6 power, an angle of 5 seconds after magnification is apparently 
 an angle of 30 seconds. Therefore, the radius of stereoscopic vision 
 with the binoculars should be the distance at which the eyes subtend 
 an angle of 5 seconds, i. e., six times 500 or 3,000 yards. Further- 
 more, if the individual telescopes of the binoculars are of the Porro 
 type, the maker turns the telescopes so that the objectives are farther 
 apart than the eye pieces. In other words, the observer views the 
 object from the two objectives which are separated, let us say, twice 
 as far as the eyes. This, then, doubles the radius of stereoscopic 
 vision so that with the service 6-power glasses it is approximately 
 6,000 yards. The formula for the radius of stereoscopic vision is: 
 
 R = a. Y 500 yards (44) 
 
 where a is the magnification of the binoculars, Z, is the distance be- 
 tween the centers of the objectives and Zj is the distance between the 
 eyes. The battery commander's telescope is a large binocular tele- 
 scope mounted on a tripod and particularly designed to give a large 
 radius of stereoscopic vision. When the halves are extended in a 
 horizontal direction, l^ is approximately 27 inches. The power is 10. 
 Substituting in equation 44, 
 
 R = 10. ^. 500 = 52,000 yards. 
 
 Such an instrument is particularly desirable for spotting distant 
 shots as the stereoscopic relief aids in estimating overs and shorts. 
 A binocular in which the distance between the objectives is much 
 greater than that between the eyes is frequently termed a stereo- 
 binocular in order to emphasize the enhanced stereoscopic efi'ect 
 which is thereby obtained. 
 
 THE COMPONENTS OF THE TELESCOPE. 
 
 It has already been noted that the telescopes illustrated in figures 
 52 and 53 are useless for most practical purposes. The simple lenses 
 
57 
 
 do not provide means for removing the aberrations and the complete 
 telescope would show chromatic and spherical aberration, coma, 
 astigmatism, distortion, and curvature. To reduce these aberra- 
 tions to such an extent that they cease to be objectionable, each 
 simple lens is commonly replaced by a system of lenses, the forms for 
 which have been carefully determined by the designer in such a way 
 that the aberrations of the different lenses neutralize each other. 
 
 The lens nearer the object is termed the objective. It is com- 
 monly made of two or three lenses cemented together or separated 
 only slightly. When the objective is composed of two components, 
 one must be positive and the other negative in order that the chro- 
 matic correction may be achieved. The positive component is made 
 of crowTi glass, a barium crown being usually selected. The negative 
 component is made of flint glass. A crown glass is more permanent 
 than the softer flint when exposed to the weather and as in many 
 types of instruments the outer surface of the objective is exposed. 
 
 -I-' 
 
 m 
 
 'A 
 
 B 
 
 Fig. 60.— Types of telescope objectives. 
 
 it is desirable that the objective be designed with the crown compo- 
 nent toward the object. For objectives under three inches in diam- 
 eter, the two components are usually cemented together \^^th 
 Canada balsam. An objective of this type is illustrated at A (fig. 
 60). A cemented objective is particularly desirable in fire-control 
 instruments, as there is then no danger of relative displacement of 
 the two parts. For objectives larger than three inches the two 
 components are not cemented. The two inner radii are then given 
 different values as shown at B (fig. 60). Since there is one more 
 radius of curvature at the disposal of the designer in type B than in 
 type A, it follows that a better correction may be secured. If, 
 however, the most favorable types of glass are selected, for small 
 objectives there is not much choice between lenses of types A and B. 
 At C is shown a cemented objective made up of a flint negative com- 
 ponent placed between two crown positive components. This type 
 of construction is adopted when it is desired to secure the best possible 
 correction of the aberrations or when it is desired to construct a 
 
58 
 
 high-grade objective from ordinary crown glass and Hint instead of 
 from barium cro^vn and flint, as is utilized in the objective of type A. 
 The single lens at the eye end of the diagrammatic telescope is 
 replaced by a system of separated lenses which collectively make up 
 the eyepiece. Several types of eyepiece are illustrated in figure 61. 
 In eacli illustration the lens to the right is nearer the eye and is 
 termed the eye lens. The other lens is called the ''field" lens or the 
 
 fi-4 
 
 HUYGHENS 
 
 RAMSDEN 
 
 KELLNER 
 
 FRENCH 
 
 SYMMETRICAL 
 
 ORTHGSCOPIC 
 
 Fig. 01 .—Types of eyepieces 
 
 " collective " lens. The Huyghens and Ramsden eyepieces are each 
 made up of two single lenses. These types of eyepiece were used 
 exclusively in the earlier telescopes and are still much used in lab- 
 oratory telescopes of low power. A telescope with a Huyghens 
 eyepiece can not be provided with a reticule (see below) and for this 
 reason is only used on telescopes designed for observation purposes. 
 The Kellner eyepiece is a modified Ramsden in wliich the single eye 
 
59 
 
 lens of the Ramsden type is replaced ])y a cemented doublet made up 
 of crown and flint glass. The Kellner eyepiece is one of the types 
 most commonly employed on fire-control apparatus. It is also 
 illustrated in figure 54. The symmetrical eyepiece is the other type 
 most common on fire-control instruments. It is always employed 
 on a telescope which is designed to have a long eye distance as for a 
 sight which is mounted directly on the gun. The French eyepiece - 
 is so named because it was generally indicated on French drawings 
 of optical instruments furnished us during the war. It is very 
 similar to the Kellner eyepiece, but the tlu-ee-component eye lens 
 permits the use of ordinary crown instead of barium crown for the 
 positive components. The orthoscopic eyepiece is used only on 
 high-power telescopes in which an eyepiece of very short focus is 
 required. It is the eyepiece which is commonly employed on the 
 self-contained range finder. This eyepiece is so named because of 
 its freedom from distortion. 
 
 A telescope designed for establisliing a line of sight as a surveying 
 instrument, a fire-control instrument, or telescopic sight is provided 
 with a reticule. In a fire-control instrument the reticule is com- 
 monly a piece of plane parallel glass with a pattern etched upon it 
 and adjusted in the telescope so that the plane of the etched side 
 is in the focal plane of the objective. In figure 52 the reticule would 
 be placed in the plane of KL. (The Galilean telescope shown in / 
 figm-e 53 can not be fitted with a reticule, as the image formed by 
 the objective is virtual and not real.) In a surveying instrument 
 the sturdy construction required for fire-control apparatus is not so 
 necessary. Accordingly, the reticule frequently consists of two 
 crossed spider webs or, where a stadia is to be employed, a more 
 elaborate network made of spider web, fine wire, or quartz fibers. 
 Since the reticule lies in the plane of the image of the object, the 
 markings of the reticule appear superposed upon the target and there 
 is no relative motion between the two. 
 
 When a telescope is intended for use simply to establish a line of 
 sight, the pattern of the reticule is very simple, usually a pair of 
 crossed lines. However, since the apparent angular dimensions of 
 the reticuh' remain constant, reticule scales represent convenient 
 means for measuring small angles. The angle subtended by any 
 two points on the reticule is tlie angle, the tangent of which is d/f, 
 where d is the linear distance between the two points in question and 
 /is the equivalent focal length of the objective. This formula only 
 holds when there is no lens between the objective and reticule but 
 is entirely unaffected by any changes in the optical system traversed 
 by the light after passing through the reticule. 
 
 Different types of reticule are illustrated in figure 02. The single 
 pair of cross wires shown at A is used in the ordinary surveying instru- 
 
60 
 
 ment. In some fire-control apparatus, particularly in that for use 
 against aircraft, typo B or C is commonly employed. The fine lines 
 in the center at B or the clear center at C enables the small image 
 of the airplane to be seen better than if it were obscured by heavy 
 crosslines. At the same time the outer heavy lines help to locate 
 the center of the field. The stadia scale in a surveyor's transit is 
 essentially a scale for establishing a constant small angle. The 
 observer reads through the transit the length of the stadia rod which 
 subtends this small angle. It is then a simple matter to determine 
 the distance to the stadia rod. A stadia scale is shown at D. At E 
 the reticule used in the service binocular is shown. The larger inter- 
 
 B 
 
 D 
 
 Fig. 62.— Types of reticule designs. 
 
 vals of the horizontal scale are 10 mils each. The vertical scale is an 
 inverted copy of the scale on the rear musket sight. The reticule 
 shown at F bears an entire range scale and is used in the telescopic 
 sight for tank guns. The numbers refer to range in hectometers. 
 If an object is ten hectometers away, the gun is trained with the 
 tenth graduation upon the target. The marks are so spaced that 
 the gun is then automatically elevated correctly for that range. The 
 line is purposely inclined so that it makes a small angle with^the 
 vertical. As a result of this, the gun is turned slightly to the right, 
 the amount increasing with increasing range. This compensates 
 for the drift of the projectile which results from the rifling. 
 
61 
 
 If the adjustment of the telescope is mcorrect, there will be rela- 
 tive motion between the target and the reticule as the eye is moved 
 from side to side before the eyepiece. During this test the instru- 
 ment must, of course, be rigidly supported. This relative shifting 
 of target and reticule is said toJ)e due to parallax. It arises because 
 the reticule and image do not lie accurately in the same plane. 
 Parallax can only be eliminated by altering the spacing between 
 objective and reticule until the reticule is brought into the plane of 
 the image. As any apparent shift due to parallax is magnified by 
 the eyepiece, a surprisingly slight maladjustment will lead to a 
 troublesome amount of parallax. 
 
 The telescope with a reticule is superior to the open sight for 
 at least three reasons. The magnification enables the target to be 
 seen much more distinctly. It enables the same accuracy to be 
 secured with a much shorter sighting base than with the open sight. 
 The two objects to be brought into coincidence, i. e., the reticule and 
 target, are in the same focal plane, and the eye accommodates itself 
 for both objects simultaneously without any difficulty, while the open 
 sight requires that the eye view simultaneously or at successive inter- 
 vals the front sight, and the target wliich are obviously at greatly 
 differing distances from the eye. Furthermore, in many cases, as 
 with the periscopes, the telescope is so designed that the observer 
 can see over a parapet and thus secure more protection than with an 
 open sight. 
 
 For perfect focusing of a telescope under all conditions, two ad- 
 justments are necessary. Means must be provided for adjusting 
 the distance between reticule and objective so that there shall be no 
 parallax. This is usually accomplished by a rack and pinion move- 
 ment. If the focal length of the objective is short and the targets 
 are in general at a great distance, the reticule may be adjusted for a 
 target at an infinite distance when the instrument is assembled in the 
 factory and no field adjustment provided for the objective. This is> 
 the case with most fire-control instruments and low power telescopes. 
 A surveyor frequently, however, has to take a sight upon an object 
 distant only a few feet, and with a telescope with a fixed objective, 
 the parallax would be so great that the requisite accuracy could not 
 be secured. 
 
 If the distance between reticule and eyepiece is not adjusted for the 
 observer's eye, the reticule will not be sharply defined. Tliis is the 
 phase of focusing illustrated in figure 57. The two lenses of the 
 eyepiece are mounted in a single tube and its distance from the reticule 
 can be adjusted by a rack and pinion, by a simple draw tube or by 
 rotating the entire eyepiece, causing it to screw in or out. This 
 adjustment is primarily designed to enable the telescope to be 
 focused for different eves and is referred to as the diopter movement. 
 
G2 
 
 Oh fire-control instruments, a scale is generally })rovided around the 
 eyepiece reading in diopters/ by which the telescope can be adjusted 
 directly if the correction required by the eye is known. The scale 
 is graduated from — 5 through to 5. For a normal eye, the eyepiece 
 is set at zero. If a positive 2-diopter spectacle lens is commonly worn, 
 then the eyepiece should be set at +2. This, of course, presupposes 
 that the spectacles are removed. It is usually preferable to use the 
 numbers on the diopter scale as reference numbers only. After 
 having carefully focused the instrument once, the reading of the 
 eyepiece should be noted and used for future focusing. 
 
 If a telescope is only provided with a focusing eyepiece, as is the 
 case in most fire-control instruments, very near objects may be 
 focused sharply by properly adjusting the eyepiece. But there will 
 then be a great deal of parallax between reticule and target which 
 can not be eliminated because of lack of means for adjusting the 
 distance between reticule and objective. This parallax, in a properly 
 assembled instrument, should, however, vanish when the instrument 
 is focused upon a distant object. 
 
 Low-power telescopes are frequently made without any means for 
 focusing. Such an instrument is termed a ''fixed-focus" telescope. 
 A fixed-focus telescope when assembled is often so adjusted that the 
 bundle of rays emerging from the instrument, instead of being 
 parallel, appears to diverge from a point 18 to 30 inches in front of 
 the eye lens. It has been learned that a telescope so focused is more 
 readily adaptable to the eye of the average observer than a telescope 
 focused so that the rays of the pencil are parallel. For a fire-control 
 instrument, the fixed-focus telescope is preferable, as the construction 
 is much simplified and the instrument can be made entirely waterproof. 
 But if the power of the telescope is greater than 3.5 or 4, the accom- 
 modation of the average eye is not sufficient to permit its use. 
 
 THE TELESCOPE WITH AN ERECTING SYSTEM. 
 
 The inverting telescope as commonly constructed with a two- 
 component cemented objective and a Kellner eyepiece, is shown in 
 figure 54. Telescopes of this type are often employed in laboratory 
 apparatus where the inversion of the image is not particularly 
 troublesome. In fire-control apparatus, however, it is essential that 
 all telescopes be of the erecting type, as the observer must not be 
 forced to devote a portion of his energy to the mental transposition 
 of an inverted image. 
 
 1 In the diopter system the "converging power" of the lens is measured instead of the focal length. A 
 positive lens having a focal length of 1 meter is a plus 1-diopter lens. A lens havijig half the focal length has 
 twice the converging power and is accordingly a 2-diopter lens. A negative lens of 1-meter focal length has a 
 power of minus-1 diopter. If P is the converging power in diopters and /is the focal length, measured in 
 
63 
 
 ^ 
 
 LJ 
 
 r 
 
 L 
 
 r 
 
 CD % 
 
 The Galilean telescope, although it gives an erect image, is not 
 suitable for use on fire-control instruments, as it can not be fitted 
 with a reticule. If, in the telescope showm in figure 54, an optical 
 sj^stem is interposed between the objective and ej^epiece, which will 
 reinvert the inverted image formed by the ob- 
 jective, the net result of the two inversions will 
 be an erect image. This is the method adopted 
 in most erecting telescopes used for fire control 
 and either a lens or a prism system may be em- 
 ployed. 
 
 A telescope in which a lens erecting system 
 has been interposed to erect the image is shown 
 in figure 63. The objective A forms an in- 
 verted image of the object at B. The lens 
 erecting system at C is composed of two ce- 
 mented achromatic lenses. It forms at D an 
 image of the original image formed at B. This 
 final image is then viewed by the eyepiece at E. 
 Often a plano-convex lens is inserted in the 
 system in the neighborhood of B. This bends 
 the outer rays proceeding from the objective in 
 toward the axis so that they will pass through 
 the erecting lenses. This serves to increase the 
 field of view for a given diameter of the lenses 
 of the erecting system. Such a lens is termed a 
 "collective" lens. A reticule may be placed at 
 B or D. It is much preferable to place the reti- 
 cule at B, as the objective and reticule then 
 form a unit, and any shift of the erecting sys- 
 tem does not disturb the line of collimation. 
 
 The objective and eyepiece shown in figure 63 
 are the same as in figure 54 and the scale of the 
 two drawings is the same. The increase in 
 length, due to the use of the erecting system, 
 is evident and this can jiot be avoided. For an 
 instrument to 'beTield in the hand, this is a dis- 
 advantage, as the increased length makes the in- 
 strument difficult to hold without vibration. If 
 an attempt is made to decrease the length of a 
 telescope of this type by using an erecting system 
 of very short focus, excessive curvature of field results and the border 
 of the field is very much blurred. For some instruments the increased 
 length due to the use of the erecting system is an advantage. This 
 is the case with all periscopes, and telescopes for this purpose have 
 b(HMi ])uilt in which the distance from objective to eyepiece is as great 
 
 ir-M^ <J. 
 
64 
 
 as 80 feet. The optical system of the periscope is similar to that 
 shown in figure 63, with the collective lens added at B. In order to 
 secure the great length, an erecting system of long focal length is 
 employed so that the distance from B to D makes up the greater 
 portion of the length of the instrument. 
 
 When a lens erecting system is used, the magnifying power is not 
 correctly given by formula 42. This formula is correct when the 
 image formed at B is viewed directly by the eyepiece, as in figure 54. 
 But the lens system at C may enlarge or reduce the image in addi- 
 tion to erecting it. Lengths h and h' are measured from B and D 
 to the corresponding principal points of the erecting system. The 
 ratio of the size of the image at D to the size of the image at B is then 
 
 — -^ where the negative sign is introduced to indicate that one image 
 
 is inverted with respect to the other. The magnification of the com- 
 plete telescope is: 
 
 The focal lengths of objective 
 and eyepiece are /i and/2, respec- 
 tively. The first quantity in 
 parentheses is the magnification 
 produced by the original invert- 
 ing telescope, the second quan- 
 tity is the magnification pro- 
 duced by the erecting system. 
 In the telescope shown in figure 
 63, the erecting system does not affect the numerical value of the 
 magnification, as h and 6' are equal, but the erecting system may, 
 according to the construction, increase or reduce the magnification. 
 
 Where compactness is a desirable feature, a prism erecting system 
 is employed. The Porro system shown in figure 64 is the type em- 
 ployed in prism field glasses and in many small telescopes. A per- 
 spective view of the prism system is shown in figure 19. The scale 
 of figure 54 and of the following figures illustrating the erecting tele- 
 scopes are the same as for figures 54 and 63. The great reduction 
 in length resulting from the use of the prism system is apparent. It 
 will be noted that the path of the light is doubled back upon itself 
 twice and it is by this means that the compactness is attained. This 
 doubling back of the ray is better shown in figure 19 than in 64. In 
 describing the lens erecting system, it was noted that its tendency 
 is to increase the curvature of the field. The prism system tends to 
 decrease curvature and its use, therefore, is particularly advantageous 
 when a compact instrument having a large field of view is desired- 
 
 i 11 
 
 
 
 \ 
 
 1 
 Ik 
 
 
 
 
 
 
 1 N 
 1 1 
 
 ^s 
 
 V J 
 
 
 i^ 
 
 Fig. 64.— Telescope with Porro erecting system. 
 
65 
 
 A prism erecting s3-stem, either of this type or of the following 
 types, does not alter the magnification of the telescope. There- 
 fore, to determine the magnification, equation 42 is used without 
 change. The reticule is placed at C and occupies the same posi- 
 tion relative to the eyepiece as in the simple inverting telescope. 
 If the line of collimation is to be preserved invariant, there must be 
 no movement of the two Porro prisms as they are interposed between 
 the objective and the reticule. If the telescope is to be mounted on 
 a gun as a telescopic sight, this is a real disadvantage, as the Porro 
 prisms are of an awkward shape and it is difficult to clamp them so 
 tightly that the}' will not shift when the gun is fired. 
 
 In figure 65 the erecting system used is the second type of Porro 
 prism. A perspective view of this is shown in figure 20. The relative 
 positions of the two upper polished faces are such that the system has 
 to be constructed in two pieces, which are either cemented together or 
 clamped when properly located with respect to each other. This 
 type of prism does not shorten the telescope as much as the first type 
 
 Fig. 65. — Telescope with second type of Porro erecting system 
 
 of Porro prism and is not so generall}^ employed. At present it is 
 found in no fii-e-control instruments of American construction, but is 
 used in the three-power observation telescopes which were purchased 
 from the French Government under the name "Longue-vue- 
 Monoculaire." 
 
 For antiaircraft fire-control apparatus and for the right-angle 
 theodolites used for observing the course of the rubber meteorological 
 sounding balloons, a right-angle erecting telescope is required. The 
 right-angle prism with roof angle is employed. By its use the right 
 angle in the axis of the telescope and the erection of the image are 
 secured. Figure 66 shows the complete telescopic system. The per- 
 spective view of the prism is illustrated in figure 21. 
 
 The Brashear-IIastings prism formerly used in our large depression 
 position finders is shown in figure 23. Figure 67 illustrates the com- 
 plete telescopic system. The sole advantage of the Brashear- 
 Hastings prism lay in the fact that erection was accomplished without 
 any lateral displacement of the axis of the telescope. Ileference to 
 48918°— 21 5 
 
66 
 
 preceding and following prism systems will show that for all other 
 
 prism types the 
 axes of the 
 objective and 
 telescope do not 
 fall along the 
 same straight 
 lines. This was 
 thought to 
 make it difii- 
 cult to quickly 
 direct the tele- 
 scope at a tar- 
 get. The ce- 
 mented surfaces 
 and the roof 
 
 angle proved to be disadvantages which outweighed the advantage of 
 
 Fig. 66. — Telescope with roof angle erecting prism. 
 
 Fig. 67.— Telescope with Brashear-Hastings erecting prism. 
 
 the common axis. Accordingly, the Brashear-Hastings system has 
 
 Telescope with Spreiiger prism. 
 
 been discontinued in fire-control instruments. Many times a small 
 open sight is added to aid in the preliminary directing of the telescope. 
 
67 
 
 Figure 68 illustrates the use of the erecting prism shown in figure 24. 
 This prism may be constructed in one piece and is of such shape that 
 it may be rigidly clamped in the instrument. This makes its use par- 
 ticularly adyantageous in a sight to be mounted directly on a gun. 
 There is a considerable lateral displacement of the ray. This may be 
 an advantage or disadvantage, depending upon the particular use to 
 which the instrument is to be put. The light path in the glass is 
 very long, and as the prism is constructed in one piece, it is difficult 
 to secure a piece of glass large enough and sufficiently homogeneous 
 except for telescopes of the smaller size. 
 
 THE FIELD OF VIEW AND BRIGHTNESS OF IMAGE. 
 
 In figure 69 the telescope shown in figure 52 is again illustrated. 
 The courses of rays AD, BD, and CD are traced. These are all inci- 
 dent at different inclinations at the center of the objective and pro- 
 ceed from different points of the object. The ray BD proceeds from 
 a point on the axis, and its course is easily traced as it is not deviated 
 
 Fig. 69.— Field of view of telescope. 
 
 by the lenses but proceeds along the straight line BDGHIJ. Tliis ray 
 should be considered as representative of a bundle of parallel rays 
 proceeding from the axial point of the distant object, brought to a 
 focus at G and again converted into a bundle of parallel rays by the 
 eye lens. 
 
 The aperture of the reticule at G is limited by the diaphragm EF. 
 Ray CD has been chosen so that it just passes through tliis diaphragm. 
 As in figure 51, the course of this ray after inciden^ upon the sec- 
 ond lens is obtained by drawing the auxiliary lineCfeHN. ^Vny ray 
 falling upon the objective at an inclination- greater than that of CD 
 will be stopped by the diaphragm EF. Ray AD is representative of 
 a bundle of rays proceeding from a point of the object and this point 
 will lie at the extreme edge of the field of view as seen through the 
 telescope. As diaphragm EF lies in the focal plane of the objective, 
 it will be projected upon the object in the same manner that the 
 reticule is projected. The field of view will therefore be enclosed by 
 the image of the diaphragm EF. Ray CD is the ray corresponding to 
 AD and symmetrically placed with respect to BD. 
 
68 
 
 The angle ADC represents the maximum angle subtended by two 
 objects which may be viewed simultaneously tlirough the telescope. 
 This is spoken of as the true field of view of the telescope. If the 
 radius of EF is r and angle ADC is u, then it is evident that — 
 
 tan^/z^JJ (46) 
 
 J\ 
 
 u - 2 tan-' Y (47) 
 
 J\ . 
 
 The rays AD and CD after passage through the telescope subtend 
 the angle KIL. This angle is spoken of as the apparent field of view. 
 If it is represented by v, it may be said that objects in directions DA 
 and DC actually subtend the angle u but apparently subtend the 
 angle v. If a is the magnification of the telescope, then — 
 
 v = au (48) 
 
 It should be noted that increasing the size of the objective does 
 not increase the field of view. It does increase the size of the indi- 
 vidual bundles of rays received by the telescope, as shown in figure 
 52, but the field of view is independent of the diameter of the objec- 
 tive. The eye lens must be large enough to receive the ray CDE or 
 it will reduce the extent of field below that given by the equation. 
 
 The apparent field of view is generally limited by the optical possi- 
 bilities of the eyepiece. An apparent field of 45° may be considered as 
 a practical maximum for a highly corrected eyepiece. Thirty-five 
 to forty degrees is a more common value and in the more simply 
 constructed eyepieces the field may be only 25° or 30°. This, with 
 equation 48, determines a maximum value of the true field for any 
 given magnification. For example, if we use an eyepiece corrected 
 for an apparent field of 40° and have a ten-power telescope, the maxi- 
 mum true field will be 4°. If the telescope has an erecting system 
 of any type, the components must be designed sufficiently large to 
 transmit all the rays indicated in figure 69, otherwise the erecting 
 system will decrease the field of view as determined by the fore- 
 going analysis. 
 
 In figure 69 the rays which are traced cross at the point I. It is 
 evident that D and I are, respectively, object and image points with 
 respect to the eyepiece. Reference to figure 52 shows that at this 
 point the cone of rays proceeding from the telescope has the minimum 
 diameter. The eye should be so placed that the pupil is brought to 
 this point. If a telescope is held ten or fifteen inches from the eye 
 and pointed at a bright source of light, a brightly illuminated disk 
 will be seen a short sistance in front of the eyepiece. This disk 
 occupies the position at Y in figure 52 and I in figure 69. It is termed 
 the exit pupil and is the image of the objective formed by the eyepiece. 
 
69 
 
 Referring to figure 53, it is seen that the emergent bundles of par- 
 allel rays cross at a point between the eye lens and the objective. 
 It is desirable to bring the pupil of the eye to this point but this 
 obviously can not be done as the location of the eye lens prevents it. 
 The- exit pupil in this case is virtual. This follows naturally as the 
 exit pupil is the image of the objective formed by the eyepiece antl, 
 if the eyepiece is negative, the image of the objective is necessarily 
 virtual. Since the eye can not be brought to the exit pupil, the field 
 of view is very much restricted and equation 47 does not apply. 
 For the Galilean telescope the field of view is increased when the 
 objective is enlarged. In order to secure a satisfactory field of view 
 with tliis type of telescope, it is necessary to make the objective 
 much larger than is conmaonly employed in telescopes of type shown 
 in figure 52. 
 
 A consideration of figure 70 will make clear the simple relationship 
 existing between the brightness of an object viewed with the naked 
 eye and through a telescope. The telescope, with the objective as 
 limited by the full lines, is so constructed that the diameter of the 
 pencil delivered to the eye is the same as that of the pupil. To make' 
 
 Fig. 70.— Restriction of the rays by the pupil of the eye. 
 
 the illustration more concrete, assume that the telescope magnifies 
 two times. Then, by reference to equation 43, it is seen that the 
 entrance pupil, i. e., the diameter of the unobstructed portion of the 
 objective, is twice that of the pupil of the eye. Since the area of the 
 objective varies as the square of its diameter, four times as much 
 light enters the telescope from a given small object as would enter 
 the naked eye. But the telescope magnifies two times and therefore 
 the light forms an image on the retina having four times as great 
 an area as that of the image formed by the unaided eye. As the 
 telescope collects four times as much light and distributes it over four 
 times the area on the retina, it follows that the image has the same 
 brightness per unit area as though the telescope were not used. This, 
 however, represents the ideal case. In practice there is always a 
 considerable loss of light by absorption in the lenses and reflection 
 at the surfaces which will seldom amount to less than 25 per cent 
 and may be as much as 75 per cent of the incident light. 
 
 It follows that where the objective is enlarged to such an extent 
 that the emergent beam fills the pupil of the eye, there is no further 
 gain in illumination to be secured by enlarging the objective. The 
 result of enlarging the objective beyond this point is illustrated in 
 
 :/ 
 
70 
 
 figure 70. The light which enters the telescope through the portion 
 of the objective bounded by the full lines enters the eye. If the lens 
 is enlarged as indicated by the dotted lines, it is true that more light 
 enters the telescope, but the additional light is prevented from enter- 
 ing the eye by the iris. 
 
 To determine the proper diameter of the objective of a telescope 
 it is necessary to know the diameter of the pupil of the eye. At 
 night the pupil may dilate until its diameter is from 0.25 to 0.3 inch. 
 A telescope for use at night should have an exit pupil of this size and 
 by equation 43 it is seen that to meet this condition the diameter of 
 the entrance pupil in inches must be from 0.25 to 0.3 times a, where 
 a is the angular magnification of the telescope. If the entrance 
 pupil is at the objective, as should be the case, this is the necessary 
 diameter of the objective. During the day, the diameter of the pupil 
 is from 0.1 to 0.2 inch and the useful diameter of the objective is 
 correspondingly smaller. Many of the higher-power telescopes used 
 in fire control are made with objectives smaller than indicated above. 
 Brightness of image has been sacrificed in order to secure a more 
 compact instrument. In comparing the brightness of image formed 
 upon the retina, with and without a telescope, the brightness of the 
 image as it would be measured b}" a physical instrument has been 
 referred to. Due to physiological or psychological causes, a small, 
 very dimly illuminated object appears brighter and details may be 
 more easily discerned when there is magnification, even though the 
 image actually is no brighter. For this reason with a night glass 
 there is a distinct gain in visibility which is due to the magnification 
 and which is often more than sufficient to ofi'set the inevitable loss 
 of light caused by the use of the telescope. In using a night glass, as 
 in all conditions when the illumination is poor, the eye should be 
 protected from all stray light in order that the pupil may dilate as 
 much as possible. If a monocular instrument is used, the eye not 
 in use should also be shielded from the light, as, if it receives too 
 much light, the pupil of the eye in use will contract sympathetically. 
 
 THE SELECTION AND USE OF A TELESCOPE. 
 
 In a telescope, high power, a brightly illuminated image, and com- 
 pactness are three elements which mutually oppose each other. The 
 advantage of great magnification is often overrated, particularly in 
 instruments designed to be held in the hand. It is well to decide 
 first on the minimum power which will be satisfactory. A compro- 
 mise can then be effected between the two remaining elements. The 
 light-gathering capacity in two similar optical systems depends 
 upon the diameter of the exit pupil. This can readily be judged by 
 pointing the instrument toward the sky and looking through it with 
 the eye 10 or 12 inches from the eyepiece. The brightly illuminated 
 
71 
 
 disk seen in front of the eyepiece is the exit pupil. Other conditions 
 being eqiirtl, the brightness of the image varies directly as the area 
 of the disk. As the area varies as the square of the diameter, a slight 
 increase in diameter of exit pupil indicates greatly increased bright- 
 ness of the image. A large exit pupil can not be secured without 
 increasing the size of the instrument or decreasing the magnification. 
 In instruments designed for use during the day, the exit pupil is 
 generally from 0.18 to 0.20 inch in diameter. For a night glass the 
 diameter should be at least 0.25 inch. Eight power is commonly 
 considered the maximum allowable for an instrument to be held in 
 the hand, while six represents the more common practice. The lower 
 power instrument usually offers the advantage of a larger field of 
 view. 
 
 The usefulness of a telescope depends primarily upon the quality 
 of definition. It is not a particularly difficult matter to determine 
 which of two instruments has the better definition. The instrument 
 should first be carefully focused. In focusing an instrument it 
 is correct to focus the eyepiece first. The telescope should be turned 
 toward the sky and the eyepiece focused until the reticule stands out 
 well defined and can be comfortably viewed without straining the 
 eyes. If this is the only adjustment the instrument has, a distant 
 object viewed through the telescope should appear sharply focused 
 and without parallax. 
 
 If means are provided for focusing the objective, the telescope 
 should now be turned toward the target and the objective focused 
 until there is no parallax. This adjustment can be tested by moving 
 the eye from side to side behind the eyepiece. Focusing should be 
 continued until reticule and target appear quite stationary with 
 respect to each other. During this process the focusing of the eye- 
 piece must not be disturbed. After the parallax has been eliminated, 
 if all has been correctly done, reticule and target should both be well 
 defined. 
 
 It should be noted that the focusing of the objective depends 
 only upon the distance of the target, the focusing of the eyepiece 
 only upon the eye of the observer. If several observers are using 
 the same instrument successively upon a stationary target, difi"erent 
 observers should alter only the eyepiece adjustment. The adjust- 
 ment of the objective should be the same for all observers so long as 
 the single target or only very distant targets are observed. 
 
 Suitable test objects are usually easily found. Telegraph wu^es, 
 the cross braces and lattice members of steel tank towers, flagstaff's 
 or church steeples with weather vanes are all convenient. The 
 definition in the center of the field should first be considered. The 
 image should be very sharp and well defined. If the optical system 
 contams prisms, it is well to select an object in which two well-defined 
 
72 
 
 lines crossing each other are available. The cross braces of a steel 
 tank tower are particularly suitable. Both lines should be sharply 
 defined. If one appears slightly hazy or out of focus, it indicates the 
 presence of astigmatism. Wliere this exists in the center of the 
 field, it may be due to the use of strained glass or a slightly spherical 
 reflecting surface instead of a plane surface on one of the prisms. 
 
 Another test frequently given for the definition in the center of 
 the field is the manner in which the image changes during the process 
 of focusing. If the image appears equally sharp for a considerable 
 range of focusing, it generally indicates that the definition is equally 
 poor for the entire range. If a truly sharp well-defined image can be 
 secured at all, it will persist only through a very slight range of 
 focusing. 
 
 After the definition in the center of the field is tested the outer 
 portions may be considered. The image should be flat; that is, 
 aU parts of the field should be well focused simultaneously. In some 
 particularly high-grade instruments the definition is remarkably 
 good over the entire field, while in the medium-grade instruments 
 the definition falls off at the edges. Astigmatism may be tested at 
 the edge of the field as well as at the center by observing the crossed 
 lines, but if the field is satisfactorily flat there will not be a detri- 
 mental amount of astigmatism. 
 
 Objects near the center of the field should not be surrounded by a 
 fringe of color. However, nearly all instruments will show slight 
 color about objects near the edge of the field. In a good instrument 
 there should not be enough color to interfere with the appreciation 
 of fine detail. To test distortion, a straight flagpole or smokestack 
 is a convenient object. The telescope should be turned so that the 
 object is near the edge of the field. If the image is curved as in 
 figure 50, it indicates the presence of distortion. 
 
 If the telescope has a reticule and is not provided with means 
 for focusing the objective, it should be tested for parallax. The 
 instrument should be placed on a fixed support and directed at an 
 object several hundred yards distant. As the eye is moved from 
 side to side behind the eyepiece there should be no relative motion 
 of reticule and object. If there is motion, there is parallax. Par- 
 allax may be removed by an adjustment which, however, requires the 
 disassembling of the instrument and should only be done by the 
 maker. If the instrument has means for focusing the objective, the 
 existence of parallax only indicates that the observer has not focused 
 the instrument correctly. 
 
 The reticule of a telescope should be carefully examined for the 
 presence of fUm. In many instruments after a few months' use a 
 film forms on the reticule, which makes necessary the disassembling 
 of the instrument for cleaning. In its advanced stages the film is 
 
73 
 
 conspicuous and will not be overlooked. If a formation is beginning 
 to develop, it may be detected in the following manner: Hold a 
 sheet of white paper or the palm of the hand one or two inches in 
 front of the objective. The interior of the instrument is then 
 illuminated by diffuse light. If now the reticule is closely examined 
 through the eyepiece, a slight granular deposit may be detected. 
 If so, the film is beginning to form and may be expected to render the 
 instrument useless in a few months until taken apart and cleaned. 
 
 If the telescope is of the binocular type, the two telescopes should 
 be tested independently as outlined above. It should be further 
 tested by fairly continuous use for an hour or two in order to deter- 
 mine whether it unduly fatigues the eyes. Sometimes in a binocular 
 the mounting is not correct and the axes of the telescope are not 
 parallel. If the instrument is of the prism type, the prisms may be 
 out of adjustment, in which case the two images of a straight line 
 formed by the two telescopes are not parallel. Such a defect is 
 known as "twisted field." Either of these two defects may not be 
 noticed if the instrument is used but a short time, but serious eye 
 strain is produced, and continued use is very fatiguing. 
 
 The mechanical parts should work smootlily and without lost 
 motion. In a high-grade binocular instrument an adjustment 
 should be provided by which the distance between the two eye- 
 pieces may be varied to accommodate the iijterpupilary distance of 
 different observers. The two telescopes should be capable of inde- 
 pendent focusing in order to secure an adjustment for differences 
 which may exist between the two eyes. 
 
 In the tests described above, laboratory methods of testing have 
 purposely been avoided. Only such tests have been described 
 for which the user ordinarily has facilities. If a complete deter- 
 mination of the optical qualities of an instrument is desired, it 
 may be sent to the optical-instrument department of the Bureau 
 of Standards for test. 
 
 THE OPTICAL CHARACTERISTICS OF SERVICE FIRE-CONTROL 
 INSTRUMENTS. 
 
 Following there is given a classification of the different fire-control 
 instruments based upon the optical systems which are used. A 
 very brief description of the more salient characteristics of the 
 different systems is given, together with tables giving field of view, 
 magification, diameter of exit pupil, and type of eyepiece for each 
 instrument. The range finder has been omitted and is described 
 in a separate section. 
 
 The telescopic system with Porro prisms. — In figure 71 a perspective 
 view is shown of the optical system commonly employed on observa- 
 tion telescopes and the larger telescopic sights. Erection is accom- 
 plished by means of a Porro prism system and the eyepiece, in most 
 cases, is of the Kellner type. In the Longue-vue-Monoculaire a 
 
74 
 
 Porro system of the type shown in figure 20 is employed. All other 
 instruments have the type illustrated in figure 19, Below is tabu- 
 lated the optical characteristics of the instruments in which this 
 optical system is employed : 
 
 Telescopic musket sight, model 1913. 
 French aiming circle 
 
 Maclune-giiii panoramic sight 
 
 Azimuth instrument, model 1918 
 
 Azimuth instrument, model 1910 
 
 Observalion telescope, Longue-vue-Monoculaire. 
 
 2-inch telescopic sight 
 
 3-iQch telescopic sight 
 
 Lewis depression position iinder. 
 
 Field of Magnifl- 
 view. cation. 
 
 4 15 
 
 11 20 
 
 7 45 
 
 3 50 
 
 2 15 
 
 G 
 
 3.7 
 
 5.9 
 10 
 20 
 10 
 15 
 15 
 23 
 30 
 
 8 
 15 
 15 
 25 
 
 Diameter 
 of exit 
 pupil. 
 
 0.14 
 .15 
 .17 
 .25 
 .12 
 .30 
 .20 
 .20 
 .13 
 .10 
 .25 
 .20 
 .20 
 .12 
 
 Type of eye- 
 piece. 
 
 Triple! lei 
 7''rencli. 
 Kellner. 
 Do. 
 
 Do. 
 
 French. 
 
 Kellner. 
 Do. 
 Do. 
 Do. 
 
 EYE LENS 
 J) LENS 
 
 PORRO PRISM 
 ERECTING SYSTEM 
 
 r"iG. 71. — Telescope with Porro erecting system. 
 
 The telescopic system with lens erecting system. — A perspective view 
 of the telescopic system with a lens erecting system is shown in 
 figure 72. This optical system is used on the telescopic tank sight, 
 the new musket sight and the telescopic sight for the 37-mm. infan- 
 try gun. For the last two sights, the system is substantially as 
 shown in the drawing. The tank sight has an objective relatively 
 much smaller than indicated in the drawing and a collective lens in 
 the plane of the reticule. The optical characteristics of the different 
 instruments are given below : 
 
 Instnmient. 
 
 Field of 
 ^^ew. 
 
 38 30 
 7 30 
 7 25 
 
 Magnifi- 
 cation. 
 
 Diameter 
 of exit 
 pupil. 
 
 Type of eye- 
 piece. 
 
 
 1.1 
 
 2.5 
 2.8 
 
 0.16 
 .20 
 
 .25 
 
 
 
 Do. 
 
 Telescopic sight for 37-mm. infantry gun. 
 
 Symmetrical. 
 
 
75 
 
 The periscopic system. — The periscopic system is very similar to the 
 telescopic system %\ath lens erecting system shown in figure 72 with 
 the addition of two right-angle prisms. The complete system is 
 shown in figure 73. The officer's trench periscope and battery com- 
 mander's periscope are the two instruments in which this optical 
 system is employed. 
 
 Instrument. 
 
 Field of 
 view. 
 
 Magnifi- 
 cation. 
 
 Diameter rr.^^^f^„„ 
 of exit ^^^°^^y^ 
 pupil. ^'^^• 
 
 OflBcer's trench periscope No. 10 
 
 Battery commander's periscope 
 
 5 15 
 
 6 25 
 
 ! i 
 
 7 ' 0.17 1 KeUner. 
 5.7 , .2 1 Do. 
 
 
 
 1 
 
 The periscopic azimuth instrument, model 1918. — This is a periscopic 
 instrument in which the prisms are so turned that the observer stands 
 with his back to the object under observation. The parts of a single 
 inverting telescope with two right-angle prisms are employed. Since 
 
 
 
 ® 
 
 \y LYE lEN; 
 
 Fig. 72. — Telescope witti leus erecting system. 
 
 the prisms of this telescope are always in position for a back sight, 
 erection is accomplished by the prisms without the addition of an 
 erecting system. This system is illustrated in figure 74. 
 
 Tnstniment. 
 
 Field of 
 view. 
 
 Magnifi- 
 cation. 
 
 Diameter 
 of exit 
 pupil. 
 
 Type of eye- 
 piece. 
 
 
 6 
 
 8 
 
 0.22 KelLner. 
 
 
 
 
 
 The right-angle telescope. — The right-angle telescope is showTi in 
 perspective in figure 75. The erection of image and the bending of 
 the axis of the telescope is accomplished by a single prism, a 90° roof 
 angle prism. 
 
 The model 1918 antiaircraft sight uses a different type of telescope, 
 of which it is probable that no more will be constructed. All other 
 types of antiaircraft fire-control apparatus use a right-angle tele- 
 scope, as shown in the drawing. The optical characteristics of the 
 
76 
 
 OBJECTIVE PRI5M 
 
 obje:ctive 
 
 COLLECTIVE LENS 
 
 ^ 
 
 ERECTING 5Y5TEM 
 
 S^^®Yye lens 
 ''^^ field lens 
 lower reflecting prism 
 
 Fig. 73.— The periscopic .system. 
 
77 
 
 OBJECTIVE PRISM 
 
 T^ OBJECTIVE 
 
 Slower reflecting prism 
 
 RETICULE 
 eye LENS HELD LENS 
 
 Fig. 74.— Optical system of the periscopic 
 
 azimuth iiistrumeut, Model I'Jl 
 
78 
 
 telescopes for all instruments are substantially the same and are 
 tabulated below. 
 
 Instrument. 
 
 Field of 
 view. 
 
 Magnifi- 
 cation. 
 
 Diameter 
 of exit 
 pupil. 
 
 Type of eye- 
 piece. 
 
 Right-angle telescope 
 
 9 45 
 
 4.3 
 
 O.IS 
 
 French. 
 
 
 
 OBJECTIVE 
 
 EYE LENS 
 riELD LENS 
 
 RETICULE 
 
 ROOF PRISM 
 
 Fig. 75.— The right anple telescope. 
 
 Tfie 'panoramic telescope. — The telescopic system employed in 
 panoramic sights is shown in figure 76. The objective, lower re- 
 flecting prism and eyepiece are similar to those sho\\ai in figure 75 
 for the right angle telescope. Taken together, the}^ constitute an 
 erecting telescope. The objective prism and rotating prism when 
 oriented, as shown, deliver to the telescope an erect image of objects 
 in front of the sight. This image remains erect when viewed through 
 the telescope. 
 
79 
 
 If the rotating prism remains fixed and the objective prism is 
 rotated in azimuth for a side or rear sight, the image viewed through 
 the instrument rotates and becomes inverted when the position for 
 a back sight is reached. To overcome this difficulty, the rotating 
 
 PLANE GLASS WINDOW 
 
 OBJECTIVE PRISM 
 
 ROTATING PRI5M 
 
 OBJECTIVE 
 
 ^' '^ RETICULE 
 
 LOVER REELECTING PRISM 
 
 EYE LENS 
 FIELD LEMS 
 
 Fig. 76.— The panoramic telescni)e. 
 
 prism is added. The rotating prism is mechanically connected to 
 the objective prism by gearing, which causes it to rotate with half 
 the angular velocity of the objective prism. The rotation of this 
 lower prism serves to keep the image in a vertical position regardless 
 of the orientation of the objective prism. 
 
80 
 
 This optical system is employed in the 4-power panoramic sight 
 and in the larger 4 and 10 power panoramic telescope used on rail- 
 way artillery. Below the optical characteristics of these two in- 
 struments are given : 
 
 , Instrument. 
 
 Field of 
 view. 
 
 Magnifi- 
 cation. 
 
 Diameter 
 of exit 
 pupil. 
 
 Type of eye- 
 piece. 
 
 Panoramic sight, 4 power 
 
 10 15 
 8 20 
 3 50 
 
 4 
 4 
 10 
 
 0.16 
 .32 
 .14 
 
 S\Tnmetrical. 
 
 Panoramic telescope 4 and 10 power . . 
 
 Do. 
 
 
 
 Binocular telescopes. — Two binocular instruments are used in the 
 Army, the service binocular (t3'pe EE), and the battery command- 
 
 EYE LENS 
 
 FIELD LENS 
 RETICULE 
 
 PORRO PRISM 
 ERECTING SYSTEM 
 
 05JECTIVE 
 
 Fig. 77.— Telescopic system of field glass. 
 
 er's telescope. A perspective view of the optical system of the type 
 EE binocular is shown in figure 77. The image is erected by Porro 
 prisms. The reticule is in the left-hand telescope. 
 
 The battery commander's telescope is a large binocular instru- 
 ment mounted on a tripod. The two telescopes are hinged together 
 and can be used in two positions. When turned as in figure 78, the 
 instrument serves as a binocular periscope and observations can be 
 made from behind a parapet. Both arms of the instrument may 
 be swung into a horizontal position. The instrument then becomes 
 an excellent stereobinocular with the objectives separated approxi- 
 matelv 27 inches. 
 
81 
 
 The erection of the image in this telescope is accompHshed by a 
 modified Porro system in which one prism is replaced by two right- 
 angle prisms in order to permit the first reflection to take place 
 before the light enters the objective. The reticule is in the left- 
 hand telescope. Below aile given the optical characteristics: 
 
 lastnimer-t. 
 
 Field of 
 view. 
 
 Magnifi- 
 cation. 
 
 Diameter 
 of exit 
 pupil. 
 
 Type of eye- 
 piece. 
 
 
 8 
 4 15 
 
 6 
 
 10 
 
 0.19 
 .17 
 
 Kellner. 
 
 Battery commander's telescope 
 
 Do. 
 
 
 
 OEUrCTiVE PRISM 
 
 OBJECTIVE 
 
 .0EYE LENS 
 FIELD LENS 
 
 ERECTING PRISMS 
 
 Fig. 7S.— Telescopic system of B. C. telescope. 
 
 The aiming circle, model 1916. — The optical system shown in figure 
 79 is only used in the aiming circle, model 1916. This peculiar type 
 of erecting system was probably employed in order that a compact 
 
 48918°— 21 6 
 
82 
 
 telescope might be constructed which takes up little space in addition 
 to that occupied by the compass. In the erecting system one reflec- 
 tion takes place at a silver surface and this, together with a relatively 
 lono- light path and large number of surfaces, results in a large absorp- 
 tion of light. An optical system of this type probably will not be 
 employed in future aiming circles. 
 
 Instrument. 
 
 Field of 
 view. 
 
 Magnifi- 
 calion. 
 
 Diameter 
 of exit 
 pupil. ^, 
 
 Tj-pe of eye- 
 piece. 
 
 
 9 30 
 
 3.9 
 
 0.10 
 
 Orthoscopic. 
 
 
 
 
 
 OBJECTIVE PRISM 
 
 FIELD LENS 
 RETICULE 
 
 ERECTING PRISMS 
 
 Fig. 79.— Optical system of aiming circle, Model 1916. 
 
 THE COLLIMATOR. 
 
 The collimator is an ingenious and inexpensive type of sight occu- 
 pying a position intermediate between the open sight and the tele- 
 scopic. It was used by the Germans and French in a great many 
 places on their fire-control apparatus. On range finders, panoramic 
 sights, and other similar instruments, it quite generally replaced the 
 auxiliary open sights used on our instruments. Furthermore, it was 
 the fundamental sighting element on the French seventy-five. The 
 collimator is inferior to the telescopic sight in effectiveness or con- 
 venience and its only advantages are its simplicity, ruggedness, and 
 low cost. These factors, however, make its use particularly desirable 
 as an auxiliary sighting element. 
 
83 
 
 The principle of the sight is ilhistrated in figure 80. The colhmator 
 consists essentially of an opaque reticule bearing a transparent cross, 
 as shown at D, which is placed in the focal plane of a converging lens. 
 The assembled view is shown at A and the course of several rays 
 traced which proceed from the intersection of the two arms of the 
 cross. Since the reticule is in the focal plane, the rays, after passing 
 through the lens, are parallel. Therefore, an eye placed at E, F, G, 
 H, or I wdll see the 
 cross in a fixed direc- 
 tion, i. e., along the 
 respective one of the 
 several parallel rays. 
 Consequentl}^, as 
 the eye is moved 
 from side to side, 
 the direction in 
 which the cross is 
 located remains in- 
 variant and there is 
 no parallax. If the 
 collimator is not 
 properly adjusted, 
 the emergent rays 
 
 diverge or converge as shown at B or C. As the eye is moved, the 
 direction of the cross shifts and there is parallax. A collimator 
 which shows parallax is incorrectly assembled and should be sent to 
 an instrument shop for readjustment. 
 
 The foreign-built collimator was usually constructed as shown in 
 figure 81. The light path is almost wholly through glass. In cross 
 section, the dilTerent elements are square. Elements A and B are of 
 
 Fig. so. — Diagrammatic sketch of colhmator. 
 
 Fig. si.— The coUimator. 
 
 flint and C is made of cro^\Tl glass. This lens system was ^^Tapped in 
 paper and tightly fitted in a square brass tube. This type of system 
 was probably»selected because the complete elimmation of distortion 
 is secured by making the two curved surfaces concentric. Some of 
 the American-made collimators were constructed exactly like the 
 French type and others were as shown in figure 80. During the war, 
 only two sights built in this country were of the collimating type, the 
 American-built French seventy-five and the quadrant siglit used on 
 
84 
 
 the 37-mm. gun. Figure 81 illustrates the optics of the French 
 sight, model 1901, used on the French 75-nim. gun, model 1897. 
 
 Sighting with a collimator sight is generally accomplished by one 
 of the three following-described methods: 
 
 Fig. 82.— The correct use of the collimator sight. 
 
 Both eyes are open and about 10 or 12 inches from the cohimator. 
 One eye sees the cross through the collimator, the other views the 
 
 target, and sighting is accomplished 
 by superposing the cross on the 
 target. This method is funda- 
 mentally incorrect, as it is only accu- 
 rate when the axes of the two eyes 
 are parallel. Normally the axes are 
 ____^__^___^_^^_ parallel when viewing a distant 
 
 /^ ob j ec t , but the intrusion of the colli- 
 
 ^^^""^ yn mator mounting in the field of view 
 
 ' frequently causes the axes to con- 
 
 verge and introduces a large error. 
 The second method of use is 
 similar to the first except one eye 
 is closed. By vertical or horizontal 
 movement of the head, the eye is 
 caused to view successively the cross 
 and the target. The sighting is 
 accomplished in a manner precisely 
 analogous to that in which one lines 
 up three stakes. 
 
 The third method is indicated in 
 figure 82 and is the most precise of 
 all, but more inconvenient to apply. 
 The eye is brought directly behind 
 the collimator and in such a posi- 
 tion that the lower half of the pupil 
 receives light from tRe collimator, 
 the upper half from the target. 
 The cross and target are seen simultaneously and aiming is satisfactory 
 when one is superposed upon the other. This method of usmg the 
 collimator is much facihtated if the upper part of 
 mounting is made as thin as possible. 
 
 .—The collimator or the Michelin sight. 
 
 the collimator 
 
85 
 
 An interesting and ingenious elaboration of the collimator sight 
 is employed in the Miehelin bomb sight. The optical system is. illus- 
 trated in figure 83. The cube of glass ABCD is made up of two pieces 
 cemented together at the diagonal surface BD. This surface carries 
 a thin deposit of silver so thin that approximately half the incident 
 light is reflected and the other half transmitted. The face BC bears 
 the reticule design transparent upon an opaque background. The 
 convex surface AD is silvered, thus forming a concave reflector. 
 
 Light proceeds from the reticule toward the reflector. A portion 
 of this light is reflected do\\Tiward by the half silvered face BD and 
 is not utilized. The other portion reaches the reflector. The curva- 
 tm-e of AD is so selected that BC is in the focal plane. Therefore, an 
 image of B^is formed at an infinite distance. An eye looking down- 
 ward into the face AB sees this image reflected in the silver surface 
 DB. Therefore, the observer sees Uie image of the reticule projected 
 and magnified in the direction E(CJor FH. Furthermore, since the 
 emergent rays are parallel, the direction in which this image is seen 
 is independent of the position of the eye and there is no parallax. 
 But at the same time that the eye views this image of the reticule 
 it also sees the target below the cube by the light which proceeds from 
 the target and passes through the silver film at DB. Lines IJ and 
 KL show the path of such rays. Therefore the observer, looking 
 through the cube, sees the target and the reticule superposed upon 
 it without parallax. The reticule appears as a network of bright 
 lines upon the relatively dark background of the target. 
 
 THE COINCIDENCE-TYPE SELF-CONTAINED RANGE FINDER. 
 
 The modern horizontal base self-contained range finder deter- 
 mines the range by a method of triangulation. The base line is con- 
 tained in the instrument and by reason of its extreme shortness in 
 comparison with the range to be determined, the accuracy and pre- 
 ^ r r 
 
 T 
 
 .90° 
 
 B 
 
 Fig. 84. — The fundamental triangle of the range finder. 
 
 cision required in the determination of the angles at the two ends of 
 the base line is much greater than that afforded by other instruments, 
 with the exception of those designed for the most exacting laboratory 
 requirements. The triangle solved by the range finder is showTi in 
 figure 84. AB is the base of length h, C the target and AC the range 
 r. In the usual construction, the base length and angle at A are 
 
86 
 
 maintained constant in value. The scale upon which the other 
 angle is read may then be graduated to read directly in range. Al- 
 thougli the instrument is graduated to read in linear units, it must 
 be constantl;^ borne in mind tliat the range finder is fundamentally 
 
 an angle-measuring instrument 
 and, in a study of the errors of the 
 instrument, important generali- 
 zations can be made and conclu- 
 sions drawn much more readily if 
 the errors of range are converted 
 into the corresponding angular 
 errors. 
 
 Since the angle m is very small, 
 
 we may write with sufficient accu- 
 
 h 
 
 racy - = 
 transposing 
 
 dr 
 
 Differentiating and 
 
 -dm 
 
 (49) 
 
 and we have the relationship ex- 
 isting between the range error dr 
 ■ and the corresponding angular 
 * error dm. In the horizontal base 
 range finder, as used by the Field 
 Artillery, the base length is 1 
 meter. At a range of 4,000 me- 
 ters the total value of the angle 
 to be measured is 50 seconds and 
 if an accuracy of 2 per cent is to 
 be secured, the value of this angle 
 must be determined to within 1 
 second. 
 
 In an attempt to obtain a 
 high degree of precision and ac- 
 curacy in an instrument expected 
 to withstand the rough usage 
 of service conditions, the design 
 and construction has been very 
 carefully worked out along 
 novel lines which differentiate 
 this instrument completely from 
 all other angle-measuring instruments. It would be impossible 
 to obtain the accuracy required if the observations from the 
 two ends of the base Ime were made by two different observers or 
 successively by a single observer. Therefore, one of the most strik- 
 ing features of the instrument lies in the fact that two observations 
 
from the two ends of the base are made simultaneously by a single 
 observer, the field of view from the two telescopes being viewed 
 through a single ocular after they have been suitably combined by 
 an ocular prism. 
 
 Figure 85 is a diagrammatic sketch showing the fundamental 
 parts of the range finder. The details of construction have been 
 changed in some instances in order that the operation may be more 
 clearly indicated for the reader, but the essential parts are all repre- 
 
 M 
 
 Fig. 86.— Erect field. 
 
 sented. A and A' are two penta prisms at the ends of the base line, 
 by wliicli the ra3's of light are directed toward the ocular prism 
 sho\\Ti diagrammatically at G. B and B' are the two objectives. 
 The ocular prism functions in such a manner that at H there is a 
 "divided field," that is, the field is divided into two parts b}- a sharp 
 boundar}- line. The image in one part of the field is that produced 
 by rays traversing penta prism A and objective B, while in the other 
 
 N 
 
 
 
 Fig. S7.— Invert field. 
 
 art the image is produced by A' and B'. This composite field is 
 viewed tlu-ough the eyepiece J. 
 
 Figures 86 and 87 show two types of such a divided field. That 
 shoAvn in figure 86 is known as the erect type. At K the field is 
 shown when "coincidence" is secured. Except for the presence of a 
 dividing line, the field appears as though formed by a single objective. 
 At L the same type of field is shown when the range finder is not 
 adjusted for coincidence. In figure 87 the inverted type of field is 
 represented in which the objects in the upper half of the field are in- 
 
verted. When coincidence is secured as at N, the image in the upper 
 half is the same as would be obtained if the image in the lower half 
 were mirrored in the dividing line. At O the field is not adjusted for 
 coincidence. 
 
 The erect type of field is usually preferred when the target is of a 
 regular geometrical shape, with well-defined boundaries, and it is, 
 therefore, commonly used by the Navy and Coast Artillery Corps. 
 The inverted type is considered to have advantages when the target 
 is poorly defined or very irregular in shape, and is consequently 
 used by the Field Artillery and Infantry where the objects sighted 
 upon may be trees, trench parapets, etc. 
 
 Returning to figure 85, if prisms C and E are not present, and if 
 the instrument is symmetrically and perfectly constructed, so that 
 the deviations of the rays at all reflecting and refracting surfaces 
 have, their nominal values, the image of an infinitely distant object 
 presented by the two objectives will be the same, and in the field 
 we shall have coincidence, as indicated at K or N (figs. 86 and 87). 
 If at C and E we insert two prisms of glass of very small angle and 
 with their thicker edges turned oppositely we shall have both images 
 shifted slightly in the focal plane in the same direction and the 
 coincidence will not be affected. If now an object is viewed which 
 is within the useful working range of the instrument, the two images 
 in the two portions of the divided field will not be in coincidence. 
 If, in particular, the range finder is rotated in a horizontal plane 
 until the image of the object presented by A' lies in the center of 
 the field, the image presented by A will be displaced from the center, 
 
 the distance ( tan - ) ./" or since the angle —is small, the displacement 
 
 will be — /, where h is the base length, r the range to be determined 
 and /the focal length of the objective B. If the prism at C bends 
 the ray through the angle i by shifting the prism in the appropriate 
 direction through a distance .<? so that 
 
 (50) 
 
 
 s 
 
 tan 
 
 -\-f 
 
 or with sufficient 
 
 exactness 
 
 
 
 
 
 si- 
 
 A-f 
 
 (51) 
 
 the image may be brought back into coincidence. The length s 
 through which the prism C is moved is then a measm-e of the range, 
 and the scale along which C moves may be graduated to read directly 
 in range. The graduations will be marked according to the formula: 
 
 s~U-i (52) 
 
 ^ r 
 
89 
 
 where s is the distance of the graduation for range /• from the original 
 position of the prism which is marked infinity. The size of the 
 instrument and rec{uirements of portabihty usually limit the yalue 
 which may be given to h and /. so that by making i the angle of 
 deviation produced by the prism C very small, the distance between 
 successive graduations may be increased and the scale made as 
 open as may be desired. 
 
 It will be noted that the scale is a reciprocal one, and as r increases, 
 the distance between successive graduations for a given increment 
 in r decreases. If the scale were graduated to read the angle sub- 
 tended at the target by the base of the instrument, the formula 
 for determining the graduations is: 
 
 s=i-m (53) 
 
 where 7n is the angle to be measured. 
 
 The wedge at E provides the so-called infinity adjustment. The 
 effect of slight deformations due to stresses originating from tem- 
 perature changes and other conditions of service is to introduce 
 errors much greater than the instrumental errors of the range finder. 
 If the instrument is trained on an object at a known range wT:th the 
 prism C set opposite the corresponding reading on the scale at D, 
 there should be coincidence between the two images in the field. 
 If this is not the case, coincidence ma}^ be secured by shifting the 
 prism E. The range finder is now adjusted so as to be accurate 
 for this particular range, the errors due to the slight deformations 
 having been compensated by the shift of E. After the scale has 
 been justified at this one point, if the instrument has been correctly 
 constructed and designed, it will indicate ranges accurately over 
 the entire scale. 
 
 The known range commonly selected is an ''artificial infinity" 
 secured by the use of a ''measuring lath" or some type of internal 
 adjustment, and for this reason the adjustment of prism E is usually 
 referred to as the infinity adjustment. On some types of instru- 
 ments the infinity adjustment is provided with a scale, as at F, 
 graduated in entirely arbitrary divisions which are numbered. 
 Tliis enables one to make successively several independent infinity 
 adjustments and to select the mean as read on the scale F of the 
 several readings as denoting the correct adjustment. 
 
 THE CONSTRUCTION OF THE RANGE FINDER. 
 
 The particular methods of construction utilized in the difTerent 
 makes of range finders vary. In general, the infinity adjustment is 
 not accomplished by a prism as shown at E shifted along the axis 
 of the instrument. Instead, the prism is commonly placed between 
 
90 
 
 B' and A' and is so placed that the plane in which the deviation 
 takes place does not lie in the plane of figure 85 but at an angle to it. 
 Then, by rotating the prism about the axis of the instrument, the 
 component of the deviation lying in the plane of the drawing (the 
 part which affects the coincidence) may be varied and the infinity 
 adjustment thus secured. In the Barr & Stroud range finder the 
 window placed in front of A', not shown in figure 85, is slightly 
 wedge-shaped and may be rotated for securing the desired correction. 
 In the Bausch & Lomb range finder, the longitudinal movement 
 of C is obtained by connecting C with a spiral thread cut on the 
 interior of a collar which revolves about the entire optical tube of the 
 instrument. The range scale is then cut in a spiral on the outside 
 of this collar. This enables the scale to be made very long and open. 
 On the Barr & Stroud instrument the scale D is attached rigidly to 
 
 A B C 
 
 Fig. SS.— Rotating prisms used in the range finder. 
 
 the prism and moves past a fixed index. A window in the outside 
 tube protected by glass enables the scale to be read. 
 
 In some range finders a prism of the type shown at C in figure 85 
 is not employed for securing coincidence. Instead, two similar 
 prisms of small angle are mounted, one before the other, as shown in 
 section in figure 88. They are connected by gearing in such a manner 
 that the observer, in securing coincidence, causes the two prisms to 
 rotate through equal angles in opposite directions. \Ylien the prisms 
 are oriented with respect to each other as at B, the deviation pro- 
 duced by one prism is neutralized by the other and there is no de- 
 viation. When they are as at C, the maximum deviation is produced 
 and for intermediate positions the deviation has intermediate values. 
 Such a combination of prisms then gives a deviation which is con- 
 stantly in one plane and which may be caused to vary from zero to a 
 maximum value. These are referred to as rotating wedges or rotating 
 compensating prisms. Such a system is not placed at C, as shown 
 
91 
 
 in figure 85, but is placed in position between tbe pent a prism A and 
 the objective B. 
 
 It is very difficult to make B and B' so that they have precisely 
 the same equivalent focal length. If they difl'er, coincidence will not 
 be independent of the position of the object on the dividing line. 
 Frequently, one objective, say B, is made of longer focal length than 
 B', and a second lens termed an "equal magnification lens" is placed 
 between B and the focal plane. By varying the distances between 
 the two lenses, the equivalent focal length of the two combined may 
 be continuously varied until equal to that of B'. 
 
 There are numerous varieties of ocular prisms in use. In figure 89, 
 there is shown a drawing of a type of prism found in the 1-meter base 
 invert-type range finder, as used by the Field Artillery. Two views 
 of the prisms are shown. In the first, the different components are 
 separated in order 
 that the different 
 reflecting surfaces 
 may be seen more 
 clearly. The part 
 Q receives the light 
 from the left-hand 
 objective and after 
 two reflections, the 
 light passes upward 
 to the eyepiece at 
 an angle of approxi- 
 mately 60°. The 
 rays of light from 
 the right-hand ob- 
 jective after two reflection: 
 ward and upward by prism S. 
 
 Fig. 89.— Ocular prism. 
 
 the rhomb R are reflected for- 
 It should be noted that one re- 
 flection takes place on the lower surface of S, a portion of which is 
 silvered. It is the boundary of this silvered region which forms the 
 dividing line, the rays which are not intercepted and reflected by the 
 silver going downward and never entering the ocular. On the other 
 hand, the rays from the left-hand objective which strike the sflver 
 are reflected do\vnward and do not enter the ocular. The prism T 
 cemented on the inclined portion of S is traversed by light from both 
 objectives. Its purpose is to equalize the length of light path for 
 different portions of any beam in such a way as to obviate spectral 
 dispersion. 
 
 In order to prevent, so far as possible, any relative motion between 
 the two objectives and the ocular prism, these are commonly mounted 
 in a substantial inner tube termed the optical tube. Tliis is designed 
 very carefully and made from especially selected homogeneous ma- 
 terial in order that the detrimental effect of temperature changes 
 may be nullified so far as possible. Slight displacements of the penta 
 
92 
 
 prisms do not so seriously affect the accuracy of the instrument and 
 they are, therefore, carried by the outer tube. The outer tube is 
 covered with heat insulating material, in order to protect the inner 
 optical tube from nonuniform temperature changes. In order that 
 slight deformation of the outer tube may not subject the optical tube 
 to mechanical stress, the two are connected by a three-point sus- 
 pension. A slight displacement of one tube relative to the other 
 may lead to an appearance in the field as shown in figures 86 and 87 
 at M and P, where the dividing line cuts the two images at different 
 heights. This is usually spoken of as an error in the halving adjust- 
 ment. Means are provided for bringing the dividing line back to its 
 correct position by shifting the inner tube relative to the outer. 
 By inspection of the figures referred to above, it is apparent that if 
 any object accui'ately perpendicular to the dividing line is used for 
 determining the coincidence, as, for example, the mast of the ship in 
 the drawing, a small error in the halving adjustment introduces no 
 error in the determination of coincidence. If, however, a sloping 
 line as the edge of the sail is used for setting coincidence, the error is 
 considerable. 
 
 The long base-range finders used by the Coast Artillery Corps are 
 provided with an attachment termed an '^ astigmatizer." Two small 
 cylindrical lenses are arranged to be swung into the paths of the rays 
 from the two objectives by means of external levers. When in posi- 
 tion, the image of any point is drawn out into a short line perpendicu- 
 lar to the dividing line, (See fig. 40.) During the advance of a hos- 
 tile fleet at night, the best target available may be a searchlight. 
 Without the astigmatizer, range taking on a searchlight would require 
 that the image be kept exactly on the dividing line, a task wliich 
 would prove difficult. With the astigmatizing lenses in position, the 
 image of the searchlight is a short line and the training of the range 
 finder does not have to be done so accurately. 
 
 In the following table there is given in tabulated form the more 
 important optical data of the different types of range finder : 
 
 Instrument. 
 
 Branch of service. 
 
 ''■Ell?' 
 
 Power. 
 
 Field. 
 
 Diameter 
 of exit 
 pupil. 
 
 80-centimeter Bauscli & Lomb azimuth 
 
 type. 
 80-centimeter Bauseh & Lomb fixed-base 
 
 type. 
 80-ccntimeter Bauseh & Lomb fixed base 
 
 with internal adjustment. 
 
 Infantry 
 
 do 
 
 do 
 
 do 
 
 Invert 
 
 ...do 
 
 ...do 
 
 ...do 
 
 11 
 
 11 
 
 11 
 
 10 
 
 15 
 30 
 30 
 
 15 
 28 
 15 
 28 
 15 
 28 
 15 
 2S 
 
 4 
 
 3 45 
 
 3 45 
 
 3 30 
 
 3 10 
 1 30 
 
 1 30 
 
 2 30 
 
 1 20 
 
 2 30 
 1 20 
 1 50 
 1 5 
 1 50 
 1 5 
 
 0.11 
 .11 
 .11 
 .11 
 
 manufactured by Keuffel & Esser. 
 1-meter Bauseh & Lomb fixed base 
 
 Field Artillery.. 
 Coast Artillery.. 
 
 ...do 
 
 Erect 
 
 ...do 
 
 .10 
 
 .07 
 
 
 .05 
 
 KcufTel & Esser. 
 12- foot Bauseh & Lomb fixed base with 
 
 do 
 
 do 
 
 .13 
 
 internal adjustment. 
 
 do 
 
 ...do 
 
 .075 
 .13 
 
 internal adjustment. 
 20-foot Bauseh <V Lomb fixed base with 
 
 internal adjustment. 
 .3fKfoot Bauseh ik Lomb fixed base with 
 
 internal adjustment. 
 
 do 
 
 do 
 
 ...do 
 
 ...do 
 
 .075 
 .13 
 .075 
 .15 
 .079 
 
93 
 
 The large number of prisms in the optical train of a range finder 
 gives rise to very great absorption of light. The variation of absorp- 
 tion in different instruments is large, as so much of the light path is 
 in glass that a relatively slight falling off of the transparency of the 
 glass increases the absorption greatly. Range finders which have 
 been measured show a transmission as great as 39 per cent in some 
 cases, while in particularly poor instruments the transmission falls 
 as low as 17 to 20 per cent. In general, the absorption for the two 
 ends of a single instrument differs greatly as is shown by the fact 
 that one of the fields is usually visibly darker than the other. 
 
 THE AZIMUTH TYPE OF COINCIDENCE RANGE FINDER. 
 
 An ingenious modification of the range finder, as described above, 
 has been invented by Eppenstein and is known in the Army as the 
 "azimuth type" of instrument, in order to distinguish it from the 
 more usual form wliich in contradistinction is termed the " fixed-base" 
 type. Suppose that both prisms indicated in figure 85 are omitted 
 and that one of the objectives has a slightly longer focus than the 
 other. As both objectives serve a common eyepiece, the magnifica- 
 tion of the two ends of this instrument will be different. Now, if 
 the instrument is so constructed that all refractions and reflections 
 take place in the nominal manner, the image of an object at an 
 infinite distance will still remain in coincidence at the center of the 
 field. In general, there will not be coincidence, except at the center, 
 as the two images are represented to different scales by reason of the 
 difference in magnification. If the entire instrument is rotated in 
 azimuth, the image in one half of the field will move faster than in 
 the other half, since the magnifications are different. Therefore, if 
 the instrument is rotated as a whole in the proper direction, one 
 image vnW overtake the other and coincidence wiU be secured. It 
 will be shown below that the position on the dividing line at which 
 coincidence is secured is a function of the range. Consequently, a 
 scale may be etched on the prism along the dividing line and gradu- 
 ated to read range directly. This scale will appear in the field of the 
 instrument, and to determine the range it is only necessary to rotate 
 the instrument as a whole in azimuth until coincidence is secured. 
 Then, on the scale opposite the point on the dividing line, where 
 coincidence is obtained, the correct range may be read. 
 
 The method of graduating the scale is as follows: Let/ and f+Af 
 be the focal lengths of the two objectives. Assume that the instru- 
 ment is so oriented that the image of the target falls in the center of 
 the field of the objective, the focal length of which is/4- A/ Then, if 
 the base length is h and the range is r the image in the other half of the 
 field is displaced from the center in the focal plane through the linear 
 
 distance -^ • If we turn the range finder m azimuth through the 
 
94 
 
 angle Tc, the one image in the focal plane moves a linear distance 
 (/+ A/) ^, the other Jlc. (The angle Ic is so small that the angle is sub- 
 stituted for tan k.) If in particular Tc is so chosen that 
 
 {f+M)l=fl-\-^ (54) 
 
 the one image will have overtaken the other and we shall have coin- 
 cidence. Transposing and collecting terms, equation 54 becomes: 
 
 (f+A{)]c is the distance from the center at which coincidence is ob- 
 tained. If we denote this distance by I we have : 
 
 l=(f+M)^_^i (56) 
 
 This then is the formula by which we can construct our scale. I is 
 the distance measured from the center of the field for the graduation 
 corresponding to any range r. It will be noted that the scale is a re- 
 ciprocal one and that for a given base length the distance between 
 graduations increases as Af is decreased. The actual length of the 
 scale is limited by the total length of the dividing line that is visible 
 through the eyepiece. In an instrument constructed as described, 
 only half of this length could be utilized for a scale as the infinity 
 point from which the graduations begin is in the center of the field. 
 In actual construction, the infinity adjustment is so set that the scale 
 begins at the extreme edge of the field. 
 
 The advantages of this instrument lie in its simplicity and com- 
 parative freedom from moving parts as the movable prism and its 
 mechanism are eliminated. This also probably enables a more water- 
 proof construction to be utilized. No estimate of its accuracy as 
 compared with the fixed base type of instrument is available. One 
 possible disadvantage lies in the fact that the center of the field can 
 not be used for all ranges. In particular, in making the infinity 
 adjustment, coincidence is secured at the extreme edge of the field 
 when conditions are most unfavorable as regards aberrations. Nearly 
 all of the 80-centimeter base-range finders supplied the Infantry are 
 of the azimuth type. 
 
 THE STEREOSCOPIC RANGE FINDER. 
 
 The stereoscopic range finder difi^ers fundamentally from the coin- 
 cidence range finder in the fact that both eyes are used and range 
 finding is accomplished by means of stereoscopic vision. In its sim- 
 plest form this range finder is a large stereobinocular with specially 
 designed reticules. 
 
95 
 
 m 
 
 O LJ 
 
 -ef 
 
 A schematic sketch is shown in figure 90. A, B, C, D, and E are 
 respectively penta prism, objective, penta prism, reticule, and eye- 
 piece of one telescope of the binocular. Corresponding parts of the 
 other telescopes are at A', B', C, D', and E'. As a result of the high 
 magnifying power and the large 
 separation of the penta prisms, a 
 very great radius of stereoscopic 
 vision is obtauied. This can be 
 computed by equation 44. 
 
 The range finder as thus far 
 described is a powerful stereo- 
 instrument but it can not prop- 
 erly be called a range finder, as 
 there is no means provided for 
 the determination of range other 
 than by estimation. It is con- 
 verted into a range finder by the 
 addition of special reticules at D 
 and D'. Each reticule bears a 
 series of vertical marks arranged 
 in a zigzag pattern. The pat- 
 terns on the two reticules differ 
 in precisely the same manner as 
 the two halves of a stereoscopic 
 picture. In other words, the two 
 reticules and the two eyepieces, 
 considered separately, constitute 
 an excellent stereoscope and the 
 images on the two reticules viewed 
 by the two eyes are fused into a 
 single picture appearing in stereo- 
 scopic relief. This image is su- 
 perposed upon the target. Con- 
 sequently, on looking into the in- 
 strument, the target is seen in en- 
 hanced stereoscopic relief due to 
 the long base AA' and superposed 
 upon it are the marks of the reti- 
 cule, each of which appears by vir- 
 tue of the stereoscopic effect at a 
 different distance from the ob- 
 server. Near each line the reticule 
 carries a mark indicating its ap- 
 parent distance. To read a range it is only necessary to estimate which 
 mark on the reticule appears to be at the same distance from the 
 observer as the target. The number near the mark sives the range. 
 
 O 
 
 % 
 
 o UJ 
 
96 
 
 In a second type of range finder, the observer sees only a single 
 mark in the field of view. On turning a roller, as in the ordinary- 
 type of instrument, the mark appears to advance or recede. Adjust, 
 ment is made until this mark apparently coincides with the target 
 in distance as well as angular position. The range is then read on a 
 scale. 
 
 At present, no branch of the service is using the stereo range finder 
 This type of instrument possesses certain inherent advantages and 
 disadvantages. It is apparently rather well established that the 
 observers who are to use a stereoscopic range finder must be care- 
 fully selected if good results are to be obtained, as the stereoscopic 
 effect is not appreciated with equal effect by all. Careful training is 
 also necessary before accurate readings can be obtained. This also 
 applies, though perhaps not with equal force, to the coincidence 
 range finder. For a sharply defined target of rectilinear form, as the 
 mast of a ship, a steeple or similar object, the coincidence range 
 finder probably gives the more accurate results. For irregularly 
 shaped, poorly defined objects, as clouds of smoke or clumps of 
 trees, the stereoscopic range finder is considered by some authori- 
 ties to be superior to the coincidence type. No reports upon recent 
 tests of the two instruments are available from which conclusions 
 can be drawn. As the field of the stereoscopic range finder is free 
 from a dividing line, it is an ideal observation instrument. It 
 should possess real advantages in determining the range of an air- 
 plane, as the difficulties of keeping a small object on the dividing 
 line, met with in the use of the coincidence type of finder, are avoided 
 
 THE ERRORS OF THE RANGE FINDER. 
 
 The more important errors to which the range finder .is subject 
 may be classified under the two following heads: (a) Accidental 
 errors arising from inability to determine coincidence precisely, 
 which introduce errors proportional to the square of the range and to 
 which the theory of errors is applicable; (b) systematic errors which 
 introduce errors proportional to the square of the range and which 
 can be completely compensated by a suitable setting of infinity 
 adjustment. 
 
 (a) Accidental errors arising from inahility to determine coincidence 
 precisely and whicJi introduce errors 'proportional to the square of the 
 range. — The accidental errors depend fundamentally upon the 
 limitations of the eye and the quality of optical performance of the 
 range finder. Under the most favorable conditions, when two 
 halves of an image separated by a dividing line, are presented to the 
 eye as by the range finder, it is commonly considered that the proba- 
 ble error in determining coincidence will be of the order of ten or 
 fifteen seconds. In order to secure round numbers, assume that 
 the angular error is 12.4 seconds which corresponds approximately 
 
97 
 
 to 0.06 mil or 0.00006 radian, respectively. This is, of course, 
 
 the angular error in the apparent field. If the magnification of the 
 
 instrument is a, the true angular error in the fundamental triangle 
 
 of measurement corresponding to the apparent angular error in the 
 
 „ , , „ . ,^, , . 0.06 ., 0.00006 ,. 
 
 field of view of the ocularis mil or radian. 
 
 a a 
 
 We shall denote this angular error by dm^. It should be emphasized 
 that the value as given above gives the value of drrii under the most 
 favorable conditions, that is, a trained observer, a well illuminated, 
 sharply defined target and an instrument giving excellent definition. 
 The value of dm^ which will be obtained in practice is a function of the 
 following elements : The training of observer, the character of tai^et, 
 the condition of atmosphere and the optical performance of the 
 range finder. A bright, well-illuminated target, presenting weU- 
 defined contrasting lines on which the setting may be made will be 
 much more favorable for range finding than a target lacking these 
 features. When the air is "boiling," the irregular refraction makes 
 exact coincidence difficult to obtain. Excellent definition, high 
 light transmission and a large exit pupil may all be considered favor- 
 able to the accurate determination of coincidence. 
 
 It will be noted that the value of dm^ is independent of the value 
 of m which, of course, depends upon the range. The error in range 
 corresponding to any particular value of dm^ is, however, not inde- 
 pendent of the range, but is related to it, as shown by the following 
 equation: 
 
 T^ d'Tfi 
 
 The 1,000 is introduced in the denominator in order that din^ may 
 be expressed in mils. The magnifying power of the smaller range 
 finders is generally from 12 to 15, depending upon the particular 
 type. The larger range finders have two eyepieces, the highest of 
 which gives a magnification of 28 and can only be used under very 
 favorable conditions. If the magnification is 15, dm^ under the most 
 favorable conditions, will be 0.004 mil or 0.000004 radian. Under 
 average conditions, dm^ may be expected to be 0.008 mil. On this 
 basis, the table following is computed showing errors to be expected 
 in practice with range finders of different base length. If the 28- 
 power eyepiece is used on the large instruments, the errors may be 
 expected to be approximately 50 per cent of the tabulated values, 
 48918°— 21 7 
 
98 
 
 
 
 
 Error in yards. 
 
 
 
 Range in 
 
 
 
 
 
 
 
 
 
 
 
 
 
 yards. 
 
 SO-centi- 
 
 1-meter 
 
 9-foot 
 
 15-foot 
 
 22-foot 
 
 30-foot 
 
 
 meter base. 
 
 base. 
 
 base. 
 
 base. 
 
 base. 
 
 
 1,000 
 
 9.1 
 
 7.3 
 
 2.7 
 
 1.6 
 
 1.1 
 
 0.8 
 
 2,000 
 
 36.0 
 
 29.0 
 
 11.0 
 
 r,.4 
 
 4.4 
 
 3.2 
 
 3,000 
 
 82.0 
 
 66.0 
 
 24.0 
 
 14.0 
 
 9.8 
 
 7.2 
 
 4.000 
 
 140.0 
 
 110.0 
 
 43.0 
 
 2,5.0 
 
 17.0 
 
 1.3.0 
 
 5,000 
 
 220.0 
 
 180.0 
 
 67.0 
 
 40.0 
 
 27.0 
 
 20.0 
 
 6,000 
 
 320.0 
 
 260.0 
 
 96.0 
 
 58. 
 
 39.0 
 
 29.0 
 
 7,000 
 
 440.0 
 
 .360. 
 
 130.0 
 
 78.0 
 
 53.0 
 
 39.0 
 
 8,000 
 
 580.0 
 
 470.0 
 
 170.0 
 
 100.0 
 
 70.0 
 
 51.0 
 
 9,000 
 
 740.0 
 
 .590. 
 
 220.0 
 
 130.0 
 
 88.0 
 
 6.5.0 
 
 10,000 
 
 910.0 
 
 730.0 
 
 270.0 
 
 160.0 
 
 110.0 
 
 80.0 
 
 11,000 
 
 1, 100. 
 
 880.0 
 
 320.0 
 
 190.0 
 
 130.0 
 
 97.0 
 
 12, 000 
 
 1,300.0 
 
 1, 100. 
 
 380.0 
 
 2.30.0 
 
 160.0 
 
 120.0 
 
 13, 000 
 
 1, 500. 
 
 1,200.0 
 
 450.0 
 
 270.0 
 
 180.0 
 
 140.0 
 
 14,000 
 
 1,700.0 
 
 1,400.0 
 
 520.0 
 
 310. 
 
 210.0 
 
 160.0 
 
 15,000 
 
 2,000.0 
 
 1,600.0 
 
 600.0 
 
 360.0 
 
 240.0 
 
 180.0 
 
 16,000 
 
 2,300.0 
 
 1,900.0 
 
 680.0 
 
 410.0 
 
 2S0.0 
 
 200.0 
 
 17,000 
 
 2,600.0 
 
 2, 100. 
 
 770.0 
 
 460.0 
 
 310.0 
 
 230. 
 
 18,000 
 
 2,900.0 
 
 2,400.0 
 
 860.0 
 
 .520. 
 
 3.53. 
 
 260.0 
 
 19,000 
 
 3,300.0 
 
 2,600.0 
 
 960.0 
 
 580.0 
 
 390.0 
 
 290.0 
 
 20,000 
 
 3,600.0 
 
 2,900.0 
 
 1, 100. 
 
 640.0 
 
 440.0 
 
 320.0 
 
 21,000 
 
 4,000.0 
 
 3,200.0 
 
 1,200.0 
 
 700.0 
 
 480.0 
 
 350.0 
 
 22,000 
 
 4,400.0 
 
 3,500.0 
 
 1,300.0 
 
 770.0 
 
 530.0 
 
 390.0 
 
 23,000 
 
 4,800.0 
 
 3,900.0 
 
 1,400.0 
 
 850.0 
 
 580.0 
 
 420.0 
 
 24,000 
 
 5,200.0 
 
 4,200.0 
 
 1,500.0 
 
 920.0 
 
 630.0 
 
 460.0 
 
 25,000 
 
 5, 700. 
 
 4,600.0 
 
 1,700.0 
 
 1,000.0 
 
 680.0 
 
 500.0 
 
 (b) Systematic errors which introduce errors proportional to the square 
 of the range and which can he completely compensated hy suitable setting 
 of the infinity adjustment. — It has been shouTi in the preceding para- 
 graphs that under favorable conditions the accidental errors of 
 observation may be as small as 0.004 mil or 0.8 second. This 
 implies a remarkable degree of accuracy and it is not sm'prising that 
 under service conditions the parts of the instrument exposed to 
 mechanical shock and heating and cooling are subject to relative 
 displacements and deformations which, although slight when con- 
 sidered according to ordinary standards, are nevertheless sufficiently 
 great to introduce systematic errors too great to be neglected. 
 Many of these deformations are such that there is introduced an error 
 dm^ in the measurement of m, which is independent of the value of 
 m and will, therefore, be compensated if a corresponding opposite 
 error is introduced by a movement of the adjusting prism E shown 
 on figure 85. For example, let it be supposed that the penta prism 
 A' is nonuniformly heated ^vnth a resultant change in the angle 
 between the two reflecting surfaces such that the deviation due to 
 penta prism A' is 90° 4 dm.^ instead of the normal value of 90°. If 
 now the prism C is brought opposite a graduation corresponding to 
 any particular range, and the range finder directed at an object at 
 that distance, we shall in general not have coincidence in the field 
 of the ocular because of the incorrect deviation produced by A'. 
 This can be compensated for by appropriately shifting E along the 
 scale -F, so that an equal and opposite deviation is arbitrarily intro- 
 
99 
 
 duced. Since the error to be compensated is solely due to A' and 
 does not vary with the range of the target, the error when com- 
 pensated for one range is perfectly compensated for all other ranges. 
 It is readily seen that if there are several errors arising from different 
 sources, each of which, like the one dealt with above, is a constant 
 angular error independent of range, the net result of all the errors 
 can be compensated by a single shift of E. Such errors may be 
 introduced by many causes, among which are flexure of the optical 
 tube, relative displacement of the objectives at right angles to their 
 common axis, displacement of ocular prism, or unequal heating of the 
 optical components. The adjustment of the prism E may, as out- 
 lined above, be made by selecting an object at a known distance and 
 making the adjustment. As, however, objects at a known distance 
 are not always available, recourse is usually had to some ingenious 
 method of obtaining two parallel pencils of light. The prism C is 
 then set opposite the infinity mark and the prism E adjusted until 
 coincidence is obtained. This adjustment is, therefore, termed the 
 infinity adjustment. If the infinity adjustment is imperfectly made, 
 there is a constant error dm^ introduced and to this there corresponds 
 a range error: 
 
 dn=—-r~ dm^ (58) 
 
 For any particular faulty infinity adjustment, introducing a 
 constant angular error dm.,, the corresponding error dr^ will always 
 have the sign and magnitude indicated by the above equation. It 
 will be noted that the error dr^, unlike dr^, is an accidental error and 
 is as likely to be positive as negative. Therefore, under the most 
 favorable conditions, the reading for a range r may be expected to 
 lie between r + dr^ + dr., and r — dr^ + dr.,. 
 
 THE INFINITY ADJUSTMENT OF THE RANGE FINDER. 
 
 In the discussion of the errors of the range finder it has been made 
 apparent that a range finder must be correctly adjusted if accurate 
 readings are to be secured. Furthermore, the accuracy of angular 
 measurement required of the range finder is so great that a method of 
 adjustment in the field is necessary which, if possible, should be 
 sufiiciently convenient to permit adjustment immediately before the 
 instrument is used. 
 
 Adjustment by means of a known range is the simplest method. 
 If the distance to a suitable target is given, the range scale on the 
 range finder is set to this known value. If the two halves of the 
 target do not coincide, they are brought into coincidence by means 
 of prism E (fig. 85). The range finder is then in correct adjustment. 
 Under service conditions a known range is not ordinarily available. 
 
100 
 
 A variant of this method depends upon the use of a celestial body, 
 for which the moon is the most favorable. The range scale of the 
 range finder is set at infinity and coincidence secured, as before, by 
 movement of prism E (fig. 85). The sun should never be used as 
 a target for adjusting a range finder. Even though one should be 
 provided with a filter sufficiently dense to protect the eye, the image 
 of the sun is focused upon a cemented surface of the ocular prism and 
 is likely to damage the instrument. If a star is selected as the 
 target, the astigmatizer should not be used unless previous tests show 
 that the range finder gives the same readings yviih or without the 
 astigmatizer in the light path. 
 
 Both of the methods of adjustment outlined require special condi- 
 tions for their application. To overcome this objection, four methods 
 
 of adjustment depending upon the 
 use of an "artificial infinity" have 
 been developed and applied to range 
 finders. 
 
 The most simple and older method 
 of adjustment is by means of the ad- 
 justing lath or stadia. One of the 
 chief disadvantages of this method 
 lies in the fact that the lath is not 
 self-contained in the instrument. 
 Also, it is not always convenient to 
 obtain the range required when the 
 
 vy rry lath is to be used. Accordingly, 
 
 ~^ ^ three types of internal adjustment 
 
 '^ have been devised which form an in- 
 
 FiG. 91.— The adjusting lath. ^ , ,(■.!_•- j. j 
 
 tegral part oi the instrument and 
 permit adjustment to be made immediately before use. 
 
 The adjusting lath is commonly used wath the smaller base range 
 finder, probably because of its portabiUty, simplicity, and low- cost. 
 It consists of a simple bar, carrying at either, end a target w-hich bears 
 a well-defined black line upon a white background. These black 
 lines should be the same distance apart as the actual base length of 
 the range finder. The method of use is illustrated in figure 91. 
 
 A and B represent the two penta prisms at the ends of the range 
 finder, and C and D the black lines on the targets at the ends of the 
 adjusting lath. If the base length of the range finder is the same as 
 the length of the adjusting lath, and if CD is parallel to AB, the paths 
 AC and BD of the beams of light are accurately parallel. Conse- 
 quently, if the scale of the range finder is set at the infinity mark and 
 the adjusting prism E (fig. 85) moved until coincidence is secured be- 
 tween the image of the right end of the lath in one field and that of the 
 left end in the other, the correction is accomplished. If the distance CD 
 
101 
 
 does not equal AB, an angular error will be introduced, the value ot 
 
 ,. ^ .„ , AB-CD ., AB-CD . t. ■ 
 
 which wdl be j^oqq^ mds or ocxTOOiTAC ^®^°"^^- ^^ ^^ apparent 
 
 that this error decreases as AC is increased, and the instructions of 
 the manufacturer are that the adjusting lath shall be set at a dis- 
 tance of 200 meters or more from the range finder. However, it is 
 not improbable that conditions may arise in service when it is im- 
 practicable to use as great a distance, and accordingly some con- 
 sideration will be given to the use of an adjusting lath at distances of 
 50 and 100 meters. 
 
 The difference in length may be an apparent one due to lack of 
 parallelism between AB and CD or to an actual difference of length 
 which may be due to an initial difference of length between AB and 
 CD or to a difference of length brought about by temperature 
 changes. In order to avoid the foreshortening, the lath should 
 be provided with an efficient sighting device by which parallelism 
 can be secured to within at least 1°. By reference to a table of 
 cosines, it can be seen that the error will then be negligible for a 
 range finder of 1 or 1^ meters base length. In addition, each 
 adjusting lath should be fitted to the particular range finder with 
 which it is to be used. To a first approximation it seems that 
 the alteration in base length of the range finder mth temperature 
 will be due to the increased separation of the penta prisms and 
 a result of the expansion of the outer tube. If this is true, 
 effects due to a common rise of the temperatm-e of range finder and 
 lath will be perfectly compensated by making outer tube of range 
 finder and the measuring lath of the same material. 
 
 Below is given a table showing the difference in length between 
 AB and CD which will produce an error of 1 second in the adjustment 
 for three distances of adjusting lath from range finder. 
 
 Distance of ad- 
 justing lath 
 from range 
 finder. 
 
 Length produc- 
 ing error of 
 0.005 mil. 
 
 Meters. 
 50 
 100 
 200 
 
 Millimeters. 
 0.25 
 .50 
 1.00 
 
 It will be noted that the angular error produced is entirely inde- 
 pendent of the base length. The range error is connected with the 
 angular error by the equation: 
 
 r^ drn^ 
 h 17)00 
 
 7 / Ulll. 
 
 ^^2= ~r Trin 
 
 (59) 
 
 The value of the range errors from 5, 000 and 1,000 yards corre- 
 sponding to an error of 0.005 mil and for a 1-meter base range finder. 
 
102 
 
 ai'e resjpecthely 125 and 500 meters. The magnitude of these errors 
 makes very apparent the need for great care in the adjustment of 
 the lath to the particular instrument with which it is to be used. 
 
 Figure 92 illustrates a type of 
 internal adjuster essentially sim- 
 ilar to that used on some of the 
 Barr & Stroud range finders. A 
 and B are two small penta prisms 
 mounted directly in front of the 
 large prism. C and D are two 
 lenses, the focal lengths of wliich 
 are identical and equal to the sep- 
 aration CD. The inner face of 
 each lens has etched upon it a 
 vertical line. The small prisms 
 E and F communicating with win- 
 dows in the outer tube serve to 
 illuminate the etched marks. 
 
 Rays proceeding from the mark 
 on lens C are rendered parallel by 
 lens D and enter the optical sys- 
 tem of the range finder proper after 
 having been bent tlirough ninety 
 degrees by the small penta prisms 
 atB. Since the bundle of rays were 
 rendered parallel by lens D, the im- 
 age of the mark is seen sharply 
 defined when viewed through the 
 range finder in the same manner as 
 though the range finder were actu- 
 ally trained on an object at a great 
 distance. In a similar manner rays 
 from the mark on lens D are ren- 
 dered parallel by lens C and enter 
 the range finder at the other end. 
 Consequently, on looking into the 
 range finder one sees the one mark 
 in the upper field, the other in the 
 lower. If prisms A and B actually 
 deviate the rays through ninety 
 degrees, the two bundles of rays en- 
 tering the end prisms of the range 
 finder are parallel. Therefore, if the 
 range finder scale is set at infinity, the images of the two marks should 
 coincide. If not in coincidence, the instrument is out of adjustment 
 and the adjusting prisms must be altered until coincidence is secured. 
 
103 
 
 ^i 
 
 It should be noted that the function of any internal adjustment is 
 to deliver to the range finder two bundles of rays which shall be par- 
 allel to each other under all conditions. By ''all conditions" it is 
 meant that the parallelism must be maintained so far as possible 
 regardless of the distortion of the system produced by mechanical 
 stresses and temperature changes to which the instrument is inevi- 
 tably subjected in service. So 
 far as possible, the optical system 
 
 of the internal adjustment must 
 
 be self-compensating. Otherwise, 
 the conditions which put the range 
 finder out of adjustment also 
 render the internal adjustment 
 useless as a standard by which to 
 readjust. 
 
 The seK-compensating feature 
 of this type of adjustment lies in 
 the fact that each lens bears the 
 mark which serves as a target for 
 the other lens. If, for example, 
 lens C is displaced slightly, the 
 direction of the bundle of rays en- 
 tering penta prism H is changed 
 due to the displacement of the 
 vertical mark carried by C. But 
 at the same time the direction of 
 the bundle entering prism G is 
 deviated the same amount since 
 the rays entering G pass through 
 C. Therefore, parallelism of the 
 two bundles of rays is maintained 
 independently of any slight lateral 
 shift of lens C or D. If then the 
 deviations produced by prisms A 
 and B do not change, the adjuster 
 will function satisfactorily. As 
 
 these are penta prisms, a small - 
 
 movement will not affect the de- 
 viation. Nonuniform heating of 
 either of these prisms might introduce an error. In regard to this 
 point, the manufacturers assert the prisms are so small that appreciable 
 nonuniform heating is unlikely to result. 
 
 Previous to the war, a small lot of 80 centimeter base range finders 
 were purchased from Bausch & Lomb having an internal adjustment 
 similar to that shown in figure 93. When the range finder is to be 
 adjusted, prisms A and B, which are normally outside the path of 
 
 ^'m-i 
 
 ^^ 
 
104 
 
 the light rays, are swung into the position shown in the drawing by 
 levers on the outside of the instrument. These prisms are of the 
 
 type shown in figure 26. 
 
 A special type of ocular prism 
 is employed which bears a small 
 index mark that appears in the 
 field of view. A window is pro- 
 vided in the outer tube by which 
 this mark is illuminated. Rays 
 proceeding from this index mark 
 pass to the right tlirough the 
 wedge and the objective. The 
 rays are then deviated through 
 180° by the triple mirror. A 
 second similar triple mirror re- 
 turns the rays to the left-hand 
 objective and an image of the 
 index mark is formed in the field 
 of view. The course of the rays 
 is indicated by the arrows. On 
 looking into the eyepiece one 
 sees the index mark in one field 
 and its image in the other. Ad- 
 justment is accomplished by set- 
 ting the range at infinity and 
 bringing image and index mark 
 into coincidence by means of 
 the adjusting prism. 
 
 The self -compensating feature 
 of this adjuster lies in the fact 
 that the deviation produced by 
 a triple mirror is always 180° ir- 
 respective of the orientation of 
 the prism. If the triple mirrors 
 were nonuniformly heated to 
 such an extent that the deviation 
 produced varies from the normal 
 value, the internal adjuster does 
 not function correctly. Further- 
 more, it will be noted that when 
 adjustment is made, the two 
 penta prisms are not in the path 
 of light. Consequently, any 
 error due to the nonuniform 
 heating of these prisms or gradual deformation due to original strain 
 would not be compensated by an adjustment accomplished by the 
 
105 
 
 internal adjuster. It is hoped that tests now in progress will show 
 whether the errors arising from these sources are appreciable in the 
 smaller range finders. 
 
 The large range finders used in the coast defenses are equipped 
 with an absolute self-contained adjusting apparatus, shoA\'n in 
 schematic form in figure 94. 
 
 B and C are two small penta prisms which are brought into position 
 in front of the penta prisms of the range finder when adjustment 
 is to be accomplished. At L there is a small electric lamp, the rays 
 from which are rendered parallel by the collimator objective at A. 
 This portion of the system is so located that haK the rays from the 
 collimator are intercepted by the small penta prism at B and enter 
 the large prism of the range finder. The other half passes over the 
 prism at B and is intercepted by the rhomboid prism. (See fig. 28.) 
 This rhomboid prism brings the beam down into the plane of the 
 
 Fig. 95.— Method of internal adjustment. 
 
 penta prism at C and the half of the beam finally enters the large 
 penta prism to the left. On looking into the eyepiece one sees an 
 image of the light at L in both fields. The range scale is set at 
 infinity and coincidence secured. If now both prisms B and C devi- 
 ated the rays through ninety degrees, adjustment would be completed. 
 But in general, this is not the case. 
 
 To meet this difficulty, a second collimator indicated by the dotted 
 Unes is placed at the other end of the range finder. The light at L' 
 is turned on and the two penta prisms are swung into the positions 
 indicated by the dotted lines. The rhomboid prism at D is also 
 swung through 180°. Light now proceeds from L' into the two ends 
 of the range finder and adjustment is again made. In general, the 
 setting of the adjusting prism for the two adjustments will not be 
 the same. The adjusting prism is provided with a scale and a 
 setting intermediate between the two settings gives the correct 
 adjustment for the range finder. 
 
106 
 
 The accuracy of this adjustment does not depend upon the magni- 
 tude of the deviation produced by the two small penta prisms nor 
 upon the parallelism of the two collimator systems. That this is so 
 is apparent on reference to figure 95. Using the right-hand collimator 
 and with the penta prisms in the position indicated by the full lines, 
 the course of the rays is indicated by the heavy lines. With the 
 left-hand collimator illuminated, the prisms are turned as sho\^^l by 
 the dotted hues and the course of the rays is indicated by the light 
 lines. The lateral separation of the two positions of the prism is 
 introduced in order that the lines of the drawing may not be con- 
 fusing. Also by intention, the path followed by rays proceeding 
 from the two collimators are represented considerably out of parallel. 
 With one setting of the range finder, adjustment is made on rays 
 OA and O'A', from the other setting on OB and O'B'. It is evident 
 that neither of these two pairs of lines are parallel. Therefore, each 
 adjustment is incorrect. But the intermediate setting of the adjust- 
 ing prism is equivalent to setting on the bisectors of angles AOB 
 and A'O'B'. These two bisectors are parallel in all cases, as is 
 evident from the geometry of the drawing. The only condition 
 which can affect the accuracy of this adjustment is a change in the 
 deviation produced by any component which should develop between 
 the two successive adjustments. As the total time required for both 
 adjustments is only a few minutes, this lies outside the bounds of 
 probability. Tliis method of adjustment can therefore be truthfully 
 termed an absolute adjustment. This self-contained adjusting 
 apparatus is much too complicated for introduction into the smaller 
 instruments but is used in the permanently emplaced long base 
 range finders used by the Coast Artillery Corps. 
 
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