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MIRRORS,  PRISMS  AND  LENSES 


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THE  MACMILLAN  COMPANY 

NEW  YORK   •    BOSTON   •    CHICAGO    •  DALLAS 
ATLANTA    •   SAN  FRANCISCO 

MACMILLAN  &  CO.,  Limited 

LONDON    •  BOMBAY   •  CALCUTTA 
MELBOURNE 

THE  MACMILLAN  CO.  OF  CANADA,  Ltd. 

TORONTO 


MIRRORS,  PRISMS  AND 
LENSES 

A  TEXT-BOOK  OF  GEOMETRICAL  OPTICS 


BY 
JAMES  P.  C.  SOUTHALL 

ASSOCIATE   PROFESSOR    OF    PHYSICS,    COLUMBIA   UNIVERSITY 

AUTHOR  OF  "  THE  PRINCIPLES  AND  METHODS 

OF  GEOMETRICAL  OPTICS  " 


THE  MACMILLAN  COMPANY 
1918 

All  rights  reserved 


Copyright,  1918 

By  THE  MACMILLAN  COMPANY 

Set  up  and  electrotyped.    Published  December,  1918. 

<2>fc  1  84 

c. 

is 


PREFACE 

In  spite  of  the  existence  of  a  number  of  excellent  works 
on  geometrical  optics,  the  need  of  a  text-book  which  will  serve 
as  an  introduction  to  the  theory  of  modern  optical  instru- 
ments appears  to  be  generally  recognized;  and  the  present 
volume,  which  is  the  outgrowth  of  a  course  of  lectures  on 
optics  given  in  Columbia  University,  has  been  written  in 
the  hope  that  it  may  answer  this  purpose.  In  a  certain 
sense  it  may  be  considered  as  an  abridgment  of  my  treatise 
on  The  Principles  and  Methods  of  Geometrical  Optics,  but 
the  reader  will  also  find  here  a  considerable  mass  of  more  or 
less  new  and  original  material  which  is  not  contained  in  the 
larger  book.  I  have  endeavored,  however,  to  keep  steadily 
in  mind  the  limitations  of  the  class  of  students  for  whom 
the  work  is  primarily  intended  and  to  employ,  therefore, 
only  the  simplest  mathematical  processes  as  far  as  possible. 
With  this  object  in  view  I  have  purposely  entered  into  much 
detail  in  the  earlier  and  more  elementary  portions  of  the 
subject,  following  in  fact  the  method  which  has  been  found 
to  be  most  satisfactory  with  my  own  pupils;  but  I  venture 
to  hope  that  the  book  may  be  not  without  interest  also  to 
readers  who  already  possess  a  certain  knowledge  of  the 
subject. 

Recent  years  have  witnessed  extraordinary  progress  in 
both  ophthalmology  and  applied  optics.  Not  many  persons 
are  aware  of  the  rapid  rate  at  which  spectacle  optics,  in  par- 
ticular, is  developing  into  a  severe  scientific  pursuit;  and 
there  are  certain  portions  of  this  volume  which  I  think  will 
be  helpful  to  the  modern  oculist  and  optometrist.  Thus, 
for  example,  I  have  been  at  some  pains  to  explain  the  funda- 
mental principles  of  ophthalmic  lenses  and  prisms. 

In  general,  however,  I  have  necessarily  had  to  omit  much 


vi  Preface 

that  is  essential  to  a  thorough  knowledge  of  the  theory  of 
optical  instruments.  In  fact,  in  the  space  at  my  disposal 
it  has  been  found  quite  impossible  to  describe  a  single  one 
of  these  instruments  in  detail.  In  the  latter  portion  of  the 
book  the  theory  of  the  chromatic  and  spherical  aberrations 
is  treated  as  briefly  as  possible;  and  I  have  given  Von  SeidePs 
formulae  for  the  five  spherical  aberrations  in  the  case  of  a 
system  of  infinitely  thin  lenses,  chiefly  because  these  formulae 
are  exceedingly  useful  in  the  preliminary  design  of  an  optical 
system.  But  a  complete  discussion  of  these  subjects  would 
lie  far  beyond  the  plan  of  this  volume. 

The  problems  appended  to  each  chapter  were  originally 
collected  for  the  use  of  my  pupils  and  are  generally  of  a  very 
elementary  description.  A  few  of  them  have  been  adapted 
from  other  text-books,  but  in  such  cases  I  have  now  lost  sight 
of  their  sources. 

If  perchance  this  book  should  help  to  stimulate  the  study 
of  optics  in  our  colleges  and  universities,  the  author  will  feel 
abundantly  repaid.  Unfortunately,  at  present  geometrical 
optics  would  seem  to  be  a  kind  of  Cinderella  in  the  curric- 
ulum of  physics,  regarded  perhaps  with  a  certain  friendly 
toleration  as  a  mathematical  discipline  not  without  value, 
but  hardly  permitted  to  take  rank  on  equal  terms  with  her 
sister  branches  of  physics.  On  the  contrary,  it  might  be  in- 
ferred that  any  system  of  knowledge  which  had  already 
placed  at  the  disposal  of  scientific  investigators  such  in- 
comparable means  of  research  as  are  provided  by  modern 
optical  instruments,  and  which  has  found  so  many  useful 
applications  in  the  arts  of  both  peace  and  war,  would  be  de- 
serving of  the  highest  recognition  and  would  be  fostered  and 
encouraged  in  all  possible  ways.  According  to  the  maxim, 
fas  est  et  ab  hoste  doceri,  the  fact  that  from  the  time  of  Fraun- 
hofer  the  Germans  have  not  ceased  to  cultivate  this  field  of 
theoretical  and  applied  science  with  notable  achievements, 
is  certainly  not  without  significance  for  us  in  this  country 
and  in  England.     Indeed,  both  in  England  and  in  France, 


Preface  vii 

apparently  due  to  the  exigencies  of  war,  schools  of  applied 
optics  have  recently  been  organized. 

Nearly  all  of  the  diagrams  in  this  volume  were  drawn  by 
my  friends,  Professor  Joseph  Hudnut,  Dr.  B.  A.  Wooten  and 
Mr.  J.  G.  Sparkes,  to  whom  I  am  much  indebted.  I  desire 
also  to  express  my  grateful  acknowledgments  to  my  col- 
league, Professor  H.  W.  Farwell,  for  numerous  valuable 
criticisms  from  time  to  time  and  especially  for  aid  in  making 
the  photographic  illustrations  in  Chapter  II. 

Any  suggestions  or  corrections  which  may  improve  and 
extend  the  usefulness  of  the  book  will  be  appreciated. 

James  P.  C.  Southall. 
Columbia  University, 

New  York,  N.  Y., 
April  4,  1918. 


CONTENTS 
CHAPTER  I 

Lights  and  Shadows 

Sections  Pages 

1-11.  1-27 

1.  Luminous  Bodies 1 

2.  Transparent  and  Opaque  Bodies 1-3 

3.  Rectilinear  Propagation  of  Light 3-5 

4.  Shadows,  Eclipses,  etc 6-9 

5.  Wave  Theory  of  Light 9,  10 

6.  Huygens's  Construction  of  the  Wave-Front 10-13 

7.  Rays  of  Light  are  Normal  to  the  Wave-Surface ....  13-15 

8.  The  Direction  and  Location  of  a  Luminous  Point. . .  15-18 

9.  Field  of  View 18,  19 

10.  Apparent  Size 20-22 

11.  The  Effective  Rays 23-25 

Problems 25-27 


CHAPTER  II 

Reflection  of  Light.    Plane  Mirrors 

Sections  Pages 

12-25.  26-63 

12.  Regular  and  Diffuse  Reflection 28-30 

13.  Law  of  Reflection 30-32 

14.  Huygens's  Construction  of  the  Wave-Front  in  case 

of  Reflection  at  a  Plane  Mirror 33-37 

15.  Image  in  a  Plane  Mirror 37-40 

16.  The  Field  of  View  of  a  Plane  Mirror 40-43 

17.  Successive  Reflections  from  Two  Plane  Mirrors 43 

18.  Images  in  a  System  of  Two  Inclined  Mirrors 43-48 

ix 


x  Contents 

Sections  Pages 

19.  Construction  of  the  Path  of  a  Ray  Reflected  into 

the  Eye  from  a  Pair  of  Inclined  Mirrors 48-50 

20.  Rectangular  Combinations  of  Plane  Mirrors 50,  51 

21.  Applications  of  the  Plane  Mirror 52,  53 

22.  Porte  Lumiere  and  Heliostat 53-55 

23.  Measurement  of  the  Angle  of  a  Prism 55 

24.  Measure   of   Angular    Deflections   by   Mirror   and 

Scale 56-58 

25.  Hadley's  Sextant 58-60 

Problems 60-63 


CHAPTER  III 

Refraction  of  Light 

Sections  Pages 

26-39.  64-94 

26.  Passage  of  Light  from  One  Medium  to  Another. ...  64,  65 

27.  Law  of  Refraction 65-67 

28.  Experimental  Proof  of  the  Law  of  Refraction 67-69 

29.  Reversibility  of  the  Light  Path 69 

30.  Limiting  Values  of  the  Index  of  Refraction 70 

31.  Huygens's  Construction  of  a  Plane  Wave  Refracted 

at  a  Plane  Surface 70-72 

32.  Mechanical  Illustration  of  the  Refraction  of  a  Plane 

Wave 72,73 

33.  Absolute  Index  of  Refraction 74-76 

34.  Construction  of  the  Refracted  Ray 76-78 

35.  Deviation  of  the  Refracted  Ray 78 

36.  Total  Reflection 78-83 

37.  Experimental  Illustrations  of  Total  Reflection.  . . .  83-86 

38.  Generalization  of  the  Laws  of  Reflection  and  Re- 

fraction.     Principle    of    Least    Time  (Fermat's 

Law) 86-89 

39.  The  Optical  Length  of  the  Light-Path  and  the  Law 

of  Malus 89-91 

Problems 92-94 


Contents  xi 

CHAPTER  IV 

Refraction  at  a  Plane  Surface  and  also  through  a  Plate 
with  Plane  Parallel  Faces 

Sections  Pages 

40-47.  95-112 

40.  Trigonometric  Calculation  of  Ray  Refracted  at  a 

Plane  Surface 95,  96 

41.  Imagery  in  a  Plane  Refracting  Surface  by  Rays 

which  Meet  the  Surface  Nearly  Normally 96-98 

42.  Image  of  a  Point  Formed  by  Rays  that  are  Ob- 

liquely Refracted  at  a  Plane  Surface 98,  99 

43.  The  Image-Lines  of  a  Narrow  Bundle  of  Rays  Re- 

fracted Obliquely  at  a  Plane 100 

44.  Path  of  a  Ray  Refracted  Through  a  Slab  with  Plane 

Parallel  Sides 101-103 

45.  Segments  of  a  Straight  Line 104,  105 

46.  Apparent   Position  of   an    Object   seen   through   a 

Transparent  Slab  whose  Parallel  Sides  are   Per- 
pendicular to  the  Line  of  Sight 105-107 

47.  Multiple  Images  in  the  two  Parallel  Faces  of  a  Plate 

Glass  Mirror 107-110 

Problems 110-112 

CHAPTER  V 

Refraction  through  a  Prism 

Sections  Pages 

48-62.  113-148 

48.  Definitions  etc 113 

49.  Construction  of  Path  of  a  Ray  Through  a  Prism. .   113-116 

50.  The  Deviation  of  a  Ray  by  a  Prism   116,  117 

51.  Grazing  Incidence  and  Grazing  Emergence 117,  118 

52.  Minimum  Deviation 119-122 

53.  Deviation  away  from  the  Edge  of  the  Prism 122,  123 

54.  Refraction  of  a  Plane  Wave  Through  a  Prism 123,  124 

55.  Trigonometric  Calculation  of  the  Path  of  a  Ray  in 

a  Principal  Section  of  a  Prism 124,  125 


xii  Contents 

Sections  Pages 

56.  Total  Reflection  at  the  Second  Face  of  the  Prism. .   125-128 

57.  Perpendicular  Emergence  at  the  Second  Face  of 

the  Prism 129 

58.  Case  when  the  Ray  Traverses  the  Prism  Symmet- 

rically    129 

59.  Minimum  Deviation 129-133 

60.  Deviation  of  Ray  by  Thin  Prism 133,  134 

61.  Power    of    an    Ophthalmic    Prism.      Centrad    and 
Prism-Dioptry 134-138 

62.  Position  and  Power  of  a  Resultant  Prism  Equiva- 

lent to  Two  Thin  Prisms 138-142 

Problems 142-148 


CHAPTER  VI 

Reflection  and  Refraction  of  Paraxial  Rays  at  a  Spherical 

Surface 

Section^  Pages 

63-86.  149-216 

63.  Introduction.    Definitions,  Notation,  etc 149-153 

64.  Reflection  of  Paraxial  Rays  at  a  Spherical  Mirror ...  .   153-156 

65.  Definition  and  Meaning  of  the  Double  Ratio 156-159 

66.  Perspective  Ranges  of  Points 159-161 

67.  The  Harmonic  Range    161-164 

68.  Application  to  the  Case  of  the  Reflection  of  Par- 

axial Rays  at  a  Spherical  Mirror 164-166 

69.  Focal  Point  and  Focal  Length  of  a  Spherical  Mirror  166-168 

70.  Graphical   Method  of   Exhibiting  the  Imagery  by 

Paraxial  Rays 168-171 

71.  Extra-Axial  Conjugate  Points 171-175 

72.  The  Lateral  Magnification 176 

73.  Field  of  View  of  a  Spherical  Mirror 176-179 

74.  Refraction  of  Paraxial  Rays  at  a  Spherical  Surface .  . .  179-182 

75.  Reflection  Considered  as  a  Special  Case  of  Refrac- 

tion    182,  183 

76.  Construction  of  the  Point  M'  Conjugate  to   the 

Axial  Point  M 183-186 


Contents  xiii 

Sections  Pages 

77.  The  Focal  Points  (F,  F')  of  a  Spherical  Refracting 

Surface 186-190 

78.  Abscissa-Equation  Referred  to  the   Vertex   of   the 

Spherical  Refracting  Surface  as  Origin 190,  191 

79.  The  Focal  Lengths  /,  f  of  a  Spherical  Refracting 

Surface 191-193 

80.  Extra-Axial    Conjugate    Points;  Conjugate    Planes 

of  a  Spherical  Refracting  Surface 193,  194 

81.  Construction  of  the  Point  Q'  which  with  respect  to 

a  Spherical  Refracting  Surface  is  Conjugate  to 

the  Extra- Axial  Point  Q 194-196 

82.  Lateral   Magnification   for   case   of   Spherical   Re- 

fracting   Surface 196 

83.  The   Focal   Planes  of  a  Spherical   Refracting   Sur- 

face    197-199 

84.  Construction  of  Paraxial  Ray  Refracted  at  a  Spher- 

ical   Surface 199,  200 

85.  The  Image-Equations  in  the  case  of  Refraction  of 

Paraxial  Rays  at  a  Spherical  Surface 200,  201 

86.  The  so-called  Smith-Helmholtz  Formula 201,  202 

Problems 203-216 


CHAPTER  VII 

Refraction  of  Paraxial  Rays  through  an  Infinitely  Thin 

Lens 

Sections  Pages 

87-98.  217-257 

87.  Forms  of  Lenses 217-223 

88.  The  Optical  Center  O  of  a  Lens  surrounded  by  the 

same  Medium  on  both  sides 223-226 

89.  The  Abscissa-Formula  of  a  Thin  Lens,  referred  to 

the  Axial  Point  of  the  Lens  as  Origin 226-229 

90.  The  Focal  Points  of  an  Infinitely  Thin  Lens 229-232 

91.  Construction   of   the   Point   M'   Conjugate   to   the 

Axial  Point  M  with  respect  to  an  Infinitely  Thin 

Lens 232-234 


xiv  Contents 

Sections  Pages 

92.  Extra-Axial    Conjugate    Points    Q,    Q';  Conjugate 

Planes 234-236 

93.  Lateral   Magnification   in   case   of   Infinitely   Thin 

Lens 236,  237 

94.  Character  of  the  Imagery  in  a  Thin  Lens 237-240 

95.  The   Focal   Lengths  /,   /'  of   an    Infinitely   Thin 

Lens 240-242 

96.  Central  Collineation  of  Object-Space  and  Image- 

Space 242-244 

97.  Central    Collineation    (cont'd).    Geometrical    Con- 

structions   244-247 

98.  Field  of  View  of  an  Infinitely  Thin  Lens 247-249 

Problems 249-257 


CHAPTER  VII 

Change  of  Curvature  of  the  Wave-front  in  Reflection  and 
Refraction.    Dioptry  System 

Sections  Pages 

99-110.  258-299 

99.  Concerning  Curvature  and  its  Measure 258-265 

100.  Refraction  of  a  Spherical  Wave  at  a  Plane  Surface.  265-269 

101.  Refraction  of  a  Spherical  Wave  at  a  Spherical  Sur- 

face   269-274 

102.  Reflection   of   a    Spherical   Wave    at   a   Spherical 

Mirror 274-276 

103.  Refraction  of  a  Spherical  Wave  through  an  In- 

finitely Thin  Lens 276-279 

104.  Reduced  Distance 279-281 

105.  The  Refracting  Power 281-284 

106.  Reduced  Abscissa  and  Reduced  "Vergence" 284-286 

107.  The  Dioptry  as  Unit  of  Curvature 286-288 

108.  Lens-Gauge 288,   289 

109.  Refraction  of  Paraxial  Rays  through  a  Thin  Lens- 

System 289-291 

110.  Prismatic  Power  of  a  Thin  Lens 291-295 

Problems 295-299 


Contents  xv 

CHAPTER  IX 

Astigmatic  Lenses 

Sections  Pages 

111-116.                                                    "  300-328 

111.  Curvature  and  Refracting  Power  of  a  Normal  Sec- 

tion of  a  Curved  Refracting  Surface 300-305 

112.  Surfaces  of  Revolution.    Cylindrical  and  Toric  Sur- 

faces   305-310 

113.  Refraction  of  a  Narrow  Bundle  of  Rays  incident 

Normally  on  a  Cylindrical  Refracting  Surface. . .  310-314 

114.  Thin  Cylindrical  and  Toric  Lenses 314-318 

115.  Transposing  of  Cylindrical  Lenses 318-320 

116.  Obliquely  Crossed  Cylinders 320-326 

Problems 326-328 

CHAPTER  X 

Geometrical  Theory  of  the  Symmetrical  Optical 
Instrument 

Sections  Pages 

117-124.  329-255 

117.  Graphical  Method  of  tracing  the  Path  of  a  Paraxial 

Ray  through  a  Centered  System  of  Spherical  Re- 
fracting Surfaces 329-331 

118.  Calculation  of  the  Path  of  a  Paraxial  Ray  through 

a  Centered  System  of  Spherical  Refracting  Sur- 
faces    332-334 

119.  The  so-called  Cardinal  Points  of  an  Optical  System    334-339 

120.  Construction  of  the  Image-point  Q'  conjugate  to  an 

Extra-Axial   Object-Point   Q 339,  340 

121.  Construction  of  the  Nodal  Points,  N,  N' 340-342 

122.  The  Focal  Lengths/,/' 342-344 

123.  The  Image-Equations  in  the  case  of  a  Symmetrical 

Optical  System 344-349 

124.  The  Magnification-Ratios  and  their  Mutual  Rela- 

tions    349-351 

Problems 351-355 


xvi  Contents 

CHAPTER  XI 

Compound  Systems.    Thick  Lenses  and  Combinations  of 
Lenses  and  Mirrors 

Sections  Pages 

125-132.  356-396 

125.  Formulae  for  Combination  of  Two  Optical  Systems  356-359 

126.  Formulae  for  Combination  of  Two  Optical  Systems 

in  terms  of  the  Refracting  Power 360-362 

127.  Thick  Lenses  Bounded  by  Spherical  Surfaces 362-365 

128.  The  so-called  "Vertex  Refraction"  of  a  ThickLens  365,  366 

129.  Combination  of  Two  Lenses 366-370 

130.  Optical  Constants  of  Gullstrand's  Schematic  Eye  370-374 

131.  Combination  of  Three  Optical  Systems 374-376 

132.  "Thick  Mirror" 376-384 

Problems 384-396 

CHAPTER  XII 

Aperture  and  Field  of  Optical  System 

Sections  Pages 

133-143.  397-424 

133.  Limitation  of  Ray-Bundles  by  Diaphragms  or  Stops  397-399 

134.  The  Aperture-Stop  and  the  Pupils  of  the  System. . .  399-401 

135.  Illustrations 401-404 

136.  Aperture-Angle.  Case  of  Two  or  More  Entrance-Pupils  404-406 

137.  Field  of  View 406-409 

138.  Field  of  View  of  System  Consisting  of  a  Thin  Lens 

and  the  Eye 409-413 

139.  The  Chief  Rays 413,  414 

140.  The  so-called  " Blur-Circles"   (or  Circles  of  Diffu- 

sion) in  the  Screen-Plane 414-416 

141.  The   Pupil-Centers   as   Centers   of   Perspective   of 

Object-Space  and  Image-Space 416,  417 

142.  Proper  Distance  of  Viewing  a  Photograph 417-419 

143.  Perspective  Elongation  of  Image 419 

144.  Telecentric  Systems 420-423 

Problems 423,  424 


Contents  xvii 

CHAPTER  XIII 

Optical  System  of  the  Eye.    Magnifying  Power  of  Optical 
Instruments 

Sections  ,  Pages 

145-159.  425-464 

145.  The  Human  Eye 425-431 

146.  Optical  Constants  of  the  Eye 431-433 

147.  Accommodation  of  the  Eye 433,  434 

148.  Far  Point  and  Near  Point  of  the  Eye 434,  435 

149.  Decrease  of  the  Power  of  Accommodation  with  In- 

creasing Age 435,  436 

150.  Changes  of  Refracting  Power  in  Accommodation.  .  436,  437 

151.  Amplitude    of    Accommodation 437-^39 

152.  Various  Expressions  for  the  Refraction  of  the  Eye . .  439 

153.  Emmetropia  and  Ametropia 439-443 

154.  Correction  Eye-Glasses 443-446 

155.  Visual  Angle 446-448 

156.  Size  of  Retinal  Image 448, 449 

157.  Apparent  Size  of  an  Object  seen  Through  an  Optical 

Instrument 449-452 

158.  Magnifying  Power  of  an  Optical  Instrument  Used 

in  Conjunction  with  the  Eye 452-455 

159.  Magnifying  Power  of  a  Telescope 455-460 

Problems 461-464 


CHAPTER  XIV 

Dispersion  and  Achromatism 

Sections  Pages 

160-174.  465-507 

160.  Dispersion  by  a  Prism 465-471 

161.  Dark  Lines  of  the  Solar  Spectrum 472 

162.  Relation  between  the  Color  of  the  Light  and  the  Fre- 

quency of  Vibration  of  the  Light- Waves 473-476 

163.  Index  of  Refraction  as  a  Function  of  the  Wave- 

Length 476,  477 


xviii  Contents 

Sections  Pages 

164.  Irrationality  of  Dispersion 477^L79 

165.  Dispersive  Power  of  a  Medium 479-481 

166.  Optical  Glass 481-487 

167.  Chromatic  Aberration  and  Achromatism 487-489 

168.  "Optical   Achromatism"    and    "Actinic   Achroma- 

tism"   489^91 

169.  Achromatic  Combination  of  Two  Thin  Prisms 491-493 

170.  Direct  Vision  Combination  of  Two  Thin  Prisms. . .  493-495 

171.  Calculation  of  Amici  Prism  with  Finite  Angles 495-497 

172.  Kessler  Direct  Vision  Quadrilateral  Prism 497-499 

173.  Achromatic  Combination  of  Two  Thin  Lenses 499-502 

174.  Achromatic  Combination  of  Two  Thin  Lenses  in 

Contact 502-505 

Problems 505-507 


CHAPTER  XV 

Rays  of  Finite  Slope.    Spherical  Aberration,  Astigmatism 
of  Oblique  Bundles,  etc. 

Sections  Pages 

175-193.  508-557 

175.  Introduction 508,  509 

176.  Construction  of  a  Ray  Refracted  at  a  Spherical  Sur- 

face    509-512 

177.  The  Aplanatic  Points  of  a  Spherical  Refracting  Sur- 

face   512,  513 

178.  Spherical  Aberration  Along  the  Axis 513-515 

179.  Spherical  Zones 515,  516 

180.  Trigonometrical  Calculation  of  a  Ray  Refracted  at  a 

Spherical  Surface 516-519 

181.  Path  of  Ray  through  a  Centered  System  of  Spheri- 

cal Refracting  Surfaces.     Numerical  Calculation  519-522 

182.  The  Sine-Condition  or  Condition  of  Aplanatism .  .  .  522-525 

183.  Caustic  Surfaces 525,  526 

184.  Meridian  and  Sagittal  Sections  of  a  Narrow  Bundle 

of  Rays  before  and  after  Refraction  at  a  Spherical 
Surface 526-529 


Contents  xix 

Sections  Pages 

185.  Formula  for  Locating  the  Position  of  the  Image- 

Point  Q'  of  a  Pencil  of  Sagittal  Rays  Refracted  at 

a  Spherical  Surface 529,  530 

186.  Position  of  the  Image-Point  P'  of  a  Pencil  of  Me- 

ridian Rays  Refracted  at  a  Spherical  Surface.  . .  .   530-533 

187.  Measure  of  the  Astigmatism  of  a  Narrow  Bundle 

of  Rays 533,  534 

188.  Image-Lines  (or  Focal  Lines)  of  a  Narrow  Astig- 

matic Bundle  of  Rays 534-536 

189.  The  Astigmatic  Image-Surfaces 536-538 

190.  Curvature  of  the  Image 538-540 

191.  Coma 540-543 

192.  Distortion;  Condition  of  Orthoscopy 543-545 

193.  Seidel's  Theory  of  the  Five  Aberrations 545-550 

Problems 551-557 

Index 559-579 


MIRRORS,  PRISMS  AND  LENSES 


MIRRORS,  PRISMS  AND  LENSES 

CHAPTER  I 

LIGHTS   AND   SHADOWS 

1.  Luminous  Bodies. — The  external  world  is  revealed  to 
the  eye  by  means  of  light.  With  the  rising  sun  night  is 
changed  into  day,  and  animals,  vegetables  and  minerals  in 
all  their  manifold  varieties  of  form  and  shade  and  color, 
which  were  quite  invisible  in  the  dark,  are  now  revealed  to 
view.  Wherever  the  eye  turns  to  gaze,  there  comes  to  it 
from  far  or  near  a  messenger  of  light  conveying  information 
about  the  object  which  is  under  inspection.  In  an  absolutely 
dark  room  everything  is  invisible,  because  the  eye  can  per- 
ceive objects  only  when  they  radiate  or  reflect  light  into  it. 
In  the  strict  sense  a  source  of  light  is  a  self-luminous  body 
which  shines  by  its  own  light,  such  as  the  sun  or  a  fixed  star 
or  a  candle-flame;  but  frequently  the  term  is  applied  to  a 
body  which  merely  reflects  or  transmits  light  which  has 
fallen  upon  it  from  some  other  body,  as,  for  example,  the 
moon  and  the  planets  which  are  illuminated  by  the  light 
from  the  sun.  In  this  latter  sense  the  blue  sky  and  the 
clouds,  which,  shining  by  light  derived  originally  from  the 
sun,  contribute  the  greater  portion  of  what  is  meant  by 
daylight,  are  to  be  regarded  as  light-sources.  A  point- 
source  of  light  or  a  luminous  point  is  in  reality  a  small  ele- 
ment of  luminous  surface  of  relatively  negligible  dimensions 
or  else  a  body  like  a  star  at  such  a  vast  distance  that  it  ap- 
pears like  a  point. 

2.  Transparent  and  Opaque  Bodies. — In  general,  when 
light  falls  on  a  body,  it  is  partly  turned  back  or  reflected  at 
or  very  near  the  surface  of  the  body,  partly  absorbed  within 

1 


2  Mirrors,  Prisms  and  Lenses  [§  2 

the  body,  and  partly  transmitted  through  it.  An  absolutely 
black  body  which  absorbs  all  the  light  that  falls  on  it  does 
not  exist;  the  best  example  we  have  is  afforded  by  a  body 
whose  surface  is  coated  with  lamp-black.  The  color  of  a 
body  as  seen  by  reflected  light  is  explained  by  the  fact  that 
part  of  the  incident  light  is  absorbed,  whereas  only  light 
characteristic  of  the  color  in  question  is  cast  off  or  reflected 
from  the  body.  Thus,  when  sunlight  falls  on  a  piece  of  red 
flannel,  it  is  robbed  of  all  its  constituent  colors  except  red, 
and  thus  it  happens  that  the  color  by  which  we  describe 
the  body  is  in  fact  due  to  the  light  which  it  rejects.  If  the 
piece  of  red  flannel  were  illuminated  by  pure  blue  light,  it 
would  appear  black  or  invisible. 

A  substance  such  as  air  or  water  or  glass,  which  is  per- 
vious to  light,  is  said  to  be  transparent.  None  of  the  light 
that  traverses  a  perfectly  transparent  body  will  be  absorbed; 
and,  on  the  other  hand,  a  perfectly  opaque  body  is  one  which 
suffers  no  light  at  all  to  be  transmitted  through  it.  No 
substance  is  either  absolutely  transparent  or  absolutely 
opaque.  These  terms,  therefore,  as  applied  to  actual  bodies 
are  merely  relative,  and  so  when  we  say  that  a  body  is  opaque, 
we  mean  only  that  the  light  transmitted  through  it  is  so 
slight  as  to  be  practically  inappreciable.  Naturally,  one 
thinks  of  clear  water  as  transparent  and  of  metallic  sub- 
stances generally  as  opaque;  but  a  sufficiently  large  mass 
of  water  will  be  found  to  be  impervious  to  light,  whereas, 
on  the  other  hand,  gold  leaf  transmits  green  light.  A  per- 
fectly transparent  body  would  be  quite  invisible  by  trans- 
mitted light,  although  its  presence  could  be  detected  by 
observing  the  distortion  in  the  appearance  of  bodies  viewed 
through  it. 

Again  there  are  some  substances  which,  while  they  are 
not  transparent  in  the  ordinary  sense,  are  far  from  being 
opaque,  such,  for  example,  as  ground  glass,  alabaster,  por- 
celain, milk,  blood,  smoke,  which  contain  imbedded  or  sus- 
pended in  them  fine  particles  of  matter  of  a  different  optical 


§  3]  Rectilinear  Propagation  of  Light  3 

quality  from  that  of  the  surrounding  mass.  Light  does 
penetrate  through  materials  of  this  nature  in  a  more  or  less 
irregular  fashion,  and  accordingly  they  are  described  as 
translucent.  In  the  interior  of  such  granular  structures  or 
"cloudy  media"  light  undergoes  a  so-called  internal  diffused 
reflection  or  scattering;  so  that  while  it  may  be  possible  to 
discern  the  presence  of  a  body  through  an  intervening  mass 
of  such  material,  the  form  of  the  object  will  be  to  some  ex- 
tent indistinct  and  unrecognizable. 

An  optical  medium  is  any  space,  whether  filled  or  not 
with  ponderable  matter,  which  is  pervious  to  light.  In  geo- 
metrical optics  it  is  generally  assumed  that  the  media  are 
not  only  homogeneous  and  isotropic  (meaning  thereby  that 
the  substance  possesses  the  same  properties  in  all  directions), 
as,  for  example,  air,  glass,  water  and  vacuum,  but  perfectly 
transparent  as  well. 

3.  Rectilinear  Propagation  of  Light. — When  an  opaque 
body  is  interposed  between  the  observer's  eye  and  a  source 
of  light,  it  is  well  known  that  all  parts  of  the  latter  which 
lie  on  straight  lines  connecting  the  pupil  of  the  eye  with 
points  of  the  opaque  obstacle  will  be  hid  from  view.  We 
cannot  see  round  a  corner;  we  can  look  through  a  straight 
tube  but  not  through  a  crooked  one.  A  child  takes  note  of 
such  facts  as  these  among  the  very  earliest  of  his  experiences 
and  recognizes  without  difficulty  the  truth  of  the  common 
saying  that  "light  travels  in  straight  lines,"  which  in  the 
language  of  science  is  called  the  law  of  the  rectilinear  propa- 
gation of  light.  The  light  that  comes  to  us  from  a  star 
traverses  the  vast  stretches  of  interstellar  space  in  straight 
fines  until  it  reaches  the  earth's  atmosphere,  which  is  com- 
posed of  layers  of  air  of  increasing  density  from  the  upper 
portions  towards  the  surface  of  the  earth;  so  that  the  me- 
dium through  which  the  light  passes  in  this  short  remainder 
of  its  downward  journey  is  no  longer  isotropic,  and,  hence, 
also  this  part  of  the  light  path  will,  in  general,  be  no  longer 
straight  but  curved  by  a  gradual  and  continuous  bending 


4  Mirrors,  Prisms  and  Lenses  [§  3 

from  the  less  dense  layers  of  air  to  the  more  dense  layers 
below.  This  explains  why  it  is  necessary  for  an  observer 
on  the  earth's  surface  looking  through  a  long  narrow  tube 
at  a  star  not  directly  overhead  to  point  the  tube  not  at  the 
star  itself  but  at  its  apparent  place  in  the  sky,  which  depends 
on  the  direction  which  the  light  has  when  it  enters  the  eye; 
and,  consequently,  in  accurate  determinations  of  the  posi- 
tion of  a  heavenly  body,  the  astronomer  is  always  careful 
to  take  account  of  the  apparent  displacement  due  to  this 
so-called  "  atmospheric  refraction,"  and  a  principal  reason 
why  astronomical  observatories  are  nearly  always  located 
on  high  mountains  is  to  obviate  as  much  as  possible  the 
disturbing  influence  of  the  atmosphere.  In  aiming  a  rifle 
or  in  any  of  the  ordinary  processes  we  call  "  sighting,"  which 
are  at  the  basis  of  some  of  the  most  delicate  methods  of 
measurement  known  to  us,  we  rely  with  absolute  confidence 
on  this  proved  law  of  experience  concerning  the  rectilinear 
propagation  of  light;  and,  in  fact,  the  most  conclusive  dem- 
onstration that  a  line  is  straight  consists  in  showing  that  it 
is  the  path  which  light  pursues.  The  notion  of  a  "ray  of 
light"  is  derived  from  this  law,  and  any  line  along  which 

light  travels  is  to  be  regarded  as 

a  ray  of  light.    According  to  this 

f^~^^^  ^^^"^  idea,  therefore,  the  rays  of  light 

X^-^      "^^^^  in    an     isotropic    medium     are 

straight  lines. 

A  very  striking  proof  of  the 

Fig.  1.— Rectilinear  Propagation    rectilinear   propagation    of   light 

is  afforded  by  placing  a  lumi- 
nous object  (Fig.  1)  in  front  of  an  opaque  screen  in  which 
there  is  a  very  small  round  aperture.  If  now  a  second  screen 
or  a  white  wall  is  placed  parallel  to  the  first  screen  on  the 
other  side  of  it,  there  will  be  cast  on  it  a  so-called  inverted 
image  of  the  object,  the  size  of  which  will  be  proportional  to 
the  distance  between  the  two  screens.  From  each  point  of 
the  luminous  object  rays  go  out  in  all  directions,  and  a  narrow 


§  3]  Pinhole  Camera  5 

cone  of  these  rays  will  traverse  the  perforated  screen  through 
the  opening  and  illuminate  a  small  area  on  the  other  screen, 
and  thus  every  part  of  the  object  will  be  depicted  in  this  way 
by  little  patches  of  light  arranged  in  a  figure  which  is  similar 
in  form  to  the  object,  but  which  is  completely  inverted,  since 
not  only  top  and  bottom  but  right  and  left  are  reversed  in 
consequence  of  the  rectilinear  paths  of  the  rays  of  light.  It 
may  be  remarked  that  this  image  is  not  an  optical  image  in 
the  strict  sense  of  the  term  (see  §  11),  but  the  phenomenon 
can  be  explained  only  on  the  supposition  that  light  proceeds 
in  straight  lines.  If  another  small  opening  were  made  in  the 
front  screen  very  near  the  first  hole,  there  would  be  two 
images  formed  which  would  partly  overlap  each  other,  so  that 
the  resultant  image  would  be  more  or  less  blurred,  and  if  we 
have  a  single  large  aperture,  we  could  no  longer  see  any 
distinct  image  at  all. 

The  pinhole  camera,  invented  by  Giambattista  Della 
Porta  (c.  1543-1615),  and  sometimes  called  Porta's  camera, 
is  constructed  on  the  principle  of  the  experiment  which  has 
just  been  described.  It  is  very  useful  in  making  accurate 
photographic  copies  of  the  architectural  details  of  buildings, 
because  the  image  which  is  obtained  is  entirely  free  from 
distortion. 

In  the  pinhole  camera  there  is  a  certain  relation  between 
the  size  of  the  pinhole  and  the  distance  of  the  sensitive  plate. 
According  to  Abney,  in  order  to  get  the  best  results  with 
an  apparatus  of  this  kind  the  diameter  of  the  pinhole  ought 
to  be  directly  proportional  to  the  square-root  of  the  distance 
of  the  plate  from  the  aperture,  that  is, 

y  =  k\Zx, 

where  x  and  y  denote  the  distance  of  the  plate  and  the  di- 
ameter of  the  pinhole,  respectively,  and  k  denotes  a  con- 
stant, the  value  of  which  will  depend  on  the  unit  of  length. 
Thus,  if  x  and  y  are  measured  in  inches,  A;  =  0.008;  in  centi- 
meters, k  =  0.01275. 


6 


Mirrors,  Prisms  and  Lenses 


[§4 


4.  Shadows,  Eclipses,  etc. — The  forms  of  shadows  are  also 
easily  explained  on  the  hypothesis  that  light  proceeds  in 
straight  lines,  for  the  outline  of  the  shadow  cast  by  a  body 
is  precisely  similar  to  that  of  the  object  as  viewed  from  the 
place  where  the  source  of  light  is.    Thus,  for  example,  the 


Fig.   2. — Shadow   (umbra)   of  opaque  globe  E  illuminated  by 
point-source  S. 

shadow  of  a  sphere  held  in  front  of  a  point-source  of  light 
has  the  form  of  a  circle,  and  the  shadow  cast  by  a  circular 
disk  will  have  the  outline  of  an  ellipse  of  greater  and  greater 
eccentricity  as  the  disk  is  turned  more  and  more  nearly 
edge-on  towards  the  light.  Passing  a  shop- window  on 
Sunday  when  the  shade  is  drawn  down,  if  the  sun  is  shining 


Fig.  3. — Shadow   (umbra   and   penumbra)   of    opaque  globe  E 
illuminated  by  two  point-sources  Si,  S2. 

on  the  window,  one  can  read  the  shadow  of  the  sign  painted 
on  the  glass  quite  as  distinctly  as  the  sign  itself.  The  in- 
terposition of  an  opaque  body  between  a  source  of  light  and 
a  wall  not  only  darkens  a  portion  of  the  wall  or  casts  its 
shadow  there,  but  it  converts  an  entire  region  of  space  be- 
tween it  and  the  wall  into  a  dark  tract  either  wholly  or  par- 


§  4]  Shadows  7 

tially  screened  from  the  light.  Thus,  for  example,  the  space 
A  (Fig.  2)  behind  the  body  E  which  is  comprised  within  the 
cone  of  rays  proceeding  from  the  point-source  S  that  are 
intercepted  by  E  gets  no  light  from  S,  and  this  wholly  un- 
illuminated  region  is  called  the  umbra  or  true  shadow.  When 
there  are  two  luminous  points  Si  and  S2  (Figs.  3  and  4),  the 
region  of  shadow  behind  the  opaque  body  E  consists  of  the 


Fig.  4. — Shadow    (umbra    and    penumbra)    of   opaque   globe   E 
illuminated  by  two  point-sources  Si,  S2. 

umbra  A  which  is  wholly  screened  from  both  sources  of  light 
and  the  so-called  penumbra  or  partially  illuminated  space 
composed  of  a  space  Bi  which  gets  light  only  from  Si  and 
a  similar  space  B2  which  gets  light  only  from  S2.  Points  lying 
beyond  the  penumbra  will  receive  light  from  both  sources. 

If  the  light-source  has  an  appreciable  size,  light  will  pro- 
ceed from  each  of  its  shining  points  in  all  directions.  Sup- 
pose, for  example,  that  an  opaque  globe  E  (Fig.  5)  is  placed 
in  front  of  a  luminous  globe  S:  then  the  dark  body  will 
intercept  all  rays  that  fall  within  the  cone  which  is  tangent 
externally  to  the  two  spheres,  and,  consequently,  the  por- 
tion A  of  this  cone  which  lies  behind  E  will  be  completely 


8  Mirrors,  Prisms  and  Lenses  [§  4 

screened  from  all  points  of  the  source  S,  so  that  this  portion 
constitutes  the  umbra  where  no  light  comes.  In  this  case 
also  there  are  two  penumbral  regions  Bi  and  B2  which  are 
partially  illuminated,  but  the  illumination  is  not  uniform, 


Fig.  5. — Shadow    (umbra   and    penumbra)    of  opaque   globe  E 
illuminated  by  luminous  globe  S. 

but  increases  gradually  from  total  darkness  at  the  outer 
borders  of  the  umbra  into  the  complete  illumination  of  the 
region  outside  the  shadow.  The  shadow  cast  on  a  screen 
by  an  opaque  body  exposed  to  an  extended  source  of  light 
has  no  sharp  outline  but  fades  by  imperceptible  gradations 
into  the  bright  space  outside.  As  to  the  umbra,  it  terminates 
in  a  point  at  a  certain  distance  x  behind  the  opaque  body, 
provided  the  diameter  of  the  latter  is  less  than  that  of  the 
luminous  globe  in  front  of  it,  that  is,  provided  R  is  greater 
than  r,  where  R,  r  denote  the  radii  of  luminous  and  opaque 
globes,  respectively.  If  the  distance  d  between  the  centers 
of  the  two  globes  is  known,  the  length  x  of  the  umbra  may 
be  calculated  from  the  proportion : 

R    d+x. 
r        x 

whence  we  find : 

d 


x  = 


r 


Thus,  for  example,  the  diameter  of  the  sun  is  109.5  times 
that  of  the  earth,  and  the  distance  between  the  two  bodies 


§  5]  Wave  Theory  of  Light  9 

is  93  millions  of  miles.  Accordingly,  the  umbra  of  the  earth 
is  found  to  extend  to  a  distance  of  more  than  857  000  miles 
behind  it.  Sometimes  the  moon  whose  distance  from  the 
earth  is  about  240  000  miles  enters  inside  the  shadow,  and 
becomes  then  totally  eclipsed.  When  the  moon  is  only 
partly  inside  the  earth's  umbra,  there  is  a  partial  eclipse  of 
the  moon.  On  the  other  hand,  if  the  earth  or  any  part  of  it 
comes  inside  the  moon's  shadow,  there  will  be  an  eclipse 
of  the  sun  visible  from  points  on  the  earth  that  are  in  the 
shadow. 

The  angular  diameter  of  the  sun  is  32'  3.3";  whence  it  is 
easy  to  calculate  that  the  length  of  the  umbra  of  an  opaque 
globe  in  sunlight  is  about  105  times  the  diameter  of  the  globe. 

On  the  other  hand,  if  the  light-source  is  smaller  than  the 
interposed  object,  the  umbra,  instead  of  contracting  to  a 
point,  widens  out  indefinitely;  and  thus,  whereas  the  shadow 
cast  on  the  opposite  wall  by  a  hand  held  in  front  of  a  broad 
fire  is  smaller  than  the  object,  the  shadow  made  by  the  same 
hand  in  front  of  a  small  source  of  light  like  a  candle-flame 
may  be  prodigious  in  extent. 

5.  Wave  Theory  of  Light. — The  term  "ray,"  as  we  have 
employed  it,  is  a  purely  geometrical  conception,  but  in  or- 
dinary usage  a  ray  of  light  implies  generally  an  exceedingly 
narrow  beam  of  light  such  as  is  supposed  to  be  obtained 
when  sunlight  is  admitted  into  a  dark  room  through  a  pin- 
hole opening  in  a  shutter.  But  when  the  experiment  is 
carefully  made  to  try  to  isolate  a  so-called  ray  of  light  in 
this  fashion,  new  and  unexpected  difficulties  arise,  and, 
contrary  to  our  preconceived  notions,  we  are  disconcerted 
by  finding  that  the  smaller  the  opening  in  the  shutter,  the 
more  difficult  it  becomes  to  realize  the  geometrical  concep- 
tion which  is  conveyed  by  the  word  "ray."  In  fact,  in  con- 
sequence of  this  experiment  and  others  of  a  similar  kind, 
we  begin  to  perceive  that  the  statement  of  the  law  of  the 
rectilinear  propagation  of  light  needs  to  be  modified;  for 
among  other  phenomena  we  discover  that  when  light  pro- 


10  Mirrors ,  Prisms  and  Lenses  [§  6 

ceeds  through  a  very  narrow  aperture  in  a  screen,  it  does 
not  pass  through  it  just  as  though  the  screen  were  not  pres- 
ent, but  it  spreads  out  laterally  from  the  point  of  perfora- 
tion in  all  directions  beyond  the  screen,  proceeding,  in  fact, 
very  much  as  it  might  do  if  the  opening  in  the  screen  were 
the  seat  of  a  new  and  independent  source  of  light. 

The  truth  is,  as  has  been  ascertained  now  for  a  long  time, 
light  is  propagated  not  by  "rays"  at  all  but  by  waves;  and 
if,  in  general,  it  is  found  that  light  does  proceed  in  straight 
lines  and  does  not  bend  around  corners  as  sound-waves  do, 
the  explanation  is  because  the  waves  of  light  are  excessively 
short,  considerably  less  than  one  ten-thousandth  of  a  centi- 
meter. Wave-lengths  of  light  are  usually  specified  in  terms 
of  a  unit  called  a  "tenth-meter"  or  an  "Angstrom  unit," 
which  is  the  hundred-millionth  part  of  a  centimeter  (see 
§  162) ;  that  is,  1  Angstrom  unit  =  10  - 10  meter  =  0.000  000  01 
cm.  The  wave-length  of  the  deepest  red  light  is  found  to 
be  about  7667  of  these  units  and  the  wave-length  of  light 
corresponding  to  the  extreme  violet  end  of  the  spectrum 
is  a  little  more  than  half  the  above  value  or  3970  units. 

According  to  the  wave-theory  the  phenomena  of  light 
are  dependent  on  an  hypothetical  medium  called  the  ether, 
which  may  be  compared  to  "an  impalpable  and  all-per- 
vading jelly"  that  not  only  fills  empty  space  but  penetrates 
freely  through  all  material  substances,  solid,  liquid  and 
gaseous,  and  through  which  particles  of  ordinary  matter 
move  easily  without  apparent  resistance,  for  it  is  impon- 
derable and  exceedingly  elastic  and  subtle,  insomuch  that 
no  one  has  ever  succeeded  in  obtaining  direct  evidence  of 
its  existence.  It  is  this  ether  which  is  the  vehicle  by  which 
light-energy  is  transmitted  and  through  which  waves  of 
light  are  incessantly  throbbing  with  prodigious  but  measur- 
able velocity,  which  in  vacuo  is  about  300  million  meters  per 
second  or  about  186  000  miles  per  second. 

6.  Huygens's  Construction  of  the  Wave-Front.— The  great 
Dutch  philosopher  Huygens  (1629-1695),  who  was  a  contem- 


§6] 


Construction  of  Wave-Front 


11 


porary  of  Newton's  (1642-1727),  and  who  is  usually  regarded 
as  the  founder  of  the  wave-theory  of  light,  encountered  his 
greatest  difficulty  in  trying  to  give  a  consistent  and  satis- 
factory explanation  of  the  apparent  rectilinear  propagation  of 
light.  His  mode  of  reasoning,  as  set  forth  in  his  "  Treatise 
on  Light "  published  in  1690,  while  by  no  means  free  from 
objection,  leads  to  a  simple  geometrical  construction  of  the 
wave-front  which  corresponds  with  the  known  facts  in  regard 
to  the  procedure  of  light. 

Let  O  (Fig.  6)  designate  the  position  of  a  point-source  of 
light  from  which  as  center  or  origin  ether  waves  proceed  in 
an  isotropic  medium  with 
equal  speeds  in  all  direc- 
tions. At  the  end  of  a 
certain  time  the  disturb- 
ances will  have  arrived 
at  all  the  points  which 
lie  on  a  spherical  surface 
Ci  described  around  O  as 
center,  and  at  the  instant 
in  question  this  surface 
will  be  the  locus  of  all  the 
particles  in  the  medium 
that  are  in  this  initial 
phase  of  excitation,  and  Fig.  6 
so  it  represents  the  wave- 
front  at  this  moment.  Now  according  to  Huygens,  every 
point  in  the  wave-front  becomes  immediately  a  new  source 
or  center  from  which  so-called  secondary  waves  or  wave- 
lets spread  out.  These  innumerable'  ripples  or  wavelets 
starting  together  from  all  the  points  affected  by  the 
principal  wave  overlap  and  interfere  with  each  other, 
and  Huygens  inferred  that  their  resultant  sensible  effects 
are  produced  only  at  the  points  of  the  surface  which  at  any 
given  instant  touches  or,  as  we  say,  envelops  all  the  secondary 
wave-fronts,  and  that  accordingly  the  new  principal  wave- 


H.uygens's  construction  of  wave- 
front. 


12 


Mirrors,  Prisms  and  Lenses 


[§6 


front  will  be  this  enveloping  surface;  so  that  the  effect  is 
the  same  as  though  the  old  wave-front  #had  expanded  into 
the  new,  the  disturbance  marching  forward  along  a  straight 
line  in  any  given  direction.    Obviously,  in  an  unobstructed 

isotropic  medium,  such 
as  is  here  supposed, 
the  enveloping  surface 
or  new  wave-front  will 
be  a  sphere  concentric 
with  the  old  wave- 
front,  and  the  straight 
lines  that  radiate  out 
from  the  center  will  be 
the  paths  of  the  dis- 
turbance. 

Now  if  a  plane  screen 
MN  (Fig.  7)  is  inter- 
posed in  front  of  the 
advancing  waves,  and 
if  there  is  an  opening 

Fig.  7. — Huygens's  construction  of  spherical    AB  in  the  Screen,  each 

Tcrlen.  PaSSing  thr°Ugh  ^^  *  &  Point    in    the    opening 

between  A,  which  is 
nearest  to  the  source  O,  and  B,  which  is  farthest  from  it, 
will  become  in  turn  a  new  center  of  disturbance  whence 
secondary  spherical  waves  will  be  propagated  into  the  re- 
gion on  the  other  side  of  the  screen.  Since  the  disturbance 
will  have  arrived  at  the  point  A  before  it  has  reached  a  point 
X  between  A  and  B,  the  secondary  wave  emanating  from 
A  will  at  the  end  of  a  given  time  t  have  been  travelling  for 
a  longer  time  than  the  secondary  wave  coming  from  X.  If 
the  radius  of  the  wavelet  around  X  at  the  time  t  is  denoted 
by  r,  and  if  the  distance  OX  is  put  equal  to  x,  then  d  =  x-\-r 
will  denote  the  distance  from  O  which  the  disturbance  will 
have  gone  at  the  end  of  the  time  t;  and  since  this  distance 
is  constant,  whereas  the  distances  denoted  by  x  and  r  are 


Rays  Normal  to  Wave-Surface 


13 


variables  depending  on  the  position  of  the  point  X,  it  is 
evident  that  the  farther  X  is  from  0,  that  is,  the  greater 
the  value  of  x,  the  smaller  will  be  the  radius  r  =  d  —  re  of 
the  secondary  wavelet  around  X.  The  enveloping  surface  in 
this  case  is  seen  to  be  that  part  of  the  spherical  surface  de- 
scribed around  O  as  center  with  radius  equal  to  d  which  is  in- 
tercepted by  the  cone 
which  has  O  for  its  vertex 
and  the  opening  AB  in 
the  screen  for  a  section. 
Within  this  cone,  accord- 
ing to  Huygens's  view, 
the  disturbance  is  propa- 
gated exactly  as  though 
the  perforated  screen  had 
not  been  interposed, 
whereas  points  on  the  far 
side  of  the  screen  and 
outside  this  limiting  cone 
are  not  affected  at  all. 

It  is  plain  that  this  mode  of  explanation  is  equivalent  to 
the  hypothesis  of  the  rectilinear  propagation  of  light. 

If  the  luminous  point  O  (Fig.  8)  is  so  far  away  that  the 
dimensions  of  the  opening  AB  in  the  screen  may  be  regarded 
as  vanishingly  small  in  comparison  with  the  distance  of  the 
source,  the  straight  lines  drawn  from  0  to  the  points  A,  X,  B 
in  the  opening  in  the  screen  may  be  regarded  as  parallel, 
and  the  wave-front  in  this  case  will  be  plane  instead  of 
spherical,  that  is,  the  wave-front  is  a  spherical  surface  with 
an  exceedingly  great  radius  as  compared  with  the  dimen- 
sions of  the  aperture  in  the  screen. 

7.  Rays  of  Light  are  Normal  to  the  Wave-Surface. — The 
most  obvious  objection  to  Huygen's  construction  is,  What 
right  has  he  to  assume  that  the  places  of  sensible  effects  are 
the  points  on  the  surface  which  is  tangent  to  or  envelops 
the  secondary  waves?    And  why  is  the  light  not  propagated 


Fig.  8. — Huygens's  construction  of  plane 
waves   passing  through  opening   in  a 


14  Mirrors,  Prisms  and  Lenses  [§  7 

backwards  from  these  new  centers  as  well  as  forwards? 
Moreover,  when  the  opening  in  the  screen  is  very  narrow, 
it  is  found,  as  has  been  already  stated  (§  5),  that  this  con- 
struction does  not  correspond  at  all  with  the  observed  facts. 

It  is  entirely  beyond  the  scope  of  this  book  to  attempt  to 
answer  these  questions  here  or  to  describe  even  briefly  the 
remarkable  and  complex  phenomena  of  diffraction  (which 
is  the  name  given  to  these  effects  due  to  the  bending  of  the 
light- waves  around  the  edges  of  opaque  obstacles).  For 
an  adequate  discussion  of  these  matters  the  reader  must 
consult  a  more  advanced  treatise  on  physical  optics.  Suffice 
it  to  say,  that  the  wave-theory  of  light  and  especially  the 
principle  of  interference  as  developed  long  after  Huygens's 
death  (1695)  by  Young  (1773-1829)  and  Fresnel  (1788- 
1827)  entirely  supports  the  idea  of  the  rectilinear  propaga- 
tion of  light  as  commonly  understood;  notwithstanding 
the  fact  that  this  law,  as  indeed  is  the  case  with  nearly  all 
so-called  natural  laws,  has  to  be  accepted  with  certain  reser- 
vations; but,  fortunately,  these  latter  do  not  concern  us  at 
present. 

Accordingly,  a  luminous  point  is  said  to  emit  light  in  all 
directions,  and  the  so-called  light-rays  in  an  isotropic  medium 
are  straight  lines  radiating  from  the  center  of  the  spheri- 
cal wave-surface.  These  rays  may  subsequently  be  bent 
abruptly  into  new  directions  in  traversing  the  boundary 
between  one  isotropic  medium  and  another,  and  under  such 
circumstances  the  wave-surfaces  may  cease  to  be  spherical; 
but  no  matter  what  may  be  the  form  of  the  wave-surface, 
the  direction  of  the  ray  at  any  point  is  to  be  considered  always  as 
normal  to  the  wave-front  that  passes  through  that  point  (see  §  39) . 
In  an  isotropic  medium  the  waves  always  march  at  right 
angles  to  their  own  front,  and  the  so-called  rays  of  light  in 
geometrical  optics  are,  in  fact,  the  shortest  optical  routes 
along  which  the  disturbances  in  the  ether  are  propagated 
from  place  to  place.  With  the  aid  of  the  principle  of  inter- 
ference (alluded  to  above)  and  by  the  use  of  the  higher 


§  8]  Apparent  Place  of  Light-Source  15 

mathematics,  it  may  indeed  be  shown  that  the  effect  pro- 
duced at  any  point  P  in  the  path  of  a  ray  of  light  is  due 
almost  exclusively  to  previous  disturbances  which  have 
occurred  successively  at  all  the  points  along  the  ray  which 
lie  between  the  source  and  the  point  P  in  question,  and  that 
disturbances  at  other  points  not  lying  on  the  ray  which  goes 
through  P  are  practically  without  influence  at  P,  that  is,  their 
effects  there  are  mutually  counteracted.  And  thus  we  arrive 
also  at  the  so-called  principle  of  the  mutual  independence  of 
rays  of  light,  which  is  also  one  of  the  fundamental  laws  of 
geometrical  optics.  From  this  point  of  view  a  ray  of  light 
is  to  be  regarded  as  something  more  than  a  mere  geomet- 
rical fiction  and  as  having  in  some  real  sense  a  certain  physi- 
cal existence,  although  it  is  not  possible  to  isolate  the  ray 
from  its  companions. 

8.  The  Direction  and  Location  of  a  Luminous  Point. — 
When  a  ray  of  light  comes  into  the  eye,  the  natural  infer- 
ence as  to  its  origin  is  that  the  source  lies  in  the  direction  from 
which  the  ray  proceeded.  There  is  no  difficulty  in  pointing  out 
correctly  the  direction  of  an  object  which  is  viewed  through 
an  isotropic  medium;  but  if  the  medium  were  not  isotropic, 
the  apparent  direction  of  A 

the  object  might  not  be, 
and  probably  would  not 
be,  its  real  direction. 
Thus,  owing  to  the  ef- 
fects of  atmospheric  re- 
fraction, to  Which  allu-  FlG-  9-Direction  and  location  of  a  lumi- 
'  nous  point. 

sion  has  been  made  al- 
ready (§  3),  the  sun  is  seen  above  the  horizon  before  it  has 
actually  risen,  and  so  also  in  the  evening  the  sun  is  still 
visible  for  a  few  moments  after  sunset.    For  the  same  reason 
a  star  appears  to  be  nearer  the  zenith  than  it  really  is. 

In  general,  however,  when  a  ray  SA  (Fig.  9)  enters  the 
eye  at  A,  it  is  correctly  inferred  that  the  source  S  lies  some- 
where on  the  straight  line  AS,  but  whether  it  is  actually 


16  Mirrors,  Prisms  and  Lenses  [§  8 

situated  at  S  or  farther  or  nearer  cannot  be  determined  by- 
means  of  a  single  ray.  If  the  eye  is  transferred  from  A  to 
another  point  B,  the  source  will  appear  now  to  lie  in  the  new 
direction  BS.  If  the  spectator  views  the  source  with  both 
eyes  simultaneously,  one  eye  at  A  and  the  other  at  B,  or  if  using 
only  one  eye  he  moves  it  quickly  from  A  to  B,  the  position 
of  the  source  at  S  will  be  located  at  the  point  of  intersection 
of  the  straight  lines  AS  and  BS;  and  this  determination  will 
be  more  accurate  in  proportion  as  the  distance  between  the 
two  points  of  observation  A  and  B  is  greater  or  the  more 
nearly  the  acute  angle  ASB  approaches  a  right  angle.  That 
is  the  reason  why  in  estimating  the  distance  of  a  remote 
object  one  tries  to  observe  it  from  two  stations  as  widely 
separated  as  possible,  and  that  explains  also  why  a  person 
shifts  his  head  from  side  to  side.  If  the  object  is  compara- 
tively near  at  hand,  a  single  movement  of  the  head  may  be 
sufficient  in  order  to  get  a  fairly  good  idea  of  its  distance, 
or  it  may  be  that  it  is  simply  necessary  to  look  at  the  object 
with  both  eyes  at  the  same  time.  It  is  amusing  to  watch 
a  person  with  one  eye  closed  attempting  to  poke  a  pencil 
through  a  finger-ring  suspended  in  the  middle  of  a  room  on  a 
level  with  his  eye;  by  chance  he  may  succeed  after  repeated 
failures,  whereas  with  both  eyes  open,  the  operation  is  per- 
formed without  the  slightest  difficulty. 

In  case  the  rays  come  into  the  eye  after  having  traversed 
two  or  more  isotropic  media,  it  is  easy  to  be  deceived  about 
the  direction  of  the  source  where  they  emanated.  In  order 
for  a  bullet  to  hit  a  fish  under  water,  the  rifle  must  be 
pointed  in  a  direction  below  that  in  which  the  fish  appears 
to  be.  At  the  boundary-surface  between  two  isotropic  media 
the  direction  of  a  ray  of  light  is  usually  changed  abruptly 
by  refraction  (§  26) ;  so  that,  in  general,  the  path  of  a  ray 
will  be  found  to  consist  of  a  series  of  line-segments.  In 
Fig.  10  the  broken  line  ABCD  represents  the  course  taken 
by  a  ray  of  light  in  proceeding  through  several  media  such  as 
water,  air  and  glass.    The  line-segments  AB,  BC  and  CD 


§8] 


Image  of  Point-Source 


17 


are  portions  of  different  straight  lines  of  indefinite  extent. 

For  example,  the  actual  route  of  the  ray  in  air  is  along  the 

straight  line  between  B  and  C,  and  if  the  point  P  lies  on 

this  line  between  B  and 

C,   we  say  that  the  ray 

BC     passes     "  really" 

through   P,   whereas    we 

say  that  this   same   ray 

passes     "virtually  " 

through  a  point  Q  or  R 

which    lies  in  the  prolon-    Fig.  10.— Points  P,  Q  and  R  considered  as 

gation  of  the  line-segment         !y!ng  on  ray  BC  *? to  be  ;egarded  *» 

°          .  °    .  lying  in  same  medium  as  BC. 

BC    in    either    direction. 

Moreover,  thinking  of  the  point  Q  or  R  as  a  point  lying  on 
the  straight  line  BC  which  the  light  pursues  in  traversing 
the  medium  between  the  water  and  the  glass,  we  must  re- 
gard such  a  point  as  being  optically  in  the  same  medium 
as  the  ray  to  which  it  belongs.  Thus,  the  points  Q  and  R 
in  the  figure  considered  as  points  on  the  ray  BC  are  to  be 
regarded  as  being  optically  in  air,  although  in  a  physical 
sense  Q  is  a  point  in  the  water  and  R  is  a  point  in  the  glass 
(see  §  104). 
Now  let  us  suppose  that  two  rays  emanating  originally 

from  a  point-source  S 
(Figs.  11  and  12)  are 
bent  at  A  and  B  into 
new  directions  AP  and 
BQ,  respectively,  so  as 
to  enter  the  two  eyes  of 
an  observer  at  P  and  Q. 
In  such  a  case  the  ob- 


Fig.   11. — S'  is    said   to   be   a    "real' 
image  of  point-source  at  S. 


server  will  infer  that  the  rays  originated  at  the  point  S' 
where  the  straight  lines  AP  and  BQ  intersect.  This  point 
S',  which  is  called  the  image  of  S,  may  lie  in  the  actual  paths 
of  the  rays  AP  and  BQ  that  enter  the  eyes,  so  that  the  light 
from  S  really  does  go  through  S',  and  in  this  case  (Fig.  11) 


18  Mirrors,  Prisms  and  Lenses  [§  9 

the  image  S'  is  said  to  be  a  real  image.  On  the  other  hand, 
if  the  straight  lines  AP  and  BQ  have  to  be  produced  back- 
wards in  order  to  find  their  point  of  intersection,  the  rays 
do  not  actually  pass  through  S',  and  in  this  case  the  image 

is    said   to   be   a   virtual 

•^x-- image    of    the    point    S 

(Fig.  12).  However,  it 
must  be  borne  in  mind 
in  connection  with  these 
diagrams  that  in  reality 

Fig.  12. — S'  is  said  to  be  a  "virtual"  image    we  do  not  See  objects  by 
of  point-source  at  S.  r        •       i 

p  means    of    single    rays; 

and,  hence,  we  shall  not  be  in  a  position  to  form  an  ac- 
curate idea  of  the  term  optical  image  until  we  come  to 
consider  bundles  of  rays  in  §  11. 

9.  Field  of  View. — The  open  or  visible  space  commanded 
by  the  eye  is  called  the  field  of  view.  Since  the  eye  can  turn 
in  its  socket,  the  field  of  view  of  the  mobile  eye  is  very  much 
more  extensive  than  that  of  the  stationary  eye,  and,  more- 
over, the  field  of  view  of  both  eyes  is  greater  than  that  of 
one  eye  by  itself.  The  spectator  may  also  widen  his  field 
of  vision  by  turning  his  head  or  indeed  by  turning  his  entire 
body.  For  the  present,  however,  we  shall  employ  the  term 
field  of  view  to  mean  that  more  limited  portion  of  space 
which  is  accessible  to  the  single  eye  turning  in  its  socket 
around  the  so-called  center  of  rotation  of  the  eye.  When 
a  person  gazes  through  a  window,  the  outside  field  of  view 
is  limited  partly  by  the  size  of  the  window  and  partly  also 
by  the  position  of  the  eye  with  reference  to  it;  so  that  only 
such  exterior  objects  will  be  visible  as  happen  to  lie  within 
the  conical  region  of  space  determined  by  drawing  straight 
lines  from  the  center  of  rotation  of  the  eye  to  all  the  points 
in  the  edge  of  the  window.  Thus,  for  example,  if  the  open- 
ing in  the  window  is  indicated  by  the  gap  GH  in  the  straight 
line  GH  in  Fig.  13,  and  if  the  point  marked  0  is  the  position 
of  the  center  of  rotation  of  the  eye,  a  luminous  object  at  P 


§9] 


Field  of  View 


19 


in  front  of  the  window  and  directly  opposite  the  eye  will 
be  plainly  in  view,  because  some  of  the  rays  from  P  may  go 
through  the  window  and  enter  the  eye.  But  if  the  object 
is  displaced  far  enough  to  one  side  to  some  position  such  as 


Fig.  13. — Field   of  view    determined   by   contour  of 
window  GH  and  position  of  the  eye  at  O. 

that  marked  R  in  the  diagram,  so  that  the  straight  line  OR 
does  not  pass  through  the  window,  the  object  will  pass  out 
of  the  field  of  view.  The  straight  line  MN  drawn  parallel 
to  GH  is  supposed  to  represent  a  vertical  wall  opposite  the 
window.  If  this  wall  is  covered  with  a  mural  painting,  the 
only  part  of  the  picture  that  can  be  seen  through  the  win- 
dow by  the  eye  at  O  is  the  section  included  between  the 
points  T  and  V  where  the  straight  lines  OG  and  OH  intersect 
the  straight  line  MN.  The  window  acts  here  as  a  so-called 
field-stop  (§  137)  to  limit  the  extent  of  the  field  of  view.  But 
the  limitation  of  the  visible  region  depends  essentially  also 
on  the  position  of  the  eye,  becoming  more  and  more  con- 
tracted the  farther  the  eye  is  from  the  window.  The  size  of 
the  window  makes  very  little  difference  when  the  eye  is 
placed  close  to  it,  and  a  person  sitting  near  an  open  window 
can  command  almost  as  wide  a  view  as  if  the  entire  wall  of 
the  room  were  removed.  If  one  is  looking  through  a  key- 
hole in  a  door,  he  must  put  his  eye  close  to  the  hole  in  order 
to  see  objects  that  are  not  directly  in  front  of  it. 


20 


Mirrors,  Prisms  and  Lenses 


[§10 


10.  Apparent  Size. — The  apparent  size  of  an  object  is 
measured  by  the  visual  angle  which  it  subtends  at  the  eye. 
Several  objects  in  the  field  of  view  which  subtend  equal 
angles  when  viewed  from  the  same  standpoint  are  said  to 
have  the  same  apparent  size;  although  their  actual  sizes  will 


Fig.   14. — Apparent  size  measured  by  visual  angle. 


be  different  if  they  are  at  unequal  distances  from  the  eye. 
The  objects  marked  1,  2  and  3  in  Fig.  14  appear  to  an  eye 
at  0  to  be  all  of  the  same  size.  Thus  an  elephant  may  ap- 
pear no  bigger  than  a  man  or  a  boy.  Looking  through  a 
single  pane  of  glass  in  a  window,  one  may  see  a  large  build- 
ing or  an  entire  tree,  because  the  apparent  extent  of  the 
small  area  of  glass  is  greater  than  that  of  the  distant  object. 
A  fly  crawling  across  the  window  may  hide  from  view  a 
large  portion  of  the  distant  landscape  outside.  A  mountain 
a  few  miles  off  may  be  viewed  through  a  finger-ring. 

The  apparent  size  of  an  object,  being  measured  by  the 
visual  angle  which  it  subtends,  is  expressed  in  degrees  or 
radians.  The  apparent  diameter  of  the  full  moon  in  the  sky, 
for  example,  is  not  quite  half  a  degree,  so  that  by  holding 
a  coin  a  little  less  than  9  mm.  in  diameter  at  a  distance  of 
one  meter  from  the  eye,  the  entire  moon  could  be  hid  from 
view.  In  fact,  instead  of  the  angle  itself  it  is  customary  to 
employ  the  tangent  of  the  angle,  especially  in  case  the  vis- 
ual angle  is  not  large.  Thus,  the  apparent  size  of  an  object 
of  height  h  at  a  distance  d  from  the  eye  (in  Fig.  15  AB  = 
h,  AO  =  d)  is  measured  by  the  tangent  of  the  angle  BOA, 
that  is, 


§  10]  Apparent  Size  21 

A  .        linear  dimension   of  the  object     h 

Apparent  size= ^r— ^ —, —  =  -• 

distance  from  the  eye  a 

Accordingly,  in  order  to  determine  the  actual  size  (h)  of 

the  object,  it  is  necessary  to  know  its  distance  (d)  as  well 

as  its  apparent  size,  because  the  actual  size  is  equal  to  the 

product  of  these  two  magnitudes.    The  apparent  size  of  an 


Fig.  15. — Apparent  size  varies  inverse^  as  distance  d  and  directly  as  actual 

size  h. 

object  at  a  distance  of  one  foot  is  an  hundred  times  greater 
than  it  is  at  a  distance  of  an  hundred  feet,  or,  as  we  say,  the 
appare?it  size  varies  inversely  as  the  distance.  As  the  object 
recedes  farther  and  farther  from  the  eye,  its  apparent  size 
diminishes  until  at  last  it  looks  like  a  mere  speck  and  the 
details  in  it  have  all  disappeared.  On  the  other  hand,  al- 
though the  object  is  quite  close  to  the  eye,  its  actual  dimen- 
sions may  be  so  minute  that  it  is  not  to  be  distinguished  from 
a  point.  There  is,  indeed,  a  limit  to  the  power  of  the  human 
eye  to  see  very  small  objects,  which  is  reached  when  the 
object  subtends  in  the  field  of  view  an  angle  that  does  not 
exceed  one  minute  of  arc.  Two  stars  whose  angular  dis- 
tance apart  is  less  than  this  limiting  value  cannot  be  seen 
as  separate  and  distinct  by  a  normal  eye  without  the  aid 
of  a  telescope.  Now  tan  l'  =  sTz-g,  and  consequently  the 
eye  cannot  distinguish  details  of  form  in  an  object  which 
is  viewed  at  a  distance  3438  times  as  great  as  its  greatest 


22  Mirrors,  Prisms  and  Lenses  [§  10 

linear  dimension.  A  silver  quarter  of  a  dollar  is  about 
24  mm.  in  diameter  and  viewed  from  a  distance  of  82.5 
meters  (3438  times  24  mm.  =  82  512  mm.  =  82.5  m.)  its  ap- 
parent size  will  be  1'  of  arc  and  it  will  appear  therefore 
like  a  mere  point.  The  apparent  width  of  a  long  straight 
street  diminishes  in  proportion  as  the  distance  increases; 
until,  finally,  if  the  street  is  long  enough,  the  two  opposite 
sidewalks  seem  to  run  together  at  the  so-called  "vanishing 
point."    . 

If  rays  of  light  coming  through  a  window  and  entering 
the  eye  could  leave  marks  in  the  glass  at  the  points  where 
they  cross  it,  and  if  these  marks  could  be  made  to  emit  the 
same  kind  of  light  as  was  sent  out  from  the  corresponding 
points  of  the  object,  there  would  be  formed  on  the  glass 
a  pictorial  representation  of  the  object  which  when  held 
before  the  eye  at  the  proper  distance  would  have  almost 
exactly  the  same  appearance  as  the  object  itself.  This 
principle  of  perspective  is  made  use  of  in  the  art  of  painting, 
and  the  artist,  with  his  lights  and  shades  and  colors,  tries 
to  portray  on  a  plane  canvas  a  scene  which  will  produce 
as  nearly  as  possible  the  same  visual  impression  on  a  spec- 
tator as  would  be  produced  by  the  natural  objects  them- 
selves. So  far  as  apparent  size  is  concerned,  such  a  repre- 
sentation may  be  perfect.  In  a  good  drawing  the  various 
figures  are  delineated  in  such  dimensions  that  when  viewed 
from  the  proper  standpoint  they  have  the  same  apparent 
sizes  as  the  realities  would  have  if  seen  under  the  aspect 
represented  in  the  picture.  No  one  looking  at  a  photo- 
graph of  a  Greek  temple  will  notice  (unless  his  attention 
is  specially  directed  to  it)  that  the  more  distant  pillars  are 
much  shorter  in  the  picture  than  the  nearer  ones.  Indeed, 
generally  we  pay  little  heed  to  the  apparent  sizes  of  things, 
but  always  try  to  conceive  their  real  dimensions.  When 
two  persons  meet  and  shake  hands,  neither  is  apt  to  observe 
that  the  other  appears  much  taller  than  he  did  when  they 
were  fifty  yards  apart. 


§  11]  Effective  Rays  23 

11.  The  Effective  Rays. — All  the  rays  that  enter  the  eye 
and  fall  on  the  retina  must  pass  through  the  circular  window 
in  the  iris  or  colored  diaphragm  of  the  eye  which  is  called 
the  pupil  of  the  eye  and  which  is  sometimes  spoken  of  as 
the  "black  of  the  eye/'  because  it  appears  black  against 
the  dark  background  of  the  posterior  chamber  of  the  eye. 
The  pupil  of  the  eye  is  about  half  a  centimeter  in  diameter, 
although  within  certain  limits  its  size  can  be  altered  to  regu- 
late the  quantity  of  light  which  is  admitted  to  the  eye.  So 
far  as  the  spectator's  vision  is  concerned,  it  is  only  these 
rays  that  go  through  the  pupil  of  his  eye  that  are  of  any 
use,  and  these  are  the  effective  rays.  When  the  pupil  dilates, 
more  rays  can  enter,  and  consequently  the  source  appears 
brighter.  The  brightness  of  the  source  will  depend  also  on 
its  distance,  because  for  a  given  diameter  of  the  pupil,  the 
aperture  of  the  cone  of  rays  from  a  nearer  source  will  be 
wider  than  that  of  the  cone  of  rays  from  a  more  distant 
source.  In  general,  therefore,  the  pupil  of  the  eye  regu- 
lates the  angular  apertures  of  the  cones  of  rays  that  enter 
the  eye  from  each  point  of  a  luminous  object  and  acts  as 
the  so-called  aperture-stop  (§  134).  Thus,  while  the  extent 
of  the  field  of  view  is  controlled  by  the  field-stop  (§  9),  the 
brightness  of  the  source  depends  essentially  on  the  size  of 
the  aperture-stop. 

A  series  of  transparent  isotropic  media  each  separated 
from  the  next  by  a  smooth,  polished  surface  constitutes  an 
optical  system.  An  optical  instrument  may  consist  of  a  single 
mirror,  prisms  or  lens,  but  generally  it  is  composed  of  a  com- 
bination of  such  elements,  which  may  be  in  contact  with 
each  other  or  separated  by  air  or  some  other  medium.  In 
the  great  majority  of  actual  constructions  the  instrument 
is  symmetrical  with  respect  to  a  straight  line  called  the 
optical  axis.  Not  all  the  rays  emitted  by  a  luminous  object 
will  be  utilized  by  the  instrument;  generally,  in  fact,  only 
a  comparatively  small  portion  of  such  rays  will  be  trans- 
mitted through  it,  in  the  first  place  because  its  lateral  di- 


24  Mirrors,  Prisms  and  Lenses  [§11 

mensions  are  limited,  and  in  the  second  place  because,  in 
addition  to  the  lens-fastenings  and  other  opaque  obstacles 
(sides  of  the  tube,  etc.),  nearly  all  optical  instruments  are 
provided  with  perforated  screens  or  diaphragms  called 
"  stops,"  specially  placed  and  designed  to  intercept  such  rays 
as  for  one  reason  or  another  it  is  not  desirable  to  let  pass 
(§133).  The  planes  of  these  stops  are  placed  at  right  angles 
to  the  optical  axis  with  the  centers  of  the  openings  on  the 
axis.  Accordingly,  each  separate  point  of  the  object  is  to 
be  regarded  as  the  vertex  of  a  limited  cone  or  bundle  of  rays, 
which,  with  respect  to  the  instrument,  are  the  so-called 
effective  rays,  because  they  are  the  only  rays  coming  from 
the  point  in  question  that  traverse  the  instrument  from 
one  end  to  the  other  without  being  intercepted  on  the  way. 

Moreover,  in  every  bundle  of  rays  there  is  always  a  cer- 
tain central  or  representative  ray,  coinciding  perhaps  with 
the  axis  of  the  cone  or  distinguished  in  some  special  way, 
called  the  chief  ray  of  the  bundle  (§  139).  In  a  symmetri- 
cal optical  instrument  the  chief  ray  of  a  bundle  of  effective 
rays  is  generally  defined  to  be  that  ray  which  in  traversing 
a  certain  one  of  the  series  of  media  crosses  the  optical  axis 
at  a  prescribed  point,  which  is  usually  at  the  center  of  that 
one  of  the  stops  which  is  the  most  effective  in  intercepting 
the  rays  and  which,  therefore,  is  called  the  aperture-stop, 
as  will  be  explained  more  fully  hereafter  (see  Chapter  XII). 
According  to  this  definition,  the  chief  rays  coming  from  all 
the  various  points  of  the  object  constitute  a  bundle  of  rays 
which  in  the  medium  where  the  aperture-stop  is  placed 
(sometimes  called  the  "stop  medium")  all  pass  through 
the  center  of  the  stop. 

We  shall  employ  the  term  pencil  of  rays  to  mean  a  section 
of  a  ray-bundle  made  by  a  plane  containing  the  chief  ray. 

The  effective  rays  in  the  first  medium  before  entering 
the  instrument  are  called  the  incident  rays  or  object  rays; 
and  these  same  rays  in  the  last  medium  on  issuing  from  the 
instrument  are  called  the  emergent  rays  or  image  rays.    If  we 


Ch.  I]  Problems  25 

select  at  random  any  point  X  lying  on  one  of  the  rays  of 
the  bundle  of  emergent  rays  which  had  its  origin  at  the  lu- 
minous object-point  P,  in  general,  no  other  ray  of  this  bundle 
will  pass  through  X,  since  in  a  given  optical  system  there 
will  usually  be  one  single  route  by  which  light  starting  from 
the  point  P  and  traversing  the  instrument  can  arrive  finally, 
either  really  or  virtually  (§  8),  at  a  selected  point  X  in  the 
last  medium.  However,  there  may  be  found  a  number  of 
singular  points  where  two  or  more  rays  of  the  bundle  of 
emergent  rays  intersect;  and  under  certain  favorable  and 
exceptional  circumstances  it  may  indeed  happen  that  there 
is  one  special  point  P'  where  all  the  emergent  rays  emanating 
originally  from  the  object-point  P  meet  again;  and  then  we 
shall  obtain  at  P'  a  perfect  or  ideal  image  of  P,  which  is 
described  by  saying  that  P'  is  the  image-point  conjugate  to 
the  object-point  at  P.  This  image  will  be  real  or  virtual 
according  as  the  actual  paths  of  the  image-rays  go 
through  P'  or  merely  the  backward  prolongations  of  these 
paths  (§  8). 

In  order  to  obtain  an  image  in  this  ideal  sense,  the  optical 
system  must  be  such  as  to  transform  a  train  of  incident 
spherical  waves  spreading  out  from  the  object-point  P  into 
a  train  of  emergent  spherical  waves  converging  to  or  di- 
verging from  a  common  center  P'  in  the  image-space.  When 
all  the  rays  of  a  bundle  meet  in  one  point,  the  bundle  of  rays 
is  said  to  be  homocentric  or  monocentvic.  In  general,  how- 
ever, a  monocentric  bundle  of  rays  in  the  object-space  will 
be  transformed  in  the  image-space  into  an  astigmatic  bundle 
of  emergent  rays,  which  no  longer  meet  all  in  one  point; 
and  in  fact  this  is  a  usual  characteristic  of  a  bundle  of  op- 
tical rays. 

PROBLEMS 

1.  Why  are  the  shadows  much  sharper  in  the  case  of  an 
arc  lamp  without  a  surrounding  globe  than  with  one? 

2.  Draw  a  diagram  to  show  how  a  total  eclipse  of  the 


26  Mirrors,  Prisms  and  Lenses  [Ch.  I 

moon  occurs;  and  another  diagram  to  illustrate  a  total 
eclipse  of  the  sun.  Give  clear  descriptions  of  the 
drawings. 

3.  An  opaque  globe,  1  foot  in  diameter,  with  its  center 
at  a  point  C,  is  interposed  between  an  arc  lamp  S  and 
a  white  wall  which  is  perpendicular  to  the  straight  line 
SC.  If  the  wall  is  12  feet  from  the  lamp,  and  if  the 
distance  SC  =  3  feet,  what  is  the  area  of  the  shadow  on 
the  wall?  Ans.  12.57  sq.  ft. 

4.  What  is  the  apparent  angular  elevation  of  the  sun 
when  a  telegraph  pole  15  feet  high  casts  a  shadow  20  feet 
long  on  a  horizontal  pavement?  Ans.  36°  52'  10". 

5.  What  is  the  height  of  a  tower  which  casts  a  shadow 
160  feet  long  when  a  vertical  rod  3  feet  high  casts  a  shadow 
4  feet  long?  Ans.  120  feet. 

6.  An  object  6  inches  high  is  placed  in  front  of  a  pinhole 
camera  at  a  distance  of  6  feet  from  the  aperture.  What  is 
the  size  of  the  inverted  image  on  the  ground  glass  screen  if 
the  length  of  the  camera-box  is  1  foot?  Ans.  1  inch. 

7.  A  small  hole  is  made  in  the  shutter  of  a  dark  room,  and 
a  screen  is  placed  at  a  distance  of  8  feet  from  the  shutter. 
The  image  on  the  screen  of  a  tree  outside  120  feet  away  is 
measured  and  found  to  be  3  feet  long.    How  high  is  the  tree? 

Ans.  45  feet. 

8.  If  the  sensitive  plate  of  a  pinhole  camera  is  20  cm. 
from  the  pinhole,  what  should  be  the  diameter  of  the  pin- 
hole, according  to  Abney's  formula?  Ans.  0.57  mm. 

9.  What  is  the  apparent  size  of  a  man  6  feet  tall  at  a  dis- 
tance of  100  yards?  How  far  away  must  he  be  not  to  be 
distinguishable  from  a  point?  Ans.  1°  8'  45";  3.9  miles. 

10.  If  the  moon  is  240  000  miles  from  the  earth  and  its 
apparent  diameter  is  31'  3",  what  is  its  actual  diameter? 

Ans.  2168  miles. 

11.  A  person  holding  a  tube  6  inches  long  and  1  inch  in 
diameter  in  front  of  his  eye  and  looking  through  it  at  a 
tree  moves  backwards  away  from  the  tree  until  the  entire 


Ch.  I]  Problems  27 

tree  is  just  visible.     What  is  the  apparent  height  of  the 
tree?  Ans.  9°  27' 44". 

12.  Assuming  that  the  resolving  power  of  the  eye  is  one 
minute  of  arc,  at  what  distance  can  a  black  circle  6  inches 
in  diameter  be  seen  on  a  white  background?     Ans.  1719  feet. 


CHAPTER  II 

REFLECTION  OF  LIGHT.   PLANE  MIRRORS 

12.  Regular  and  Diffuse  Reflection. — When  a  beam  of 
sunlight,  admitted  through  an  opening  in  a  shutter  in  a 
dark  room,  falls  on  a  piece  of  smoothly  polished  glass,  al- 
though the  glass  itself  may  be  almost  or  wholly  invisible,  a 
brilliant  patch  of  light  will  be  reflected  from  the  glass  on  the 
walls  of  the  room  or  the  ceiling  or  on  some  other  adjacent  ob- 
ject. If  a  person  in  the  room  happens  to  be  looking  towards 
the  piece  of  glass  along  one  special  direction,  he  will  be  al- 
most blinded  by  the  light  that  is  reflected  into  his  eyes.  The 
glass  acts  like  a  mirror  and  reflects  the  sunlight  falling  on 
it  in  a  definite  direction  which  depends  only  on  the  direc- 
tion of  the  incident  rays  and  on  the  orientation  of  the  re- 
flecting surface,  and  in  such  a  case  the  light  is  said  to  be 
regularly  reflected.  Thus,  for  example,  signals  may  be  com- 
municated to  distant  and  inaccessible  stations  by  reflecting 
thither  the  rays  of  the  sun  by  a  plane  mirror  adjusted  in  a 
suitable  position. 

If  the  surface  is  not  smooth,  the  light  will  be  reflected  in 
many  directions  at  the  same  time.  The  long  sparkling  trail 
of  sunlight  seen  on  the  surface  of  a  lake  or  a  river  on  a  bright 
day  is  caused  by  the  reflections  of  the  sun's  rays  into  the 
eyes  of  the  spectator  from  countless  little  ripples  on  the 
surface  of  the  water. 

The  bright  spot  of  light  on  the  wall  of  a  dark  room  at 
the  place  where  a  beam  of  sunlight  falls,  which  shines  almost 
as  though  this  portion  of  the  wall  were  itself  a  self-luminous 
body,  is  visible  from  any  part  of  the  room  by  means  of  the 
light  which  is  reflected  from  it;  and  although  the  incident 
rays  have  a  perfectly  definite  direction,  the  reflected  light 

28 


§  12]  Diffuse  Reflection  29 

is  scattered  in  all  directions.  Some  of  this  reflected  light 
will  fall  on  other  bodies  in  the  room,  which  will  be  more  or 
less  feebly  illuminated  thereby  and  rendered  dimly  visible 
by  the  light  which  they  reflect  in  their  turn;  until  at  last 
the  light  after  undergoing  in  this  way  repeated  reflections 
from  one  body  to  another  becomes  too  faint  to  be  percep- 
tible. Light  which  is  reflected  or  scattered  in  this  way  is 
said  to  be  diffusely  reflected  or  irregularly  reflected,  although, 
strictly  speaking,  there  is  nothing  irregular  about  it.  Ordi- 
narily it  is  in  this  way  that  bodies  illuminated  by  day- 
light or  by  artificial  light  are  rendered  visible  to  a  whole 
group  of  spectators  at  the  same  time. 

The  paper  on  the  walls  of  an  apartment  which  gets  very 
little  light  through  the  windows  should  be  a  dull  white  in 
order  to  scatter  and  diffuse  as  much  as  possible  the  light 
that  comes  into  the  room.  The  walls  of  a  dark  chamber 
used  for  developing  photographic  plates  should  be  painted 
a  dull  black  in  order  to  absorb  the  light  that  falls  on  them. 
An  absolutely  black  body  (§  2)  exposed  to  the  direct  rays 
of  the  sun  will  be  completely  invisible,  except  by  contrast 
with  its  surroundings.  If  the  walls  of  a  dark  room  and  all 
the  objects  within  it  were  coated  with  lampblack,  and  if 
the  air  inside  were  entirely  free  from  dust  and  moisture, 
a  beam  of  sunlight  traversing  the  room  could  not  be  seen 
and  the  only  way  to  detect  its  presence  would  be  by  placing 
the  eye  squarely  in  its  path.  But  if  a  little  finely  divided 
powder  were  scattered  in  the  air  or  if  a  cloud  of  smoke  were 
blown  across  the  beam  of  light,  the  course  of  the  rays  would 
immediately  become  manifest  to  a  spectator  in  any  part  of 
the  room,  because  some  of  the  light  reflected  from  the  float- 
ing particles  of  matter  in  practically  every  direction  would 
enter  the  eye.    But  the  light  itself  is  quite  invisible. 

Any  surface  that  is  not  too  rough,  that  is,  whose  scratches 
or  ridges  are  not  wider  than  about  a  quarter  of  a  wave- 
length of  light,  will  reflect  light  in  a  greater  or  less  degree 
depending  on  the  smoothness  of  the  surface.     Waves  of 


30  Mirrors,  Prisms  and  Lenses  [§  13 

light  falling  on  a  sheet  of  white  paper  are  broken  up  or 
scattered  in  all  directions,  and  we  can  get  some  idea  of  the 
quantity  of  light  that  is  diffusely  reflected  from  such  a  sur- 
face by  letting  the  light  of  a  lamp  shine  on  the  paper  when 
it  is  held  near  an  object  that  is  in  shadow.  It  is  almost 
startling  to  see  how  under  the  influence  of  this  indirect 
illumination  the  details  of  the  obscure  body  suddenly  ap- 
pear as  if  summoned  forth  by  magic.  A  highly  polished 
metallic  surface  makes  the  best  mirror,  reflecting  some- 
times as  much  as  three-fourths  of  the  incident  light.  Our 
ordinary  looking-glasses  are  really  metallic  mirrors,  because 
they  are  coated  at  the  back  with  silver,  and  the  glass  merely 
serves  as  a  protection  for  the  reflecting  surface. 

13.  Law  of  Reflection. — A  ray  of  light  represented  in 
Fig.  16  by  the  straight  line  AB  is  incident  at  B  on  a  smooth 
reflecting  surface  whose  trace  in  the  plane  of  the  diagram 
is  the  line  ZZ.    The  straight  line  BN  normal  to  the  surface 


Fig.    16. — Law  of   reflection: 
Z  NBA  =  -Z  NBC  =  Z  CBN. 


at  B  is  called  the  incidence-normal,  and  the  plane  ABN  which 
contains  the  incident  ray  AB  and  the  normal  BN  is  called 
the  plane  of  incidence,  which  corresponds  here  with  the 
plane  of  the  diagram.  The  angle  of  incidence  is  the  angle 
between  the  incident  ray  and  the  incidence-normal;  or,  to 


§  13]  Law  of  Reflection  31 

define  this  angle  more  exactly,  the  angle  of  incidence  is  the  acute 
angle  ( a )  through  which  the  incidence-normal  has  to  be  turned 
about  the  point  of  incidence  in  order  to  make  it  coincide  with 
the  incident  ray;  thus,  a  ~  Z  NBA.  Counter-clockwise  rota- 
tion is  to  be  reckoned  as  positive  and  clockwise  rotation  as 
negative.  This  rule  will  be  consistently  observed  in  the 
case  of  all  angular  measurements. 

The  reflected  ray  corresponding  to  the  incident  ray  AB  is 
represented  by  the  straight  line  BC;  and  if  in  the  above 
definition  of  the  angle  of  incidence  we  substitute  "reflected 
ray"  for  " incident  ray,"  we  shall  obtain  the  definition  of 
the  angle  of  reflection  (/3);  that  is,  /3  =  ZNBC.  The  sense 
of  the  rotation  is  indicated  by  the  order  in  which  the 
letters  specifying  the  angle  are  named;  thus,  ZABC  is  the 
angle  described  by  rotating  the  straight  line  AB  around  the 
point  B  until  it  coincides  with  the  straight  line  BC;  whereas 
ZCBA=-ZABC  denotes  the  equal  but  opposite  rotation 
from  CB  to  BA.  The  student  should  take  note  of  this 
usage,  which  will  be  uniformly  employed  throughout  this 
book. 

The  law  of  the  reflection  of  light,  which  has  been  known 
for  more  than  2200  years,  is  contained  in  the  following 
statement : 

The  inflected  ray  lies  in  the  plane  of  incidence,  and  the  in- 
cident and  reflected  rays  make  equal  angles  with  the  normal 
on  opposite  sides  of  it;  that  is, /S  =-a. 

A  very  accurate  experimental  proof  of  this  law  may  be 
obtained  by  employing  a  meridian  circle  to  observe  the  light 
reflected  from  an  artificial  mercury-horizon,  that  is,  from 
the  horizontal  surface  of  mercury  contained  in  a  basin.  In 
fact,  this  is  the  actual  method  used  by  astronomers  in  meas- 
uring the  altitude  of  a  star.  The  telescope  is  pointed  at  the 
star  and  then  at  the  image  of  the  star  in  the  mercury  mirror, 
and  it  will  be  found  that  the  axis  of  the  telescope  in  these 
two  observations  will  be  equally  inclined  to  the  vertical  on 
opposite  sides  of  it. 


32  Mirrors,  Prisms  and  Lenses  [§  13 

A  simple  lecture-table  apparatus  for  verifying  the  law 
of  reflection  of  light  consists  of  a  circular  disk  (Fig.  17)  made 
of  ground  glass,  about  one  foot  or  more  in  diameter,  and 
graduated  around  the  circumference  in  degrees.  This  disk 
is  mounted  so  as  to  be  capable  of  rotation  in  a  vertical  plane 

about  a  horizontal  axis 
perpendicular  to  this 
plane  and  passing  through 
the  center  of  the  disk.  A 
small  piece  of  a  plane 
mirror  B  with  its  plane 
perpendicular  to  that  of 
the  disk  is  fastened  to 
the  disk  at  its  center,  and 
the  mirror  is  adjusted  so 
that  it  is  perpendicular 
to  the  radius  BN  drawn 

Fig.  17. — Optical  disk  used  to  verify  law    on  the  disk.      A  beam  of 

of  reflection.  sunlight   falling    on    the 

mirror  in  the  direction  NB  will  be  reflected  back  from  the 
mirror  in  the  opposite  direction  BN,  so  that  in  this  adjust- 
ment of  the  disk  the  paths  of  the  incident  and  reflected  rays 
coincide  (/?=  -a=0).  Now  if  the  disk  is  turned  so  that 
the  incident  ray  AB  makes  with  the  normal  BN  an  angle 
NBA,  the  reflected  ray  will  proceed  in  a  direction  BC  such 
that  ZNBC  =  ZABN=-a. 

If,  without  changing  the  direction  of  the  incident  ray,  the 
disk  is  turned  through  an  angle  6,  the  plane  of  the  mirror  to- 
gether with  the  incidence-normal  will  likewise  be  turned 
through  this  same  angle,  and  the  angles  of  incidence  and  re- 
flection will  each  be  changed  in  opposite  senses  by  the  amount 
6,  so  that  the  angle  between  the  incident  and  reflected  rays 
will  be  changed  by  2  6.  Accordingly,  when  a  plane  mirror 
is  turned  through  a  certain  angle,  the  reflected  ray  will  be  turned 
through  an  angle  twice  as  great. 


14] 


Waves  Reflected  at  Plane  Mirror 


33 


14.  Huygens's  Construction  of  the  Wave-Front  in  Case 
of  Reflection  at  a  Plane  Mirror. 

1.  The  case  of  a  plane  wave  reflected  from  a  plane  mirror. 
The  rebound  of  waves  from  a  polished  surface  affords  a  very 
simple  and  instructive 
illustration  of  Huygens's 
Principle  (§  5) .  In  Fig.  18 
the  straight  line  AD 
represents  the  trace  in 
the  plane  of  the  diagram 
of  a  plane  mirror,  and 
the  straight  line  AB  rep- 
resents the  trace  of  a 
portion  of  the  front  of 
an  incident  plane  wave 
(§  6)  advancing  in  the 
direction  of  the  wave- 
normal  BD.  At  the  first 
instant  under  considera- 
tion the  wave-front  is 
supposed  to  be  in  the 
position  AB  when  the 
disturbance  has  just 
reached  the  point  A  of 
the  reflecting  surface, 
and  from  this  time  for- 
ward, according  to  Huy- 
gens's theory,  the  point 
A  is  to  be  regarded  as 
itself  a  center  of  dis- 
turbance from  which 
secondary  hemispherical 
waves  are  reflected  back  into  the  medium  in  front  of  the 
mirror.  Exactly  the  same  state  of  things  will  prevail  at 
this  instant  (t  =  0)  at  all  points  of  the  plane  reflecting  sur- 
face lying  on  a  portion  of  the  straight  line  perpendicular 


Fig.  18. — Huygens's  construction  of  plane 
wave  reflected  at  plane  mirror. 


34  Mirrors,  Prisms  and  Lenses  [§  14 

to  the  plane  of  the  paper  at  the  point  A,  and  the  envelop 
of  the  hemispherical  wavelets  originating  from  these  points 
will  be  a  semicylindrical  surface  whose  axis  is  the  straight 
line  just  mentioned.  If  the  speed  with  which  the  waves 
travel  is  denoted  by  v,  then  at  the  end  of  the  time  t  =  YQ/v 
the  disturbance  that  was  initially  at  the  point  P  in  the  wave- 
front  AB  will  have  advanced  to  a  point  Q  on  the  reflecting 
plane  between  A  and  D;  and  from  this  moment  a  new  set 
of  hemispherical  wavelets  having  their  centers  all  on  a 
straight  line  perpendicular  to  the  plane  of  the  diagram  at 
the  point  Q  will  begin  to  develop,  and  their  envelop  will 
also  be  a  semicylinder.  And  so  at  successively  later  and 
later  instants  the  disturbance  will  arrive  in  turn  at  each 
point  along  AD;  until,  finally,  after  the  time  t  =  BT)/v  the 
farthermost  point  D  will  be  reached.  Meanwhile,  around 
all  the  straight  lines  perpendicular  to  the  plane  of  the 
paper  at  points  lying  along  AD  semicylindrical  elementary 
wave-surfaces  will  have  been  spreading  out  from  the  re- 
flecting surface,  the  radii  of  these  cylinders  diminishing 
from  A  towards  D.  At  the  time  when  the  disturbance 
reaches  D,  the  semicylindrical  wavelet  whose  axis  passes 
through  A  will  have  expanded  until  its  radius  is  equal  to 
BD,  and  at  this  same  instant  the  semicylindrical  wavelet 
corresponding  to  a  point  Q  between  A  and  D  will  have  been 
expanding  for  a  time  (BD — PQ)/v,  and  hence  its  radius  will 
be  egual  to  (BD— PQ)  =  (BD— BK)  =KD. 

Now,  according  to  Huygens's  Principle,  the  surface  which 
at  any  instant  is  tangent  to  all  these  elementary  semi- 
cylindrical  waves  will  be  the  required  reflected  wave-front 
at  that  instant.  We  shall  show  that  the  reflected  wave-front 
is  a  plane  surface  which  at  the  moment  when  the  disturb- 
ance reaches  the  point  D  contains  this  point;  or,  what 
amounts  to  the  same  thing,  we  shall  show  that  if  a  straight 
line  DC  in  the  plane  of  the  diagram  is  tangent  at  C  to  the 
semicircle  in  which  this  plane  cuts  the  semicylinder  whose 
axis  passes  through  A,  it  will  be  a  common  tangent  to  all 


§  14]  Waves  Reflected  at  Plane  Mirror  35 

such  semicircles;  for  example,  it  will  also  be  tangent  to 
the  semicircle  in  which  the  plane  of  the  diagram  cuts  the 
semicylinder  belonging  to  the  point  Q.  From  D  draw  DC 
tangent  at  C  to  the  semicircle  described  around  A  as  center 
with  radius  AC  =  BD  and  DR  tangent  at  R  to  the  semi- 
circle described  around  Q  as  center  with  radius  QR  =  KD. 
The  right  triangles  ABD  and  ACD  are  congruent,  and  hence 
ZDAB  =  ZCDA;  and,  similarly,  in  the  congruent  right  tri- 
angles QKD  and  QRD  ZDQK  =  ZRDQ.  But  ZDQK  = 
ZDAB,  and  therefore  ZRDQ=ZCDA,  and  hence  the  two 
tangents  DR  and  DC  coincide.  Accordingly,  the  trace  of 
the  reflected  wave-front  in  the  plane  of  the  diagram  is  the 
straight  line  CD.  This  reflected  plane  wave  will  be  prop- 
agated onwards,  parallel  with  itself,  in  the  direction  shown 
by  the  reflected  rays  AC,  QR,  etc.  It  is  evident  from  the 
construction  that  the  ray  incident  at  A,  the  normal  AN  to 
the  reflecting  surface  at  the  incidence-point  A,  and  the  re- 
flected ray  AC  lie  all  in  the  same  plane;  and  the  equality  of 
the  angles  of  incidence  and  reflection  is  an  immediate  con- 
sequence of  the  congruence  of  the  triangles  ABD  and  ACD. 
2.  The  case  of  a  spherical  wave  reflected  at  a  plane  mirror. 
In  Fig.  19  the  light  is  represented  as  originating  from  a 
point-source  L  and  spreading  out  from  it  in  the  form  of 
spherical  waves  which  presently  impinge  on  the  plane  re- 
flecting surface  represented  in  the  diagram  by  the  straight 
line  AD.  The  nearest  point  of  the  reflecting  plane  to  the 
source  at  L  is  the  foot  A  of  the  perpendicular  let  fall  from 
L  on  the  straight  line  AD,  and  this,  therefore,  is  the  first 
point  of  the  mirror  to  be  affected.  Obviously,  on  account 
of  symmetry  with  respect  to  LA,  it  will  be  quite  sufficient 
to  investigate  the  procedure  of  the  waves  in  the  plane  of 
the  figure.  The  wave-front  at  the  time  the  disturbance 
reaches  A  will  be  represented  by  the  arc  of  a  circle  described 
around  L  as  center  with  radius  equal  to  LA;  let  P  desig- 
nate the  position  of  a  point  on  this  arc,  and  draw  the  straight 
line  LP  meeting  AD  at  Q.    After  a  time  t  =  FQ/v  the  dis- 


36 


Mirrors,  Prisms  and  Lenses 


[§14 


turbance  will  have  advanced  from  P  to  Q,  and  from  this 
moment  the  point  Q  will  begin  to  send  back  wavelets  from 
the  reflecting  surface.  And  so  in  succession  one  point  of 
the  mirror  after  another  will  be  affected  until  presently  the 
disturbance  reaches  the  farthest  point  D.     Meanwhile,  all 

the  points  along  AD  on 
one  side  of  AL  and 
along  AF  on  the  other 
side  (AF  =  DA)  will 
have  been  sending  out 
wavelets  whose  radii  will 
be  greater  and  greater 
the  nearer  these  new 
centers  are  to  the  point 
A  midway  between  D 
and  F.  Draw  the 
straight  line  LD  meet- 
ing the  arc  AP  in  the 
point  B:  then  at  the 
moment  t  =  BT>/v  when 
the  disturbance  from  L 
has  just  arrived  at  D, 
the    reflected     wavelet 

Fig.  19. — Huygens's  construction  of  spheri-   proceeding    from    A    as 
cal  wave  reflected  at  plane  mirror.  <  hi      u 

center  will  have  ex- 
panded until  its  radius  is  equal  to  BD,  and  at  this  same 
instant  there  will  also  be  a  wavelet  around  Q  as  center 
of  radius  (BD— PQ)  =  (BD— BK)=KD.  According  to 
Huygens,  the  problem  consists,  therefore,  in  finding  the 
surface  which  is  tangent  at  a  given  instant  to  all  these 
secondary  waves.  Produce  the  straight  line  LA  on  the 
other  side  of  the  reflecting  surface  to  a  point  L'  such 
that  AL'  =  LA,  and  draw  the  straight  line  L'Q,  and  mark 
the  point  R  where  this  straight  line  produced  meets 
the  semicircle  described  around  Q  as  center  with  radius 
KD  =  QR.      Since    LQ  +  QR  =  LK  +  KD  =  LD,    obviously, 


§15] 


Image  in  Plane  Mirror 


37 


L/R  =  L/D;  and  therefore  a  circle  described  around  L'  as  cen- 
ter with  radius  equal  to  L'D  will  touch  at  R  the  semicircle 
described  around  Q  as  center  with  radius  equal  to  QR. 
Moreover,  it  will  also  touch  at  a  point  C  on  the  straight 
line  LA  the  semicircle  described  around  A  as  center  with 
radius  AC  =  BD.  Consequently,  this  circle  will  be  the. 
envelop  of  all  these  semicircles.  The  reflected  wave-front, 
therefore,  is  obtained  by  revolving  the  arc  DCF  around  LL' 
as  axis.  The  straight  line  QR  is  the  path  of  the  reflected 
ray  corresponding  to  the  incident  ray  PQ;  the  angle  of  in- 
cidence at  Q  is  equal  to  the  angle  ALQ  and  the  angle  of  re- 
flection is  equal  to  AI/Q,  and  these  angles  are  evidently 
equal,  in  agreement,  therefore,  with  the  law  of  reflection. 
15.  Image  in  a  Plane  Mirror. — In  Fig.  19  the  plane  mirror 
bisects  at  right  angles  the  straight  line  LL',  and  since  the 


Fig.  20. — L'  is  image  of  object-point   L    in  plane  mirror   AD; 

AL  =  Im- 
position of  the  point  L'  is  independent  of  the  position  of 
the  incidence-point  Q  (Fig.  20),  all  the  rays  coming  from 
the  luminous  point  L  and  falling  on  the  plane  mirror  will 
be  reflected  along  paths  which,  when  prolonged  backwards, 


38  Mirrors,  Prisms  and  Lenses  [§  15 

all  meet  in  the  point  L'.  Thus,  to  a  homocentric  bundle  of 
incident  rays  reflected  at  a  plane  mirror  there  corresponds  also 
a  homocentric  bundle  of  reflected  rays.  This  remarkable 
property  of  converting  a  homocentric  bundle  of  rays  into 
another  homocentric  bundle  is  characteristic  of  a  plane 
mirror,  because  no  other  optical  device  is  capable  of  it.  ex- 
cept under  conditions  that  are  more  or  less  unrealizable  in 
practice.  Thus,  the  image  1/  of  an  object  at  L  is  found  by 
drawing  a  straight  line  from  L  perpendicular  to  the  plane 
mirror,  and  producing  this  line  on  the  other  side  of  the 
mirror  to  a  point  L'  such  that  the  line-segment  LI/  is  bi- 
sected by  the  plane  of  the  mirror;  so  that  an  object  in  front 
of  a  plane  mirror  is  seen  in  the  mirror  at  the  same  distance 
behind  it.  The  image  in  this  case  is  virtual  (§  8).  The  late 
Professor  Silvanus  Thompson  in  his  popular  lectures 
published  under  the  title  Light  Visible  and  Invisible  de- 
scribes the  following  simple  method  of  showing  how  the 
rays  from  a  candle  flame  are  reflected  at  a  plane  mirror 
(Fig.  21).  If  a  vertical  pin  mounted  on  a  horizontal  base- 
board is  illuminated  by  a  lighted  candle,  the  position  of 
the  shadow  is  determined  by  the  line  joining  the  top  of  the 
pin  with  the  source  of  light.  If  the  pin  and  the  candle  are 
both  in  front  of  a  plane  mirror  placed  at  right  angles  to  the 
base-board,  a  second  shadow  will  be  cast  by  the  pin  on  ac- 
count of  the  reflected  rays  from  the  candle  that  are  inter- 
cepted by  it,  and  this  shadow  will  be  precisely  such  as  would 
be  produced  by  a  candle  flame  placed  behind  the  mirror 
at  the  place  where  the  image  of  the  actual  flame  is  formed, 
as  may  be  proved  by  removing  the  mirror  and  transferring 
the  candle  to  the  place  where  its  image  was. 

If  the  bundle  of  incident  rays  instead  of  diverging  from 
a  point  L  in  front  of  the  plane  mirror  converged  towards 
a  point  L  behind  it  (as  could  easily  be  effected  with  the  aid 
of  a  convergent  lens),  a  real  image  (§  8)  will  be  produced  at 
a  point  L'  at  the  same  distance  in  front  of  the  mirror  as  the 
virtual  object-point  L  was  beyond  it. 


Fig.  21. — Shadows  cast  by  an  object  in  front  of  a  plane  mirror  when  object 
is  illuminated  by  point-source  (from  actual  photograph),  showing  that 
the  source  and  its  image  are  at  equal  distances  from  the  mirror. 


15] 


Image  in  Plane  Mirror 


39 


The  image  of  an  extended  object  is  the  figure  formed  by 
the  images  of  all  of  its  points  separately.  The  diagram 
(Fig.  22)  shows,  for  example,  how  an  eye  at  E  would  see 
the  image  L'M'  of  an  object  LM  reflected  in  a  plane  mirror. 
The  series  of  parallel  lines  joining  corresponding  points  of 


Fig.  22. — Image  L'M'  of  object  LM  in  plane  mirror  ZZ. 

object  and  image  will  be  bisected  at  right  angles  by  the 
plane  of  the  mirror. 

The  dimensions  of  the  image  in  a  plane  mirror  are  ex- 
actly the  same  as  those  of  the  object.  Moreover,  the  top 
and  bottom  of  the  image  correspond  with  the  top  and  bot- 
tom of  the  object,  that  is,  the  image  is  erect.  Also,  the 
right  side  of  the  image  corresponds  with  the  right  side  of 
the  object,  and  the  left  side  of  the  image  with  the  left  side 
of  the  object  (Fig.  23),  although  it  is  frequently  stated  in 
books  on  optics  that  when  a  man  stands  in  front  of  a  mirror 
the  right  side  of  the  image  shows  the  left  side  of  the  person, 
and  that  if  the  man  extends  his  right  hand,  the  image  will 
extend  its  left  hand.  The  true  explanation  of  the  so-called 
"perversion"  of  the  image  in  a  plane  mirror,  which  is  strik- 
ingly seen  when  a  printed  page  is  held  in  front  of  the  mirror, 
is  that  it  is  the  rear  side  of  the  image  that  is  opposite  the  front 


40  Mirrors,  Prisms  and  Lenses  [§  16 

side  of  the  object.  The  image  of  a  printed  page  in  a  mirror 
has  exactly  the  same  appearance  as  it  would  have  if  the 
page  were  held  in  front  of  a  bright  light  and  it  was  viewed 
from  behind  through  the  paper.  When  a  person  looks  in 
a  mirror  at  his  own  image,  his  image  appears  to  be  looking 
back  at  him  in  the  opposite  direction,  if  he  faces  east,  his 
image  faces  west,  and  if  we  call  the  east  side  of  object  or 
image  its  front  side  and  the  west  side  its  rear  side,  then  the 
rear  side  of  the  image  is  turned  towards  the  front  side  of 
the  object;  although,  because  this  side  of  the  image  cor- 
responds to  the  front  side  of  the  object,  it  is  a  natural  mis- 
take to  regard  it  as  also  the  front  side  of  the  image.  The 
explanation  of  the  common  impression  that,  whereas  up 
and  down  remain  unchanged  in  the  image  of  an  object  in 
a  plane  mirror,  right  and  left  are  reversed,  is  probably  be- 
cause a  person  regarding  his  own  image  under  such  circum- 
stances is  unconsciously  disposed  to  transfer  himself  men- 
tally into  coincidence  with  his  image  by  a  rotation  of  180°, 
not  around  a  horizontal,  but  around  a  vertical  axis,  thus 
producing  a  confusion  of  mind  as  to  right  and  left  but  not 
as  to  top  and  bottom.  The  reason  why  this  mental  revolu- 
tion is  performed  around  the  vertical  axis  seems  to  be  due 
partly  to  the  circumstance  that  this  movement  can  be 
readily  executed  in  reality,  and  partly  also  perhaps  to  the 
fact  that  the  human  body  happens  to  be  very  nearly 
symmetrical  with  respect  to  a  vertical  plane. 

16.  The  Field  of  View  of  a  Plane  Mirror. — In  the  adjoin- 
ing diagram  (Fig.  24)  the  straight  line  GH  represents  the 
trace  in  the  plane  of  the  paper  of  the  surface  of  a  plane  mir- 
ror, and  the  point  marked  O'  shows  the  position  of  the  center 
of  the  pupil  of  the  eye  of  a  person  who  is  supposed  to  be 
looking  towards  the  mirror.  Evidently,  the  straight  lines 
HO',  GO'  drawn  to  0'  from  the  points  G,  H  in  the  edge  of 
the  mirror  will  represent  the  paths  of  the  outermost  reflected 
rays  that  can  enter  the  eye  at  0',  and  therefore  the  field  of 
view   (§  9)   is  limited  by  the  contour  of  the  mirror  just 


Fig.  23. — Image  of  object  in  plane  mirror  (from  actual  photograph). 


16] 


Field  of  View  of  Plane  Mirror 


41 


as  if  the  observer  were  looking  into  the  image-space  through 
a  hole  in  the  wall  that  exactly  coincided  with  the  place  oc- 
cupied by  the  mirror.  Corresponding  to  the  pair  of  re- 
flected rays  HO'  and  GO'  intersecting  at  O'  there  would  be 
a  pair  of  incident  rays  directed  along  the  straight  lines  HO 


: &** 


Fig.  24. — Field  of  view  of  plane  mirror  for  given  position  of  eye. 

and  GO  towards  a  point  0  on  the  other  side  of  the  mirror, 
and  it  is  evident  that  0'  will  be  the  real  image  of  a  virtual 
object-point  at  O  (§  15).  Any  luminous  point  lying  in  front 
of  the  plane  mirror  within  the  conical  surface  formed  by 
drawing  straight  lines  such  as  OG,  OH  from  0  to  all  the 
points  in  the  edge  of  the  mirror  will  be  visible  by  reflected 
light  to  an  eye  placed  at  0',  and  hence  this  cone  limits  the 
field  of  view  of  the  object-space. 

Through  O'  draw  a  straight  line  parallel  to  GH,  and  take 
on  it  two  points  C',  B'  at  equal  distances  from  O'  on  oppo- 
site sides  of  it,  and  let  us  suppose  that  B'C  represents  the 
diameter  in  the  plane  of  the  diagram  of  the  pupil  of  the  eye. 
Construct  the  image  BOC  of  the  eye-pupil  B'O'C.  Then 
if  P  designates  the  position  of  a  luminous  point  lying  any- 


42 


Mirrors,  Prisms  and  Lenses 


[§16 


where  within  the  field  of  view  of  the  object-space,  it  is  clear 
that  the  incident  rays  PO,  PC  and  PB  will  be  reflected  at 


Fig.  25. — Deviation  of  a  ray  reflected  twice  in  suc- 
cession from  a  pair  of  inclined  mirrors. 

the.  mirror  into  the  pupil  of  the  eye  in  the  directions  P'O', 
P'C  and  P'B',  as  though  they  had  all  come  from  the  point 
P'  which  is  the  image  of  P.    This  imaginary  opening  or  vir- 


Fiq.  26. — Deviation  of  a  ray  reflected  twice  in  suc- 
cession from  a  pair  of  inclined  mirrors. 

tual  stop  BOC  towards  which  the  incident  rays  must  all  be 
directed  in  order  to  be  reflected  into  the  eye-pupil  B'O'C  is 


§  18]  Inclined  Mirrors  43 

called  the  entrance-pupil  of  the  optical  system  consisting 
of  the  plane  mirror  and  the  eye  of  the  observer;  and  the 
pupil  of  the  eye  itself  is  called  here  the  exit-pupil  (see  Chap- 
ter XII).  Since  the  entrance-pupil  limits  the  apertures  of 
the  bundles  of  rays  that  ultimately  enter  the  eye,  it  acts 
as  the  aperture-stop  of  the  system  (§  11). 

17.  Successive  Reflections  from  two  Plane  Mirrors. — 
Any  section  made  by  a  plane  perpendicular  to  the  line  of  in- 
tersection of  the  planes  of  a  pair  of  inclined  mirrors  is  called 
a  principal  section  of  the  system.  If  a  ray  lying  in  a  prin- 
cipal section  is  reflected  successively  at  two  plane  mirrors}  it 
will  be  deviated  from  its  original  direction  by  an  angle  equal  to 
twice  the  dihedral  angle  between  the  mirrors. 

Let  the  plane  of  the  principal  section  intersect  the  planes 
of  the  mirrors  in  the  straight  lines  OM,  ON  (Figs.  25  and 
26) ;  and  let  7  =  Z  MON  denote  the  angle  between  the  mir- 
rors. The  ray  PQ  lying  in  the  plane  MON  is  incident  on 
the  mirror  OM  at  the  point  Q,  whence  it  is  reflected  along 
the  straight  line  QR,  meeting  the  mirror  ON  at  the  point 
R,  where  it  is  again  reflected,  proceeding  in  the  direction 
RS.  Let  the  point  of  intersection  of  the  straight  lines  PQ 
and  RS  be  designated  by  K.  Then  Z  PKR  is  the  angle  be- 
tween the  original  direction  of  the  ray  and  its  direction  after 
undergoing  two  reflections,  and  we  must  show  that  this 
angle  is  equal  to  2  7. 

Draw  the  incidence-normals  at  Q  and  R,  and  prolong 
them  until  they  meet  at  J.    Then  by  the  law  of  reflection 
the  straight  lines  QJ  and  RJ  bisect  the  angles  PQR  and 
QRS,  respectively. 
In  Fig.  25,       ZPKR=ZPQR+ZQRS  =  2(ZJQR+ZQRJ) 

=  2(180°  -ZRJQ)  =  27; 
and  in  Fig.  26,  Z  PKR  =  Z  PQR  -  Z  SRQ 

=  2(180°-ZRQJ-ZJRQ) 

=  2ZQJR-27. 

18.  Images  in  a  System  of  Two  Inclined  Mirrors. — When 
a  luminous  point  lies  in  the  dihedral  angle  between  two 


44 


Mirrors,  Prisms  and  Lenses 


[§18 


plane  mirrors,  some  of  its  rays  will  fall  on  one  mirror  and 
some  on  the  other,  and  consequently  there  will  be  two  sets 
of  images.  In  Fig.  27  the  plane  of  the  diagram  is  the  prin- 
cipal section  which  contains  the  point-source  S,  and  the 
straight  lines  OM,  ON  represent  the  traces  of  the  mirrors 


Fig.  27. — Images  of  a  luminous  point  S  in  a  pair  of  inclined 
mirrors  OM  and  ON. 

in  this  plane.  The  rays  which  fall  first  on  the  mirror  OM 
will  be  reflected  as  though  they  came  from  the  image  Px  of 
the  luminous  point  S  in  this  mirror.  Some  of  these  rays 
falling  on  the  mirror  ON  will  be  again  reflected  and  proceed 
thence  as  though  they  came  from  the  point  P2  which  is 
the  image  of  Px  in  the  mirror  ON.  Thus,  by  successive  re- 
flections, first  at  one  of  the  mirrors  and  then  at  the  other, 
a  series  of  images  Pi,  P2,  etc.,  will  be  formed  by  those  rays 
which  fall  first  on  the  mirror  OM ;  let  us  call  this  the  P-series 
of  images.  Similarly,  the  rays  that  fall  first  on  the  mirror 
ON  will  produce  another  series  of  images  Qx,  Q2,  etc.,  which 
will  be  called  the  Q-series.  Each  of  these  series  will  termi- 
nate with  an  image  which  lies  behind  both  mirrors  in  the 


§  18]  Images  in  Inclined  Mirrors  45 

dihedral  angle  COD  opposite  the  angle  MON  between  the 
mirrors  themselves;  because  rays  which,  after  reflection 
at  one  of  the  mirrors,  appear  to  come  from  a  point  thus  sit- 
uated cannot  fall  on  the  other  mirror,  and  so  there  will  be 
no  more  images  after  this  one. 

Since  the  straight  line  OM  is  the  perpendicular  bisector 
of  the  line-segment  SPX,  the  points  S  and  Px  are  equidistant 
from  every  point  in  the  straight  line  OM;  and,  similarly, 
since  P2  is  the  image  of  Pi  in  the  plane  mirror  ON,  these  two 
points  are  likewise  equidistant  from  every  point  in  the 
straight  line  ON.  Accordingly,  the  three  points  S,  P^  P2 
are  all  equidistant  from  the  point  0  where  the  straight  lines 
OM  and  ON  intersect.  Applying  the  same  reasoning  to 
all  the  other  images,  we  perceive  that  the  images  of  both 
series  are  ranged  on  the  circumference  of  a  circle  whose  center 
is  at  0  and  whose  radius  is  OS. 

In  the  following  discussion  of  the  angular  distances  of 
the  images  from  the  luminous  point  S,  the  angles  will  be 
reckoned  always  in  the  same  sense,  either  all  clockwise  or 
all  counter-clockwise.  Let  y  =  Z  AOB  denote  the  angle  be- 
tween the  two  mirrors,  the  letters  A  and  B  referring  to  the 
points  where  the  circle  crosses  the  planes  of  the  mirrors  OM 
and  ON,  respectively.  Also,  let  a=Z  AOS,  j8  =ZSOB  de- 
note the  angular  distances  of  S  from  A,  B,  respectively,  so 
that  a+/3=y.    Then 

ZPiOS  =  2a; 

ZSOP2  =  ZSOB+ZBOP2=0+ZPiOB  =  2(a+/3)  =  2Y; 

Z  P3OS  =  Z  P3OA+  a=Z  AOP2+  a  =  Z  SOP2+ 2  a 
=  27+2a; 

Z  SOP4  =  Z  SOB+  Z  BOP4  =  P+  Z  P3OB 

=  2/3+ZP3OS  =  2(a+/3+7)=47; 

Z  P5OS  =  Z  P5OA+  a  =  Z  AOP4+  a  =  Z  SOP4+  2  a 
=  47+2a. 
In  general,  therefore, 

ZSOP2k  =  2/c7,  ZP2k+1OS  =  2/b7+2a, 

where  P2k,  P2k+i  designate  the  positions  of  the  2fcth  and 


46  Mirrors,  Prisms  and  Lenses  [§  18 

(2fc+l)th  images  of  the  P-series,  k  denoting  any  integer, 
and  where  the  angles  SOP2k,  P2k+iOS  are  the  angles  sub- 
tended by  the  arcs  SBP2k,  P2k+iAS,  respectively.  Similarly, 
for  the  Q-series  of  images  we  find : 

ZQ2k0S  =  2A;7,  Z80Q2k+1  =  2ky+2p, 

where  these  angles  are  the  angles  subtended  by  the  arcs 
Q2kAS,  SBQ2k+1,  respectively. 

Evidently,  the  image  P2k+i  will  fall  on  the  arc  CD  be- 
hind both  mirrors,  if  arc  P2k+1AS>arc  DAS,  that  is,  if 

2&Y+2  a>  180°  -0; 
and,  by  adding   (/3-a)  to  both  sides  of  this  inequation, 
and  dividing  through  by  y,  this  condition  may  be  expressed 
as  follows: 

,  ,     '  180°-a 

In  the  same  way  we  find  that  the  image  P2k  will  fall  between 
C  and  D  if 

7     180°  -a 
2k>—j—' 

Thus,  the  total  number  of  images  of  the  "P-series  t  whether  it 
be  odd  or  even,  will  be  given  by  the  integer  next  higher  than 
(180°—  a)/ y;  and,  similarly,  the  total  number  of  images 
of  the  Q-series  will  be  given  by  the  integer  next  higher  than 
(180°-  PHy. 

The  only  exception  to  this  rule  is  when  the  angle  y  is 
contained  in  (180° — a)  or  (180° — /3)  an  exact  whole  num- 
ber of  times;  in  the  former  case  the  last  image  of  the  P-series 
falls  at  C,  and  in  the  latter  case  the  last  image  of  the  Q-series 
falls  at  D;  and  instead  of  taking  the  integers  next  above 
the  quotient  (180°—  a)/y  or  (180°—  fi)/y,  we  must  take 
the  actual  integer  obtained  by  the  division.  An  example 
will  make  the  matter  clear.  Thus,  suppose  7  =  27°,  a  =  8°, 
then  /?=  19°,  and  the  integers  next  higher  than  (180° — a)jy 
and  (180° — f})/y  will  be  7  and  6,  respectively;  hence  in 
this  case  there  will  be  7  images  of  the  P-series  and  6  images 
o  the  Q-series  or  13  images  in  all.    But  if  a  =  10°  and  fi  =  17°, 


§  18]  Kaleidoscope  47 

each  series  will  be  found  to  have  7  images,  14  images  in 
all.  The  exceptional  case  occurs  when  a  =  9°  and  /3  =  18°, 
for  then  (180°—  /3)/y  =  6,  and  hence  there  will  be  7  P-images 
and  6  Q-images. 

If  the  angle  y  between  the  mirrors  is  an  exact  multiple 
of  180°,  that  is,  if  180°/ y  =  p,  where  p  denotes  an  integer, 
the  integers  next  higher  than  (180°-  a)/ J  and  (180°-  P)/y 
will  both  be  equal  to  p, 
no  matter  what  may  be 
the    special    position    of 
the  object   between  the 
two  mirrors;   so  that  in 
such  a  case  the  number 
of  images  in  each  series 
will  be  equal,  but  the  last 
image  of  one  set  will  co- 
incide with  the  last   of 
the  other.     In  fact,  the 
points    S,    P2,    P4,  .  .  . 

Q47  Q2  and  the  points  Fig.  28.— Images  of  a  luminous  point  in  a 
p       p  Q       Q      arp  pair  of  plane  mirrors  inclined  to  each 

1,        3,    •    •    •    v*3>    ^1  other  at  an  angle  of  60°. 

the  vertices  01  two  equal 

regular  polygons,  of  p  sides  each;  and  if  p  is  odd,  the  polygon 
P1P3  .  .  .  Q3Q1  will  have  one  of  its  corners  between  C  and 
D,  whereas  if  p  is  even,  one  of  the  corners  of  the  polygon 
SP2P4  .  .  .  Q4Q2  will  fall  between  C  and  D;  in  either  case 
this  vertex  is  the  position  of  the  last  image  of  both  series. 
Thus,  for  example,  if  7  =  60°  (Fig.  28),  then  p  =  3,  and  the 
two  polygons  are  the  equilateral  triangles  SP2Q2  and  P1P3Q1 
(orPAQx). 

The  toy  called  a  kaleidoscope,  devised  by  Sir  David 
Brewster  (1781-1868),  consists  essentially  of  two  long  nar- 
row strips  of  mirror-glass  inclined  to  each  other  at  an  angle 
of  60°  and  inclosed  in  a  cylindrical  tube.  One  end  of  the 
tube  is  closed  by  a  circular  piece  of  ground  glass  whereon 
are  loosely  disposed  a  lot  of  fragments  of  colored  glass  or 


48 


Mirrors,  Prisms  and  Lenses 


[§19 


beads,  and  at  the  other  end  of  the  tube  there  is  a  peep-hole. 
When  the  instrument  is  held  towards  the  light,  an  observer 
looking  in  it  will  see  an  exquisitely  beautiful  and  symmetrical 
pattern  formed  by  the  colored  objects  and  their  images,  the 
form  of  which  may  be  almost  endlessly  varied  by  revolving 
the  tube  around  its  axis  so  that  the  bits  of  glass  assume  new 


Fig.  29. — Path  of  ray  reflected  into  eye  from  a  pair  of  inclined 
mirrors. 

positions.    In  fact,  this  device  has  been  turned  to  practical 
use  in  making  designs  for  carpets  and  wall-papers. 

19.  Construction  of  the  Path  of  a  Ray  Reflected  into  the 
Eye  from  a  Pair  of  Inclined  Mirrors. — In  order  to  trace  the 
paths  of  the  rays  by  which  a  spectator  standing  in  front  of  a 
pair  of  inclined  mirrors  sees  the  image  of  a  luminous  point, 
it  is  convenient  to  assume,  for  the  sake  of  simplicity,  that 
the  eye  at  E  in  Fig.  29  lies  in  the  plane  of  the  paper.  The 
first  step  in  the  construction  of  the  path  of  the  ray  is  to  draw 
the  straight  line  from  the  given  image-point  to  the  eye, 
because  if  the  eye  sees  this  point  the  light  that  enters  the 


19] 


Inclined  Mirrors 


49 


eye  must  arrive  along  this  line.  If  this  line  does  not  cross 
the  mirror  in  which  the  image  is  produced,  this  particular 
image  will  not  be  visible  from  the  point  E.  Now  join  the 
point  where  this  line  meets  the  mirror  with  the  preceding 
image  in  the  same  series;  the  part  of  this  line  that  lies  be- 
tween the  two  mirrors  will  evidently  show  the  route  of  the 


Fig.  30. — Showing  how  an  eye  at  E  sees  the  images  of  a  lumi- 
nous point  S  in  a  rectangular  pair  of  plane  mirrors. 

light  before  its  last  reflection.  Proceeding  in  this  fashion 
from  one  mirror  to  the  other,  we  shall  trace  backwards  the 
zigzag  path  of  the  ray  until  we  arrive  finally  at  the  luminous 
source  at  S.  Consider,  for  example,  the  image  P3  formed 
in  the  mirror  OA  in  Fig.  29.  This  image  is  visible  to  the 
eye  at  E  because  the  straight  line  P3E  cuts  at  K  the  mirror 
OA.  If  J  and  H  designate  the  points  where  the  straight 
lines  KP2  and  JPi  meet  the  mirrors  OB  and  OA,  respectively, 
the  broken  line  SHJKE  will  represent  the  path  of  the  ray 
from  the  source  at  S  into  the  eye  at  E.    Fig.  30  shows  how 


50  Mirrors,  Prisms  and  Lenses  [§  20 

an  eye  at  E  in  front  of  two  perpendicular  plane  mirrors  can 
see  the  images  Pi,  P2,  Qi  and  Q2. 

20.  Rectangular  Combinations  of  Plane  Mirrors. — In  a 

rectangular  combination  of  two  plane  mirrors  (7  =  90°)  the 
image  formed  by  two  successive  reflections  will  be  in- 
verted in  the  principal  section  of  the  system,  but  in  any 
plane  at  right  angles  to  a  principal  section  the  image  will 
be  erect.  For  example,  if  an  object  is  placed  in  front  of  two 
vertical  plane  mirrors  at  right  angles  to  each  other,  the 
image  produced  by  two  reflections  will  have  the  same  posi- 
tion and  appearance  as  if  the  object  had  been  revolved  bodily 
through  an  angle  of  180°  about  a  vertical  axis  coinciding 
with  the  line  of  intersection  of  the  planes  of  the  mirrors,  as 
represented  in  Fig.  31.  In  this  case  the  image  remains  ver- 
tically erect,  whereas  it  is  horizontally  inverted.  On  the 
other  hand,  if  one  of  the  mirrors  is  vertical  and  the  other 
horizontal,  the  image  by  twofold  reflection  will  have  the 
same  position  and  appearance  as  if  the  object  had  been 
revolved  through  180°  around  a  horizontal  axis  coinciding 
with  the  line  of  intersection  of  the  two  mirrors  (Fig.  32); 
that  is,  the  image  now  will  be  upside  down  but  not  inverted 
horizontally. 

Therefore,  in  order  to  obtain  an  image  that  is  completely 
reversed  in  every  respect,  two  rectangular  combinations  of 
plane  mirrors  may  be  employed  with  their  principal  sections 
mutually  at  right  angles,  so  disposed  that  the  rays  coming 
from  the  object  will  be  reflected  in  succession  from  each  of 
the  four  plane  surfaces.  An  auxiliary  system  of  this  descrip- 
tion is  sometimes  used  in  connection  with  an  optical  instru- 
ment for  the  purpose  of  rectifying  the  image  which  otherwise 
would  be  seen  inverted.  A  rectifying  device  depending  on  this 
principle  is  the  so-called  Porro  prism-system  (1852),  utilized 
by  Abbe  (1840-1905)  in  the  design  of  the  famous  prism  binocu- 
lar telescope  or  field-glasses  (c.  1883) .  A  sketch  of  the  arrange- 
ment is  shown  in  Fig.  33.  Two  rectangular  prisms  are  placed 
in  the  tube  of  the  instrument,  between  the  objective  and  the 


Fig.  32. — Image  of  an  object  in  a  rectangular  pair  of  plane  mirrors  (from 
actual  photograph) ;  showing  how  the  last  image  is  obtained  by  rotating 
the  object  through  180*  around  the  line  of  intersection  of  the  mirrors. 
One  mirror  vertical,  the  other  horizontal. 


20] 


Porro  Prism  System 


51 


ocular,  with  their  principal  sections  at  right  angles  to  each 
other.  The  axial  ray,  after  traversing  the  objective,  crosses 
normally  the  hypothenuse-face  of  the  first  prism  and  is 
totally  reflected  (see  §  36),  in  the  plane  of  a  principal  section, 
at  each  of  its  two  per- 
pendicular faces  so  as  to 
emerge  from  the  hypoth- 
enuse-face in  a  direction 
precisely  opposite  to  that 
which  it  had  when  it  first 
crossed  this  surface.  This 
ray  now  undergoes  a  simi- 
lar cycle  of  experiences  in 
a  principal  section  of  the 
second  prism,  and  finally 
emerges  from  this  prism 

in  the  Same  direction  as  FlG-  33.— Porro  prism-system  in  prism 
•  ,     i     j        i  .,  ,     .i  binocular  field  glasses. 

it  had  when  it  met  the 

first  prism.  A  ray  parallel  to  the  axial  ray  and  lying  above 
a  horizontal  plane  containing  the  axis  will  be  converted  by 
virtue  of  the  two  reflections  in  the  first  prism  into  a  ray 
whose  path  lies  below  this  plane;  and,  similarly,  a  ray  par- 
allel to  the  axis  and  lying  on  one  side  of  a  vertical  plane 
containing  the  axis  will,  in  consequence  of  the  two  reflec- 
tions within  the  second  prism,  be  converted  into  a  ray  whose 
path  lies  on  the  opposite  side  of  this  vertical  plane.  Thus, 
the  combined  effect  of  the  two  reflecting  prisms  together 
will  be  to  reverse  completely  the  position  of  the  ray  with 
respect  to  the  horizontal  and  vertical  meridian  planes,  so 
that  the  ray  will  issue  from  the  system  on  opposite  sides  of 
both  these  planes.  If  the  system  of  prisms  were  removed, 
the  image  in  the  instrument  would  appear  inverted,  but  by 
interposing  the  prisms  in  this  fashion  the  image  will  be 
rectified  and  oriented  exactly  in  the  same  way  as  the  object; 
which  in  the  case  of  many  optical  instruments  is  an  essential 
consideration. 


52 


Mirrors,  Prisms  and  Lenses 


[§21 


21.  Applications  of  the  Plane  Mirror. — It  is  hardly- 
necessary  to  say  that  the  plane  mirror  for  various  pur- 
poses has  been  in  use  among  civilized  peoples  of  all  ages; 
although  the  use  of  mirrors  as  articles  of  household  fur- 
niture and  decoration  does  not  go  back  farther  than  the 
early  part  of  the  16th  century.  By  a  combination  of  two 
or  more  plane  mirrors  a  lady  can  arrange  the   back   of 


Fig.  34. — Porte  lumi&re. 


her  dress  and  in  fact  see  herself  as  others  see  her.  With 
the  aid  of  a  mirror  or  combination  of  mirrors  many  in- 
genious " magical  effects"  are  produced  in  theaters.  The 
plane  mirror  also  constitutes  an  essential  part  of  numerous 
useful  scientific  instruments  in  some  of  which  its  only  duty 
is  to  alter  the  course  of  a  beam  of  light,  whereas  in  various 
forms  of  goniometrical  instruments  and  contrivances  for  de- 
termining an  angular  magnitude  that  is  not  easily  measured 


§  22]  Porte  Lumiere  53 

directly  the  angle  in  question  is  ascertained  indirectly  by 
observing  the  angle  turned  through  by  a  ray  of  light  which 
is  reflected  from  a  plane  mirror. 

22.  Porte  Lumiere  and  Heliostat. — As  good  an  illustra- 
tion as  can  be  given  of  the  use  of  a  plane  mirror  for  chang- 
ing the  direction  of  a  beam  of  sunlight  is  afforded  by  the 


Fig.  35. — Heliostat. 

porte  lumiere  (Fig.  34),  which  consists  essentially  of  a  plane 
mirror  ingeniously  mounted  so  as  to  be  capable  of  rotation 
about  two  rectangular  axes,  whereby  it  may  be  readily  ad- 
justed in  any  desired  azimuth  and  reflect  a  beam  of  sun- 
light through  a  suitable  opening  in  the  wall  of  the  building 
to  any  part  of  the  interior  of  the  room. 

However,  owing  to  the  diurnal  movement  of  the  sun, 


54  Mirrors,  Prisms  and  Lenses  [§  22 

a  continual  adjustment  of  the  mirror  is  necessary  in  order 
to  keep  the  spot  of  light  for  any  length  of  time  at  the  place 
in  the  room  where  it  is  needed,  and  sometimes  this  manipu- 
lation is  very  inconvenient  and  annoying,  especially  in  the 
case  of  a  laboratory  experiment  extending  perhaps  over 
a  considerable  part  of  a  day.  Thus,  for  example,  in  study- 
ing the  solar  spectrum  it  is  often  desirable  to  illuminate  the 
slit  in  the  collimator  tube  of  the  spectrometer  for  hours  at 
a  time.  For  such  purposes  it  is  better  to  use  a  heliostat 
(Fig.  35),  which  is  contrived  so  that  the  plane  mirror  is  con- 
tinuously revolved  by  clockwork  around  an  axis  parallel 
to  the  earth's  axis  so  as  to  preserve  always  the  same  relative 
position  with  respect  to  the  sun  in  its  apparent  diurnal 
motion  in  the  sky.     The  mirror  can  also  be  turned  about 

a  horizontal  axis,  and  it  has  first 
to  be  adjusted  about  this  axis  so 
that  the  rays  of  the  sun  are  re- 
flected towards  the  north  pole 
of  the  celestial  sphere,  that  is, 
parallel  to  the  axis  of  the  earth. 
The  mirror  being  adjusted  at 
this  angle,  which  will  depend  on 
the  declination  of  the  sun  above 
Fig.  36.-PrinciPle  of  heliostat.    or  below^  the  celestial  equator, 

and  turning  at  the  rate  of  15° 
per  hour  around  an  axis  parallel  to  the  axis  of  rotation  of 
the  earth,  it  is  evident  that  the  rays  of  the  sun  will  continue 
to  be  reflected  constantly  in  the  same  direction.  Suppose, 
for  example,  that  the  mirror  is  adjusted  in  the  position 
ZZ  (Fig.  36)  so  that  the  ray  SB  coming  from  the  sun  at  S  is 
reflected  at  B  in  the  direction  BP  parallel  to  the  axis  of  the 
earth  and  therefore  parallel  to  the  axis  of  rotation  AB  of  the 
mirror.  If  the  polar  distance  of  the  sun  is  denoted  by 
2a  =  ZPBS,  and  if  the  angle  between  the  normal  to  the 
mirror  and  the  axis  of  rotation  is  denoted  by  r),  then,  evi- 
dently, rj  =  a.     If  the  sun's  declination  on  a  certain  day 


§  23]  Measurement  of  Angle  of  Prism  55 

is  +10°,  then  2  a  =  90°- 10°  =  80°,  and  77  =  40°.  If,  on  the 
other  hand,  the  sun  is  10°  below  the  equator,  2a=100°  and 
77  =  50°. 

The  heliostat  is  provided  also  with  a  fixed  mirror  which 
reflects  the  rays  from  the  rotating  mirror  in  a  definite  di- 
rection, as  desired,  usually  in  a  horizontal  direction  into 
the  room  where  the  sunlight  is  to  be  used.  Generally,  the 
instrument  is  mounted  on  a  permanent  ledge  outside  the 
window;  sometimes  it  is  placed  on  the  roof  of  the  building 
and  the  fixed  mirror  adjusted  so  as  to  send  the  sun's  rays 
down  a  vertical  tube  at 
the  bottom  of  which  there 
is  another  mirror  placed 

at  an  angle  of  45°  with  S^~ — '     ^^^C^ 

the  vertical  where  the 
rays  are  once  more  re- 
flected so  that  the  beam 
of  sunlight  which  enters 
the  room  will  be  hori- 
zontal. 

23.  Measurement  of 
the  Angle  of  a  Prism. — 
Another  laboratory  ap- 
plication of  the  principle 
of  a  plane  mirror  is  seen 
in  the  method  of  using  a 

goniometer    to    ascertain     FlG-  37-Measurement  of  angle  of  prism. 

the  dihedral  angle  between  two  plane  faces  of  a  glass  prism 
(§  48) .  The  angle  that  is  actually  measured  by  the  goniom- 
eter is  the  angular  distance  between  the  images  of  a  distant 
object  as  seen  in  the  two  faces  of  the  prism.  Parallel  rays 
coming  from  a  far-off  source  at  S  (Fig.  37)  and  incident  on 
the  two  faces  of  the  prism  that  meet  in  the  edge  V  are  re- 
flected as  shown  in  the  diagram,  and  the  angle  between  the 
two  directions  of  the  reflected  rays  is  obviously  equal  to 
twice  the  dihedral  angle  /3. 


56 


Mirrors,  Prisms  and  Lenses 


[§24 


24.  Measure  of  Angular  Deflections  by  Mirror  and 
Scale. — The  angular  rotation  of  a  body,  for  example,  the 
deflection  of  the  magnetic  needle  of  a  galvanometer,  is  fre- 
quently measured  by  attaching  a  mirror  to  the  rotating 
body  from  which  a  beam  of  light  is  reflected.  This  reflected 
light  acts  as  a  long  weightless  pointer  whereby  the  actual 


Fig.  38. — Mirror,  telescope  and  scale  for  measure- 
ment of  angles. 

movement  of  the  body  can  be  magnified  to  any  extent  with- 
out in  the  least  affecting  the  sensitiveness  of  the  apparatus. 

In  Fig.  38  the  plane  mirror  which  is  capable  of  rotation 
about  an  axis  perpendicular  at  A  to  the  plane  of  the  paper 
is  represented  in  its  initial  position  by  the  line-segment 
marked  1.  The  straight  line  MN  in  front  of  the  mirror  and 
at  a  known  distance  (c?  =  AB)  from  it  represents  a  scale 
graduated  in  equal  divisions.  An  eye  at  E  looking  through 
a  telescope  pointed  towards  A  will  see  the  image  in  the  mirror 
of  the  scale-division  at  S,  the  so-called  "  zero-reading,"  be- 
cause the  light  from  S  incident  at  A  on  the  mirror  in  the 
position  1  (" equilibrium-position")  is  reflected  along  AE 
into  the  eye  at  E.  If  now  the  mirror  is  turned  through  an 
angle  6  into  the  position  marked  2,  another  scale-division 
will  come  into  the  field  of  view  of  the  telescope  and  coin- 


§  24j  Mirror  and  Scale  57 

cide  with  the  cross-hair  in  the  eye-piece.  If  this  scale- 
division  corresponds  to  the  point  marked  P,  it  is  the  light 
that  comes  along  PA  that  is  now  reflected  along  AE  into 
the  eye  at  E;  and  evidently,  according  to  §  13,  Z  PAS  =  2  0. 
In  making  a  measurement  by  this  method,  the  three  points 
designated  by  S,  B  and  E  are  generally  adjusted  so  as  to 
be  very  near  together,  if  not  actually  coincident.  If  they 
were  coincident,  the  planes  of  the  mirror  and  scale  would 
be  parallel,  and  the  axis  of  the  telescope  would  coincide 
with  the  straight  line  BA  perpendicular  to  the  scale  at  B. 
But  in  any  case  the  Z  B AS  =  e  will  be  a  constant,  depending 
partly  on  the  initial  position  of  the  mirror  and  partly  on 
the  direction  of  the  axis  of  the  telescope;  thus, 

tan  €  =  a/d, 
where  a  =  BS.    If,  therefore,  we  put  x  =  SP,  we  have: 

x 

-j  =  tan  ( e+  2  6)—  tan e  ; 

whence,  since  the  value  of  x  can  be  read  off  on  the  scale,  it 
will  be  easy  to  calculate  the  value  of  the  required  angle  6 
through  which  the  mirror  has  been  turned.  In  many  cases 
where  this  method  is  employed  the  angles  denoted  by  6  and 
€  are  both  so  small  that  there  will  be  little  error  in  sub- 
stituting the  angles  themselves  in  place  of  their  tangents. 
Under  these  circumstances  the  above  formula  will  be  greatly 
simplified,  for  the  angle  e  will  disappear  entirely,  and  we 
shall  obtain  : 

0      x 

e=Td> 

where,  however,  it  must  be  noted  that  this  expression  gives 
the  value  of  the  angle  6  in  radians.  The  value  of  6  in  de- 
grees is  found  by  multiplying  the  right-hand  side  of  this 
formula  by  180/ 7T,  so  that  we  obtain: 

u  =  — j  degrees. 
Tr.a 

A  lamp  and  scale  is  sometimes  used  instead  of  a  telescope 

and  scale,  the  light  of  the  lamp  being  reflected  from  the 


58 


Mirrors,  Prisms  and  Lenses 


[§25 


mirror  on  to  the  scale  which  is  usually  made  of  translucent 
glass,  so  that  it  is  easy  to  read  the  position  of  the  spot  of 
light. 

25.  Hadley's  Sextant. — Another  instrument  which  utilizes 
the  principle  of  §  17  is  the  sextant,  which  is  employed  for 


Fig.  39. — Principle  of  sextant. 

measuring  the  angular  distance  between  two  bodies,  for  ex- 
ample, the  altitude  of  the  sun  above  the  sea-horizon.  The 
plan  and  essential  features  of  this  apparatus  are  shown  in 
Fig.  39.  At  the  center  A  of  a  graduated  circular  arc  ON 
a  small  mirror  is  set  up  in  a  plane  at  right  angles  to  that  of 
the  arc.  This  mirror  can  be  turned  about  an  axis  perpendic- 
ular to  the  plane  of  the  paper  and  passing  through  A.  Rig- 
idly connected  to  this  mirror  and  turning  with  it  is  a  long 
solid  arm  AP  whose  other  end  P,  provided  with  a  vernier 
scale,  moves  over  the  arc  ON,  whereby  the  angle  through 
which  the  mirror  turns  can  be  accurately  measured.  A  little 
beyond  the  extremity  N  of  the  graduated  part  of  the  arc, 
a  second  mirror  B  is  erected  facing  the  first  mirror.  The 
plane  of  this  mirror  is  likewise  perpendicular  to  that  of  the 
circle,  but  from  the  upper  half  of  it  the  silver  has  been 
removed,  so  that  this  portion  of  the  mirror  B  is  transparent. 
Moreover,  this  mirror  is  fixed  with  respect  to  the  instru- 
ment.    An  observer  looking  through  a  peep-hole  or  tele- 


§  25]  Mirror  Sextant  59 

scope  attached  to  the  instrument  towards  the  mirror  B  may- 
see  a  distant  object  through  the  upper  transparent  part  of 
this  mirror,  and  at  the  same  time  he  may  also  see  just  below 
it  the  image  of  a  second  object  reflected  in  the  lower  half 
of  the  glass.  When  the  planes  of  the  two  mirrors  A  and  B 
are  parallel,  the  zero-mark  of  the  vernier  on  the  movable 
arm  coincides  with  the  zero-mark  0  of  the  graduated  arc. 
Suppose,  for  example,  that  when  the  two  mirrors  are  par- 
allel to  each  other,  the  instrument  is  pointed  at  a  distant 


Fig.  40. — Model  of  mirror  sextant. 

object,  say,  a  star  at  Si,  which  will  be  seen  directly  through 
the  upper  half  of  the  fixed  mirror  B.  At  the  same  time  the 
observer  will  see  an  image  of  the  object  Si  by  rays  which 
have  been  reflected  from  the  mirror  A  to  the  mirror  B  and 
thence  into  the  eye  at  E;  for  if  the  two  mirrors  are  parallel, 
the  direction  of  a  ray  after  two  reflections  will  be  the  same 
as  its  initial  direction.  If  now  the  mirror  A  is  turned  until 
the  image  of  another  object  at  S2  comes  into  the  field  of 
vision,  the  two  objects  Si  and  S2  will  be  seen  simultaneously, 
for  with  the  mirror  at  A  in  this  new  position  the  incident  ray 
S2A  will  be  the  ray  that  is  reflected  from  A  to  B  and  thence, 
as  before,  into  the  eye  at  E.  Moreover,  since  the  angle  between 
the  original  direction  S2A  of  this  ray  and  its  final  direction 


60  Mirrors,  Prisms  and  Lenses  [Ch.  II 

SXA  is  equal  to  double  the  angle  between  the  planes  of  the 
two  mirrors,  that  is,  is  equal  to  2  6,  where  8  =  Z  OAP,  the 
angular  distance  between  the  objects  at  Si  and  S2  must  be 
equal  to  2  6,  that  is,  Z  SiAS2  =  2  0.  In  order  to  save  trouble 
in  making  the  readings,  half-degrees  on  the  graduated  arc 
are  reckoned  as  degrees,  so  that  the  value  of  the  angle  2  6  is 
read  directly  on  the  scale.  As  the  angular  distance  between 
the  objects  will  seldom  exceed  120°,  and  since,  in  fact,  the 
method  is  not  very  accurate  for  angles  greater  than  this, 
the  actual  length  of  the  graduated  arc  need  not  be  greater 
than  about  60°  or  one-sixth  of  the  circumference;  whence 
the  name  sextant  is  derived. 

A  simple  model  of  a  mirror  sextant  is  shown  in  Fig.  40. 
For  accurate  measurements  the  instrument  is  made  of  metal 
with  a  scale  etched  on  a  silver  strip.  Moreover,  a  telescope 
is  used  instead  of  a  peep-hole;  so  that  with  a  fine  sextant  it 
is  comparatively  easy  to  measure  the  angular  distance  be- 
tween two  points  to  within  one-half  minute  of  arc.  One 
great  advantage  of  this  instrument  is  its  portability,  and 
since  it  does  not  have  to  be  mounted  on  a  stand,  it  is  very 
serviceable  on  shipboard  for  measurements  of  altitude  and 
determinations  of  latitude,  etc. 

PROBLEMS 

1.  The  top  of  a  vertical  plane  mirror  2  feet  high  is  4  feet 
from  the  floor.  The  eye  of  a  person  standing  in  front  of  the 
mirror  is  6  feet  from  the  floor  and  3  feet  from  the  mirror. 
What  are  the  distances  from  the  wall  on  which  the  mirror 
hangs  of  the  farthest  and  nearest  points  on  the  floor  that 
are  visible  in  the  mirror?  Ans.  6  ft. ;  18  in. 

2.  A  ray  of  light  is  reflected  at  a  plane  mirror.  Show  that 
if  the  mirror  is  turned  through  an  angle  6,  the  reflected  ray 
will  be  turned  through  an  angle  2  0. 

3.  Show  that  the  deviation  of  a  ray  reflected  once  at  each 
of  two  plane  mirrors  is  equal  to  twice  the  angle  between  the 
mirrors. 


Ch.  II]  Problems  61 

4.  If  a  plane  mirror  is  turned  through  an  angle  of  5°, 
what  is  the  deflection  indicated  by  the  reading  on  a  straight 
scale  100  cm.  from  the  mirror?  Ans.    About  17.6  cm. 

5.  Find  the  angle  turned  through  by  the  mirror  when  the 
deflection  on  the  scale  in  the  preceding  example  is  10  cm.? 

Ans.  2°  52'. 

6.  What  must  be  the  length  of  a  vertical  plane  mirror  in 
order  that  a  man  standing  in  front  of  it  may  see  a  full  length 
image  of  himself?  Ans.  The  length  of  the  mirror  must  be 
equal  to  half  the  height  of  the  man. 

7.  Show  that  a  plane  mirror  bisects  at  right  angles  the 
line  joining  an  object-point  with  its  image. 

8.  A  ray  of  light  proceeding  from  a  point  A  is  reflected 
from  a  plane  mirror  to  a  point  B.  Show  that  the  path  pur- 
sued by  the  light  is  shorter  than  any  other  path  from  A  to 
the  mirror  and  thence  to  B. 

9.  Give  Huygens's  construction,  (1)  for  the  reflection 
of  a  p'ane  wave  at  a  plane  mirror,  and  (2)  for  the  reflection 
of  a  spherical  wave  at  a  plane  mirror. 

10.  Explain  clearly  how  to  determine  the  limits  of  the 
field  of  view  in  a  plane  mirror  for  a  given  position  of  the  eye 
of  the  spectator. 

11.  A  candle  is  placed  between  two  parallel  plane  mirrors. 
Show  how  an  observer  can  see  the  image  of  the  candle  pro- 
duced by  rays  which  have  been  twice  reflected  at  one  mirror 
and  three  times  at  the  other.  Draw  accurate  diagram  show- 
ing the  paths  of  the  rays,  the  positions  of  the  images,  etc.; 
and  give  clear  explanation  of  the  figure. 

12.  OA  and  OB  are  two  plane  mirrors  inclined  at  an  angle 
of  15°,  and  P  is  a  point  in  OA.  At  what  angle  must  a  ray 
of  light  from  P  be  incident  on  OB  in  order  that  after  three 
reflections  it  may  be  parallel  to  OA?  Ans.  45°. 

13.  Show  that  the  image  of  a  luminous  object  placed 
between  two  plane  mirrors  all  He  on  a  circle. 

14.  Show  how  by  means  of  two  plane  mirrors  a  man 
standing  in  front  of  one  of  them  can  see  the  image  of  the  back 


62  Mirrors,  Prisms  and  Lenses  [Ch.  II 

of  his  head.    Trace  the  course  of  the  rays  from  the  back  of 
his  head  into  his  eye  and  explain  clearly. 

15.  Show  by  a  diagram,  with  clear  explanations,  how 
one  sees  the  image  of  an  arrow  in  a  plane  mirror. 

16.  Construct  the  image  of  an  arrow  formed  by  two  re- 
flections in  a  pair  of  inclined  mirrors,  (1)  when  the  mirrors 
are  at  right  angles,  and  (2)  when  the  angle  between  the 
mirrors  is  60°. 

17.  Show  how  a  horizontal  shadow  of  a  vertical  rod  can 
be  thrown  on  a  vertical  screen  by  a  point-source  of  light  with 
the  aid  of  a  plane  mirror.    Draw  a  diagram. 

18.  An  object  is  placed  between  two  plane  mirrors  in- 
clined at  an  angle  of  45°.  Show  by  a  figure  how  a  spectator 
may  see  the  image  after  four  successive  reflections.  Give 
clear  explanation. 

19.  Two  plane  mirrors  are  inclined  at  an  angle  of  50°. 
Show  that  there  will  be  7  or  8  images  of  a  luminous  point 
placed  between  them,  according  as  its  angular  distance  from 
the  nearer  mirror  is  or  is  not  less  than  20°. 

20.  Find  the  number  of  images  formed  when  a  bright  point 
is  placed  between  two  plane  mirrors  inclined  to  each  other 
at  an  angle  of  25°.  Ans.  14  or  15  images  according  as  the 
angular  distance  of  the  luminous  point  from  the  nearer  mir- 
ror is  or  is  not  less  than  5°. 

21.  A  luminous  object  moves  about  between  two  plane 
mirrors,  which  are  inclined  at  an  angle  of  27°.  Prove  that 
at  any  moment  the  number  of  images  is  13  or  14  according 
as  the  angular  distance  of  the  luminous  point  from  the  nearer 
mirror  is  or  is  not  less  than  9°. 

22.  The  angle  between  a  pair  of  inclined  mirrors  is  80°. 
Find  the  position  of  an  object  which  is  reproduced  by  5 
images.  Ans.  The  object  must  be  less  than  20°  from  the 
nearer  mirror. 

23.  Describe  a  sextant  with  the  aid  of  a  diagram,  and  ex- 
plain its  use. 

24.  Describe  and  explain  the  heliostat. 


Ch.  II]  Problems  63 

25.  Construct  the  image  of  the  capital  letter  F  as  seen  in 
a  plane  mirror. 

26.  When  a  candle-flame  is  placed  in  front  of  a  screen 
with  a  pin-hole  opening,  an  image  of  the  flame  is  formed  on 
a  second  screen  placed  parallel  to  the  first.  But  if  the  second 
screen  is  replaced  by  a  plane  mirror,  the  image  will  be  formed 
on  the  back  of  the  first  screen.    Explain  how  this  happens. 

27.  Explain  clearly  (with  diagram,  formula,  etc.)  the 
method  of  using  a  mirror  and  scale  for  measurement  of 
angles. 

28.  Describe  how  the  dihedral  angle  of  a  glass  prism  is 
measured  on  a  goniometer-circle. 


CHAPTER  III 


REFRACTION  OF  LIGHT 


Fig.  41. — Coin  at  bottom  of  bowl  rendered 
visible  by  filling  bowl  with  water. 


26.  Passage  of  Light  from  One  Medium  to  Another. — 

Hardly  any  one  can  have  failed  to  observe  that  the  course  of 
light  in  passing  obliquely  from  water  to  air  is  abruptly  changed 
at  the  surface  of  the  water.  For  example,  if  a  coin  is  placed 
at  A  in  the  bottom  of  a  china  bowl  (Fig.  41),  and  if  the  eye 

is  adjusted  at  a  point  C 
so  that  the  coin  is  hid 
from  view  by  the  side  of 
the  vessel,  then,  without 
altering  the  position  of 
the  eye,  the  coin  can  be 
made  visible  merely  by 
pouring  water  in  the 
bowl  up  to  a  certain  level. 
The  broken  line  ACB  illustrates  how  a  ray  proceeding  from 
A  may  be  bent  at  the  surface  of  the  water  so  as  to  pass  over 
the  edge  of  the  bowl  and  enter  the  eye  at  C.  It  is  true  the 
coin  will  will  not  appear  to  be  at  A  but  at  a  point  A'  nearer 
the  surface  of  the  water  and  displaced  a  little  sideways  to- 
wards the  eye,  because  the  rays  that  come  to  the  eye  inter- 
sect at  this  point  A'  (§  42).  A  clear  pool  of  water  seems  to 
be  shallower  than  it  really  is,  and  this  illusion  is  greater  in 
proportion  as  the  line  of  sight  is  more  oblique,  so  that  bright 
objects  at  the  bottom  of  the  pool  appear  to  be  crowded  to- 
gether towards  the  surface.  When  a  stick  is  partly  immersed 
in  water,  the  part  under  water  appears  to  be  bent  up  to- 
wards the  surface  (§  42). 

This  bending  of  the  rays  which  takes  place  when  light 
crosses  the  boundary  between  two  media  is  called  refraction. 

64 


Fig.  42. — Law  of  Refraction. 


§  27]  Law  of  Refraction  65 

The  path  of  a  beam  of  sunlight  through  water  can  easily 
be  shown  by  mixing  a  little  milk  in  the  water  or  by  stir- 
ring in  it  a  minute  quantity  of  chalk-dust,  while  a  puff  of 
smoke  will  at  once  reveal  the  track  of  the  beam  in  the  air, 
so  that  the  phenomena  of 
refraction  can  readily  be 
exhibited  to  the  eye.  In 
every  case  it  will  be  found 
that  the  ray  is  bent  farther 
from  the  incidence-normal 
in  the  rarer  or  less  dense 
medium  (see  §  30) ;  and 
here  also,  as  in  the  case 
of  reflection,  there  is  a 
perfectly  definite  connec- 
tion between  the  direction  of  the  incident  ray  and  that  of 
the  corresponding  refracted  ray. 

27.  Law  of  Refraction. — In  Fig.  42  the  straight  line  AB 
represents  the  path  of  a  ray  incident  at  the  point  B  on  a 
smooth  refracting  surface  separating  two  media  which  for 
the  present  will  be  designated  by  the  letters  a  and  b.  The 
straight  line  NN'  drawn  perpendicular  to  the  plane  which 
is  tangent  to  the  refracting  surface  at  B  represents  the 
incidence-normal;  and  the  plane  of  the  paper  which  con- 
tains the  incident  ray  and  the  incidence-normal  is  the  plane 
of  incidence,  as  already  defined  (§  13).  The  line  ZZ  repre- 
sents the  trace  of  the  refracting  surface  in  this  plane.  And, 
finally,  the  path  of  the  refracted  ray  is  shown  by  the  straight 
line  BC.  The  angles  of  incidence  and  refraction  are  defined 
to  be  the  acute  angles  through  which  the  incidence-normal 
has  to  be  turned  in  order  to  bring  it  into  coincidence  with 
the  incident  and  refracted  rays,  respectively.  Thus,  if 
these  angles  are  denoted  by  a,  a',  then 

a  =  ZNBA,  a'  =  ZN'BC. 

In  the  figure  as  drawn  the  angle  a  is  represented  as  greater 
than  the  angle  a',  so  that,  according  to  the  statement  at 


66  Mirrors,  Prisms  and  Lenses  [§  27 

the  end  of  §  26,  the  medium  a  is  less  dense  or  "rarer"  than 
the  medium  b. 

Before  stating  the  relation  which  is  found  to  exist  be- 
tween the  angles  a  and  a! ,  it  is  necessary  to  allude  to 
Newton's  great  discovery  that  sunlight  and  indeed  so- 
called  " white  light"  of  any  kind,  as,  for  example,  the  light 
of  an  arc  lamp,  is  composed  of  light  of  an  innumerable 
variety  of  colors  (see  Chapter  XIV),  as  may  be  shown  by 
passing  a  beam  of  sunlight  through  a  glass  prism,  whereby 
it  will  be  seen  that  white  light  is  a  mixture  of  all  the  colors 
of  the  spectrum  in  all  their  infinite  varieties  of  hues.  On 
the  other  hand,  monochro?natic  light,  as  it  is  called,  is  light 
of  some  one  definite  color,  as,  for  example,  the  yellow  light 
emitted  by  a  sodium  flame  which  may  be  obtained  by 
burning  common  salt  in  the  flame  of  a  Bunsen  burner. 
In  geometrical  optics,  unless  we  are  specially  concerned 
with  the  investigation  of  color-phenomena  (as  in  Chapter 
XIV),  it  is  nearly  always  tacitly  assumed  that  the  source 
of  the  light  is  monochromatic. 

The  law  of  refraction,  as  found  by  experiment,  may  now 
be  stated  as  follows : 

The  refracted  ray  lies  in  the  plane  of  incidence  on  the  op- 
posite side  of  the  normal  in  the  second  medium  from  the  incident 
ray  in  the  first  medium;  and  the  sines  of  the  angles  of  incidence 
and  refraction  are  to  each  other  in  a  constant  ratio,  the  value  of 
which  depends  only  on  the  nature  of  the  two  media  and  on  the 
color  (or  wave-length)  of  the  light. 

This  constant  ratio,  denoted  by  the  symbol  nah,  is  called 
the  relative  index  of  refraction  from  the  first  medium  (a)  to 
the  second  medium  (b)  for  light  of  the  given  color;  thus, 

sin  a 

£^~%; 

the  value  of  this  constant,  as  a  rule,  being  greatest  for  violet 
and  least  for  red  light,  so  that  the  violet  rays  are  the  most 
" refrangible"  of  all.  When  light  is  refracted  from  air  (a)  to 
water  (w)  the  relative  index  of  refraction  is,  approximately, 


§  28]  Proof  of  Law  of  Refraction  67 

naw=4/3,  and  hence  under  these  circumstances  sina/  = 
%  sin  a.  Although  there  are  many  different  varieties  of 
optical  glass,  for  rough  calculations  the  value  of  the  rela- 
tive index  of  refraction  from  air  (a)  to  glass  (g)  may  be 
taken  as  nag  =  3/2;  which  means  that  the  sine  of  the  angle 
which  the  ray  makes  with  the  normal  in  glass  is  about  two- 
thirds  of  the  sine  of  the  angle  which  the  corresponding  ray 
makes  with  the  normal  in  air. 

Although  the  law  of  refraction  is  quite  simple,  it  some- 
how eluded  discovery  until  early  in  the  seventeenth  century 
when  the  true  relation  between  the  angle  of  refraction  and 
the  angle  of  incidence  was  first  ascertained  by  Willebrord 
Snell  (1591-1626)  or  Snellius,  of  Leyden,  and  the  law  is, 
therefore,  often  referred  to  as  Snell' s  Law  of  Refraction. 
The  law  was  first  published  by  the  French  philosopher 
Descartes  (1596-1650),  who  had  probably  seen  Snell's 
papers,  although  he  does  not  allude  to  him  by  name. 

28.  Experimental  Proof  of  the  Law  of  Refraction. — The 
relation  between  the  angles  of  incidence  and  refraction  can 
be  very  strikingly  exhibited  with  the  aid  of  the  optical  disk 
that  was  mentioned  in  §  13  in  connection  with  a  lecture- 
table  experiment  for  verifying  the  law  of  reflection  of  light. 
The  vertical  ground  glass  disk  is  adjusted  in  the  track  of 
a  narrow  beam  of  sunlight  (or  parallel  rays  from  a  lantern) 
in  such  a  position  that  the  path  of  the  light  is  shown  by  a 
band  of  light  crossing  the  face  of  the  disk  along  one  of  its 
diameters.  The  glass  body  through  which  the  light  is  re- 
fracted has  the  form  of  a  semicylinder,  the  two  plane  par- 
allel sides  being  ground  rough  so  as  to  be  more  or  less  opaque, 
whereas  the  curved  surface  and  the  diametral  plane  face 
are  both  highly  polished.  This  half -disk  has  a  radius  of 
about  2  inches  and  is  about  one-half  inch  thick  or  more.  It 
can  be  fastened  against  the  vertical  face  of  the  optical  disk 
with  its  axis  horizontal  and  coinciding  with  the  axis  of  rotation 
of  the  disk,  as  represented  in  the  diagram  Fig.  43.  If  this 
adjustment  is  made,  and  the  disk  turned  so  that  the  inci- 


68  Mirrors,  Prisms  and  Lenses  [§  28 

dent  ray  AB  meets  the  polished  plane  face  of  the  glass  body 
at  its  center  B,  the  refracted  ray  BC  will  proceed  through 
the  glass  along  a  radius  of  the  semicylinder,  and  therefore 
meeting  the  curved  surface  normally,  it  will  emerge  again 

into  the  air  without  being 
further  deviated.  The  di- 
ameter NN'  which  is 
marked  on  the  face  of  the 
optical  disk  is  normal  to 
the  plane  surface  of  the 
glass  body,  and  if  from  the 
points  A  and  C  where  the 
incident  and  refracted  rays 
cross  the  circumference  of 
the  disk  perpendiculars  are 

Fig.  43. — Optical    Disk    used    to   verify    i    ,      *  -m     ^_      ,1         «^«^„i 
law  of  refraction.  let     fal1     0n     the     normal 

NN',  the  lengths  of  these 
perpendiculars  AX  and  CY  will  be  proportional  to  the  sines 
of  the  angles  of  incidence  and  refraction  NBA  and  N'BC, 
respectievly.  Now  it  will  be  found  that,  no  matter  how  we 
turn  the  disk,  the  perpendicular  AX  will  always  be  about 
one-and-a-half  times  as  long  as  the  perpendicular  CY.  If 
we  substitute  for  the  half -disk  of  solid  glass  a  hollow  vessel 
of  the  same  form  and  size  with  thin  glass  walls,  and  if  we  fill 
this  vessel  with  water,  we  shall  find  now  that  the  length  of 
the  perpendicular  AX  will  always  be  about  one-and-one- 
third  times  that  of  the  perpendicular  CY,  because  the  relative 
index  of  refraction  from  air  to  water  is  4/3,  as  above  stated. 
But  the  best  proof  of  the  law  of  the  refraction  of  fight 
is  to  be  found  in  the  fact  that  this  law  is  at  the  basis  of  the 
theory  and  construction  of  nearly  all  optical  instruments, 
and  it  has  been  subjected,  therefore,  to  the  most  searching 
tests.  The  law  of  refraction  may  also  be  regarded  as  com- 
pletely verified  by  the  methods  that  are  employed  in  the 
determination  of  the  indices  of  refraction  of  transparent 
bodies,  solid,  liquid  and  gaseous;  which  are  described  in 


§  29]  Reversibility  of  Light  Path  69 

treatises  on  experimental  optics  usually  under  the  title  of 
"refractometry." 

29.  Reversibility  of  the  Light  Path. — When  a  ray  of  light 
AB  is  reflected  at  B  in  the  direction  BD,  a  plane  mirror 
placed  at  D  at  right  angles  to  BD  will  turn  the  reflected 
ray  back  on  itself;  arriving  again  at  B,  the  light  will  ob- 
viously be  reflected  there  so  as  to  return  finally  to  the  point 
A  where  it  started.  This  is  a  simple  instance  of  a  general 
law  of  optics  known  as  the  principle  of  the  reversibility  of 
the  light  path.  Experiment  shows  that  the  same  rule  holds 
likewise  in  the  case  of  the  refraction  of  light,  and  that  if 
ABC  is  the  route  pursued  by  light  in  going  from  a  point 
A  in  one  medium  to  a  point  C  in  an  adjoining  medium  by 
way  of  the  incidence-point  B,  and  if  then  the  light  is  re- 
versed by  some  means  so  as  to  be  started  back  along  the 
path  CB,  it  will  be  refracted  at  B  into  the  first  medium 
along  the  path  BA.  And,  in  general,  if  the  final  direction 
of  the  ray  is  reversed,  for  example,  by  falling  normally  on 
a  plane  mirror,  the  light  will  retrace  its  entire  path,  no 
matter  how  many  reflections  or  refractions  it  may  have 
suffered.  Thus,  in  any  optical  diagram,  in  which  the  di- 
rections of  the  rays  of  light  are  indicated  by  arrow-heads, 
these  pointers  may  all  be  reversed,  if  we  wish  to  ascertain 
how  the  rays  would  go  through  the  system  if  they  were  to 
enter  it  from  the  other  end. 

It  follows,  therefore,  since 

sin  a  sin  a' 

sin  a'       ab'         sin  a        ba' 

that  we  have  the  relation : 

^ab-^ba  =  1 ; 

that  is,  the  relative  indices  of  refraction  from  (a)  to  (b)  and 
from  (b)  to  (a)  are  reciprocals  of  each  other.  Thus,  for  ex- 
ample, since  naw  =  4/3  is  the  index  from  air  to  water,  the 
index  from  water  to  air  is  nwa  =  3/4.  Similarly,  if  ?iag  =  3/2, 
the  index  from  glass  to  air  is  nga  =  2/3. 


70  Mirrors,  Prisms  and  Lenses  [§  31 

30.  Limiting  Values  of  the  Index  of  Refraction. — Accord- 
ingly, we  see  that  the  value  of  the  relative  index  of  refrac- 
tion may  be  greater  or  less  than  unity.  If  nab>l,  the 
second  medium  (b)  is  said  to  be  more  highly  refracting  or 
(optically)  denser  than  the  first  medium  (a);  and  since  in 
this  case  sin  a  >sina',  it  follows  that  a  >  a' ',  which  means 
that  the  refracted  ray  is  bent  towards  the  normal,  as  happens 
when  light  is  refracted  from  air  to  water  (wab  =  1.33).  On 
the  other  hand,  if  nab<l,  the  second  medium  (b)  is  said  to 
be  less  highly  refracting  or  (optically)  rarer  than  the  first 
medium  (a),  and  now  the  angle  of  refraction  (a')  will  be 
greater  than  the  angle  of  incidence  (a),  so  that  in  this  case 
the  refracted  ray  will  be  bent  away  from  the  normal,  as,  for 
example,  when  light  is  refracted  from  water  into  air  (nwa  = 
0.75).  Glass  is  more  highly  refracting  than  water,  and 
diamond  has  the  greatest  light-bending  power  of  all  optical 
media,  the  index  of  refraction  from  air  to  diamond  being 
about  2.5.  The  values  of  the  constant  nab  for  pairs  of 
media  a,  b  that  are  available  for  optical  purposes  are  com- 
prised within  comparatively  narrow  limits,  say,  between 
1/2  and  2.  In  the  exceptional  case  when  nab=l,  the  angles 
of  incidence  and  refraction  will  be  equal,  and  the  rays  pass 
from  a  to  b  without  change  of  direction.  This  is  the  reason 
why  a  glass  rod  is  invisible  in  oil  of  cedar.  Sometimes  ac- 
cidental differences  of  refrangibility  between  two  adjacent 
layers  of  the  same  medium  enable  us  to  distinguish  one 
part  of  a  transparent  medium  from  another.  Similarly, 
also,  the  presence  of  air-bubbles  in  water  or  glass  is  made 
manifest  by  the  refractions  that  take  place  at  the  boundaries. 
A  fish  swimming  in  water  does  not  see  the  water  around  him, 
but  the  phenomena  of  refraction  may  make  him  aware  of  the 
existence  of  a  different  medium  above  the  surface  of  the  water. 

31.  Huygens's  Construction  of  a  Plane  Wave  Refracted 
at  a  Plane  Surface. — The  straight  lines  AB  and  AD  (Fig.  44) 
show  the  traces  in  the  plane  of  the  diagram  of  the  plane 
wave-front  advancing  in  the  first  medium  (a)  in  the  direc- 


31] 


Waves  Refracted  at  Plane  Surface 


71 


tion  BD  and  the  plane  refracting  surface,  respectively.  The 
disturbance  is  supposed  to  have  just  arrived,  at  the  point  A 
of  the-  refracting  plane,  which  from  this  moment  (£  =  0) 
becomes  a  new  origin 
from  which  secondary 
hemispherical  wavelets 
are  propagated  into  the 
second  medium  (b) .  Now 
light  is  propagated  with 
different  velocities  in  dif- 
ferent media;  thus,  for 
example,  the  velocity  of 
light  in  water  is  only 
about  three-fourths  of 
what  it  is  in  air  and  the 
velocity  in  glass  is  about 
two-thirds  of  the  velocity 
in  air.  Consequently, 
when  waves  of  light  pass 
from  air  into  water  or 
glass,  the  part  of  the  wave-front  that  is  in  the  denser  medium 
advances  more  slowly  than  the  part  that  is  still  in  the  air, 
so  that  the  direction  of  the  wave-front  is  changed  in  passing 
from  one  medium  to  another.  Let  the  velocities  of  light 
in  the  media  a  and  b  be  denoted  by  va  and  v^,  respectively. 
Then  after  a  time  i  =  BD/ya,  when  the  disturbance  which 
was  at  B  has  just  arrived  at  D  on  the  boundary  between 
the  two  media,  the  secondary  wavelets  which  have  been 
spreading  out  from  A  as  center  will  have  been  propagated 


Fig.  44. — Huygens's  construction  of  plane 
wave  refracted  at  plane  surface. 


in  the  second  medium  (6)  to  a  distance  AC 


M  =  --BD; 


and,  similarly,  at  the  same  instant  from  any  intermediate 
point  Q  lying  on  AD  between  A  and  D  the  disturbance  will 
have  proceeded  into  the  second  medium  (b)  to  a  distance 

QR=-(BD— PQ)=-KD, 


72  Mirrors,  Prisms  and  Lenses  [§  32 

where  K  (not  shown  in  the  figure)  designates  the  foot  of 
the  perpendicular  let  fall  from  Q  on  BD.  Thus,  the  radii 
of  the  elementary  cylindrical  refracted  waves  whose  axes  are 
perpendicular  to  the  plane  of  the  diagram  at  A  and  Q  are 

SfcD,  "-bKD, 

respectively;  and,  according  to  Huygens's  principle,  the 
refracted  wave-front  at  this  instant  will  be  the  surface  which 
is  tangent  to  all  these  elementary  cylindrical  surfaces.  Ex- 
actly the  same  method  as  was  used  in  the  similar  problem 
of  reflection  (§  14)  can  be  applied  here;  and  thus  it  may  be 
shown  that  at  the  moment  when  the  disturbance  reaches 
the  point  D  of  the  plane  refracting  surface,  the  refracted 
wave-front  will  be  the  plane  CD  containing  this  point,  which 
is  perpendicular  to  the  plane  of  the  figure  and  tangent  at  C 
to  the  elementary  wave  represented  by  the  spherical  sur- 
face described  about  C  as  center  with  radius  equal  to  AC. 
In  the  first  medium  the  wave  marches  forward  in  the  di- 
rection LA  and  in  the  second  medium  in  the  direction  AC. 

Snell's  law  of  refraction  (§  27)  may  be  deduced  from 
the  figure  by  observing  that  BD  =  AD.sina,  where  a  = 
Z  NAL  =  Z  DAB  denotes  the  angle  of  incidence,  and  AC  = 
AD. sin  a',  where  a'  =  Z  N'AC  =  Z  ADC  denotes  the  angle  of 
refraction.    Consequently, 

sin  a     BD     va 

/  =  ttt  =  —  =  a  constant, 

sin  a'    AC     Vb 

which  constant  must,  therefore,  be  identical  with  the  relative 

index  of  refraction  nab. 

The  diagram  is  drawn  for  the  case  when  the  light  travels 
faster  in  the  first  medium  than  it  does  in  the  second  (va>vh), 
that  is,  when  the  second  medium  is  more  retarding  or  "op- 
tically denser"  (§  30)  than  the  first. 

32.  Mechanical  Illustration  of  the  Refraction  of  a  Plane 
Wave. — A  simple  mechanical  illustration  of  the  refraction 
of  a  plane  wave  at  a  plane  surface  may  be  devised  as 
follows : 


32] 


Mechanical  Illustration 


73 


Two  boxwood  wheels  each  about  two  inches  in  diameter 
are  connected  by  an  iron  axle  about  4  inches  long  passing 
through  the  centers  of  the  wheels  at  right  angles  to  their 
planes  of  rotation  (Fig.  45).  If  this  body  is  placed  on  a 
smooth  rectangular  board,  about  a  yard  long  and  about 
18  inches  wide,  which  is 
slightly  tilted,  and  allowed 
to  roll  diagonally  down  the 
board,  its  path  will  be 
along  a  straight  line.  But 
if  a  piece  of  felt  cloth  or 
velveteen  cut  in  the  form 
of  a  rectangle  is  glued  in 
the  middle  of  the  board, 
with  its  long  side  parallel 
to  the  edge  of  the  board, 
then  when  the  body  de- 
scends the  inclined  plane 
obliquely,  one  of  the  wheels 
will  arrive  at  the  edge  of  FlG 
the  cloth  before  the  other, 
so  that  it  will  be  suddenly  slowed  up  while  the  other  wheel 
continues  to  move  on  the  bare  board  under  the  same  condi- 
tions as  before.  Consequently,  the  axle  will  be  made  to  swing 
round  until  both  wheels  get  on  to  the  cloth  piece,  the  direc- 
tion of  motion  having  been  abruptly  changed  in  this  process. 
At  the  opposite  edge  of  the  cloth  rectangle,  a  similar  change 
of  the  direction  of  motion  takes  place  in  an  opposite  sense, 
so  that  when  the  roller  leaves  the  retarding  surface  and 
emerges  again  on  to  the  bare  board,  it  will  be  found  to  be 
going  approximately  in  the  same  direction  as  at  first.  These 
bendings  in  the  course  of  the  roller  descending  the  inclined 
plane  at  the  places  where  it  crosses  the  parallel  sides  of  the 
cloth  rectangle  are  analogous  to  the  deviations  in  the  line 
of  march  of  a  plane  wave  of  light  in  traversing  a  glass  slab 
surrounded  by  air. 


45. — Mechanical     illustration 
refraction. 


of 


74  Mirrors,  Prisms  and  Lenses  [§  33 

33.  Absolute  Index  of  Refraction. — If  v&,  vh  and  vc  denote 
the  velocities  of  light  in  the  media  a,  b  and  c,  respectively, 
then,  as  we  have  just  seen  (§  31),  according  to  the  wave- 
theory  of  light,  the  relative  indices  of  refraction  will  be: 


iti  naC~Vnbc"~Vc; 

and,  hence,  we  find : 

nac. 

nab=— -; 

Wbc 

so  that  in  case  we  know  the  values  nac,  nbc  of  the  indices 
of  a  medium  c  with  respect  to  each  of  the  two  media  a  and 
b,  the  value  nab  of  the  index  of  medium  b  with  respect  to 
medium  a  can  be  obtained  at  once  by  means  of  the  above 
relation.    Moreover,  since  (§  29) 

1 

Wbc 

the  preceding  equation  may  be  written  as  follows : 

Thus,  for  example,  suppose  the  three  media  a,  b  and  c  are 
water,  glass  and  air,  respectively;  since  nac  =  3/4  and  nch=-- 
3/2,  the  index  of  refraction  from  water  to  glass  is  found  by 
the  above  formula  to  be  nab  =  9/8. 

In  fact,  if  there  are  a  number  of  media  a,  &,  c,  .  .  .  ,  i>  j,  h 
it  is  obvious  that  we  shall  have  the  following  relation  be- 
tween the  relative  indices  of  refractions: 

nab-nbc  .  .  .  nij.njk  =  nak, 
which  is  easily  remembered  by  observing  the  order  in  which 
the  letters  occur  in  the  subscripts.  In  particular,  if  the  last 
medium  k  is  identical  with  the  first  medium  a,  as  is  the  case 
in  an  optical  instrument  surrounded  by  air,  then  nak  =  naa=  1, 
and  accordingly  we  obtain : 

^ab-^bc  ....  nij.nja  =  l. 
A  special  case  of  this  general  relation,  viz., 

has  already  been  remarked  (§  29). 


§  33]  Absolute  Index  of  Refraction  75 

Since  wac.nca=7ibc.ncb  =  l  and  nab.n\iC  =  nac,  we  may  write 
also: 

"'ca 

and  this  formula  suggests  immediately  the  idea  of  employ- 
ing some  suitable  medium  c  as  a  standard  optical  medium  with 
respect  to  which  the  indices  of  refraction  of  all  other  media 
may  be  expressed.  The  natural  medium  to  choose  for  this  pur- 
pose is  the  ether  itself  which  light  traverses  in  coming  to  the 
earth  from  the  sun  and  stars;  and  so  the  index  of  refraction  of 
a  medium  with  respect  to  empty  space  or  vacuum  is  called 
its  absolute  index  of  refraction  or  simply  its  refractive  index. 
Thus,  the  absolute  index  of  refraction  of  vacuum  (c)  is  equal 
to  unity,  that  is,  nc  =  l.  Similarly,  the  symbols  na,  nh  will 
be  employed  to  denote  the  absolute  indices  of  the  media 
a,  b,  respectively;  so  that  here  they  are  really  equivalent 
to  the  magnitudes  denoted  by  nCSL,  nch  in  the  preceding 
formula,  which,  therefore,  may  be  written : 

nh 

that  is,  the  relative  index  of  refraction  of  medium  b  with  respect 
to  medium  a  is  equal  to  the  ratio  of  the  absolute  index  of  medium 
b  to  that  of  medium  a. 

The  absolute  indices  of  refraction  of  all  known  transparent 
substances  are  greater  than  unity.  The  velocity  of  light 
in  ordinary  atmospheric  air  is  so  nearly  equal  to  its  velocity 
in  vacuo  that  for  all  practical  purposes  we  may  generally 
take  the  absolute  index  of  refraction  of  air  as  also 
equal  to  unity.  The  actual  value  for  air  at  0°C.  and 
under  a  pressure  of  76  cm.  of  mercury,  for  sodium  light, 
is  1.000293. 

With  every  isotropic  medium  there  is  associated,  there- 
fore, a  certain  numerical  constant  n  called  its  (absolute) 
index  of  refraction;  and,  hence,  when  a  ray  of  light  is  re- 
fracted from  a  medium  of  index  n  into  another  of  index  nf, 


76 


Mirrors,  and  Prisms  Lenses 


34 


the  trigonometric  formula  for  the  law  of  refraction  may  be 
written  thus: 

sina       n' 


sina'     n 
which  may  also  be  put  in  the  following  symmetric  form : 

n'.sina'  =  ft.sina. 
This  latter  mode  of  writing  this  relation  suggests  also  an- 
other way  of  stating  the  fundamental  fact  in  regard  to  the 


Fig.  46. — Construction  of  refracted  ray  (n'>n) 

refraction  of  light,  as  follows:  Whenever  a  ray  of  light  is  re- 
fracted from  one  medium  to  another ,  the  product  of  the  index 
of  refraction  and  the  sine  of  the  angle  between  the  ray  and  the 
normal  to  the  refracting  surface  has  the  same  value  after  re- 
fraction (n'.sina/)  as  before  refraction  (n.sina).  This  prod- 
uct K  =  n.sin  a  =  n'.sina'  which  does  not  vary  when  the 
light  crosses  a  surface  separating  a  pair  of  isotropic  media 
is  called  the  optical  invariant  of  refraction. 

34.  Construction  of  the  Refracted  Ray. — Let  the  absolute 
indices  of  refraction  of  two  media  separated  from  each  other 
by  a  smooth  refracting  surface  be  denoted  by  n,  n',  and  let 
the  straight  line  AB  (Figs.  46  and  47)  represent  the  path 
in  the  first  medium  (n)  of  a  ray  incident  on  the  boundary- 


§34] 


Construction  of  Refracted  Ray 


77 


surface  at  the  point  B.  The  straight  line  NN'  represents 
the  normal  to  the  refracting  surface  at  this  point,  and  hence 
the  plane  of  the  diagram  is  the  plane  of  incidence.  The 
straight  line  ZZ  shows  the  trace  in  this  plane  of  the  plane 
tangent  to  the  refracting  surface  at  the  incidence-point  B; 
in  the  special  case  when  the  refracting  surface  is  itself  plane, 
this  straight  line  will  be  the  trace  of  the  surface  of  separa- 
tion between  the  two  media.    With  the  point  B  as  center 


Fig.  47. — Construction  of  refracted  ray  (n'<n) 

and  with  any  radius  r  describe  in  the  plane  of  incidence  the 
arc  of  a  circle  cutting  the  incident  ray  AB  in  a  point  P  lying 
in  the  first  medium;  and  in  the  same  plane,  with  radius 
n'jn  times  as  great,  that  is,  with  radius  n'r/n,  describe  also 
the  arc  of  a  concentric  circle  intersecting  at  P'  the  straight 
line  HP  drawn  through  P  perpendicular  to  ZZ  at  H.  If 
the  second  medium  is  more  highly  refracting  than  the  first, 
that  is,  if  n'>n,  the  radius  of  the  second  circle  will  be  greater 
than  that  of  the  first,  as  represented  in  Fig.  46;  whereas 
when  n'<n,  the  second  circle  is  inside  the  first,  as  in  Fig.  47. 
The  path  of  the  refracted  ray  correspodinng  to  the  given 


78  Mirrors,  Prisms  and  Lenses  [§  36 

incident  ray  AB  will  be  represented  by  the  prolongation 
BC  in  the  second  medium  of  the  straight  line  P'B. 

The  proof  of  this  construction  consists  simply  in  showing 
that  the  ZN'BC  between  the  normal  and  the  straight  line 
BC  is  equal  to  the  angle  of  refraction  a'  as  given  by  the 
formula  n'.sin a'  =  n.sin a,  where  a=ZNBA  denotes  the 
given  angle  of  incidence.    Evidently,  from  the  figure,  we  have : 

sinZHPB  _BP/_n' 

sin/HP/B~BP  "»' 
and  since  ZHPB  =  ZNBA=a,  and  Z  HP'B  =  Z  N'BC,  we 
obtain  immediately  the  relation:  n'.  sinZN'BC  =  ft.sina  and 
therefore  ZN'BC=a'. 

35.  Deviation  of  the  Refracted  Ray. — The  acute  angle 
through  which  the  direction  of  the  refracted  ray  has  to  be 
turned  to  bring  it  into  the  same  direction  as  that  of  the  in- 
cident ray  is  called  the  angle  of  deviation  of  the  refracted  ray 
and  is  denoted  by  e;  thus,  €  =  ZP/BP  (Figs.  46  and  47). 
Obviously, 

e=  a  —  a'. 
The  only  ray  incident  at  B  whose  direction  will  remain  un- 
changed after  the  ray  enters  the  second  medium  is  the  one 
that  proceeds  along  the  normal  NB  (a  =  a'=e  =  0).  The 
more  obliquely  the  ray  AB  meets  the  refracting  surface, 
that  is,  the  greater  the  angle  of  incidence,  the  greater  also  will 
be  the  deviation-angle.  The  truth  of  this  statement  will  be 
apparent  from  an  inspection  of  the  relation  between  the 
angles  a  and  e  as  exhibited  in  Fig.  46  or  Fig.  47.  The  inter- 
cept PP'  included  between  the  circumferences  of  the  two 
construction-circles,  which  remains  constantly  parallel  to  the 
incidence-normal,  increases  in  length  as  the  angle  of  inci- 
dence increases,  whereas  the  other  two  sides  BP,  BP'  of 
the  triangle  BPP',  being  always  equal  to  the  radii  of  the 
circles,  remain  constant  in  length;  and  hence  the  angle  € 
must  increase  in  absolute  value  as  the  angle  a  increases. 

36.  Total  Reflection. — In  ordinary  refraction,  as  we  have 
seen,  there  can  only  be  one  refracted  ray  corresponding  to 


§36] 


Total  Reflection 


79 


a  given  incident  ray,  but  the  question  may  be  asked:  Is  it 
possible  that,  under  certain  circumstances,  there  will  be 
no  refracted  ray,  so  that  the  incident  light  will  be  totally 
reflected  at  the  surface  without  being  refracted  at  all?  Evi- 
dently such  will  be  the  case  whenever  in  the  foregoing  con- 
struction (§  34)  the  point  P'  (Figs.  46  and  47)  cannot  be 
located,  because  the  path  of  the  refracted  ray  is  determined 
by  the  straight  line  P'B. 

Let  us  examine,  first,  the  case  when  the  second  medium 
is  more  highly  refracting  than  the  first,  n'>n   (Fig.  46). 


Fig.  48. — Limiting  refracted  ray   (nf>ri) 

Suppose  that  the  straight  line  AB  which  represents  the 
path  of  the  incident  ray  is  initially  in  the  position  NB,  and 
that  it  is  rotated  from  this  position  around  the  point  B  as 
a  pivot  until  it  has  turned  through  a  right  angle  in  the  plane 
of  the  figure.  While  the  point  P  on  AB  describes  a  quadrant 
of  the  circumference  of  the  circle  of  radius  BP,  the  point 
P'  will  trace  out  an  arc  of  the  concentric  circle  of  radius 
BP',  which,  however,  will  never  be  equal  to  a  quadrant  of 
this  circumference;  for  when  the  point  P  has  completed  its 
quadrant  and  arrived  at  the  point  D  (Fig.  48)  on  the  tan- 
gent plane  drawn  to  the  refracting  surface  at  B,  the  point 


80  Mirrors,  Prisms  and  Lenses  [§  36 

P'  will  likewise  have  reached  the  extremity  of  its  arc  where 
the  tangent  to  the  inner  circle  at  D  meets  the  circumference 
of  the  outer  circle.  The  incident  ray  ZB  just  grazes  the  re- 
fracting surface  at  B  or  skims  along  it,  and  most  of  the 
light  is  reflected  and  does  not  enter  the  second  medium  at 
all,  but  the  portion  that  is  refracted  pursues  the  path  BQ 
corresponding  to  this  extreme  position  of  the  point  P',  and 
this  will  be  the  outermost  of  all  the  refracted  rays  that 
enter  the  second  medium  at  the  point  B.  The  ZN'BQ=A 
which  is  the  greatest  value  that  the  angle  of  refraction  can 
have  in  the  case  when  n'>n  is  called  the  limiting  or  critical 
angle  with  respect  to  the  two  media.    Since 

sinZ  N'BQ  =  sinZ  PP'B  =  BD/BP' = n\n\ 
the  magnitude  of  the  angle  A  may  be  found  from  the  rela- 
tion: 

sinA = n/n',  (n  <nf) ; 
which  may  likewise  be  derived  by  substituting  the  values 
a  =  90°,  a/  =  A  in  the  refraction-formula.  Thus,  if  the 
first  medium  is  air.(n  =  l)  and  the  second  medium  is  glass 
(n' =  3/2),  sinA  =  2/3,  so  that  the  critical  angle  for  air-glass 
is  found  to  be  A =41°  49'.  For  air-water  sinA  =  3/4,  A  = 
48°  35';  and,  consequently,  a  ray  of  light  whose  path  lies 
partly  in  air  and  partly  in  water  cannot  possibly  make 
an  angle  with  the  normal  in  the  water  greater  than  about 
48°  30'.  For  example,  when  a  star  is  just  rising  or  setting, 
the  rays  coming  from  it  will  fall  very  nearly  horizontally 
on  the  surface  of  tranquil  water  and  will  be  refracted  into 
the  water,  therefore,  at  an  angle  of  approximately  48°  30' 
with  the  vertical,  so  that  if  these  rays  entered  an  eye  under 
the  water,  the  star  would  appear  to  be  nearly  halfway  to 
the  zenith.  In  fact,  all  the  rays  coming  into  an  eye  placed 
under  water  from  the  entire  overhanging  arch  of  the  sky 
would  be  comprised  in  the  water  within  a  cone  whose  axis 
points  to  the  zenith  and  whose  angular  aperture  is  about 
97°.  In  this  connection  it  is  interesting  and  instructive  to 
examine  a  photograph  of  an  air-scene  made  with  a  so-called 


36] 


Total  Reflection 


81 


"fish-eye"  camera  immersed  below  the  level  of  a  clear  pool 
of  water,  which  affords  some  idea  of  how  the  world  outside 
the  pond  must  look  to  a  fish.  Professor  Wood,  of  the  Johns 
Hopkins  University,  has  obtained  a  number  of  pictures  of 
this  kind,  some  of  which  are  reproduced  in  illustrations  in 
his  very  original  book  on  Physical  Optics,  where  also  a  brief 
description  of  the  essential  features  of  the  ingenious  pin- 


Fig.  49. — Limiting  incident  ray  (n'<n) 

hole  camera  which  was  used  in  making  these  pictures  is 
also  given. 

Accordingly,  when  light  is  refracted  from  a  rarer  to  a 
denser  medium,  there  will  always  be  a  refracted  ray  cor- 
responding to  a  given  incident  ray,  because  it  is  always 
possible  under  these  circumstances  to  locate  the  position 
of  the  point  P'  opposite  P,  or,  to  express  it  in  another  way, 
because  when  n<nr  there  will  always  be  a  certain  acute 
angle  a'  that  will  satisfy  the  equation  sina'  =  n.sina/tt'  for 
values  of  a  comprised  between  0°  and  90°.  But  in  the  op- 
posite case  when,  the  first  medium  is  denser  than  the  second 
(n>nf),  for  example,  when  the  light  is  refracted  from  water 
to  air,  the  statement  just  made  is  no  longer  true.    The  es- 


82 


Mirrors,  Prisms  and  Lenses 


[§36 


1 

\  \ 

r 

z 

AM         \       \ 

/ 

WATER        \   \      \ 

s 

/  /^i///\ 

Fig.   50.- 


-Refraction  from  water  to   air; 
total  reflection. 


sential  difference  in  the  two  cases  may  be  seen  at  once  by 
reversing  the  arrow-heads  in  the  diagram  Fig.  48,  at  the 
same  time  making  corresponding  changes  in  the  letters  and 

symbols.  Fig.  49  is  a 
special  diagram  to  illus- 
trate this  case.  The  re- 
fracted ray  BQ  which 
grazes  the  surface  at  the 
point  B  corresponds  to 
the  limiting  incident  ray 
PB  which  is  incident  at 
B  at  the  critical  angle 
A  =  ZNBP;  and,  conse- 
quently, any  ray,  such 
as  RB,  which  meets  the 
surface  at  an  angle  of  incidence  greater  than  the  angle  A 
will  be  totally  reflected  in  the  direction  BS.  Thus,  for  values 
of  a  which  are  greater  than  the  value  A  of  the  critical  angle 
of  incidence,  there  will  be  no  value  of  a'  that  will  satisfy 
the  equation  sina'  =  7i.sina/w' 
when  n>n'.  Only  those  rays 
incident  at  B  which  lie  within 
the  cone  generated  by  the 
revolution  of  the  limiting  in- 
ident  ray  around  the  inci- 
dence-normal as  axis  will  be 
refracted  into  the  second 
medium;  and  all  rays  falling 
on  the  refracting  surface  at 
B  and  lying  outside  this  cone 
will  be  totally  reflected. 

Fig.    50   shows    how    rays 
proceed  from  a  radiant  point  below  the  horizontal  free  sur- 
face of  still  water. 

If  a  pin  is  stuck  in  the  under  side  of  a  flat  circular  cork 
floating  on   water,  as  represented   in  Fig.    51,   and  if  the 


Fig.  51. 


-Experiment     illustrating 
total  reflection. 


37] 


Total  Reflection 


83 


diameter  of  the  cork  is  (say)  6  inches  and  the  head  of 
the  pin  is  not  more  than  2.5  inches  below  the  water-level 
and  vertically  beneath  the  center  of  the  cork,  an  eye  placed 
anywhere  above  the  level  of  the  water  will  be  unable  to  see 
the  pin,  because  all  the  rays  coming  from  it  that  meet  the 
surface  of  the  water  beyond  the  edge  of  the  cork  will  be 
totally  reflected  back  into  the  water. 

In  Fig.  49  since  sin  ZNBP  =  sinZP'PB  =  BP'/BP,  we  find 
in  this  case  when  n'<n  that  sinA  =  n'/n,  which  will  also  be 
obtained  by  putting  a  =  A,  a' =  90°  in  the  refraction- 
formula  n.sina  =  n/.sina'.  Comparing  this  result  with 
the  formula  sin  A  =  n/n'  obtained  for  the  case  when  n'>n, 
and  recalling  the  fact  that  the  sine  of  an  angle  is  never 
greater  than  unity,  we  may  formulate  the  following  rule: 

The  sine  of  the  so-called 
critical  angle  (A)  with  re- 
spect to  two  media  is  the 
ratio  of  the  index  of  refrac- 
tion of  the  rarer  to  that  of 
the  denser  medium.  Or, 
the  sine  of  the  critical  angle 
(A)  of  a  substance  is  the 
reciprocal  of  the  absolute 
index  of  refraction  of  the 
substance:  thus, 


A-* 


Fig.  52. — Optica  Disk  used  to  show  total 
reflection. 


sinA  =  -. 
n 

37.  Experimental  Il- 
lustrations of  Total  Re- 
flection.— The  phenomenon  of  total  reflection  may  be  ex- 
hibited with  the  aid  of  the  optical  disk  and  the  semicylinder 
of  glass  described  in  §  28.  If  the  disk  is  turned  so  that  the 
beam  of  incident  parallel  rays  falls  first  on  the  curved  surface 
of  the  semicylinder,  as  shown  in  Fig.  52,  the  rays  meet  this 
surface  normally  and  proceed  through  the  glass  to  the  plane 
face  without  being  deviated.     At  the  plane  surface  a  por- 


84 


Mirrors,  Prisms  and  Lenses 


37 


tion  of  the  beam  is  reflected  and,  in  general,  a  portion  is  re- 
fracted from  glass  to  air.  If  the  disk  is  turned  until  the 
angle  of  incidence  at  the  plane  surface  is  just  equal  to  the 
critical  angle  (A),  the  rays  emerging  into  the  air  will  pro- 
ceed along  the  plane  face,  and  if  the  disk  is  turned  a  little 
farther  in  the  same  sense,  so  that  the  angle  of  incidence 
exceeds  the  critical  angle,  the  light  will  be  totally  reflected. 
An  ingenious  contrivance  for  exhibiting  the  procedure  of 
light  in  passing  from  water  to  air  consists  of  a  compara- 


Fig.  53. — Demonstration  of  refraction  from  water  to 
air  and  total  reflection. 

tively  large  glass  tank  (Fig.  53)  filled  with  water  and  pro- 
vided with  a  plane  vertical  metallic  screen  the  lower  half  of 
which  is  under  water  while  the  upper  half  extends  into  the 
air  above.  A  cylindrical  beam  of  light  is  directed  horizon- 
tally and  normally  against  the  lower  part  of  the  vertical 
glass  wall  of  the  tank,  which  is  behind  the  screen  and  par- 
allel to  it.  The  rays  entering  the  water  are  received  first 
on  the  surface  of  a  solid  reflecting  cone  of  aperture-angle 
90°  placed  in  the  water  under  the  screen  and  mostly  in  front 
of  it,  the  axis  of  the  cone  being  horizontal  and  its  apex 
turned  towards  the  on-coming  light.  From  the  surface  of 
this  cone  the  rays  are  reflected  through  the  water  in  all  di- 
rections in  a  vertical  plane  coinciding  as  nearly  as  possible 
with  the  front  side  of  the  screen  turned  towards  the  spec- 
tators.   Surrounding  the  conical  reflector  and  co-axial  with 


§37] 


Total  Reflection  Prism 


85 


it,  there  is  a  cylindrical  cavity  of  diameter  very  little  larger 
than  that  of  the  base  of  the  cone.  The  surface  of  this  cylin- 
der is  made  of  thin  sheet-metal  blackened  on  the  inside, 
wherein  a  number  of  equal  horizontal  slits  are  cut  at  equal 
angular  distances  apart,  and  through  these  slits  narrow 
beams  of  light  reflected  from  the  surface  of  the  cone  are 
permitted  to  pass  upwards  towards  the  surface  of  the  water, 
their  courses  being  shown  by  the  bright  traces  on  the  screen. 
Some  of  these  beams  will  be  refracted  out  into  the  air, 
whereas  others,  meeting  the  water-surface  more  obliquely, 
will  be  totally  reflected. 

If  rays  are  incident  normally  on  one  of  the  two  perpendic- 
ular faces  of  a  glass  prism  (§  48)  whose  principal  section  is  an 
isosceles  right-triangle  (Fig.  54), 
they  will  enter  the  prism  with- 
out deviation,  and  falling  on  *~~ 
the  hypothenuse-face  at  an  angle  z — ^_ 
of  45°,  which  is  greater  than  the 
critical  angle  of  glass,  they  will  *"~ 
be  totally  reflected  there  and 
turned  through  a  right  angle,  so 
that  they  will  emerge  in  a  direc- 
tion normal  to  the  other  of  the  two 
perpendicular  faces  of  the  prism.  FlG-  54.— Total  reflection  prism 
A  prism  of  this  kind  is  frequently  employed  in  optical  sys- 
tems. It  is  used,  for  example,  in  connection  with  a  photo- 
graphic lens  to  rectify  the  image  focused  on  the  sensitive 
plate  of  the  camera,  so  that  the  right  and  left  sides  of  the 
negative  will  correspond  to  the  right  and  left  sides  of  the 
object.  None  of  the  light  is  lost  by  the  total  reflection  in 
the  prism,  and  if  the  prism  is  made  of  good  optical  glass 
of  high  transparency  there  will  be  comparatively  little  loss 
of  light  by  absorption  in  the  prism  or  by  reflection  on  enter- 
ing and  leaving  it.  The  same  optical  effect  can  be  produced 
by  a  simple  plane  mirror,  but  as  a  rule  a  polished  metallic 
surface  absorbs  the  incident  light  to  a  considerable  extent. 


86  Mirrors,  Prisms  and  Lenses  [§  38 

However,  the  loss  of  light  in  the  case  of  a  mirror  silvered 
on  glass  is  very  slight;  but  on  the  other  hand,  the  fine  layer 
of  silver  may  easily  be  injured  mechanically  or  tarnished 
by  exposure  to  the  air.  If  the  glass  mirror  is  silvered  on 
the  back  side,  the  light  will  be  reflected  from  both  surfaces 
of  the  glass  and  there  will  be  confusion.  Moreover,  a  glass 
mirror  may  easily  get  broken  or  become  dislocated  in  an 
optical  instrument;  whereas  a  prism  made  of  a  solid  piece 
of  glass  is  much  more  substantial  and  durable. 

Optical  prisms  consisting  of  solid  pieces  of  highly  trans- 
parent homogeneous  glass  with  three  or  more  polished  plane 
faces  are  very  extensively  used  in  the  construction  of  modern 
optical  instruments  for  rectifying  images  which  would  other- 
wise be  inverted  or  for  bending  the  rays  of  light  into  new 
directions,  etc.  Usually  the  light  undergoes  several  interior 
reflections  before  it  issues  from  the  prism,  and  these  reflec- 
tions are  often  total  reflections.  If  the  reflection  is  not 
total,  it  is  best  to  silver  the  surface. 

38.  Generalization  of  the  Laws  of  Reflection  and  Re- 
fraction. Principle  of  Least  Time  (Fermat's  Law). — The 
laws  of  reflection  and  refraction,  which  merely  describe  the 
observed  effects  when  light  falls  on  the  common  surface  of 
separation  of  two  homogeneous  media,  and  which  are  cap- 
able of  simple  explanation  on  the  basis  of  the  wave-theory, 
as  has  been  illustrated  in  certain  special  cases  (§§  14  and  31), 
may  be  combined  into  a  general  law  which  was  first  an- 
nounced about  1665  by  the  French  philosopher  Fermat, 
and  which  may  be  stated  as  follows:  The  actual  "path  pur- 
sued by  light  in  going  from  one  point  to  another  is  the  route 
that,  under  the  given  conditions,  requires  the  least  time. 

In  case  the  reflections  and  refractions  take  place  only 
at  plane  surfaces,  the  truth  of  the  above  statement  is 
easily  proved.  Consider,  first,  the  case  when  the  light  is  re- 
flected from  a  plane  mirror.  The  straight  line  ZZ  (Fig.  55) 
represents  the  trace  of  the  plane  mirror  in  the  plane  of  the 
diagram,  and  A  and  C  designate  the  positions  of  a  pair  of 


Cv 

B 

V 

A 

A' 

§  38]  Principle  of  Least  Time  87 

points  lying  in  this  plane  in  front  of  the  mirror.  Now  if 
a  point  X  in  the  plane  of  the  mirror  is  connected  with  A 
and  C  by  the  straight  lines  XA,  XB,  the  route  AXC  will 
be  shortest  when  the  normal  to  the  mirror  at  X  lies  in  the 
plane  AXC  and  bisects  the  angle 
AXC.  The  point  X  must  lie,  there- 
fore, in  the  plane  of  the  diagram  at 
the  point  B,  so  that  when  AB  is  the 
direction  of  an  incident  ray,  BC  will 
be  the  direction  of  the  reflected  ray. 
Obviously,  if  A'  is  the  image  of  A  in 
the  mirror,  then  AB-f-BC=A'B+BC 

-AT     and    *\nop>   thp    strfliVht     linp  FlG*  55'— Fermat's  Prfnci- 

-a^,  ana  since  tne  straignt  line  pie  of  least  time  in  case 
A'C  is  shorter,  for  example,  than  of  reflection  at  a  plane 
(A'D+DC)  =  (AD+DC),  where  D  is  mirron 
another  point  on  the  mirror  different  from  the  point  B,  it 
is  evident  that  the  route  from  A  to  C  by  way  of  B  is  shorter 
than  the  route  via  any  other  point  on  the  mirror.  More- 
over, if  the  ray  is  reflected  at  a  number  of  plane  mirrors  in 
succession,  its  entire  path  will  be  the  shortest  possible  route 
from  the  starting  point  to  the  terminal  point,  subject  to  the 
condition  that  it  must  touch  at  each  mirror  in  turn.  The 
principle  of  least  time  in  the  case  of  reflection  of  light  at  a 
plane  mirror  dates  back  to  the  time  of  Hero  of  Alexandria 
(150  B.  C). 

When  light  is  refracted  at  a  plane  surface,  the  route  pur- 
sued between  a  point  A  in  one  medium  to  a  point  C  in  the 
other  is  indeed  the  quickest  way  but  generally  not  the 
shortest.  The  following  illustration  will  help  to  make  the 
problem  clear  in  this  case.  Suppose  a  level  field  is  divided 
into  two  parts  by  a  straight  line  ZZ  (Fig.  56),  on  one  side 
of  which  the  ground  is  bare  and  smooth  while  on  the  other 
side  it  is  plowed  and  rough;  and  let  us  also  suppose  that 
a  man  can  walk  only  half  as  fast  over  the  rough  part  of  the 
field  as  over  the  smooth  part,  and  that  he  desires  to  march 
as  quickly  as  possible  from  a  point  A  in  the  smooth  ground 


88 


Mirrors,  Prisms  and  Lenses 


38 


Fig.    56. — Quickest    route 
from  A  to  C  via 
ABC. 


to  a  certain  other  point  C  in  the  plowed  ground.  The 
question  is,  Where  should  he  cross  the  dividing  line  ZZ? 
Of  course,  his  shortest  route  would  be  along  the  straight 
line  from  A  to  C  which  intersects  ZZ 
at  the  point  marked  E  in  the  figure, 
but  unless  the  straight  line  AC  hap- 
pens to  be  perpendicular  to  ZZ  this 
will  not  be  his  quickest  way.  In- 
stead of  crossing  at  E,  suppose  he 
selects  a  point  F  on  ZZ  which  is  a 
little  nearer  to  his  objective  at  C; 
then  although  the  length  FC  in  the 
plowed  ground  is  shorter  than  be- 
fore, on  the  other  hand  the  distance 
path  AF  over  the  smooth  ground  is  longer, 
but  on  the  whole  we  may  assume  that 
the  route  AFC  will  take  less  time  than  the  shortest  route 
AEC.  But  if  the  point  of  crossing  ZZ  is  taken  too  far  from 
E,  the  advantage  of  the  shorter  dis- 
tance in  the  rough  ground  will  pres- 
ently be  more  than  offset  by  the 
increasing  length  of  the  distance  that 
has  to  be  traversed  in  the  smooth 
ground.  Accordingly,  there  is  a  cer- 
tain point  B  on  ZZ  such  that  the 
time  taken  along  the  route  ABC 
will  be  the  quickest  of  all  routes. 
Now  we  shall  see  that  this  is  also  the 
very  path  that  light  would  take  if  it 

p  j    £  »     ,       ^  Fig.   57. — Fermat's    princi- 

were  refracted  from  A  to  C  across         ple  of  least  time  in  case 

ZZ,  supposing  that  the  ratio  of  the 

velocities  of  light  on  the  two  sides 

of  ZZ  were  the  same  as  the  ratio  of  the  velocities  of  walking 

in  the  two  parts  of  the  field. 

In  the  accompanying  diagram  (Fig.  57)  the  broken  line 
ABC  represents  the  actual  path  of  a  ray  of  light  from  a 


of   refraction    at   plane 
surface. 


§  39]  Optical  Length  89 

point  A  in  the  first  medium  (n)  to  a  point  C  on  the  other 
side  of  the  plane  refracting  surface  ZZ  in  the  second  medium 
(nf) ;  so  that  if  NBN'  is  the  normal  to  the  surface  at  B,  then 
by  the  law  of  refraction : 

sin/NBA  _rt_v 
sinZN'BA~n     v'' 
where  v,  v'  denote  the  speeds  with  which  light  travels  in 
the  media  n,  n',  respectively.     The  time  taken  to  go  over 
the  route  ABC  is 

AB     BCl 
*~   v  +  v'  * 
and  we  wish  to  show  that  this  time  t  is  less  than  the  time 

AD    DC 
v         V 
along  any  other  route  ADC,  where  D  designates  the  posi- 
tion of  any  point  on  ZZ  different  from  the  point  B.     Draw 
DG,  DH  perpendicular  to  AB,  BC,  respectively;  then,  since 

Z  BDG  =  Z  NBA,       Z  BDH  =  Z  N'BC, 
evidently  we  have: 

sinZBDG     GB     v  GB     HB 


Now 


sinZBDH    HB~V  '  0r      v         V 


AB    BC_AG+GB    BC  =  AG    HC. 
v        v'  v  v'       v         v'  ' 


and  since  AG<AD  and  HC<DC,  therefore 
/AB  ,  BC\        /AD  ,  DC\ 

(v+vj  <  \ir+y)> 

and  hence  the  time  via  ABC  is  less  than  it  would  be  via  any 
other  route  from  A  to  C. 

It  should  be  remarked,  however,  that  when  the  boundary- 
surface  between  two  media  is  curved,  the  time  taken  by 
light  to  go  from  a  point  A  across  the  surface  to  another  point 
C  is  not  always  a  minimum.  It  may,  indeed,  be  a  maximum, 
but  it  is  always  one  or  the  other. 

39.  The  Optical  Length  of  the  Light-path,  and  the  Law 
of  Malus. — In  the  time  t  that  light  takes  to  go  along  the  path 


90  Mirrors,  Prisms  and  Lenses  [§  39 

ABC  from  a  point  A  in  one  medium  (n)  to  a  point  C  in  an 
adjacent  medium  (nf)  it  would  traverse  in  vacuo  the  distance 

-="(AVB+BTC). 

where  V  denotes  the  velocity  of  light  in  vacuo.    But  by  the 
definition  of  the  absolute  index  of  refraction  (§  33),  n  =  V/v, 

n'=V/v';  and  hence  the 
equivalent  distance  in 
vacuo  is : 

nJUB+n'JBC. 
The  optical  length  of  the 
path  of  a  ray  in  a  medium 

Fig.  5S.-Optical|ngth  of    ray-path     ig  defined  tQ  be  ^   prod_ 

uct  of  the  actual  length  (I) 
of  the  ray-path  by  the  index  of  the  medium  (n)  that  is,  n.  I. 
Suppose,  for  example,  that  light  traverses  a  series  of  media 
wi,  ri2,  etc.,  as  represented  in  Fig.  58;  the  total  optical  length 
along  a  ray  will  be: 

k  =m 

ni.li+rh.k+ +nm.Zm=2Jnk.Zk; 

k=l 

where  ?k  denotes  the  actual  length  of  the  ray-path  in  the 
fcth  medium. 

Now  the  wave-front  at  any  instant  due  to  a  disturbance 
emanating  from  a  point-source  is  the  surface  which  con- 
tains all  the  farthest  points  to  which  the  disturbance  has 
been  propagated  at  that  instant.  Thus,  the  wave-surface 
may  be  defined  as  the  totality  of  all  those  points  which  are 
reached  in  a  given  time  by  a  disturbance  originating  at  a  point. 
In  a  single  isotropic  medium  the  wave-surfaces,  as  we  have 
seen,  will  be  concentric  spheres  described  around  the  point- 
source  as  center;  but  if  the  wave-front  arrives  at  a  reflect- 
ing or  refracting  surface  /*,  at  which  the  directions  of  the 
so-called  rays  of  light  are  changed,  the  form  of  the  wave- 
surface  thereafter  will,  in  general,  no  longer  be  spherical;  and 
even  in  those  exceptional  cases  when  the  reflected  or  refracted 
wave-front  is  spherical,  the  waves  will  spread  out  from 


§  39]  Law  of  Malus  91 

a  new  center  which  is  seldom  identical  with  the  original 
center.  The  function  2nl  has  the  same  value  for  all  ac- 
tual ray-paths  between  one  position  of  the  wave-surface  and 
another  position  of  it;  so  that  when  the  form  and  position 
of  the  wave-front  and  the  paths  of  the  rays  at  any  instant 
are  known,  the  wave-front  at  any  subsequent  instant  may 
be  constructed  by  laying  off  equal  optical  lengths  along  the 
path  of  each  ray. 

A  consequence  of  this  definition  of  the  wave-surface  is 
that  the  ray  is  always  normal  to  the  wave-surface  (§7),  as  will 
be  evident  from  the  following 
reasoning.  Suppose  that  the 
straight  line  AB  (Fig.  59)  repre- 
sents the  path  of  a  ray  incident 
on  the  refracting  surface  ZZ  at 
the  point  B,  and  that  the  straight 
line  BC  represents  the  path  of 
the  corresponding  refracted  ray. 

Moreover,    let    the     wave-SUrface    FlG-  59.— Law  of  Malus:  Ray 

.  normal  to  wave-front. 

which  passes  through   the  pomt 

C  be  designated  by  <r.  From  the  incidence-point  B  draw 
any  other  straight  line,  as  BD,  meeting  the  wave-surface  a 
in  the  point  D.  Then  by  the  principle  of  least  time,  the 
route  ABC  is  quicker,  that  is,  optically  shorter,  than  the 
route  ABD,  because  the  natural  or  actual  route  between  the 
points  A  and  D  would  not  be  by  way  of  the  incidence-point 
B.  Hence,  the  straight  line  BC  must  be  shorter  than  BD, 
and  therefore  BC  is  the  shortest  line  that  can  be  drawn  from 
the  incidence-point  B  to  the  wave-surface  a. 

The  same  reasoning  is  applicable  to  all  cases  of  reflection 
and  refraction,  and  hence  we  may  make  the  following  gen- 
eral statement: 

Rays  of  light  meet  the  wave-surface  normally;  and,  con- 
versely, The  system  of  surfaces  which  intersect  at  right  angles 
rays  emanating  originally  from  a  point-source  is  a  system  of 
wave-surfaces. 

This  law  was  published  by  Malus  in  1808. 


92  Mirrors,  Prisms  and  Lenses  [Ch.  Ill 

PROBLEMS 

1.  (a)  A  ray  is  refracted  from  vacuum  into  a  medium 
whose  index  of  refraction  is\/2,  the  angle  of  incidence  being 
45°:  find  the  angle  of  refraction. 

(b)  Find  the  angle  of  incidence  of  a  ray  which  is  re- 
fracted at  an  angle  of  30°  from  vacuum  into  a  medium  of 
index  equal  to  \/3. 

(c)  Find  the  relative  index  of  refraction  when  the 
angles  of  incidence  and  refraction  are  30°  and  60°,  respec- 
tively. Ans.  (a)  30°;  (b)  60°;  (c)  y/%:  3. 

2.  Assuming  that  the  indices  of  refraction  of  air,  water, 
glass  and  diamond  have  the  values  1,  -|,  f  and  -§,  respec- 
tively, calculate  the  angle  of  refraction  in  each  of  the 
following  cases: 

(a)  Refraction  from  air  to  glass,  angle  of  incidence  40°; 
(b)  from  air  to  water,  angle  of  incidence  60°;  (c)  from  air 
to  diamond,  angle  of  incidence  75°;  (d)  from  glass  to  water, 
angle  of  incidence  30°;  (e)  from  diamond  to  glass,  angle  of 
incidence  36°  52'  11.6".  Ans.  (a)  25°  22'  26";  (6)  40°  30'  19"; 
(c)  22°  43'  44";  (d)  34°  13'  44";  (e)  90°. 

3.  The  height  of  a  cylindrical  cup  is  4  inches  and  its  di- 
ameter is  3  inches.  A  person  looking  over  the  rim  can  just 
see  a  point  on  the  opposite  side  2.25  inches  below  the  rim. 
But  when  the  cup  is  filled  with  water,  looking  in  the  same 
direction  as  before,  he  can  just  see  the  point  of  the  base 
farthest  from  him.    Find  the  index  of  refraction  of  water. 

Ans.  4:3. 

4.  The  index  of  a  refracting  sphere  is\/3;  it  is  surrounded 
by  air.  A  ray  of  light,  entering  the  sphere  at  an  angle  of 
incidence  of  60°  and  passing  over  to  the  other  side,  is 
there  partly  reflected  and  partly  refracted.  Show  that  the 
reflected  ray  and  the  emergent  ray  are  at  right  angles  to 
each  other. 

5.  In  the  preceding  problem,  show  that  the  reflected  ray 
will  cross  the  sphere  again  and  be  refracted  back  into  the 


Ch.  Ill]  Problems  93 

air  in  a  direction  exactly  opposite  to  that  which  the  ray  had 
before  it  entered  the  sphere. 

6.  A  straight  line  drawn  through  the  center  C  of  a  spher- 
ical refracting  surface  meets  the  surface  in  a  point  desig- 
nated by  A.  If  J,  J'  designate  the  points  where  an  inci- 
dent ray  and  the  corresponding  refracted  ray  intersect  the 

TL  71 

straight  line  AC,  and  if  CJ  =  — .AC,  show  that  CJ'=— .AC, 

n  n' 

where  n,  n'  denote  the  indices  of  refraction  of  the  first 

and  second  media,  respectively. 

7.  Construct  the  path  of  a  ray  refracted  at  a  plane  sur- 
face. Draw  diagrams  for  the  cases  when  n'  is  greater  and 
less  than  n.    Construct  the  critical  angle  in  each  figure. 

8.  The  velocity  of  light  in  air  is  approximately  186000 
miles  per  second.  How  fast  does  it  travel  in  alcohol  of 
index  1.363?         Ans..  Approximately,  136  460  miles  per  sec. 

9.  A  fish  is  8  feet  below  the  surface  of  a  pool  of  clear  water. 
A  man  shooting  at  the  place  where  the  fish  appears  to  be 
points  his  gun  at  an  angle  of  45°.  Where  will  the  bullet 
cross  the  vertical  line  that  passes  through  the  fish?  (Take 
index  of  water  as  1.33,  and  neglect  any  deflection  of  the 
bullet  caused  by  impact  with  the  water.) 

Ans.  3  feet  above  the  fish. 

10.  Assuming  that  the  velocity  of  light  in  air  is 
30  000  000  000  cm.  per  sec,  calculate  its  velocity  in  water 
and  in  glass. 

11.  Prove  that  nab  =  ncb:  nac. 

12.  Show  that  the  sine  of  the  critical  angle  of  an  optical 
medium  is  equal  to  the  reciprocal  of  the  absolute  index  of 
refraction. 

13.  Assuming  same  values  of  the  indices  of  refraction  as 
in  problem  No.  2,  calculate  the  values  of  the  critical  angle 
for  each  of  the  following  pairs  of  media:  (a)  air  and  glass, 
(b)  air  and  water,  (c)  air  and  diamond. 

Ans.  (a)  41°  48'  40";  (b)  48°  35'  25";  (c)  23°  34' 41". 

14.  A  45°  prism  is  used  to  turn  a  beam  of  light  by  total 
internal  reflection  through  a  right  angle.     What  must  be 


94  Mirrors,  Prisms  and  Lenses  [Ch.  Ill 

the  least  possible  value  of  the  index  of  refraction  of  the 
glass?  Ans.  \/2. 

15.  Show  that  when  a  ray  of  light  passes  from  air  into 
a  medium  whose  index  of  refraction  is  equal  to\/2>  the  de- 
viation cannot  be  greater  than  45°. 

16.  The  absolute  index  of  refraction  of  a  certain  trans- 
parent substance  is  -§.  Show  that  a  luminous  point  at  the 
center  of  a  cube  of  this  material  cannot  be  seen  by  an 
eye  in  the  air  outside,  if  at  the  center  of  each  face  of  the 
cube  a  circular  piece  of  opaque  paper  is  pasted  whose  radius 
is  equal  to  three-eighths  of  the  edge  of  the  cube. 

17.  What  will  be  the  greatest  apparent  zenith  distance  of 
a  star  to  an  eye  under  water? 

18.  Explain  why  it  is  that  it  is  not  possible  for  a  person 
by  merely  opening  his  eyes  under  water  to  see  distinctly 
objects  in  the  water  around  him  or  in  the  air  above  the 
water;  whereas,  if  he  is  provided  with  a  diver's  helmet  with 
a  plate  glass  window  in  it,  he  will  experience  no  difficulty 
in  distinguishing  such  objects  clearly. 

19.  Rays  of  light  are  emitted  upwards  in  all  directions 
from  a  luminous  point  at  the  bottom  of  a  trough  contain- 
ing a  layer  of  a  transparent  liquid  3  inches  in  depth  and  of 
refractive  index  1.25.  Show  that  all  rays  which  meet  the 
surface  outside  a  certain  circle  whose  center  is  vertically 
above  the  point  will  be  totally  reflected;  and  find  the  radius 
of  this  circle.  Ans.  4  inches. 

20.  A  pin  with  a  white  head  is  stuck  perpendicularly  in 
the  center  of  one  side  of  a  flat  circular  cork,  and  the  cork 
is  floated  on  water  with  the  pin  downwards.  Assuming 
that  the  head  of  the  pin  is  2  inches  below  the  surface  of  the 
water,  find  the  smallest  diameter  the  cork  can  have  so  that 
a  person  looking  down  through  the  water  (index  -|)  from 
the  air  above  (index  unity)  could  not  see  the  head  of  the  pin. 

Ans.  4.535  inches. 

21.  Plot  a  curve  showing  the  deviation  e  as  a  function 
of  the  angle  of  incidence  a  for  the  case  when  the  refraction 
is  from  water  (n  =  4/3)  to  air  (nf  =  1) . 


CHAPTER  IV 

REFRACTION  AT  A  PLANE  SURFACE,  AND  ALSO  THROUGH  A 
PLATE  WITH  PLANE  PARALLEL  FACES 


40.  Trigonometric  Calculation  of  Ray  Refracted  at  a 
Plane  Surface. — A  geometrical  construction  of  the  path  of 
the  refracted  ray  was 
given  in  §  34.  The  path 
of  a  ray  refracted  at  a 
plane  surface  may  also 
be  easily  determined  by 
trigonometric  calculation. 
The  straight  line  yy  in 
Figs.  60  and  61  represents 
the  plane  refracting  sur- 
face Separating  the  two  FlG-  60.-RefractioD .of  ray  at  plane  sur- 
1       ,       .&  face:  a  =  AL,  v  =AL    (n  >n). 

media   of   indices   n,    n', 

and  the  straight  line  LB  shows  the  path  of  a  ray  which  is  in- 
cident on  yy  at  the  point  marked  B.    The  straight  line  LA 

perpendicular  to  yy  at  A 
is  the  axis  of  the  refract- 
ing plane  with  respect 
to  the  position  of  the 
point  L.  The  magni- 
tudes ?;  =  AL  a  = 
Z  ALB  which  determine 
completely  the  position 

Fig.  61. — Refraction  of  ray  at  plane  surface:    q£   ^\q   incident   rav   are 

v  =  AL,  v  =AL'  (n'<n).  J 

sometimes  called  the 
ray-coordinates.  Let  L'  designate  the  point  where  the  re- 
fracted ray  L'B  intersects  the  axis  xx,  and  let  z/  =  AL', 
a'  =  Z  AL'B  denote  the  coordinates  of  the  refracted  ray.    The 

95 


B 

V 

x    ^\a 

-"vV 

X 

L 

n 

n' 
y 

96 


Mirrors,  Prisms  and  Lenses 


41 


problem  is:  Given  the  incident  ray  (v,  a),  determine  the  re- 
fracted ray  (*/,  a'). 

From  either  diagram  we  obtain  immediately  the  relation: 

z/_tan  a 

v  "tana" 

and  since  n.sma  =  n'.sma',  we  obtain  finally  the  following 
formulae  for  calculating  the  refracted  ray: 


v  V? 


n 


cos  a 


??2.sm2a     .      ,    n    . 

-,  sin  a  =— .sin  a. 

n' 


Now  if  the  point  L  is  a  luminous  point,  rays  will  emanate 
from  it  in  all  directions,  and,  whereas  the  magnitude  v  will 

remain  the  same  for  all  these 
rays,  the  angle  a  will  vary  from 
ray  to  ray.  But  for  different 
values  of  a,  in  general  we  shall 
obtain  different  values  of  the 
magnitude  v',  and,  consequently, 
the  position  of  the*  point  1/  on 
the  axis  will  be  different  for  dif- 
ferent incident  rays  coming  from 
L.  Accordingly,  the  bundle  of 
Fig.  62.— Refraction  of  paraxial  refracted  rays   corresponding  to 

rays  at  plane  surface:  u=  ,  . " 

a  homocentnc  bundle  of  mcident 


AM,  w'  =  AM' 

(n'>n). 


u  :  n  =u:n, 


rays  will  not  be  homocentric. 
41.  Imagery  in  a  Plane  Refracting  Surface  by  Rays 
which  Meet  the  Surface  Nearly  Normally. — The  more 
or  less  blurred  and  distorted  appearance  of  objects  seen 
under  water  is  familiar  to  everybody.  When  the  rays 
that  enter  the  eye  meet  the  surface  of  the  water  very 
obliquely,  the  distortion  is  almost  grotesque.  If  the  pupil 
of  the  eye  were  not  comparatively  small,  it  would  indeed 
be  practically  almost  impossible  to  recognize  an  object  under 
water,  even  if  the  eye  were  placed  in  the  most  favorable 
position  vertically  over  the  object.  It  is  only  because  the 
apertures  of  the  bundles  of  effective  rays  that  enter  the  eye 


41] 


Plane  Refracting  Surface 


97 


are  quite  narrow,  that  there  is  any  true  image-effect  at  all 
in  the  case  of  refraction  at  a  plane  surface. 

When  the  eye  looks  directly  along  the  normal  to  the 
plane  refracting  surface  at  an  object-point  M  on  the  other 
side  of  the  surface  (Figs.  62  and 
63),  the  effective  rays  coming 
from  M  will  meet  the  surface 
very  nearly  perpendicularly, 
and  the  incidence-points  will 
all  be  so  close  to  the  point  A 
that  there  will  be  practically  no 
difference  between  the  lengths 
of  the  straight  lines  MA  and 
MB,  and  accordingly  under 
these  circumstances  we  may 
write  sin  a  in  place  of  tan  a. 
Similarly,  also,  with  respect  to 
the  refracted  ray,  sin  a'  can  be  substituted  here  for  tan  a/. 
And  if  in  this  case  we  put  KM  =  u,  AM,  =  ii',  where  M,  M' 
designate  the  points  where  a  ray  which  is  very  nearly  nor- 
mal to  the  refracting  plane  crosses  the  normal  before  and 
after  refraction,  we  have  therefore, 


Fig.  63. — Refraction  of  paraxial 
rays  at  plane  surface:  m  =  AM, 
u'  =  AM',  u' :  n'  =  u:n  ,  (n'<n). 


tan  a      sin  a 


u      tan  a'     sin  a' 
and,  hence,  by  the  law  of  refraction : 

n'     n  .     n' 

—  —  -  ,     or     u  =  —.u. 
u'     u  n 

The  angle  a  has  disappeared  entirely  from  this  formula,  and 
the  value  of  v!  may  be  found  as  soon  as  the  value  of  u  is  given. 
This  means  that  corresponding  to  a  given  position  of  the 
object-point  M  there  is  a  perfectly  definite  image-point  M', 
and  the  points  M,  M'  are  said  to  be  a  pair  of  conjugate  points. 
Accordingly,  when  a  narrow  bundle  of  homocentric  rays  is 
incident  nearly  normally  on  a  refracting  plane,  the  correspond- 
ing bundle  of  refracted  rays  will  be  homocentric  also.    And  if 


98 


Mirrors,  Prisms  and  Lenses 


[§42 


the  aperture  of  the  bundle  is  infinitely  narrow,  the  imagery 
will  be  ideal. 

For  example,  a  pebble  at  the  bottom  of  a  pool  of  water 
12  inches  deep  will  be  seen  distinctly  from  a  point  in  the  air 
vertically  above  it,  but  it  will  appear  to  be  only  9  inches 
below  the  surface  of  the  water,  since  n'/w  =  3/4.  On  the  other 
hand,  an  object  9  inches  above  the  surface  will  seem  to  be 

12  inches  above  it  to  an  eye  in 
the  water  vertically  beneath  the 
object,  beca-use  in  this  case 
n'/n= 4/3. 

42.  Image  of  a  Point  Formed 
by  Rays  that  are  Obliquely  Re- 
fracted at  a  Plane  Surface. — 
But  if  the  bundle  of  rays  com- 
ing from  the  luminous  point  S 
(Fig.  64)  is  a  wide-angle  bundle 
of  considerable  aperture,  no  dis- 

Fig.  64,-Caustic  by  refraction  at    tinct   imaSe    wil1  be    formed   b^ 

plane  surface  from  water  to  these  rays  after  refraction  at  a 
air'  plane,  but  the  points  of  inter- 

section of  the  refracted  rays  will  be  spread  over  a  so- 
called  caustic  surface,  which  in  this  case  is  a  surface  of 
revolution  around  the  normal  SA  drawn  from  S  to  the  re- 
fracting plane.  The  figure  shows  a  meridian  section  of  this 
surface  for  the  case  when  the  rays  are  refracted  from  a 
denser  to  a  rarer  medium  (n'<  n),  the  curve  in  this  case  being 
the  evolute  of  an  ellipse.  Each  refracted  ray  produced  back- 
wards touches  the  caustic  surface.  The  cusp  of  the  meridian 
curve  is  on  the  normal  SA  at  the  point  M'  where  the  image  of 
S  is  formed  by  rays  that  meet  the  refracting  plane  nearly  per- 
pendicularly, as  explained  in  the  preceding  section.  Wherever 
the  eye  is  placed  in  the  second  medium,  only  a  narrow 
bundle  of  rays  coming  from  S  can  enter  it  through  the  pupil 
of  the  eye.  The  nearest  approach  to  an  image  of  the  source 
at  S  as  seen  by  rays  that  are  refracted  more  or  less  obliquely 


§  42]  Caustic  Surface  99 

will  be  the  little  element  of  the  caustic  surface  which  is  the 
assemblage  of  the  points  where  the  effective  rays  that  enter 
the  eye  touch  this  surface.  Thus,  rays  entering  the  eye  at  E 
appear  to  come  from  the  point  S'  where  the  tangent  from  E 
touches  the  caustic.    It  is  evident  now  why  an  object  S  under 


WATER 


Fig.  65. — Rod   partly   immersed  in  water  appears  to   be  bent 
upwards. 

water  appears  to  be  raised  towards  the  surface  and  at  the 
same  time  also  to  be  shifted  towards  the  spectator  more  and 
more  as  the  eye  at  E  is  brought  nearer  to  the  surface  of  the 
water,  until  finally  when  the  eye  is  on  a  level  with  the  surface 
of  the  water,  the  image  of  S  appears  now  to  be  at  V  on  the 
refracting  plane.  Rays  from  S  that  meet  the  surface  beyond 
this  limiting  point  V  where  the  caustic  curve  is  tangent  to 
the  straight  line  ZZ  will  be  totally  reflected.  The  image  of  S 
seen  by  the  eye  at  E  is  blurred  and  distorted,  because 
the  image-point  S'  is  the  point  of  intersection  of  a  very 
limited  portion  of  the  bundle  of  refracted  rays  that  enter 
the  eye. 

The  above  explanation  makes  it  clear  why  a  straight  line 
ABC  (Fig.  65)  which  is  partly  in  air  and  partly  in  water  will 
appear  to  an  eye  at  E  to  be  bent  at  B  into  the  broken  line 
ABC'.  The  image  BC  of  the  part  BC  under  water  can  be 
plotted  point  by  point  for  any  position  of  the  eye. 


100 


Mirrors,  Prisms  and  Lenses 


[§43 


43.  The  Image-lines  of  a  Narrow  Bundle  of  Rays  Re- 
fracted Obliquely  at  a  Plane. — The  diagram  (Fig.  66)  shows 
the  paths  of  two  rays  SBD  and  SCE  which  originating  at  S 
and  falling  on  the  refracting  plane  ZZ  at  the  points  B  and  C 
are  refracted  in  the  directions  CE  and  BD  into  the  eye  of  an 
observer.    The  refracted  rays  produced  backwards  intersect 

at  S'  and  cross  the  normal 
SA  at  the  points  marked 
WandV.  Evidently,  all 
the  rays  from  S  that  fall 
on  the  refracting  plane  at 
points  between  B  and  C 
will,  after  refraction,  in- 
tersect SA  at  points  be- 
tween V  and  W.  Sup- 
pose   that  the  figure    is 

Fig.  66. — Oblique  refraction  at  plane  sur-    revolved    around    SA    as 
face  (n'<n).  ^    then    each    my   ^yj 

generate  a  conical  surface,  and  the  vertices  of  these  cones  will 
be  at  the  points  S,  V,  and  W  for  the  rays  that  are  actually 
drawn  in  the  diagram.  The  bundle  of  rays  that  enter  the  eye 
at  DE  will  be  a  small  portion  of  the  refracted  rays  that  are 
contained  between  the  conical  surfaces  whose  vertices  are 
at  V  and  W.  These  conical  surfaces  intersect  each  other  in 
the  circle  which  is  described  by  the  point  S'  when  the  figure 
is  rotated  around  the  axis  SA,  and  it  is  a  little  element  of 
arc  of  this  circle  perpendicular  to  the  plane  of  the  diagram 
at  S'  that  contains  the  points  of  intersection  of  the  rays  that 
enter  the  eye.  This  is  called  the  primary  image-line  (§188)  of 
the  narrow  bundle  of  refracted  rays.  There  is  another 
image  line  at  V  called  the  secondary  image-line,  which  lies  in 
the  plane  of  the  paper,  and  which  is  generally  taken  as  per- 
pendicular to  the  axis  of  the  bundle  of  refracted  rays,  though 
sometimes  it  is  considered  as  the  segment  VW  of  the  axis 
of  revolution.  But  these  are  intricate  matters  that  can  be 
only  alluded  to  in  this  place.    (See  Chapter  XV.) 


§44] 


Path  of  Ray  through  Plate 


101 


44.  Path  of  a  Ray  Refracted  Through  a  Slab  with  Plane 
Parallel  Sides. — When  a  ray  of  light  traverses  several  media 
in  succession,  then 

ui .  sin  ai  =  rbi .  sin  a/,     n^ .  sin  a2  =  n3 .  sin  a2' ',  etc. , 
where  nh  n^,  n3,  etc.,  denote  the  indices  of  refraction  of  the 
media,  and  ah  oi';  a2,  02';  etc.,  denote  the  angles  of  incidence 
and  refraction  at  the  various  surfaces  of  separation.    In  the 


Fig.  67. — Path  of  ray  refracted  through  plate  with  plane  parallel  sides. 

special  case  when  these  refracting  surfaces  are  a  series  of 
parallel  planes,  the  angle  of  incidence  at  one  plane  will  be 
equal  to  the  angle  of  refraction  at  the  preceding  plane 
( ak  +i  =  a/,  where  the  integer  k  denotes  the  number  of  the 
plane). 

The  simplest  case  of  this  kind  occurs  when  there  are  only 
two  parallel  refracting  planes,  and  when  the  last  medium  is 


102  Mirrors,  Prisms  and  Lenses  [§  44 

the  same  as  the  first,  as,  for  example,  in  the  case  of  a  slab 
of  glass  bounded  by  plane  parallel  sides  and  surrounded  by 
air,  as  represented  in  Fig.  67.    Then 

n<i  =  ni  =  n,  ri2  =  n', 
and  ai=a2=a'. 

Accordingly,  we  have  the  following  pair  of  equations: 

n .  sin  ai  =  n' .  sin  af,  n' .  sin  a'  =  n .  sin  a2f ; 
and,  therefore: 

0.2  —  CLi  =  a; 
which  means  that  the  ray  emerges  from  the  slab  in  the  same 
direction  as  it  entered  it.  Thus,  when  a  ray  of  light  traverses 
a  slab  with  plane  parallel  sides  which  is  bounded  by  the  same 
medium  on  both  sides,  the  emergent  ray  will  be  parallel  to  the 
incident  ray.  Obviously,  this  statement  may  be  amplified 
as  follows:  When  a  ray  of  light  traverses  a  series  of  media  each 
separated  from  the  next  by  one  of  a  series  of  parallel  refracting 
planes,  the  final  and  original  directions  of  the  ray  will  be 
parallel,  provided  the  first  and  last  media  have  the  same  index 
of  refraction. 

The  only  effect  of  the  interposition  of  the  glass  plate 
(Fig.  67)  in  the  path  of  the  ray  is  to  shift  the  path  to  one 
side  without  altering  the  direction  of  the  ray.  It  might  be 
inferred,  therefore,  that  the  apparent  position  of  an  object 
as  seen  through  such  a  plate  of  glass  would  not  be  altered, 
but  this  is  not  true  in  general,  as  we  shall  proceed  to  explain. 
Every  ray  that  traverses  the  plate  will  be  found  to  be  dis- 
placed at  right  angles  to  its  original  position  through  a  dis- 
tance 

sin(a-aO 
cos  a'       ' 
where  d  denotes  the  thickness  of  the  plate.    Since 

\/n'2-n2.sin2a 

cos  a  = , 

n' 

the  formula  above  may  be  put  also  in  the  following  form : 
"R  t>_  sma  (~\/^/2-^2sin2a-n.cosa  )  , 
\/V2-ft2.sin2a 


§  44]  Plate  with  Plane  Parallel  Faces  103 

Accordingly,  the  shift  B2D  varies  with  the  slope  of  the  in- 
cident ray.  If  the  object  is  very  far  away,  the  rays  that 
enter  the  eye  will  be  parallel,  so  that  the  apparent  position 
of  a  distant  object  will  not  be  altered  in  the  slightest  by  view- 
ing it  through  a  plate  of  glass  with  plane  parallel  sides,  no 
matter  what  may  be  the  angle  of  incidence  of  the  rays,  and 
consequently  the  plate  may  be  turned  to  the  rays  at  dif- 


0&^ 

^k* 


Fig.  68. — Apparent  position  of  object  seen  through  plate  with 
plane  parallel  sides 

ferent  angles  without  producing  any  change  in  the  appear- 
ance of  the  object  as  seen  through  it.  But  if  the  object- 
point  S  (Fig.  68)  is  near  at  hand,  an  eye  at  E  will  see  it  in 
the  direction  ES,  but  when  the  glass  is  interposed,  it  will 
appear  to  lie  in  the  direction  ES'  which  is  sensibly  different 
from  ES,  and  this  difference  can  be  increased  or  diminished 
by  rotating  the  plate  around  an  axis  perpendicular  to  the 
plane  of  the  figure.  This  principle  is  utilized  very  ingeniously 
in  the  original  form  of  ophthalmometer  designed  by  Helm- 
holtz  (1821-1894)  for  measuring  the  curvatures  of  the  re- 
fracting surfaces  of  the  eye.  It  is  employed  also  in  an  instru- 
ment for  measuring  the  diameter  of  a  microscopic  object, 
which  Professor  Poynting  has  called  the  "  parallel  plate  mi- 
crometer" (see  Proc.  Opt.  Convention,  London,  1905,  p.  79). 


104  Mirrors,  Prisms  and  Lenses  [§  45 

45.  Segments  of  a  Straight  Line. — The  finite  portion  of 
a  straight  line  included  between  two  points  is  called  a  segment 
of  the  line,  while  each  of  the  other  two  parts  of  the  line  is  to 
be  regarded  as  a  prolongation  of  the  segment.  Considered 
as  generated  by  the  motion  of  a  point  along  a  straight  line 
from  a  starting-point  or  origin  A  to  an  end-point  or  terminus 

A& B,   the   segment   AB   is 

_ g frequently  spoken  of  also 

<  bZ  m  as  the  step  from  A  to  B 

Fig.    60.— Segments  of  a  straight  line:     Or    the     Step     AB.       The 

AB  =  -ba.  order  of  naming  the  two 

capital  letters  placed  at  the  ends  of  a  segment  describes 
the  sense  of  the  motion  or  the  direction  of  the  segment.  Thus, 
with  respect  to  direction  the  step  BA  (Fig.  69)  is  exactly  the 
reverse  of  the  step  AB. 

Two  steps  AB  and  CD  are  said  to  be  congruent,  that  is, 
AB  =  CD, 
provided  these  steps  are  not  only  equal  in  length  but  ex- 
ecuted in  the  same  sense. 

If  A,  B,  C  are  three  points  ranged  along  a  straight  line  in 
any  order,  that  is,  if  AB  and  CD  are  two  steps  along  the  same 
straight  line  such  that  the  end  of  one  step  is  the  starting 
point  of  the  other,  then  the  step  AC  is  said  to  be  equal  to 
the  sum  of  the  steps  AB  and  BC;  thus, 

AB+BC  =  AC; 
and  hence  also : 

AB  =  AC-BC,  BC  =  AC-AB. 
Moreover,  if  we  suppose  that  the  point  C  is  identical  with 
the  point  A,  it  follows  that 

AB+BA  =  0orAB=  -BA. 
Thus,  if  one  of  the  two  directions  along  a  straight  line  is 
regarded  as  the  positive  direction,  the  opposite  direction  is 
to  be  reckoned  as  negative.  For  example,  if  the  distance 
between  A  and  B  is  equal  to  12  linear  units,  and  if  we  put 
AB=  +12,  thenBA=  -12. 


46] 


Apparent  Position 


105 


Similarly,  also,  we  may  write: 

AB+BC+CA  =  0; 
or  if  X  designates  the  position  of  any  fourth  point  on  the 
straight  line,  then 

AB+BC+CX  =  AX. 

These  ideas  will  be  found  to  be  of  great  service  in  treating 
a  certain  class  of  problems  in  geometrical  optics;  and  an 
application  of  this  method  of  adding  line-segments  occurs 
in  the  following  section. 

46.  Apparent  Position  of  an  object  seen  through  a 
transparent  Slab  whose  Parallel  Sides  are  perpendicular 
to  the  Line  of  Sight. — In 
Fig.  70  the  line  of  sight 
joining  the  object-point 
Mi  with  the  spectator's 
eye  at  E  is  perpendicular 
at  Ai  and  A2  to  the  paral- 
lel faces  of  the  transpar- 
ent slab,  and  all  the  rays 
that  enter  the  eye  will 
pass  through  the  slab 
close  to  this  axial  line. 
Inside  the  slab  they  will  proceed  as  if  they  had  originated  at 
a  point  Mi'  on  the  line  of  sight,  but  being  again  refracted, 
they  will  emerge  into  the  surrounding  medium  as  if  they  had 
come  from  a  point  M2',  which  is  the  apparent  position  of  the 
object-point  as  seen  by  rays  that  are  very  nearly  perpen- 
dicular to  the  faces  of  the  slab.  If  n,  n'  denote  the  indices 
of  refraction  of  the  two  media,  then,  according  to  §§  41  and 
45,  we  may  write  the  following  equations: 


n 

-  '.*'  "' 

n 

k\       M,   M^       A, 

y-.       .ja*  '          E 

Fig.  70. — Displacement  of  object  viewed 
perpendicularly  through  plate  with 
plane  parallel  sides. 


AiMi    AiMi' 


A2Mi,  =  A2A1+A1M1/, 


A2Mi'    A2M2' 


106  Mirrors,  Prisms  and  Lenses  [§  46 

Hence,  the  apparent  displacement  of  the  object  is: 
MiM2'  =  MiAi+AiA2+A2M2' 

77 

=  MiA1+A1A2+-A2M1/ 

lb 

77 

=  M1A1+A1A2+-  (A2A1+A1M1') 

lb 

=M1A1+A1A2(l-^)+A1M1=^AIA2; 

lb  lb 

accordingly,  if  the  thickness  of  the  plate  is  denoted  by 
d  =  AiA2, 

MlM2'  =  ^d. 

Thus,  we  see  that  the  apparent  displacement  in  the  line  of 
sight  depends  only  on  the  thickness  of  the  plate  and  on  the 
relative  index  of  refraction  (V:  n),  and  is  entirely  independent 
of  the  distance  of  the  object-point  from  the  slab.  Hence, 
also,  the  size  of  the  image  of  a  small  object  viewed  directly 
through  a  glass  plate  is  the  same  as  that  of  the  object,  but 
its  apparent  size  will  be  different,  because  since  the  image 
and  object  are  at  different  distances  from  the  eye,  the  angles 
which  they  subtend  will  be  different. 

An  object  viewed  perpendicularly  through  a  glass  plate 
surrounded  by  air  (nr :  n  =  3 :  2)  will  appear  to  be  one-third 
the  thickness  of  the  plate  nearer  the  eye  than  it  really  is. 

If  the  displacement  of  the  object  is  denoted  by  x,  that  is, 
if  we  put  MiM'2  =  x,  then 

n'_  d 
n  d-x' 
This  relation  has  been  utilized  in  a  method  of  determining 
the  relative  index  of  refraction  {n'\  n).  A  microscope  S 
pointed  vertically  downwards  is  focused  on  a  fine  scratch 
or  object-point  O.  A  plate  of  the  material  whose  index 
is  to  be  determined  is  then  inserted  horizontally  between 
the  object  and  the  objective  of  the  microscope.  The  inter- 
position of  the  plate  necessitates  a  re-focusing  of  the  micro- 
scope  in   order   to  see  the   object   distinctly,   which   will 


§47] 


Multiple  Images 


107 


now  appear  to  be  at  a  point  0'  nearer  the  microscope  by 
the  distance  #  =  00'.  This  distance  x  is  easily  ascertained 
in  terms  of  the  distance  through  which  the  objective  of  the 
microscope  has  to  be  raised  in  order  to  obtain  a  distinct  image 
of  the  object.  The  thickness  of  the  plate  is  easily  measured, 
and,  consequently,  we 
have  all  the  data  for  de- 
termining the  value  of 
n'jn.  This  method  is 
especially  convenient  for 
obtaining  the  index  of 
refraction  of  a  liquid 
(Fig.  71). 

47.  Multiple  Images 
in  th  e  two  Parallel  Faces 
of  a  plate  glass  Mir- 
ror.— An  object  is  repro- 
duced in  a  metallic  mir- 


Fig.  71.- 


-Measurement  of  index  of  refrac- 
tion of  a  liquid. 


ror  by  a  single  image,  but  in  a  glass  mirror  which  is  silvered 
on  the  back  side  there  will  be  a  series  of  images  of  an  object 
in  front  of  the  glass,  which  may  be  readily  seen  by  looking  a 
little  obliquely  at  the  reflection  of  a  candle-flame  in  an  or- 
dinary looking  glass.  The  first  image  will  be  comparatively 
faint,  the  second  one  the  brightest  and  most  distinct  of  all, 
and  behind  these  two  principal  images  other  images  more 
or  less  shadowy  may  also  be  discerned  whose  intensities 
diminish  rapidly  until  they  fade  from  view  entirely.  These 
multiple  images  by  reflection  may  also  be  seen  in  a  trans- 
parent block  of  glass  with  plane  parallel  sides. 

The  light  falling  on  the  first  surface  is  partly  reflected  and 
partly  refracted.  It  is  this  reflected  portion  that  gives  rise 
to  the  first  image  of  the  series.  The  rays  that  are  refracted 
across  the  plate  will  be  partly  reflected  at  the  second  face, 
and,  returning  to  the  first  face,  a  portion  of  this  light  will  be 
refracted  back  into  the  air  and  give  rise  to  the  second  image 
of  the  series;  while  the  other  portion  of  the  light  will  be  re- 


108 


Mirrors,  Prisms  and  Lenses 


[§47 


fleeted  back  into  the  glass  to  be  again  reflected  at  the  back 
face,  and  so  on.  In  the  diagram  (Fig.  72)  the  source  of  the 
light  is  supposed  to  be  at  the  point  marked  S,  and  the  straight 


Fig.  72. — Multiple  images  by  reflection  from 
the  two  parallel  faces  of  a  plate  of  glass. 

line  drawn  from  S  perpendicular  to  the  parallel  faces  of  the 
glass  slab  meets  these  faces  in  the  points  marked  Ai  and  A2. 
The  path  of  one  of  the  rays  coming  from  S  is  indicated  in 


§47]  Multiple  Images  109 

the  figure,  and  it  can  be  seen  how  it  zigzags  back  and  forth 
between  the  two  sides  of  the  slab,  becoming  feebler  and 
feebler  in  intensity  at  each  reflection.  We  consider  here  only 
such  rays  from  S  as  meet  the  surface  very  nearly  normally. 
The  series  of  images  of  S  will  be  formed  at  S',  S",  S"',  etc., 
all  lying  on  the  prolongation  of  the  normal  SAiA2,  and  it  is 
because  these  images  are  all  ranged  in  a  row  one  behind  the 
other,  that  ordinarily  when  we  look  in  a  mirror  we  do  not  see 
the  images  separated. 

The  reflected  ray  1  proceeds  as  if  it  had  come  from  S',  the 
position  of  this  point  being  determined  by  the  relation  A]S'  = 
SAi.  But  the  refracted  ray  crosses  the  slab  as  if  it  had  come 
from  the  point  T,  the  position  of  which  is  determined  by  the 
relation  TAi  =  n.SAi,  where  n  denotes  the  index  of  refraction 
of  the  glass  (the  other  medium  being  assumed  to  be  air  of 
index  unity).  Arriving  at  the  second  face,  this  ray  will  be 
reflected  as  if  it  had  come  from  a  point  U  such  that  A2U  = 
TA2.  Returning  to  the  first  surface,  it  will  be  partly  re- 
fracted out  into  the  air  as  the  ray  marked  2  proceeding  as 
if  it  came  from  the  second  image-point  S",  the  position  of 
which  is  determined  by  the  relation  AiS"  =  AiU/n;  and  also 
partly  reflected  as  if  it  had  come  from  a  point  V  such  that 
VAi  =  AiU.  The  ray  is  reflected  a  second  time  at  the  second 
face,  as  if  it  came  from  the  point  W,  where  A2W  =  VA2 ;  and 
being  once  more  refracted  at  the  first  face,  emerges  into  the 
air  as  the  ray  marked  3,  appearing  now  to  come  from  the 
image-point  marked  S'"  determined  by  the  relation  AiS'"  = 
AiW/n. 

What  is  the  interval  between  one  image  and  the  next? 
For  example,  let  us  try  to  obtain  an  expression  for  the  inter- 
val S"S'".    This  may  be  done  as  follows: 
S"S'"  =  S"Ai+AiS'"; 

AiS"'  =  AiW/n=  (AiA2+A2W)/n=  (AiA2+VA2)/n 
=  (A!A2+VAi+AiA2)/tt=  (AiU+2AiA2)/n 
=  AiS"+2d/n; 


110  Mirrors,  Prisms  and  Lenses  [Ch.  IV 

where  d  =  AiA2  denotes  the  thickness  of  the  glass  plate. 
Hence,  we  find : 

n ' 

It  appears,  therefore,  that  the  distance  between  one  image 

2 
and  the  next  is  constant  and  equal  to  -  times  the  thickness 

of  the  plate.  Thus,  for  a  glass  plate  for  which  n  =  3/2  the 
distance  from  one  image  to  the  next  is  equal  to  4/3  the  thick- 
ness of  the  plate. 

PROBLEMS 

1.  A  ray  of  light  traverses  in  succession  a  series  of  isotropic 
media  bounded  by  parallel  planes,  and  emerges  finally  into 
a  medium  with  the  same  index  of  refraction  as  that  of  the 
first  medium.  Show  that  the  final  path  of  the  ray  is  parallel 
to  its  original  direction. 

2.  Construct  accurately  the  paths  of  six  rays  proceeding 
from  a  point  below  the  horizontal  surface  of  water  and  re- 
fracted into  air;  and  show  where  the  object-point  will  appear 
to  be  as  seen  by  an  eye  above  the  surface  of  the  water,  for 
three  different  positions  of  the  eye. 

3.  Why  does  the  part  of  a  stick  obliquely  immersed  in 
water  appear  to  be  bent  up  towards  the  surface  of  the  water? 
Explain  clearly. 

4.  Derive  the  formula  —,  =  -  for  the  refraction  of  paraxial 

u     u 

rays  (§63)  at  a  plane  surface. 

5.  A  ray  of  light  incident  on  a  plane  refracting  surface  at 
an  angle  a  crosses  a  straight  line  drawn  perpendicular  to  the 
surface  at  a  distance  v  from  this  surface.  How  far  from  the 
surface  does  the  refracted  ray  cross  this  line? 

6.  If  a  bird  is  36  feet  above  the  surface  of  a  pond,  how  high 
does  it  look  to  a  diver  who  is  under  the  water?  What  is  the 
apparent  depth  of  a  pool  of  water  8  feet  deep? 

Ans.  48  feet  above  the  surface;  6  feet. 


Ch.  IV]  Problems  111 

7.  What  will  be  the  effect  on  the  apparent  distance  of  an 
object  if  a  slab  of  transparent  material  with  plane  parallel 
sides  is  interposed  at  right  angles  to  the  line  of  vision? 

Ans.  It  will  appear  to  be  nearer  the  eye  by  the  amount 
(n—  I)  jd,  where  d  denotes  the  thickness  of  the  slab  and  n  de- 
notes the  index  of  refraction  of  the  material. 

8.  A  cube  of  glass  of  index  of  refraction  1.6  is  placed  on  a 
fiat,  horizontal  picture;  where  does  the  picture  appear  to  be 
to  an  eye  looking  perpendicularly  down  on  it? 

Ans.  It  will  appear  to  be  raised  three-eighths  of  the  thick- 
ness of  the  cube. 

9.  A  microscope  is  placed  vertically  above  a  small  vessel 
and  focused  on  a  mark  on  the  base  of  the  vessel.  A  layer  of 
transparent  liquid  of  depth  d  is  poured  in  the  vessel,  and  then 
it  is  found  that  the  image  of  the  mark  has  been  displaced 
through  a  distance  x  which  is  determined  by  re-focusing  the 
microscope.  Show  that  the  index  of  refraction  of  the  liquid 
is  equal  to  d/(d  —  x). 

10.  In  an  actual  experiment  made  by  the  above  method  to 
determine  the  index  of  refraction  of  alcohol,  the  depth  of  the 
liquid  was  4  cm.,  and  the  displacement  of  the  image  was 
found  to  be  1.06  cm.  What  value  was  found  for  the  index  of 
alcohol?     Ans.  1.36. 

11.  A  candle  is  observed  through  a  tank  of  water  with 
vertical  plane  glass  walls.  The  line  of  sight  is  perpendicular 
to  the  sides  of  the  tank,  the  candle  being  15  cm.  from  one 
side  and  39  cm.  from  the  opposite  side.  What  is  the  apparent 
position  of  the  candle?  (Neglect  the  effect  of  the  thin  glass 
walls.)     Ans.  It  appears  to  be  9  cm.  from  the  near  side. 

12.  If  an  object  viewed  normally  through  a  plate  of  glass 
with  plane  parallel  faces  seems  to  be  five-sixths  of  an  inch 
nearer  than  it  really  is,  how  thick  is  the  glass? 

Ans.  2.5  inches. 

13.  A  layer  of  ether  2  cm.  deep  floats  on  a  layer  of  water 
3  cm.  deep.    What  is  the  apparent  distance  of  the  bottom  of 


112  Mirrors,  Prisms  and  Lenses  [Ch.  IV 

the  vessel  below  the  free  surface  of  the  ether?    (Take  index  of 
refraction  of  water  =1.33  and  of  ether  =  1.36.) 

Ans.  3.73  cm. 

14.  A  person  looks  perpendicularly  into  a  mirror  made 
of  plate  glass  of  thickness  one-half  inch  silvered  on  the  back. 
If  his  eye  is  at  a  distance  of  15  inches  from  the  front  face, 
where  will  his  image  appear  to  be? 

Ans.  152/3  inches  from  the  front  face. 

15.  When  a  stick  is  partly  immersed  in  a  transparent 
liquid  of  index  n  at  an  angle  6  with  the  free  horizontal  sur- 
face, what  is  the  angle  6 '  which  the  part  of  the  stick  below 
the  surface  appears  to  make  with  the  horizon  as  seen  by  an 
eye  looking  vertically  down  on  it  from  the  air  above  the 
liquid? 

tan0 
Ans.  tancr  = . 


CHAPTER  V 

REFRACTION    THROUGH   A    PRISM 

48.  Definitions,  etc. — An  optical  prism  is  a  limited  portion 
of  a  highly  transparent  substance  with  polished  plane  faces 
where  the  light  is  reflected  or  refracted.  Prisms  in  a 
great  variety  of  geometrical  forms  and  combinations  are 
employed  in  many  types  of  modern  optical  instruments  (cf. 
§§  20,  37) ;  but  in  this  chapter  the  term  prism  will  be  re- 
stricted to  mean  a  portion  of  a  transparent,  isotropic  sub- 
stance included  between  two  polished  plane  faces  that  are 
not  parallel.  The  straight  line  in  which  the  planes  of  the 
two  faces  meet  is  called  the  edge  of  the  prism,  and  the  di- 
hedral angle  between  these  planes  is  called  the  refracting  angle. 
This  angle,  which  will  be  denoted  by  the  symbol  /3,  may  be 
more  precisely  defined  as  the  convex  angle  through  which  the 
first  face  of  the  prism  has  to  be  turned  around  the  edge  of  the 
prism  as  axis  in  order  to  bring  this  face  into  coincidence  with 
the  second  face.  The  first  face  of  the  prism  is  that  side  where 
the  rays  enter  and  the  second  face  is  the  side  from  which  the 
rays  emerge.  Every  section  made  by  a  plane  perpendicular 
to  the  edge  of  the  prism  is  a  principal  section,  and  we  shall 
consider  only  such  rays  as  traverse  the  prism  in  a  principal 
section,  not  only  because  the  problem  of  oblique  refraction 
through  a  prism  presents  some  difficulties  which  are  beyond 
the  scope  of  this  volume,  but  especially  because  in  actual 
practice  the  principal  rays  are  usually  confined  to  a  principal 
section  of  the  prism.  It  will  also  be  assumed,  for  simplicity, 
that  the  prism  is  surrounded  by  the  same  medium  on  both 
sides. 

I.  Geometrical  Investigation 

49.  Construction  of  Path  of  a  Ray  Through  a  Prism. — 
The  plane  of  the  diagram  (Fig.  73)  represents  the  principal 

113 


114  Mirrors,  Prisms  and  Lenses  [§  49 

section  of  a  prism  whose  edge  meets  this  plane  perpendicu- 
larly at  the  point  marked  V.  The  traces  of  the  two  plane 
faces  are  shown  by  the  straight  lines  ZiV,  Z2V  intersecting  at 
V.  The  straight  line  ABi  represents  the  path  of  the  given 
incident  ray  lying  in  the  plane  of  the  principal  section  and 


Fig.  73. — Construction  of  path  of  ray  through  principal  section 
of  prism  (n'>n). 

falling  on  the  first  face  of  the  prism  at  the  incidence-point  Bi. 
The  problem  of  constructing  the  path  of  the  ray  both  within 
the  prism  and  after  emergence  from  it  is  solved  by  a  method 
essentially  the  same  as  that  employed  in  §  34. 

Let  n  denote  the  index  of  refraction  of  the  medium  sur- 
rounding the  prism  and  n'  the  index  of  refraction  of  the  prism- 
medium  itself.    With  the  point  V  as  center,  and  with  radii 

equal  to  r  and  —  .r,  where  the  radius  r  may  have  any  con- 
venient length,  describe  the  arcs  of  two  concentric  circles  both 
lying  within  the  angle  Z2VE,  where  E  designates  a  point  on 
the  prolongation  of  the  straight  line  ZiV  beyond  V.    Through 


§49]  Construction  of  Ray  through  Prism  115 

V  draw  a  straight  line  VG  parallel  to  ABi  meeting  the  arc 
of  radius  nr/n'  in  the  point  designated  by  G;  and  through 
the  point  G  draw  a  straight  line  GE  perpendicular  at  E  to 
the  first  face  of  the  prism  (produced  if  necessary),  and  let  H 
designate  the  point  where  the  straight  line  GE  (likewise 
produced  if  necessary)  meets  the  circumference  of  the  other 
of  the  two  circular  arcs.  Then  the  straight  line  BiB2  drawn 
parallel  to  the  straight  line  VH  will  represent  the  path  of 
the  ray  within  the  prism.  For  if  the  straight  line  NiN/  is  the 
incidence-normal  to  the  first  face  of  the  prism  at  the  point 
Bi,  and  if  the  angles  of  incidence  and  refraction  at  this  face 
are  denoted  by  ai  =  ZNiBiA,ai,=  ZNi/BiB2,  then  by  the 
law  of  refraction : 

n.sinai  =  n'.sinai'.- 
But  by  the  construction  : 

sinZEGV    VH    n' 

sinZEHVVG-n' 

and  since  Z  EGV=  Z  NiBiA  =  ai,  it  follows  that  Z  EHV  =  a/; 
and  hence  the  path  of  the  ray  within  the  prism  must  be 
parallel  to  VH. 

Again,  from  the  point  H  let  fall  a  perpendicular  HF  on 
the  second  face  of  the  prism,  where  F  designates  the  foot  of 
this  perpendicular;  and  let  J  designate  the  point  where  HF 
intersects  the  arc  of  radius  nr/n'.  Then  the  straight  line 
B2C  drawn  from  the  incidence-point  B2  parallel  to  the  straight 
line  VJ  will  represent  the  path  of  the  emergent  ray.  For  if 
we  draw  N2N2'  perpendicular  to  the  second  face  of  the  prism 
at  B2,  and  if  the  angles  of  incidence  and  refraction  at  this 
face  are  denoted  by  a2  =  ZN2B2Bi,  a2'  =  ZN2'B2C,  re- 
spectively, then  n' .  sin  a2  =  n .  sin  a2'.    But 

sinZFJV      VH     n' 

sinZFHV"  VJ~n' 

and  since  by  construction  ZFHV=  a2,  it  follows  that 
ZFJV=  a2',  and  hence  the  path  of  the  emergent  ray  will 
be  parallel  to  VJ. 


116  Mirrors,  Prisms  and  Lenses  [§  50 

The  diagram  (Fig.  73)  is  drawn  for  the  case  when  nf>n,  as 
in  the  ordinary  case  of  a  glass  prism  surrounded  by  air.  The 
student  should  draw  also  a  diagram  for  the  other  case  when 
n'<n,  showing  the  procedure  of  a  ray  through  a  prism  of  less 
highly  refracting  substance  than  that  of  the  surrounding 
medium,  for  example,  an  air  prism  surrounded  by  glass,  such 
as  is  formed  by  the  air-space  between  two  separated  glass 
prisms. 

50.  The  Deviation  of  a  Ray  by  a  Prism. — The  total  de- 
viation of  a  ray  refracted  through  a  prism,  which  is  equal  to 
the  algebraic  sum  of  the  deviations  produced  by  the  two 
refractions  (§35),  may  be  defined  as  the  angle  e=  ei~\r  e2 
through  which  the  direction  of  the  emergent  ray  must  be 
turned  in  order  to  bring  it  into  the  direction  of  the  incident 
ray;  thus,  in  Fig.  73,  e  =  ZJVG;  and  if  the  angle  e  is  meas- 
ured in  radians,  the  arc  JG  =  e .  J V.  In  order  to  specify 
completely  an  angular  displacement,  it  is  necessary  to  give 
not  only  the  magnitude  of  the  angle  and  the  sense  of  rotation 
of  the  radius  vector,  but  also  the  plane  in  which  the  displace- 
ment occurs.  This  plane  may  be  specified  by  giving  the 
direction  of  a  line  perpendicular  to  it,  which  in  the  case  of 
the  angle  here  under  consideration  may  be  the  edge  of  the 
prism  or  any  line  parallel  to  it;  because  any  such  line  will 
be  perpendicular  to  the  principal  section  of  the  prism  in 
which  the  ray  lies.  In  fact,  the  angle  e  may  be  completely 
represented  in  a  diagram  by  a  straight  line  drawn  parallel 
to  the  edge  of  the  prism,  which  by  its  length  indicates  the 
magnitude  of  the  angle  and  by  its  direction  shows  the  sense 
of  rotation.  Thus,  for  example,  the  line  may  be  drawn  along 
the  edge  of  the  prism  itself  from  a  point  V  in  the  plane  of  the 
principal  section  and  always  in  such  a  direction  that  on 
looking  along  the  line  towards  that  plane  Z  JVG  =  e  will 
be  seen  to  be  a  counter-clockwise  rotation.  A  deviation  of 
20°  in  a  principal  section  coinciding,  say,  with  the  plane  of 
the  paper  would  be  represented,  therefore,  by  a  straight  line 
perpendicular  to  this  plane  of  length  20  cm.,  if  each  degree 


§  51]  Ray  "Grazes"  one  Face  of  Prism  117 

were  to  be  represented  by  one  centimeter.  If  e=  +20°,  this 
line  would  point  out  from  the  paper  towards  the  reader,  and 
if  €  =  -20°,  it  would  point  away  from  him.  Thus,  if  the 
prism,  originally  "base  down,"  is  turned  "base  up"  (as  the 
opticians  say),  everything  else  remaining  the  same,  the  sign 
of  the  angle  e  will  be  reversed,  and  so  also  will  be  the  direc- 
tion of  the  vector  which  represents  this  angle. 

51.  Grazing  Incidence  and  Grazing  Emergence. — The 
angle  GHJ  between  the  normals  to  the  two  faces  of  the  prism 
is  equal  to  the  refracting  angle  (3;  and  hence  for  a  given  prism 
this  angle  will  remain  always  constant.  No  matter  how  the 
direction  of  the  incident  ray  ABi  (or  VG)  may  be  varied, 
the  vertex  H  of  this  angle  will  lie  always  on  a  certain  portion 
of  the  circumference  of  the  construction-circle  of  radius  r, 
and  the  sides  HG,  HJ  will  remain  always  in  the  same  fixed 
directions  perpendicular  to  the  faces  of  the  prism.  Obviously, 
there  will  be  two  extreme  or  limiting  positions  of  the  point  H 
marking  the  ends  of  the  arc  on  which  it  is  confined,  namely, 
the  positions  winch  H  has  when  one  of  the  sides  of  the  angle 
GHJ  is  tangent  to  the  circle  of  radius  nrjn';  which  can  occur 
only  for  the  case  when  n'>n,  because  otherwise  the  point  H 
will  lie  inside  the  circumference  of  this  circle  and  therefore 
it  will  be  impossible  for  either  HG  or  HJ  to  be  tangent  to  it. 

If  the  side  HG  is  tangent  to  the  inner  circle  at  G,  as  shown 
in  Fig.  74,  the  point  G  will  lie  in  the  plane  of  the  first  face 
of  the  prism,  and  accordingly  the  corresponding  ray  incident 
on  the  first  face  of  the  prism  at  the  point  Bi,  which  must 
have  the  direction  VG,  will  be  the  ray  ZiBi  which,  entering 
the  prism  at  "grazing"  incidence  (ai  =  90°),  traverses  the 
prism  as  shown  in  the  figure. 

On  the  other  hand,  when  the  side  HJ  of  the  angle  GHJ  is 
tangent  at  J  to  the  construction-circle  of  radius  nr\n'  (Fig.  75), 
the  point  J  will  lie  in  the  second  face  of  the  prism,  and  the 
straight  line  VJ  will  coincide  with  the  straight  line  VZ2. 
Under  these  circumstances  the  ray  emerges  from  the  prism  at 
B2  along  the  second  face  in  the  direction  B2Z2(a2/ '=  -90°). 


118 


Mirrors,  Prisms  and  Lenses 


51 


The  straight  line  KBX  shows  the  path  of  the  ray  incident  on 
the  first  face  of  the  prism  at  Bi  which  " grazes"  the  second 
face  on  emerging  from  the  prism.  Any  ray  incident  at  Bi  and 
lying  in  the  principal  section  of  the  prism  within  the  angle 
KB1Z1  will  succeed  in  getting  through  the  prism  and  emerging 


Fig.  74. — Case  when  ray  "grazes"  first  face  of  prism. 


into  the  surrounding  medium  again;  whereas  if  the  ray  in- 
cident at  Bi  lies  anywhere  within  the  angle  VBiK,  it  will 
be  totally  reflected  at  the  second  face  of  the  prism.  The 
ray  KBi  is  called  the  limiting  incident  ray  and  ZNiBiK  =  t 
is  the  limiting  angle  of  incidence.  These  relations  will  be 
discussed  more  fully  in  the  analytical  investigation  of  the 
path  of  a  ray  through  a  prism  (§§  55,  foil.) ;  but  it  may  be 
remarked  that  ZGHV=  a/  in  Fig.  74  and  ZJHV=  a2  in 
Fig.  75  are  both  equal  to  the  critical  angle  A  (§  36)  with 
respect  to  the  two  media  n,  n'  (sinA=n/n'). 


52] 


Minimum  Deviation 


119 


52.  Minimum  Deviation. — Between  the  two  extreme  or 
terminal  positions  of  the  vertex  H  of  ZGHJ  shown  in 
Figs.  74  and  75,  there  is  also  an  intermediate  place  which  is 
of  special  interest  and  importance  and  to  which,  therefore, 
attention  must  be  called.    In  general,  the  sides  HG,  HJ  inter- 


Fig.  75. — Case  when  ray  "grazes"  second  face  of  prism. 

cepted  between  the  two  construction-circles  (Fig.  73)  will 
be  unequal  in  length,  but  if  HG  =  H J,  as  in  Fig.  76,  the  angles 
GVJ,  GHJ  and  EVF  will  evidently  all  be  bisected  by  the 
diagonal  VH  of  the  quadrangle  VGHJ.  When  this  happens, 
the  path  BiB2  of  the  ray  inside  the  prism,  which  is  parallel 
to  VH,  crosses  the  prism  symmetrically,  that  is,  the  triangle 
VBiB2  is  isosceles.  In  fact,  the  points  designated  in  the  dia- 
gram by  the  letters  V,  D  and  O  will  be  the  summits  of  isos- 
celes triangles  having  the  common  base  BiB2,  and  they  will 
all  lie  therefore  on  the  bisector  of  the  refracting  angle  /3  = 


120  Mirrors,  Prisms  and  Lenses  [§  52 

ZZiVZ2,  which  is  perpendicular  to  VH.  The  angle  of  in- 
cidence at  the  first  face  and  the  angle  of  emergence  at  the 
second  face  are  equal  in  magnitude,  although  they  are  de- 
scribed in  opposite  senses,  so  that  012'  =  —  ai.  The  same  is 
true  also  in  regard  to  the  angles  which  the  ray  makes  inside 


Fig.  76. — Ray  traverses  prism  symmetrically  (VBi  =  VB2) ;  case 
of  minimum  deviation. 

the  prism  with  the  normals  to  the  two  faces,  that  is,   ci2  = 
-a/. 

Now  when  the  ray  traverses  the  prism  symmetrically,  as 
represented  in  Fig.  76,  the  deviation  e  has  its  least  value 
€Q.  In  order  to  show  that  this  is  true,  it  will  be  convenient 
to  reproduce  the  symmetrical  quadrangle  VGHJ  in  Fig.  76 
in  a  separate  diagram,  as  in  Fig.  77.  Suppose  that  H'  desig- 
nates the  position  of  a  point  infinitely  near  to  H  lying  likewise 
on  the  arc  of  the  circle  of  radius  r,  and  draw  H'G',  H'J' 
parallel  to  HG,  HJ  and  meeting  the  arc  of  the  other  circle 
in  the  points  G',  J',  respectively.  In  the  figure  the  point  H' 
is  taken  below  the  point  H,  and  in  this  case  it  is  plain  that 


Fig.  77. — Case  of  minimum  deviation. 


§  52]  Minimum  Deviation  121 

the  two  parallels  HJ,  H'J'  will  meet  the  circumference  of 

the  inner  circle  more  obliquely  than  the  other  pair  of  parallel 

lines  HG,  H'G',  and,  consequently,  the  infinitely  small  arc 

J' J   intercepted    between 

the    first   pair    will    be 

greater  than  the  arc  G'G 

intercepted    between  the 

second  pair.     Hence,  the 

small  angle  J'VJ  will  be 

greater  than  Z  G'VG,  and 

therefore 

ZJ'VG'>ZJVG. 
The  angle  JVG  here  is  the 
angle  of  deviation  ( e0)  of 
the  ray  that  goes  sym- 
metrically through  the  prism;  whereas  Z  J' VG/=  €  is  the  angle 
of  deviation  of  a  ray  which  traverses  the  prism  along  a  very 
slightly  different  path.  And  according  to  the  above  reason- 
ing (for  we  shall  arrive  at  the  same  result  if  we  take  the 
point  H'  also  above  H),  we  find: 

e>  e0. 
Accordingly,  we  see  that  the  ray  which  traverses  the  prism  sym- 
metrically in  the  plane  of  a  principal  section  is  also  the  ray 
which  is  least  deviated. 

It  is  easy  to  verify  this  statement  experimentally.  Thus, 
for  example,  if  a  bundle  of  parallel  rays  is  allowed  to  fall  on 
an  isosceles  triangular  prism,  so  that  while  some  of  the  rays 
are  incident  on  one  of  the  equal  faces  and  are  transmitted 
through  the  prism,  the  other  rays  of  the  bundle  are  reflected 
from  the  base  of  the  prism,  as  represented  in  (1)  in  Fig.  78; 
and  if  then  the  prism  is  gradually  turned  around  an  axis 
parallel  to  its  edge,  first,  into  position  (2),  which  is  the  posi- 
tion of  minimum  deviation,  and  then  past  this  position  into 
a  third  position  (3),  it  will  be  observed  that  when  the  prism 
is  in  the  position  of  minimum  deviation  the  rays  reflected 
from  the  base  will  be  parallel  to  the  rays  which  emerge  at 


122  Mirrors,  Prisms  and  Lenses  [§  53 

the  second  face  of  the  prism;  which  can  only  be  the  case 

when  the  rays  cross  the  prism  symmetrically. 

In  spectroscopic  work  and  in  many  other  scientific  uses  of 

the  prism,  the  position  of  mini- 
mum deviation,  which  is  easily 
found,  is  frequently  the  most 
convenient  and  advantageous  ad- 
justment of  the  prism  for  purposes 
of  observation. 

53.  Deviation  away  from  the 
Edge  of  the  Prism. — When  a  ray 
of  light  passes  through  a  prism  of 
more  highly  refracting  material 
than  that  of  the  surrounding  me- 
dium (n'>n),  the  deviation  is  al- 
ways away  from  the  edge  towards 
the  thicker  part  of  Ihe  prism. 

If  the  angles  of  the  triangle 
VBiB2  (Fig.  79)  at  Bx  and  B2  are 
both  acute,  the  incident  and 
emergent  rays  lie  on  the  sides  of 
the  normals  at  Bi  and  B2  away 

Fig.    78. — Experimental      proof    e  r*  •  j  ,1     ,       , 

that    ray   which    traverses    fr0m  the  P^sm-edge,   SO   that  at 

prism  symmetrically  is  ray  both  refractions  the  ray  will  be 

of  minimum  deviation.  u       ,  £  ,-,  •,  T£ 

bent  away  trom  the  edge.  It 
one  of  the  angles,  say,  the  angle  at  B2,  is  a  right  angle, 
the  ray  will  not  be  deviated  at  all  by  the  refraction  at 
this  point,  but  at  the  other  incidence-point  it  will  be  bent 
away  from  the  edge.  And,  finally,  if  one  of  the  angles  at  Bi 
or  B2  is  obtuse,  for  example,  the  angle  at  Bi  (Fig.  80),  the 
deviation  on  entering  the  prism  will,  it  is  true,  be  towards 
the  edge  of  the  prism,  but  this  deviation  will  not  be  so  great 
as  the  subsequent  deviation  away  from  the  edge  which  is 
produced  at  the  second  refraction  when  the  ray  issues  from 
the  prism,  as  may  be  easily  seen  from  the  diagram.    Thus, 


§  54]        Plane  Wave  Refracted  Through  Prism         123 

in  every  case  when  nf>n,  the  total  deviation  will  be  away 
from  the  prism-edge. 

If  n'<n,  all  these  effects  will  be  reversed. 


Fig.  79. — Deviation  away  from  edge  of  prism. 

54.  Refraction  of  a  Plane  Wave  Through  a  Prism. — 

The  diagram  (Fig.  81)  shows  a  principal  section  of  the  prism, 
and  the  straight  line  BiD  represents  the  trace  of  a  plane 
wave  (supposed  to  be  perpendicular  to  the  plane  of  the 
paper  and  parallel  therefore  to  the 
edge  of  the  prism)  advancing  to- 
wards the  first  face  of  the  prism  in 
the  direction  DV  at  right  angles  to 
BiD.  If  around  the  point  Bi,  which 
lies  in  the  first  face  of  the  prism,  the 
arc  of  a  circle  is  described  with  ra- 

71 

dius  BE  =  -,DV,  then,  according  to 

HUYGENS'S      principle       (§     5),     the    Fig.   80  —  Deviation     away 

straight  line  VE  tangent  to  this  circle         from  edge  of  prism' 

at  E  will  represent  the  trace  of  the  wave-front  inside  the  prism. 

Let  the  straight  line  BiE  meet  the  second  face  of  the  prism 

n' 
at  B2.    Around  V  as  center  with  radius  VF=  -  EB2  describe 


the  arc  of  a  circle;  then  the  straight  line  B2F  tangent  to  this 


124 


Mirrors,  Prisms  and  Lenses 


55 


circle  at  F  will  represent  the  trace  of  the  emergent  wave- 
front. 

The  disturbance  at  any  point  C  will  have  emanated  from 
some  point  on  ABi,  and  the  time 
taken  by  the  light  to  go  from  Bi  to 
B2  inside  the  prism  will  be  the 
same  as  that  required  to  go  from 
D  to  F  in  the  surrounding  me- 
dium (§  39) ;  that  is,  the  optical 
lengths  along  these  two  routes 
are  equal.    For,  as  appears  from 

Fig.    81.— Refraction   of  plane    ^he  Construction, 

wave  through  prism.  n(DV+ VF)  =  n'.BiB2. 

An  excellent  and  most  instructive  mechanical  illustration 
of  the  refraction  of  a  plane  wave  through  a  prism  can  be  ob- 
tained by  using  the  roller  and  tilted  board  described  in  §  32 
with  a  triangular  piece  of  plush  cloth  glued  in  the  middle 
of  the  board  to  represent  the  prism  (see  Fig.  45). 


II.  Analytical  Investigation 

55.  Trigonometric  Calculation  of  the  Path  of  a  Ray  in  a 
Principal  Section  of  a  Prism. — The  angles  of  incidence  and 
refraction  at  the  first  and  second  faces  of  the  prism,  denoted 
by  ai,  ai'  and  a2,  a2',  are,  by  definition  (§  27),  the  acute  angles 
through  which  the  normals  to  the  refracting  surfaces  at  the 
incidence-points  have  to  be  turned  in  order  to  bring  them  into 
coincidence  with  the  incident  and  refracted  rays  at  the  two 
faces  of  the  prism;  thus,  in  Fig.  73,  ZNiBiA=  rii,  ZNi'BiB2 
=  ai',  ZN2B2Bi=  a2,  ZN2/B2C=  a2'. 

Assuming  that  the  prism  is  surrounded  by  the  same  me- 
dium on  both  sides,  and  being  careful  to  note  the  sense  of 
rotation  of  each  of  the  angles,  we  obtain  by  the  law  of  re- 
fraction, taken  in  conjunction  with  the  obvious  geometrical 
relations  as  shown  in  the  figure,  the  following  system  of 


§  56]  Total  Reflection  in  Prism  125 

equations  for  calculating  the  path  of  a  ray  through  a  prin- 
cipal section  of  a  prism : 

n'.sinai'  =  7i.sinai,   a2=  a/—  fi,  n . sin a2'  =  n' . sin a2. 
Combining  these  formulse  so  as  to  ehminate  ai'  and  a2,  we 
may  derive  the  following  convenient  expression  for  deter- 
mining the  angle  of  emergence  (a2')  at  the  second  face  of  the 
prism : 

,  o  nVn'2-n2 .  sin2  a\ 

sin  a2  =  sm  ai .  cos  p  -  sin  p . 

n 

Thus,  if  we  know  the  value  of  the  relative  index  of  refraction 

(n'/ri)  and  the  refracting  angle  of  the  prism  (/3=ZZiVZ2), 

we  can  calculate  the  angle  of  emergence  ( a2')  corresponding 

to  any  given  direction  (ai)  of  the  ray  incident  on  the  first 

face  of  the  prism. 

The  total  deviation  ( e )  of  a  ray  refracted  through  a  prism 

is  measured,  as  defined  above  (§50),  by  ZJVG,  and  since 

this  angle  is  equal  to  the  external  angle  at  D  in  the  triangle 

DBiB2,  we  have: 

€=ZB2BiD+ZDB2Bi 

=  Z  Ni'BiD  -  Z  NiBiB2-f  Z  DB2N2  -  Z  BiB2N2 

=  ai—  a/—  a2'+  a2; 
and  since   a\  —  a2=/3,  we  obtain  finally  the  following  ex- 
pression for  the  angle  of  deviation: 

e=  ai-  a2'-/3. 
These  formulae  contain  the  whole  theory  of  the  refraction  of 
a  ray  through  a  prism  in  a  principal  section.  It  will  be  in- 
teresting to  discuss  analytically  some  of  the  special  cases 
which  we  have  already  studied  in  the  preceding  sections  of 
this  chapter. 

56.  Total  Reflection  at  the  Second  Face  of  the  Prism. — 
If  the  angle  of  emergence  at  the  second  face  of"  the  prism  is 
a  right  angle,  that  is,  if  a2'  =  -90°,  the  emergent  ray  B2C 
will  issue  from  the  prism  along  the  second  face  in  the  direc- 

Tl  Tl 

tion  B2Zi  (Fig.  75) .  Hence,  sin  <x2  =  — .  sin  a2'  = -„  and  there- 

Tl  Tl 

fore  a2  =  -  A,  where  A  denotes  the  critical  angle  (§  36)  of  the 


126  Mirrors,  Prisms  and  Lenses  [§  56 


n 


media  n,  n',  denned  by  the  relation  sinA  =  — .    If  the  absolute 

value  of  the  angle  a2  is  greater  than  A,  the  ray  will  be  totally 
reflected  at  the  second  face  of  the  prism,  and  there  will  be  no 
emergent  ray.    This  case  may  be  discussed  in  some  detail. 

For  a  prism  of  given  refracting  angle  (/J),  there  is  a  certain 
limiting  value  (t)  of  the  angle  of  incidence  ( cti)  at  the  first 
face  of  the  prism  (§  51)  for  which  we  shall  have  at  the  second 


/ 
/ 
/ 
1 
1 

\ 

\ 

X 

1 

\ 

^V.       1 

t 

&r 

Xl 

t 

n   t 

— 7& 

1 
/ 

/        1 

'•tk 

/       y 

r            1 

i    jS 

1 

1 
1 

,     t*4  = 


^=A. 


-K 


Fig.  82. — Prism  with  refracting  angle  /3  =  2A. 

face  the  values  a2= -A,  a2'  =  -90°;  so  that  a  ray  which  is 

incident  on  the  first  face  of  the  prism  at  an  angle  less  than  the 

limiting  angle  i  will  not  pass  through  the  prism  but  will  be 

totally  reflected  at  the  second  face.    Putting  a2=  —  A,  we  find 

a\  =  /3  -  A,  and  therefore,  since  ai  =  i, 

n' 
.    sin  i=— sin  'jfl~A), 

which  is  the  trigonometric  formula  for  computing  the  value 
of  the  limiting  angle  of  incidence  for  a  given  prism.    It  will 


§  56]  Limiting  Incident  Ray  127 

be  worth  while  to  examine  this  formula  for  certain  particular 
values  of  the  refracting  angle  /3. 

(1)  If  /3>2A,  then,  since  sinA  =  -?,  the  formula  shows  that 

sin  l  will  be  greater  than  unity,  so  that  for  a  prism  of  this 
form  there  is  no  angle  corresponding  to  the  limiting  angle  t. 
No  ray  can  be  transmitted  through  a  prism  whose  refracting 
angle  is  more  than  twice  as  great  as  the  critical  angle  of  the  two 
media  in  question.     A  prism  of  this  size  is  called  a  totally 


Fig.  83. — Prism  with  refracting  angle  /3  =  A. 

reflecting  prism;  if  it  is  made  of  glass  of  index  1 . 5  and  sur- 
rounded by  air,  the  refracting  angle  should  be  about  84°  at 
least. 

(2)  If  jft  =  2A,  we  find  that  t  =  90°;  which  is  the  case  repre- 
sented in  Fig.  82.  The  only  ray  that  can  get  through  this 
prism  is  the  ray  that  traverses  it  symmetrically,  entering  the 
prism  along  one  face  and  leaving  it  along  the  other. 

(3)  If  /3>  A  but  <2A  (that  is,  if  2A>  (3>  A),  the  value  of 
the  angle  i  as  determined  by  the  formula  above  will  be  com- 
prised between  90°  and  0°.  This  is  the  case  which  was  shown 
in  Fig.  73.    The  direction  of  the  limiting  incident  ray  is  be- 


128 


Mirrors,  Prisms  and  Lenses 


56 


Fig.  84. 


tween  ZiBi  and  NiBi;  that  is,  ZViBK  will  be  an  obtuse 
angle. 

(4)  If  j8  =  A,  we  find  i  =  0°,  and  then  the  limiting  incident 

ray  will  proceed  along  the 
normal  NiBi,  as  shown  in 
Fig.  83,  and  ZVBiK  (or 
ZVBiA)  will  be  a  right 
angle. 

(5)  FinaUy,  if  /5<A,the 
limiting  angle  of  inci- 
dence (i)  will  be  negative 
in  sign;  and  therefore  in  a 
more  or  less  thin  prism  of 
this  description  the  limit- 
ing incident  ray  KBi  will 
fall  on  the  side  of  the 
normal  NiBi  towards  the 
apex  V  of  the  prism,  so  that  the  angle  VBiK  will  be  an 
acute  angle  (Fig.  84). 

Any  ray  incident  on  the  first  face  of  the  prism  at  Bi  and 
lying  within  the  angle  KBiZi  will  be  transmitted  through 
the  prism;  whereas  if  the  ray  falls  within  the  supple- 
mentary angle  VBiK,  it  will  be  totally  reflected  at  the 
second  face. 

In  Kohlrausch's  method  of  measuring  the  relative  index 

n' 

of  refraction  (— ),  the  prism  is  adjusted  so  that  the  incident 

lb 

ray  " grazes"  the  first  face,  and  then  if  the  refracting  angle 
of  the  prism  (/3)  is  known,  and  if  the  angle  of  emergence 
(a2r)  is  measured,  the  value  of  n'\  n  may  be  calculated  by 
means  of  the  formula : 

cos/3-sina2' 


-Prism  with  a  refracting  angle 
/5<A. 


v/ 


;n 


&   "I 


■,  (ai  =  90°). 


n  ■  sin  P 

The  principle  of  total  reflection  is  also  employed  in  the 
prism  refractometers  of  Abbe  and  Pulfrich  for  measure- 
ment of  the  index  of  refraction. 


§  59]  Minimum  Deviation  129 

57.  Perpendicular  Emergence  at  the  Second  Face  of  the 
Prism. — For  this  case  we  have  d2  =  d2'  =  0°,  and  therefore 
di'=  fi,  cii=  £-€,  and  hence: 

n'_sin(ft-  e)  . 

n         sin/3 
which  is  also  a  convenient  formula  for  the  experimental  de- 
termination of  the  value  of  the  relative  index  of  refraction. 
A  description  of  the  apparatus  and  the  method  of  procedure 
may  be  found  in  the  standard  treatises  on  physics. 

58.  Case  when  the  Ray  Traverses  the  Prism  Symmet- 
rically.— As  has  been  pointed  out  already  (§  52),  a  special 
case  of  great  interest  occurs  when  the  ray  traverses  the  prism 
symmetrically.  Under  these  circumstances,  the  general 
prism-equations  given  in  §  55  take  the  following  forms: 

di=  -  d2  = — 2 — '   ai  =  ~"  a2=  2> 

.      /3+€o 
sin  — - — 


sin- 


where  eQ  denotes  the  angle  of  deviation  of  this  symmetric 
ray.  The  last  of  these  formulae  is  the  basis  of  the  Fraun- 
hofer  method  of  determining  the  relative  index  of  refrac- 
tion, the  angles  fi  and  e0  being  both  capable  of  easy  measure- 
ment. 

This  last  formula  may  also  be  transformed  into  the  fol- 
lowing form: 

n.sin| 
tan-^  = 


2       ,  e0' 

n  —  n.cos-^ 


whereby  the  refracting  angle  f3  can  be  calculated  in  terms 
of  n,  n'  and  e0. 

59.  Minimum  Deviation. — The  prism  itself  is  defined  by 
its  refracting  angle  (/3)  and  the  relative  index  of  refraction 
(n'/ri).    The  total  deviation  (e)  of  a  ray  refracted  through 


130  Mirrors,  Prisms  and  Lenses  [§  59 

a  given  prism  depends  only  on  the  angle  of  incidence  (ai), 
according  to  the  formula: 

€=  ai-  a2'-  P; 
for  the  angle  a2'  may  be  expressed  in  terms  of  ah  /3  and  n'[n, 
as  we  have  seen  (§  55).  Hence,  for  a  given  value  of  these 
three  magnitudes  the  angle  e  will  be  uniquely  determined. 
On  the  other  hand,  for  a  given  value  of  the  angle  e  there 
will  always  be  two  corresponding  values  of  the  angle  of  in- 
cidence ai;  for  it  is  obvious  from  the  principle  of  the  reversi- 
bility of  the  light-path  (§  29)  that  a  second  ray  incident  on 
the  first  face  of  the  prism  at  an  angle  equal  to  the  angle  of 
emergence  of  the  first  ray  will  emerge  at  the  second  face  at 
an  angle  equal  to  the  angle  of  incidence  of  the  first  ray  at 
the  first  face,  and  these  two  rays  will  be  equally  deviated  in 
passing  through  the  prism.  For  example,  suppose  that  the 
values  of  the  angles  of  incidence  and  emergence  in  the  case 
of  the  first  ray  are  ai=  7,  a2'  =  7':  a  second  ray  incident  on 
the  first  face  of  the  prism  at  the  angle  ai=  —  y'  will  emerge 
at  the  second  face  at  an  angle  a2'  = — y,  and  each  of  these  rays 
will  suffer  precisely  the  same  deviation,  viz.,  e=  7  —  7'  —  /3. 
Thus,  corresponding  to  any  given  value  of  the  angle  €, 
within  certain  limits,  there  will  always  be  a  pair  of  rays  which 
are  deviated  by  this  same  amount.  One  pair  of  such  rays 
consists  of  the  two  identical  rays  determined  by  the  relation 

di=7=-  a2'. 
In  fact,  this  is  the  ray  which  traverses  the  prism  symmet- 
rically, and  a  little  reflection  will  show  that  the  deviation  of 
this  ray  must  be  either  a  maximum  or  a  minimum. 

But  while  the  best  way  of  demonstrating  that  the  ray 
which  goes  symmetrically  through  the  prism  is  the  ray  of  mini- 
mum deviation  (§  52)  involves  the  employment  of  the  methods 
of  the  differential  calculus,  the  following  analytical  proof 
demands  of  the  student  a  knowledge  of  only  elementary 
mathematics. 

The  deviation  at  the  first  face  of  the  prism  is  ei  =  Hi—  a/, 


§  59]  Minimum  Deviation  131 

and  that  at  the  second  face  is    e2=a2-a2'    (§35),   and 
hence  the  total  deviation  is 

e=  €i-f-e2  =  (ai-  a/)  +  (a2-  a2'), 
or,  since  a/  -  a2  =  13,  e  =  ai  —  a2'  -  /3,  as  has  been  already 
remarked,  for  example,  in  §  55.  Assume  now  that  n'>n,  and, 
consequently,  that  the  angle  e  is  positive,  as  is  always  the 
case  when  the  ray  is  bent  away  from  the  edge  of  the  prism 
(§  53) ;  then  it  is  evident  that  the  angle  e  will  have  its  least 
value  (e0)  in  the  case  of  that  ray  for  which  the  function 
(ai—  a2')  is  least.    Now  since 

n .  sin  di  =  n' .  sin  a/,     n .  sin  a2'  =  n' .  sin  a2, 
we  obtain  by  subtraction : 

n(sin  ai  —  sin  a2')  =  7i'(sin  a/ — sin  a2), 
and  hence  by  an  obvious  trigonometric  transformation : 
.   d\-  a2'        ai+tt/       ,    .   ai'— a2        ai'+a2 
n.  sin — ^ — -cos — o — =w         — 9 — ,cos — 2 —  ' 

which  may  be  written  as  follows: 

.    ai-a2      n     .   p  2 

sm — ^ — =„  .sm-^.- 


According  as  ai=  —  a2r,  the  deviation  €i  at  the  first  face  of 

the  prism  will  (see  §  35)  be  greater  than,  equal  to,  or  less  than, 
the  deviation   e2  at  the  second  face;  that  is,  according  as 

ai=—  a2',  we  shall  have  ( ai—  di')=(  a2—  a2'),  and  hence  also 

ai+a2/>  a2+ai/ 
2       <       2       ' 

If  we  suppose,  first,  that  ai>  —  a2r,  then  a/>  —  a2  and 
(a2+a/)>0;  and  since  the  cosine  of  a  positive  angle  de- 
creases as  the  angle  increases,  it  follows  that  here  we  must 
have: 

a/+a2  ai-f-a2' 

cos x >cos — ^ —  • 

On  the  other  hand,  if  we  suppose,  second,  that  ai<  —  a2',  then 


132  Mirrors,  Prisms  and  Lenses  [§  60 

ai'<-a2  and  (a2+ai')<0;  but  in  this  case  (a2+ai')> 

( ai+  a2'),  so  that  although  ( a2+  a/)  and  ( cti+  a2')  are  both 

negative,  the  absolute  value  of  the  former  is  greater  than  that 

of  the  latter,  and  hence  here  also  we  find  exactly  the  same 

result  as  before. 

Thus,  whether  ai  is  greater  or  less  than  —  a2',  the  ratio 

ai'+ft2 

cos — ^ — 

aH-a2'       ' 
cos- 


2 

and  only  in  the  case  when  ai  =  —  a2'  will  this  ratio  equal  to 

unity.    Hence,  sin  — s has  its  least  value  when  a\  =  —  a/, 

and  then  also  the  deviation  ( e)  is  a  minimum  and  equal  to 
€0  =  2ai-/3. 
The  same  process  of  reasoning  applied  to  the  case  when 
n'<n  leads  to  the  conclusion  that  the  angle  €  will  be  a  maxi- 
mum for  the  ray  which  traverses  such  a  prism  symmetrically, 
for  example,  an  air-prism  surrounded  by  glass;  but  in  this  case 
the  angle  €  will  be  negative  in  sign,  and  since  a  maximum  value 
of  a  negative  magnitude  corresponds  to  a  minimum  absolute 
value,  the  actual  deviation  of  the  ray  is  least  in  this  case  also. 
60.  Deviation  of  Ray  by  Thin  Prism. — If  the  refracting 
angle  of  the  prism  (/3)  is  small,  as  represented,  for  example, 
in  Fig.  85,  the  deviation  (e)  will  likewise  be  a  small  angle 
of  the  same  order  of  smallness;  for  if  /5  =  a/  -  a2  is  small,  then 
( ai  —  a2')  will  be  small  also,  and  the  angle  e  is  the  difference 
between  these  two  small  magnitudes.    In  fact,  the  deviation 
€  produced  by  a  thin  prism  will  not  only  always  be  small, 
but  it  will  never  be  very  different  from  its  minimum  value 
e0.     Accordingly,  in  the  case  of  a  thin  prism,  we  may  put 
e  =  €0  without  much  error;  and   therefore  very  approxi- 
mately (see  §  58) : 

n  2 


n        .      & 
sin    ^ 


60] 


Deviation  in  Thin  Prism 


133 


Consequently,  the  deviation  e,  as  calculated  by  this  formula, 
will  depend  only  on  the  prism-constants  ( /3,  n' :  n)  and  not 
on  the  angle  of  incidence  ( ai).  The  smaller  the  angle  /5,  the 
more  nearly  correct  this  formula  will  be;  and  if  the  angle  /3 

is  so  small  that  we  may  substitute  ~  and  — ^ —  in  place  of 
sin  2  and  sin  — ^—,  respectively,  we  obtain  the  exceedingly 


Fig.  85. — Prism  with  comparatively  small  refracting 
angle. 


useful  and  convenient  practical  relation  for  the  angle  of 
deviation  of  a  ray  refracted  through  a  thin  prism,  viz.: 

which,  however,  is  more  frequently  written: 

«-(»-Dft 

where  n  is  employed  now  to  denote  the  relative  index  of 
refraction.    Accordingly,  in  a  thin  prism  the  deviation  is  di- 


134  Mirrors,  Prisms  and  Lenses  [§  61 

redly  proportional  to  the  refracting  angle.  For  example,  the 
deviation  in  the  case  of  a  thin  glass  prism  surrounded  by 
air  for  which  n  =  1 . 5  is  one-half  the  refracting  angle. 

61.  Power  of  an  Ophthalmic  Prism.  Centrad  and  Prism- 
Dioptry. — An  ophthalmic  prism  is  a  thin  glass  prism,  whose 
index  of  refraction  is  usually  about  1.52,  which  is  used  to 
correct  faulty  tendencies  and  weaknesses  of  the  ocular 
muscles  which  turn  the  eye  in  its  socket  about  the  center  of 
rotation  of  the  eye-ball.  In  an  ordinary  laboratory  prism 
the  two  faces  are  usually  cut  in  the  form  of  rectangles  having 
the  edge  of  the  prism  as  a  common  side;  but  the  contour  of 
an  ophthalmic  prism  which  has  to  be  worn  in  front  of  the  eye 
in  a  spectacle-frame  is  circular  or  elliptical  like  that  of  any 
other  eye-glass,  and  its  edge  is  the  line  drawn  tangent  to  this 
curve  at  the  thinnest  part  of  the  glass.  The  line  drawn 
perpendicular  to  this  tangent  at  the  point  of  contact  and 
lying  in  the  plane  of  one  of  the  faces  of  the  prism  is  the  so- 
called  "base-apex"  line,  which  is  a  term  frequently  employed 
by  writers  on  spectacle-optics. 

The  formula 

e=(n-l)/3 
obtained  in  §  69  is  peculiarly  applicable  to  the  weak  prisms 
used  in  spectacles.  As  long  as  the  refracting  angle  of  the 
prism  does  not  exceed,  say,  10°,  the  error  in  the  value  of  e 
as  calculated  by  this  approximate  formula  will  be  less  than 
5  per  cent. 

Formerly  it  was  customary  to  give  the  strength  or  power 
of  an  ophthalmic  prism  in  terms  of  its  refracting  angle  ft 
expressed  in  degrees;  but  the  proper  measure  of  this  power 
is  the  deviation  produced  by  the  prism.  However,  instead 
of  measuring  this  angle  in  degrees,  Dennett  has  suggested 
that  the  deviation  of  an  ophthalmic  prism  shall  be  measured 
in  terms  of  a  unit  angle  called  a  centrad,  which  is  the  one- 
hundredth  part  of  a  radian  and  equal  therefore  to  the  angle 
subtended  at  the  center  of  a  circle  of  radius  one  meter  by  an 
arc  of  length  one  centimeter.    Since  7r  radians  =  180°,  the 


§  61]  Centrad  and  Prism-Dioptry  135 

relation  between  the  centrad  and  the  degree  is  given  as 
follows : 

1°=    ,ft0  centrads, 

or 

1°  =  1 .  745  ctrd.,  1  ctrd.  =  0 .  573°. 
Prior  to  this  suggestion,  Mr.  C.  F.  Prentice,  of  New 
York,  had  proposed  in 
1888  to  measure  the  de- 
viation of  an  ophthalmic 
prism  in  terms  of  the 
linear  or  tangential  dis- 
placement  in    centime- 

i         j  Fig.  86. — Deviation  of  prism: 

ters  on  a  screen  placed  tan  €=ab:OA. 

at    a   distance    of    one 

meter  from  the  prism.  If  the  straight  lines  OA,  OB  (Fig.  86) 
represent  the  directions  of  the  incident  and  emergent 
rays,  respectively,  then  ZAOB  will  be  the  angle  of  devi- 
ation of  the  prism;  and  if  a  plane  screen  placed  at  right  angles 

AB 

to  OA  at  A  is  intersected  by  OB  at  B,  then  tanZ  AOB=^-. 

Now  if  the  distance  OA  =  100  cm.  and  if  AB  =  z  cm., 
then,  according  to  Prentice's  method,  the  ZAOB  would 
be  an  angle  of  x  units  and  the  power  of  the  prism  would  be 
denoted  by  x.  Dr.  S.  M.  Burnett  suggested  that  the  name 
prism-diopter  or  prism-dioptry  be  given  to  this  unit.  (The  term 
"prismoptrie"  was  proposed  by  Professor  S.  P.  Thompson.) 
The  prism-dioptry  is  the  angle  corresponding  to  a  deviation  of 
one  centimeter  on  a  tangent  line  at  a  distance  of  one  meter; 
and,  accordingly,  when  the  angle  of  deviation  is  equal  to  the 
angle  whose  trigonometric  tangent  is  x/100,  the  power  of  the 
prism  is  said  to  be  x  prism-dioptries  or  z  A,  where  the  symbol 
A  stands  for  prism-dioptry.  The  chief  objection  to  be  urged 
against  this  unit  of  angular  measurement  is  that  the  angle 
subtended  at  a  given  point  O  (Fig.  87)  by  equal  line-segments 
on  a  line  Ay  perpendicular  to  Ox  at  A  diminishes  as  the 


136  Mirrors,  Prisms  and  Lenses  [§  61 

segment  on  Ay  is  taken  farther  and  farther  from  A.  In 
other  words,  since  tan  — *  z/100  is  less  than  x .  tan  -  x  1/100,  x 
prism-dioptries  is  less  than  x  times  one  prism-dioptry.  Or- 
dinarily, the  variability  in  the  magnitude  of  a  unit  would 
constitute  an  insuperable  objection  to  it;  but  so  long  as  the 


Fig.  87. — Unequal   angles   subtended  at  O   by  equal  intervals  on  straight 
line  Ay  drawn  perpendicular  to  OA. 


angles  to  be  measured  are  always  small,  as  is  the  case  with 
ophthalmic  prisms,  the  prism-dioptry  may  be  regarded  as  in- 
variably equal  to  the  tan  - 1 1/100  or  about  34'  22. 6"  without 
sensible  error;  and  hence  we  may  say,  for  example,  that 
2A+3A  =  5A,  although  this  statement  is  not  quite  accu- 
rate. At  any  rate,  whatever  may  be  the  theoretical  objec- 
tions, this  unit  of  measurement  of  the  strength  of  a  thin 
prism  is  so  convenient  and  satisfactory  that  it  has  been  gen- 
erally adopted  in  ophthalmic  practice. 

In  point  of  fact,  with  the  small  angular  magnitudes  which 
are  here  pre-supposed  (the  power  of  an  ophthalmic  prism 
seldom  exceeds  6  ctrd.),  there  is  practically  no  distinction  to 
be  made  between  the  angle  itself  and  the  tangent  of  the  angle, 


§  61]  Centrad  and  Prism-  Dioptry  137 

so  that  we  may  regard  the  centrad  and  the  prism-dioptry  as 
identical  in  most  cases;  that  is, 

1A  =  1  ctrd.  =0.573°. 
Accordingly,  we  obtain  the  following  relation  between  the 
power    (p)    of   an   ophthalmic    prism   expressed   in   prism- 
diop tries  or  centrads  and  the  refracting  angle  (/3)  given  in 
degrees: 

P=^f  (n-l)/3=l. 745(n-l)ft 

where  n  denotes  the  relative  index  of  refraction.  If  n=  1.5, 
then  the  power  of  a  prism  of  refracting  angle  /3  degrees  is 
0.873  prism-dioptries. 

However,  in  order  to  exhibit  the  actual  relations  still 
more  clearly,  the  following  table  gives  the  values  in  degrees, 
minutes  and  seconds  of  all  integral  numbers  of  prism-dioptries 
and  centrads  from  1  to  20;  and  incidentally  it  will  be  seen 
that  whereas  an  angle  of  k  centrads  contains  k  times  as  many 
degrees,  minutes  and  seconds  as  an  angle  of  1  centrad,  where 
k  denotes  any  integer  from  1  to  20,  the  same  statement  is 
not  strictly  true  of  the  prism-dioptry. 


138 


Mirrors,  Prisms  and  Lenses 


[§62 


Prism- 
Dioptries 

Equivalent  in  degrees, 
minutes  and  seconds 

Centrads 

Equivalent  in  degrees, 
minutes  and  seconds 

1 

0°  34' 22.6" 

1 

0°  34'  22.7" 

2 

1°    8'  44. 8" 

2 

1°    8'  45.3" 

3 

1°43'    6.1" 

3 

1°43'    8.0" 

4 

2°  17' 26.2" 

4 

2°  17'  30.6" 

5 

2°  51'  44.7" 

5 

2°  51' 53.3" 

6 

3°  26'    1.1" 

6 

3°  26'  15.9" 

7 

4°    0'15.0" 

7 

4°    0'38.6" 

8 

4°  34' 26.1" 

8 

4°  35'    1.2" 

9 

5°    8' 33.9" 

9 

5°    9'  23.9" 

10 

5°  42' 38.1" 

10 

5°  43'  46.5" 

11 

6°  16' 38.3" 

11 

6°  18'    9.2" 

12 

6°  50' 34.0" 

12 

6°  52' 31.8" 

13 

7°  24' 24.9" 

13 

7°  26' 54.5" 

14 

7°  58' 10.6" 

14 

8°    1'17.1" 

15 

8°  31' 50.8" 

15 

8°  35' 39.8" 

16 

9°    5'  25.0" 

16 

9°  10'    2.4" 

17 

9°  38' 53.0" 

17 

9°  44'  25.1" 

18 

10°  12' 14.3" 

18 

10°  18' 47.7" 

19 

10°  45' 28.7" 

19 

10°  53' 10.4" 

20 

11°  18' 35.8" 

20 

11°  27'  33.0" 

62.  Position  and  Power  of  a  Resultant  Prism  Equivalent 
to  Two  Thin  Prisms. — In  ascertaining  the  prismatic  cor- 
rection of  the  eye  of  a  patient,  the  oculist  or  optometrist 
sometimes  finds  it  convenient  and  advantageous  to  employ 
a  combination  of  two  thin  prisms  placed  one  in  front  of  the 


62] 


Combination  of  Two  Thin  Prisms 


139 


other  with  their  edges  inclined  to  each  other  at  an  angle  7 

which  can  be  measured;  and  having  obtained  the  necessary 

correction  in  this  way,  he  has  to  prescribe  a  single  prism  which 

will  produce  precisely  the 

same  resultant  effect  as 

the     two     superposed 

prisms  of  the  trial-case. 

In  general,  it  would  be 

exceedingly  laborious  and 

difficult  to  calculate  the 

power  of  this   resultant 

prism,    but,  fortunately, 

the  problem  in  this  case 

is  enormously  simplified 


Fig.  88,  a. — Parallelogram  law  for  find- 
ing single  prism  equivalent  to  a  com- 
bination of  two  thin  prisms. 


by  the  fact  that  the  refracting  angles  are  so  small  that  it  is 
quite  simple  to  obtain  an  approximate  solution  which  is 
sufficiently  accurate  and  reliable  for  ordinary  practical 
purposes. 

Let  the  deviation-angles  or  powers  of  the  two  prisms,  de- 
noted by  pi  and  p2,  be  represented,  according  to  the  method 
•r/  explained    in    §  50,    by 

the  vectors  OA,  OB, 
respectively  (Fig.  88), 
which  are  drawn  parallel 
to  the  edges  of  the  prism, 
so  that  Z  AOB  =  7.  Com- 
plete the  parallelogram 
OACB  and  draw  the  di- 
agonal OC.  The  vector 
OC  will  represent  on 
the  same  scale  the  deviation-angle  or  power  p  of  the  resultant 
prism,  as  we  shall  proceed  to  show. 

If  a  point  P  is  taken  anywhere  in  the  plane  of  the  parallelo- 
gram OACB,  it  may  easily  be  proved  that  the  area  of  the 
triangle  POC  is  equal  to  the  sum  or  difference  of  the  areas  of 
the  triangles  POA  and  POB  according  as  the  point  P  lies 


Fig.  88,  b. — Parallelogram  law  for  finding 
single  prism  equivalent  to  a  combina- 
tion of  two  thin  prisms. 


140  Mirrors,  Prisms  and  Lenses  [§  62 

outside  the  Z  AOB,  as  in  Fig.  88  (a),  or  inside  this  angle,  as  in 

Fig.  88  (b) ,  respectively.     And,  therefore,  if  PQ,  PR  and  PS  are 

drawn  perpendicular  to  OA,  OB  and  OC,  respectively,  then 

SP.OC  =  QP.OA=*=RP.OB. 

For  simplicity,  let  us  assume  that  the  deviations  p\,  p2 
produced  by  the  two  component  prisms  are  indefinitely 
small.  Now  suppose  that  the  point  P  is  turned,  first,  about 
OA  as  axis  through  a  very  small  angle  pi  and  then  about  OB 
as  axis  through  the  small  angle  p2.  In  consequence  of  the 
first  rotation  it  will  move  perpendicularly  out  from  the  plane 
of  the  paper  towards  the  reader  through  a  tiny  distance 
corresponding  to  the  arc  of  a  circle  described  around  Q  as 
center  with  radius  QP,  the  length  of  this  arc  being  equal 
to  the  product  of  the  radius  by  the  angle,  that  is,  equal  to 
QP .  OA,  since  the  length  of  OA  is  made  equal  to  the  magni- 
tude of  the  angle  p±.  If  now  in  this  slightly  altered  position 
the  point  P  is  again  rotated,  this  time,  however,  around  OB 
as  axis,  through  another  small  angular  displacement  pi  =  OB, 
either  it  will  move  a  little  farther  out  from  the  plane  AOB, 
as  in  the  case  shown  in  Fig.  88  (a),  or  it  will  move  back 
away  from  the  reader,  as  in  the  case  shown  in  Fig.  88  (b), 
by  an  additional  amount  equal  to  RP.OB.  And  as  this 
latter  displacement  will  also  be  very  nearly  at  right  angles  to 
the  plane  of  the  paper,  the  resultant  angular  displacement 
of  the  point  P  may  be  regarded  as  equal  to  the  algebraic 
sum  of  its  two  successive  displacements  and  numerically 
equal,  therefore,  to 

QP.OA±  RP.OB, 
where  the  upper  sign  is  to  be  taken  in  case  the  point  P  lies 
outside  the  angle  AOB  and  the  lower  sign  in  case  it  lies  inside 
this  angle.  In  either  case,  therefore,  the  resultant  displace- 
ment of  P  will  be  equal  to  SP .  OC.  But  this  product  is  equal 
to  the  linear  displacement  which  the  point  P  would  have  if 
it  experienced  an  angular  displacement  represented  by  the 
vector  OC. 

Hence,  if  the  straight  lines  OA,  OB  drawn  parallel  to  the 


§  62]  Combination  of  Two  Thin  Prisms  141 

edges  of  the  two  thin  prisms  represent  the  components  of  the 
total  deviation  of  a  ray  which  traverses  both  prisms,  the 
diagonal  OC  of  the  parallelogram  OABC  will  represent  the 
resultant  or  total  deviation,  and  this  effect  will  be  produced 
by  a  single  prism  of  power  p  =  OC  placed  with  its  edge  in- 
clined to  the  edge  of  the  prism  of  power  pi  (  =  OA)  at  an  angle 
6  =  Z.  OAC.  If  the  powers  ph  p2  of  the  two  component  prisms 
are  given  in  prism-dioptries  (or  in  terms  of  any  other  suit- 
able unit,  for  example,  degree,  centrad,  etc.),  and  if  also  the 
angle  y  between  the  edges  of  the  prisms  is  given  in  degrees, 
the  power  p  of  the  resultant  prism  may,  therefore,  be  com- 
puted by  the  formula: 

P=Vpi2+P22+2pi. Pi- cost    , 
and  the  angle  6  which  shows  how  the  resultant  prism  is  to 
be  placed  may  be  calculated  by  the  formula: 

tanfl=    ^Sin7    ■ 
Pi+P2.cosy 

In  particular,  if  7  =  90°,  then  p=  -\/pi2+P22>  tan  6  =— . 

As  an  illustration  of  the  use  of  these  formulae,  suppose 

that  the  deviations  produced  by  the  two  prisms  separately 

are  3°  and  5°,  and  that  the  edges  of  the  prisms  are  inclined  to 

each  other  at  an  angle  of  60°.    Then  pi  =  3°,  p2  =  5°,  y  =  60°, 

and  hence  the  deviation  produced  by  the  two  prisms  together 

5V3 

will   be  p  =  \/9+  25+ 15  =  7°;   and  since  tan#=  — — ,   the 

resultant  prism  in  this  case  is  found  to  be  a  prism  of  power 
7°  placed  with  its  edge  at  an  angle  of  nearly  38°  13'  with  that 
of  the  weaker  of  the  two  component  prisms. 

A  " rotary  prism"  used  for  finding  the  necessary  prismatic 
correction  of  a  patient's  eye  is  an  instrument,  circular  in  form, 
which  consists  of  two  ophthalmic  prisms  of  equal  power 
(pi  =  7?2)  conveniently  mounted  so  that  the  prisms  can  be 
rotated  about  an  axis  perpendicular  to  the  plane  of  the  in- 
strument, one  in  front  of  the  other,  the  angle  between  the 
prism-edges  being  shown  by  the  positions  of  two  marks  which 


142  Mirrors,  Prisms  and  Lenses  [Ch.  V 

move  as  the  prisms  are  turned  over  a  circular  arc  graduated 
in  degrees.  In  the  initial  position  when  the  two  marks  are 
at  opposite  ends  of  a  diameter  of  the  circular  scale  the  base 
of  one  prism  corresponds  with  the  edge  of  the  other,  so  that 
in  this  position  the  two  prisms  are  equivalent  to  a  glass 
plate  with  plane  parallel  faces  (7  =  180°,  p  =  pi—  P2  =  0). 
The  maximum  effect  is  obtained  when  the  edges  of  the  prism 
correspond  (7  =  0°,  p  =  pi~{-p2  =  2pi).  With  a  device  of  this 
kind,  we  can  obtain,  therefore,  any  prismatic  power  from 
p  =  0  to  p  =  2pi. 

On  the  other  hand,  we  can  resolve  the  effect  of  a  given 
prism  of  power  p  into  a  component  p .  cos  6  in  one  direction 
and  a  component  p .  sin  6  in  a  direction  perpendicular  to  the 
first.  Thus,  a  prism  of  power  5  centrads  with  its  edge  at  an 
angle  of  30°  to  the  horizontal  is  equivalent  to  a  combination 

of  two  prisms  of  powers  -—-  and  ~  centrads,  with  their 

edges  horizontal  and  vertical,  respectively. 

PROBLEMS 

1.  Show  how  to  construct  the  path  of  a  ray  refracted 
through  a  prism  in  a  principal  section;  and  prove  the  con- 
struction. Discuss  the  following  special  cases,  and  draw 
separate  diagrams  for  each  of  them :  (a)  Incident  ray  normal 
to  first  face  of  prism,  (b)  Emergent  ray  "  grazes  "  second 
face;  (c)  Ray  traverses  prism  symmetrically;  (d)  Ray  is  in- 
cident on  first  face  on  side  of  normal  towards  the  edge  of 
the  prism.  ^ 

2.  Show  that  the  total  deviation  of  a  ray  in  a  principal 
section  of  a  prism  of  more  highly  refracting  material  than 
the  surrounding  medium  is  always  away  from  the  prism- 
edge.  Discuss  each  of  the  three  possible  cases,  viz.,  When 
the  point  where  the  two  incidence-normals  intersect  falls 
(a)  inside  the  prism,  (b)  outside  the  prism,  and  (c)  on  one  of 
the  two  faces  of  the  prism.    Draw  diagram  for  each  case. 

3.  Obtain  a  formula  for  calculating  the  magnitude  of  the 


Ch.  V]  Problems  143 

angle  of  incidence  at  the  first  face  of  the  prism  of  the  ray 
which  emerges  from  the  prism  along  the  second  face;  and  dis- 
cuss this  formula  for  the  cases  when  the  refracting  angle  of 
the  prism  is  (a)  greater  than  2 A,  (b)  equal  to  2 A,  (c)  less 
than  2A  but  greater  than  A,  (d)  equal  to  A,  and  (e)  less  than 
A;  where  A  denotes  the  so-called  critical  angle  of  the  two 
media  concerned.    Draw  diagram  for  each  case. 

4.  Show  that  the  deviation  of  a  ray  which  goes  symmet- 
rically through  a  prism  in  a  principal  section  is  less  than 
that  of  any  other  ray. 

5.  Show  that  the  point  of  intersection  of  the  incidence- 
normals  to  the  two  faces  of  a  prism  is  equidistant  from  the 
incident  ray  and  its  corresponding  emergent  ray. 

6.  Construct  the  path  of  a  ray  refracted  through  a  prism 
of  small  refracting  angle;  and  show  that  the  angle  of  deviation 
will  also  be  a  small  angle  of  the  same  order  of  smallness,  no 
matter  how  the  ray  falls  on  the  prism. 

7.  What  is  the  smallest  angle  that  a  glass  prism  (n  =  1 . 5) 
can  have  so  that  no  ray  can  be  transmitted  through  it? 
What  is  the  magnitude  of  this  angle  for  a  water  prism 
(n  =  1.33)?  (Assume  in  each  case  that  the  prism  is  sur- 
rounded by  air  of  index  unity.) 

Ans.  83°  37'  14";  97°  10'  52". 

8.  What  must  be  the  refracting  angle  of  a  prism  whose 
index  of  refraction  is  equal  to  \/2  in  order  that  rays  that 
are  incident  on  one  of  its  faces  at  angles  less  than  45°  will 
be  totalfy  reflected  at  the  other  face?  Ans.  75°. 

9.  The  refracting  angle  of  a  prism  is  60°  and  the  index  of 
refraction  is  equal  to  \/2.  Show  that  the  angle  of  minimum 
deviation  is  30°,  and  draw  accurate  diagram  showing  the 
construction  of  the  path  of  this  ray  through  the  prism. 

10.  The  refracting  angle  of  a  glass  prism  (n  =  1.5)  is  60°, 
and  the  angle  of  incidence  is  45°.  Find  the  angle  of  deviation. 
What  is  the  angle  of  minimum  deviation  for  this  prism? 

Ans.  37°  22'  52.5";  37°  10'  50". 

11.  If  the  angle  of  minimum  deviation  of  a  ray  traversing 


144  Mirrors,  Prisms  and  Lenses  [Ch.  V 

a  principal  section  of  a  prism  is  90°,  show  that  the  index  of 
refraction  cannot  be  less  than  s/2. 

12.  Find  the  angle  of  minimum  deviation  in  the  case  of  a 
glass  prism  (n  =  1 .  54)  of  refracting  angle  60°. 

Ans.  40°  42'  28". 

13.  The  minimum  deviation  for  a  prism  of  refracting  angle 
40°  is  found  to  be  32°  40'.  Find  the  value  of  the  index  of 
refraction.  Ans.  1.7323. 

14.  A  glass  prism  of  refracting  angle  60°  is  adjusted  so 
that  the  ray  "grazes"  the  first  face,  and  in  this  position  the 
angle  of  emergence  is  found  to  be  29°  25'  49".  Determine 
the  index  of  refraction.  Ans.  1 .  52. 

15.  A  prism  is  made  of  glass  of  index  1.6,  and  the  angle 
of  minimum  deviation  is  found  to  be  28°  31'  20".  Calculate 
the  refracting  angle.  Ans.  42°  39'  44". 

16.  The  efracting  angle  of  a  water  prism  (n  =  -|)  is  30°. 
How  must  a  ray  be  sent  into  this  prism  so  that  it  will  emerge 
along  the  second  face? 

Ans.  Ray  must  He  on  the  side  of  the  normal  towards  the 
edge  of  the  prism,  and  make  with  the  normal  an  angle  of 
25°  9'  15". 

17.  The  angle  of  incidence  for  minimum  deviation  in  the 
case  of  a  prism  of  refracting  angle  60°  is  60°.  Find  the 
index  of  refraction.  Ans.  v3. 

18.  Find  the  index  of  refraction  of  a  glass  prism  for  sodium 
light  for  the  following  measurements:  Refracting  angle  of 
prism  =  45°  4';  angle  of  minimum  deviation  =  26°  40'. 

Ans.  1.53. 
19  The  refracting  angle  of  a  prism  is  30°  and  its  index  of 
refraction  is  1.6.    Find  the  angles  of  emergence  and  deviation 
for  each  of  the  following  rays:  (a)  Ray  meets  first  face  nor- 
mally; (b)  Angle  of  incidence  at  first  face  is  equal  to  24°  28'; 

(c)  Angle  of  incidence  at  first  face  is  equal  to  53°  8';  and 

(d)  Ray  " grazes"  first  face. 

Ans.  (a)  53°  8';  23°  8';  (6)  24°  28';  18°  56';  (c)  0°;  23°  8'; 
(d)  13°  59';  46°  1'. 


Ch.  V]  Problems  145 

20.  Find  the  refracting  angle  of  a  glass  prism  (n  =  1.52) 
for  which  the  minimum  deviation  is  15°.     Ans.  27°  24'  15". 

21.  The  refracting  angle  of  a  flint  glass  prism  is  measured 
and  found  to  be  59°  56'  22.4";  and  the  angles  of  minimum 
deviation  for  rays  of  light  corresponding  to  the  Fraunhofer 
lines  D,  F  and  H  are  also  measured  and  found  to  have  the 
following  values:  46°  31'  4.15";  47°  35'  59.2";  and  49°  30' 
5.7",  respectively.  Calculate  the  values  of  the  indices  of 
refraction  nB,  nF,  and  nH- 

Ans.  rcD  =  1 .  603528;  nF  =  1 .  614771 ;  nH  =  1 .  634183. 

22.  The  refracting  angle  of  a  crown  glass  prism  is  measured 
and  found  to  be  60°  2'  10.8";  and  the  angles  of  minimum 
deviation  for  rays  of  light  corresponding  to  the  Fraunhofer 
lines  D,  F  and  H  are  also  measured  and  found  to  have  the 
following  values:  38°  38'  14.3";  39°  10'  51.8";  and  40°  3' 
49.4",  respectively.  Calculate  the  values  of  the  indices  of 
refraction  nD,  nF,  and  nK. 

Ans.  nD  =  1 .  516274 ;  nF  =  1 .  522437 ;  nu  =  1 .  532370. 

23.  A  prism  is  to  be  made  of  crown  glass  of  index  1.526, 
and  it  is  required  to  produce  a  minimum  deviation  of  17°  20'. 
To  what  angle  must  it  be  ground?  Ans.  31°  20'. 

24.  A  ray  of  light  falls  on  one  face  of  a  prism  in  a  direction 
perpendicular  to  the  opposite  face.  Assuming  that  the  re- 
fracting angle  of  the  prism  (/3)  is  an  acute  angle,  show  that 
the  ray  will  emerge  along  the  opposite  face  if 

cot/3  =  cotA—  1, 
where  A  denotes  the  critical  angle  of  the  prism-medium. 

25.  A  ray  "grazes"  the  first  face  of  a  prism  and  emerges 
at  the  second  face  in  a  direction  perpendicular  to  the  first 
face:  show  that  the  refracting  angle  (/3)  is  such  that 

cot/3=Vw2-l-l, 
where  n  denotes  the  index  of  refraction  of  the  prism-medium. 

26.  The  refracting  angle  of  a  prism  is  60°  and  the  index  of 
refraction  is  s/7/3.  What  is  the  limiting  angle  of  incidence 
of  a  ray  that  will  be  transmitted  through  the  prism? 

Ans.  30°. 


146  Mirrors,  Prisms  and  Lenses  [Ch.  V 

27.  Show  that  if  €0  denotes  the  angle  of  minimum  devia- 
tion of  a  prism  of  refracting  angle  /3,  the  angle  fi  cannot  be 
greater  than  (7r—  e0)  and  the  index  of  refraction  cannot  be 

less  than  sec-^-- 

28.  Show  that  the  minimum  deviation  of  a  prism  of  given 
index  of  refraction  increases  with  increase  of  the  refracting 
angle  of  the  prism. 

29.  Derive  the  formula  for  the  angle  of  deviation  of  a  thin 
prism,  and  show  that  the  deviation  is  approximately  con- 
stant for  all  angles  of  incidence. 

30.  Show  that  when  a  thin  glass  prism  of  index  f  is  im- 
mersed in  water  of  index  |-  the  deviation  of  a  ray  will  be 
only  one-fourth  of  what  it  would  be  if  the  prism  were  sur- 
rounded by  air. 

31.  The  refracting  angle  of  a  prism  of  rock  salt  is  1°  30'. 
How  much  will  a  ray  be  deviated  in  passing  through  it? 
And  what  should  be  the  refracting  angle  of  a  rock  salt  prism 
which  is  to  produce  a  deviation  of  48'?  (Index  of  refraction 
of  rock  salt  =  1 .  54.)  Ans.  48'  36" ;  1°  29\ 

32.  What  must  be  the  refracting  angle  of  a  water  prism  of 
index  |-  to  produce  the  same  deviation  as  is  obtained  with 
a  glass  prism  of  index  f  whose  refracting  angle  is  equal  to 
2°?  Ans.  3°. 

33.  A  glass  prism  of  index  1.5  has  a  refracting  angle  of 
2°.    What  is  the  power  of  the  prism  in  prism-dioptries? 

Ans.  1 .  745  prism-dioptries. 

34.  The  power  of  a  prism  is  2  prism-dioptries  and  n=  1 .5. 
Find  the  refracting  angle.  Ans.  2 .  29°. 

35.  A  prism  of  refracting  angle  1°  25'  bends  a  beam  of 
light  through  an  angle  of  1°  15'.  Calculate  the  index  of 
refraction  and  the  power  of  the  prism  in  prism-dioptries. 

Ans.  n  =  1 .  882;  2 .  18  prism-dioptries. 

36.  Two  thin  prisms  are  crossed  with  their  edges  at  an  an- 
gle of  30°.  The  first  prism  produces  a  deviation  of  6°  and 
the  second  a  deviation  of  8°.    Find  the  deviation  produced 


Ch.  V]  Problems  147 

• 
by  the  single  prism  which  is  equivalent  to  this  combination 

and  the  angle  which  the  edge  of  the  resultant  prism  must 
make  with  the  edge  of  the  first  prism. 

Ans.  Deviation  of  resultant  prism  =  13.53°;  angle  be- 
tween its  edge  and  that  of  the  6°-prism=17°  11'. 

37.  Two  prisms,  each  of  power  5  prism-dioptries,  are 
combined  base  down  with  their  base-apex  lines  inclined  to 
the  horizontal  at  angles  of  45°  and  135°.  Find  the  equivalent 
single  prism. 

Ans.  A  prism  of  power  a  little  more  than  7  prism-dioptries, 
base  down,  vertical  meridian  (edge  horizontal). 

38.  What  will  be  the  horizontal  effect  of  a  prism  of  power 
10  placed  with  its  base-apex  line  at  an  angle  of  20°  with  the 
horizontal? 

Ans.  It  will  be  the  same  as  the  effect  of  a  prism  of  power 
nearly  9 . 4  in  horizontal  meridian  (edge  vertical) . 

39.  The  base-apex  line  of  a  prism  of  power  4  centrads  makes 
an  angle  of  120°  with  the  horizontal.  Show  that  it  is  equiva- 
lent to  a  combination  of  two  prisms,  one  of  power  2  centrads  in 
the  vertical  meridian  (edge  horizontal)  and  the  other  of  power 
3.46  centrads  in  the  horizontal  meridian  (edge  vertical). 

40.  Find  the  single  prism  equivalent  to  a  combination  of 
two  prisms  superposed  with  their  base-apex  lines  at  right 
angles  to  each  other,  the  power  of  one  being  3  and  that  of 
the  other  4. 

Ans.  A  prism  of  power  5  with  its  base-apex  line  inclined  to 
that  of  the  weaker  prism  at  an  angle  of  nearly  53°  8'. 

41.  Two  equal  prisms,  each  of  power  3,  are  superposed 
in  meridians  inclined  to  each  other  at  an  angle  of  120°. 
Find  the  equivalent  single  prism. 

Ans.  A  prism  of  power  3  in  a  meridian  halfway  between 
the  meridians  of  the  two  components. 

42.  The  angle  between  the  base-apex  lines  of  a  combina- 
tion of  two  unit  prisms  is  82°  50',  and  the  bisector  of  this 
angle  is  horizontal.  What  is  the  horizontal  effect  of  the 
combination?  Ans.  1 . 5  units. 


148  Mirrors,  Prisms  and  Lenses  [Ch.  V 

• 

43.  ABCDE  is  the  principal  section  of  a  pentagonal  prism. 
AB  =  BC,  AE  =  CD,  ZABC  =  90°,  ZEAB  =  Z  BCD  =  112.5°. 
A  ray  of  light  RS  lying  in  the  principal  section  is  incident  on 
the  face  BC  at  the  point  S.  The  ray  enters  the  prism  at  this 
face,  and  is  reflected,  first,  from  the  face  AE,  and  then  from 
the  face  DC,  and  emerges  finally  at  a  point  P  in  the  face  AB 
in  the  direction  PQ.  Show  that  PQ  makes  a  right  angle 
with  RS. 

44.  ABC  is  a  principal  section  of  a  triangular  prism, 
Z  B  =  2Z  A.  A  ray  of  light  lying  in  the  plane  ABC  is  refracted 
into  the  prism  at  the  side  BC,  and  after  undergoing  two 
internal  reflections,  first,  from  the  side  AB  and  then  from 
the  side  CA,  emerges  into  the  surrounding  medium  at  the 
side  AB.  Show  that  the  total  deviation  of  the  ray  will  be 
equal  to  the  angle  at  B. 


CHAPTER  VI 

EEFLECTION     AND     REFRACTION     OF     PARAXIAL     RAYS     AT     A 
SPHERICAL    SURFACE 


63.  Introduction.  Definitions,  Notation,  etc. — The  center 
of  the  spherical  refracting  or  reflecting  surface  ZZ  (Fig.  89) 
will  be  designated  by  C.  The  axis  of  the  surface  with  respect 
to  a  given  point  M  is  the 
straight  line  joining  M 
with  C,  and  the  point  A 
where  the  straight  line 
MC  (produced  if  neces- 
sary) meets  ZZ  is  called  the 
pole  or  vertex  of  the  surface 
with  respect  to  the  point 
M.  Evidently,  the  spheri- 
cal surface  will  be  sym- 
metrical around  MC  as 
axis,  and  the  plane  of  the 
diagram  which  contains  the  axis  is  a  meridian  section  of  the 
surface. 

It  will  be  convenient  to  take  the  vertex  A  as  the  origin 
of  a  system  of  plane  rectangular  coordinates;  the  axis  of 
the  surface  being  chosen  as  the  z-axis  and  the  tangent  to  the 
surface  at  its  vertex,  in  the  meridian  plane  of  the  diagram, 
being  taken  as  the  ?/-axis.  The  positive  direction  of  the  x-axis 
is  the  direction  of  the  incident  ray  which  coincides  with  this 
line,  and  since  the  diagrams  are  all  drawn  on  the  supposition 
that  the  incident  light  goes  from  left  to  right,  a  point  lying  on 
the  z-axis  to  the  right  of  A  will  be  on  the  positive  half  of 
the  axis.  The  positive  direction  of  the  y-axis  is  the  direction 
found  by  rotating  the  positive  half  of  the  x-axis  through  a 

149 


Fig.  89,  a. — Ray  incident  on  convex 
spherical  surface  crosses  axis  at 
point  M  in  front  of  surface. 


150 


Mirrors,  Prisms  and  Lenses 


l§  63 


right  angle  in  a  sense  opposite  to  that  of  the  motion  of  the 
hands  of  a  clock  in  the  meridian  plane  of  the  diagram.  Ac- 
cordingly, if  the  positive  direction  of  the  x-axis  is  along  a 


Fig. 


89,  b. — Ray  incident  on  convex  spherical  surface  crosses  axis  at  point 
M  on  the  other  side  of  the  surface. 


horizontal  line  from  left  to  right,  the  positive  direction  of 
the  i/-axis  will  be  vertically  upwards. 

According  as  the  center  C  lies  on  the  same  side  of  the 
spherical  surface  as  that  from  which  the  incident  light  comes 
or  on  the  opposite  side,  it  is  said  to  be  concave  (Fig.  89,  c 
and  d)  or  convex  (Fig.  89,  a  and  6),  respectively.  The  radius 
r  of  the  spherical  surface  is  the  abscissa  of  the  center  C,  that 
is,  r  =  AC.  It  is  the  step  from  A  to  C,  and  this  is  always  a 
positive  step  for  a  convex  surface  (Fig.  89,  a  and  b)  and  a 
negative  step  for  a  concave  surface  (Fig.  89,  c  and  d).  The 
radius  of  a  convex  surface  whose  center  is  60  cm.  from  its 
vertex  is  r  =  +60  cm.,  and  the  radius  of  a  concave  surface  of 
the  same  size  is  r—  —  60  cm. 


§63] 


Ray  Incident  on  Spherical  Surface 


151 


Fig 


c. — Ray  incident  on  concave 
spherical  surface  crosses  axis  at  point 
M  in  front  of  the  surface. 


It  will  be  assumed  in  this  chapter  that  any  ray  with  which 
we  are  concerned  lies  in  a  meridian  plane  of  the  spherical 
surface;  so  that  any  straight  line  such  as  RB  which  repre- 
sents the  path  of  an  inci-  y 
dent  ray  will  intersect  the 
axis  either  " really"  (Fig. 
89,  a  and  c)  or  "  virtually  " 
(Fig.  89,  b  and  d)  at  some 
point  designated  here  by 
M  (see  §  8).  The  point 
designated  by  R  is  any 
point  on  the  incident  ray 
RB  at  which  the  light 
arrives  before  it  gets  to 
either  M  or  the  incidence- 
point  B.  The  straight  line  BC  which  joins  the  point  of 
incidence  with  the  center  of  the  surface  will  be  the  incidence- 
normal,  and  if  N  designates  a  point  on  this  normal  lying  in 
front  of  the  spherical  surface,  then   Z  NBR  =  a  will  be  the 

angle  of  incidence  (§§  13  & 
27).  The  plane  of  this 
angle  is  the  plane  of  inci- 
dence, which  is  the  merid- 
ian plane  of  the  diagram. 

From  the  incidence-point 
B  draw  BD  perpendicular 
to  the  x-axis  at  D ;  the  or- 
dinate h  =  DB  is  called  the 

Fig.    89,   d.-Ray  incident  on  concave    incidence-height  of  the  ray. 

spherical  surface  crosses  axis  at  point    The  slope  of  the  Tdy  is  the 
M  on  the  other  side  of  the  surface.      acute  angle  thrQugh  wMch 

the  rr-axis  has  to  be  turned  around  the  point  M  in  order 
that  it  may  coincide  in  position  (but  not  necessarily  in 
direction)  with  the  rectilinear  path  of  the  ray.  If  this  angle 
is  denoted  by  6,  then  ZAMB=  6.  Here,  as  always  in  the 
case  of  angular  magnitudes  (§  13),  counter-clockwise  rotation 


152  Mirrors,  Prisms  and  Lenses  [§  63 

is  to  be  reckoned  as  positive.  And,  finally,  the  acute  angle 
at  the  center  C  of  the  spherical  surface  subtended  by  the 
arc  BA  will  be  denoted  by  (f>.  This  angle,  sometimes  called 
the  "central  angle,"  is  denned  as  the  angle  through  which 
the  radius  CB  must  be  turned  around  C  in  order  to  bring  B 
into  coincidence  with  the  vertex  A;  thus,  0  =  ZBCA.  The 
angles  A,  6  and  <£,  defined  as  above,  are  given  by  the  fol- 
lowing relations: 

h  h 

tan#  =  —  t^Tt>     sm<f>  =  ~,     a=  6-\-<j). 

These  formulae  should  be  verified  for  each  of  the  diagrams 
Fig.  89,  (a),  (&),  (c),  {d). 

Moreover,  since  BM  = -,  and  since  (see  §  45) 

COS0'  v        *      / 

DM  =  DC+CA+AM  =  r.cos</>-r+AM, 
we  find: 

EM  =  r(cos<ft-l)+AM 

COS0 

Now  in  the  special  case  when  the  incidence-point  B  is  very 
close  to  the  vertex  A  of  the  spherical  surface,  the  angle  of  in- 
cidence a  will  be  exceedingly  small  as  will  be  also  the  angles 
denoted  by  6  and  </> ;  and  if  these  angles  expressed  in  radians 
are  all  such  small  fractions  that  we  may  neglect  their  second 
and  higher  powers,  so  that  in  place  of  the  sines  (or  tangents) 
we  can  write  the  angles  themselves  and  put  cos  6  =  cos  <f>  = 
cos  a  =  1.  Obviously,  in  such  a  case  we  shall  have  BM  =  AM. 
Under  these  circumstances  the  ray  RB  is  called  a  paraxial 
ray,  sometimes  also  a  "central"  or  "zero"  ray,  a=  d  =  4>  =  0, 
approximately. 

A  paraxial  ray  is  one  whose  path  lies  very  near  the  axis  of 
the  spherical  surface  and  which  therefore  meets  this  surface  at 
a  point  close  to  the  vertex  and  at  nearly  normal  incidence:  the 
angles  denoted  by  a,  6  and  <j>  being  all  so  small  that  their  second 
powers  may  be  neglected. 

In  this  chapter  and  for  several  subsequent  chapters  we 
shall  be  concerned  entirely  with  the  procedure  of  paraxial 


64] 


Paraxial  Rays:  Spherical  Mirror 


153 


Fig. 


90,   O. — Reflection  of  ray  at  con- 
cave mirror. 


rays;  that  is,  we  shall  consider  only  such  rays  as  are  com- 
prised within  a  very  narrow  cylindrical  region  immediately 
surrounding  the  axis  of  the  spherical  surface  which  is  like- 
wise the  axis  of  the  cylinder.  Accordingly,  the  only  portion 
of  the  spherical  surface  that  will  be  utilized  for  reflection  or 
refraction  will  be  a  small  zone  whose  summit  is  at  A;  so  that, 
so  far  as  paraxial  rays  are 
concerned,  the  rest  of  the 
spherical  surface  may  be 
regarded  as  if  it  had  no 
optical  existence  or  at  any 
rate  as  if  it  were  opaque 
and  non-reflecting.  Thus, 
for  example,  the  surface 
might  be  painted  over 
with  lampblack  leaving 
bare  and  exposed  only 
the  small  effective  zone 
in  the*  immediate  vicinity  of  the  vertex;  or  a  screen  might 
be  set  up  at  right  angles  to  the  axis  close  to  the  vertex  with 
a  small  circular  opening  in  it.    Even  then  a  source  of  light 

lying  at  a  considerable  dis- 
tance off  the  axis  would 
send  rays  which  notwith- 
standing that  they  were 
incident  near  the  vertex 
would  not  be  paraxial  rays. 
64.  Reflection  of  Par- 
axial Rays  at  a  Spherical 
Mirror. — In  the  accom- 
panying diagrams  (Fig.  90, 
a  and  6)  the  straight  line 
RB  represents  the  path  of 
an  incident  ray  crossing  the  axis  of  a  spherical  mirror  ZZ  at  the 
point  M  and  incident  on  the  mirror  at  the  point  B,  and  the 
straight  line  BS  shows  the  path  of  the  corresponding    re- 


FlG. 


90,   6. — Reflection  of  ray  at  con- 
vex mirror. 


154  Mirrors,  Prisms  and  Lenses  [§  64 

fleeted  ray  crossing  the  axis,  "really"  (Fig.  90,  a)  or  "virtu- 
ally" (Fig.  90,  b),  at  the  point  marked  M'.  By  the  law  of 
reflection  Z  NBR  =  Z  SBN  where  BN  is  the  incidence-normal 
and  N  designates  a  point  on  it  which  lies  in  front  of  the 
mirror.  Since  the  normal  bisects  the  interior  or  exterior 
angle  at  B  of  the  triangle  MBM',  the  following  proportion 
may  be  written : 

CM_M/C 

BM    BM'# 
Now  if  the  ray  RB  is  a  paraxial  ray,  the  letter  A  may  be  sub- 
stituted in  the  above  equation  in  place  of  B,  and  thus  *  we 
obtain : 

CMM'C 

■    AM     AM'' 

Denoting  the  abscissae,  with  respect  to  the  vertex  A,  of 

the  axial  points  M,  M'  by  u,  u',  respectively,  that  is,  putting 

AM  =  u,  AM'  =  u',  and  also,  as  stated  in  §  63,  putting  AC  =  r, 

we  may  write: 

CM  =  CA+AM=  -r+u  =  u-r, 
M'C  =  M'A+AC= -u'+r=-(u'-r); 
so  that,  introducing  these  symbols  in  the  equation  above, 
we  obtain: 

u  —  r_     u' —  r 

u  u' 

which  may  be  put  in  the  form  (see  §  67) : 

u      u       r 

If,  therefore,  the  form  and  dimensions  of  the  mirror  are 
known  (that  is,  if  the  value  of  r  is  assigned  as  to  both  mag- 
nitude and  sign),  and  if  also  the  position  of  the  point  M 

*  In  writing  this  proportion,  care  must  be  taken  to  see  that  the  two 
members  of  it  shall  have  the  same  sign.  For  example,  in  each  of  the 
diagrams  in  Fig.  90,  as  they  are  drawn,  the  segments  CM  and  AM 
have  the  same  direction  along  the  axis,  so  that  for  each  of  these  figures 
the  ratio  CM  :  AM  is  positive.  Now  if  the  ratio  M'C  :  AM'  is  to  be 
put  equal  to  this  ratio,  it  must  be  positive  also,  that  is,  the  segments 
M'C  and  AM'  in  each  diagram  must  have  the  same  direction. 


§64]  Paraxial  Rays:  Spherical  Mirror  155 

where   the   incident  paraxial   ray  crosses  the  axis  of   the 

spherical  mirror  is  given,  the  abscissa  u'  of  the  point  M' 

where  the  corresponding  reflected  ray  crosses  the  axis  may 

be  calculated  by  means  of  the  expression: 

,       r.u 

u  = . 

2u-r 

But  the  most  noteworthy  conclusion  to  be  drawn  from  this 
formula  is  the  fact  that,  provided  the  rays  are  paraxial,  their 
actual  slopes  do  not  matter,  for  none  of  the  angular  magni- 
tudes a,  6,  or  cf>  appears  in  the  formula;  which  means  that 
all  paraxial  rays  which  cross  the  axis  at  the  point  M  before 
reflection  will  cross  the  axis  after  reflection  in  the  spherical 
mirror  at  one  and  the  same  point  M'.  Thus,  a  homocentric 
bundle  of  paraxial  rays  incident  on  a  spherical  mirror  remains 
homocentric  after  reflection.  If,  therefore,  M  designates  the 
position  of  a  luminous  point  in  front  of  the  mirror,  and  if 
the  mirror  is  screened  so  that  only  such  rays  as  proceed  close 
to  the  axis  are  incident  on  it,  the  bundle  of  reflected  rays 
will  form  at  a  point  M'  on  the  straight  line  MC  an  ideal 
image  of  the  luminous  point  M.  According  as  the  image- 
point  M'  lies  in  front  of  the  mirror  (Fig.  90,  a)  or  beyond  it 
(Fig.  90,  6),  the  image  will  be  real  or  virtual,  respectively. 
Thus,  for  a  real  image  in  a  spherical  mirror,  the  value  of  u' 
as  found  by  the  formula  above  will  be  negative,  whereas 
for  a  virtual  image  it  will  be  positive. 

It  may  be  noted  also  that  the  formula  is  symmetrical  with 
respect  to  u  and  nf,  so  that  the  equation  will  not  be  altered 
by  interchanging  the  symbols  u  and  u' ;  and  hence  it  follows 
that  if  M'  is  the  image  of  M,  then  likewise  M  may  be  regarded 
as  the  image  of  M'.  This  is  indeed  merely  an  illustration  of 
the  general  law  known  in  optics  as  the  "principle  of  the 
reversibility  of  the  light-path"  (§29).  But  the  symmetry 
of  the  equation  implies  more  than  is  involved  in  this  prin- 
ciple; for  it  indicates  that  in  the  case  of  reflection  object- 
space  and  image-space  coincide  completely,  the  actual  paths 
of  the  incident  and  reflected  rays  both  lying  in  the  space  in 


156  Mirrors,  Prisms  and  Lenses  [§  65 

front  of  the  mirror.  Accordingly,  an  incident  ray  and  its 
corresponding  reflected  ray  are  always  so  related  that  when 
either  is  regarded  as  object-ray  the  other  will  be  an  image-ray. 

The  Double  Ratio  of  Four  Points  on  a  Straight  Line 

65.  Definition  and  Meaning  of  the  Double  Ratio. — It 
will  be  convenient  and  profitable  at  this  place  to  turn  aside 
from  the  special  problem  which  is  here  under  investigation 
in  order  to  devote  a  few  paragraphs  to  a  brief  explanation 
of  the  simpler  metrical  processes  of  modern  projective 
geometry,  which  are  of  great  utility  in  geometrical  optics, 
especially  when  we  are  concerned  with  imagery  by  means 
of  the  so-called  paraxial  rays. 

A  B 

• , 1 , 

C  D 

(ft/) 


■ 1 1 

Fig.  91. — Line-segment  AB  divided  (a)  internally  at 
C  and  externally  at  D,  and  (b)  internally  at  C 
and  D. 

If  L  designates  the  position  of  a  point  on  a  straight  line 
determined  by  the  two  points  A,  B,  the  line-segment  AB  is 
said  to  be  divided  at  L  in  the  ratio  AL  :  BL.  If  the  point  L 
lies  between  A  and  B,  the  steps  (see  §  45)  AL  and  BL  are 
in  opposite  senses  along  the  line,  and  the  ratio  AL  :  BL  will 
be  negative,  and  in  this  case  we  say  that  the  segment  AB  is 
" divided  internally"  at  L.  On  the  other  hand,  if  the  point  L 
does  not  lie  between  A  and  B,  the  ratio  AL  :  BL  will  be 
positive,  and  we  say  that  the  segment  AB  is  "  divided  ex- 
ternally" at  L. 

Accordingly,  if  A,  B,  C,  D  (Fig.  91,  a  and  6)  designate  a 


§  65]  Double  Ratio  157 

series  of  four  points  all  ranged  along  a  straight  line  in  any 
order  of  sequence,  the  segment  AB  will  be  divided  at  C  and 
D  in  the  ratios  AC  :  BC  and  AD  :  BD,  respectively;  and 
the  quotient  of  these  two  ratios  is  called  the  double  ratio  (or 
" cross  ratio")  of  the  four  points  A,  B,  C,  D.  This  double 
ratio  is  denoted  symbolically  by  inclosing  the  four  letters 
ABCD  in  parentheses;  thus,  according  to  the  above  def- 
inition, 

where  the  first  two  letters  in  the  parentheses  mark  the  end- 
points  of  the  segment  and  the  last  two  letters  designate  the 
points  of  division.  The  line-segment  CD  is  divided  in  the 
same  way  by  the  points  A  and  B ;  for 

According  as  the  two  ratios  AC  :  BC  and  AD  :  BD  have 
the  same  sign  or  opposite  signs,  the  value  of  the  double  ratio 
(ABCD)  will  be  positive  or  negative,  respectively.  Suppose, 
for  example,  that  the  segment  AB  is  divided  internally  at  C, 
as  represented  in  both  a  and  b  of  Fig.  91.  Then  the  ratio 
AC  :  BC  will  be  negative.  Now  if  AB  is  divided  also  in- 
ternally at  D,  as  in  Fig.  91,  a,  the  ratio  AD  :  DB  will  likewise 
be  negative.  Accordingly,  if  C  and  D  are  both  points  of  in- 
ternal division  (or  both  points  of  external  division),  the 
double  ratio  (ABCD)  will  be  positive.  But  if  one  of  these 
points  divides  AB  internally  while  the  other  divides  it  ex- 
ternally (Fig.  91,  b),  the  double  ratio  (ABCD)  will  be  nega- 
tive. 

In  order  to  form  a  clear  idea  of  the  values  which  (ABCD) 
may  assume,  let  us  suppose  that  the  points  designated  by 
A,  B  and  C  in  Fig.  92  represent  three  stationary  points  on  a 
straight  line  x,  and  that  0  designates  another  fixed  point  not 
on  this  line.  The  straight  line  x  and  the  point  O  together 
determine  a  plane  which  is  the  plane  of  the  diagram.  Now 
let  y  designate  a  second  straight  line  lying  in  this  plane  and 


158  Mirrors,  Prisms  and  Lenses  [§  65 

passing  through  O,  and  let  the  point  of  intersection  of  the 
straight  lines  x  and  y  be  designated  by  Y.  And  if  the 
straight  line  y  is  supposed  to  turn  around  0  as  a  pivot  in  a 

sense,  say,  opposite  to 
that  of  the  motion  of  the 
hands  of  a  clock,  the 
point  Y  will  be  a  variable 
point  moving  along  the 
straight  line  x  constantly 
in  the  same  sense,  namely, 
in  Fig.  92  from  left  to 

Fig.  92.— Central  projection  from  Oof  the  right-  Assume,  for  ex- 
point-range  ABCDE  lying  on  the  ample,  that  the  three 
straight  line  x.  ...  -    •       k     r~\   -n 

stationary  points  A,  C,  B 
are  ranged  along  the  straight  line  x  from  left  to  right  in  the 
order  named,  as  shown  in  the  figure;  and  suppose  that  the 
variable  point  Y  starts  originally  at  B,  so  that  the  revolving 
line  OY  or  y  coincides  initially  with  the  "ray"  marked  b  in 
the  figure  and  BY  =  BB  =  0,  and,  consequently,  the  ratio 
AY  :  BY  =  oo  ,  Hence,  under  these  circumstances  the  initial 
value  of  the  double  ratio  of  the  four  points  A,  B,  C,  Y  will  be: 

/ABcY>=i:S=°- 

When  the  revolving  ray  has  turned  through  ZBOD,  where 
D  designates  a  point  lying  on  the  straight  line  x  to  the 
right  beyond  B,  the  point  Y  will  be  at  D  outside  the  segment 
AB  and  the  double  ratio  (ABCY)  will  be  negative,  as  ex- 
plained above.    As  y  continues  to  revolve  around  O,  the  point 

Y  will  move  farther  and  farther  to  the  right  along  the  straight 
line  x,  until  when  y  is  parallel  to  x,  and  in  the  position  of  the 
ray  marked  e  in  the  figure,  the  point  Y  will  then  coincide 
with  the  infinitely  distant  point  E  of  the  straight  line  x.  Now 
AE  =  BE  =  oo ,  and  hence  AE  :  BE  =  1 ;  and  therefore  when 

Y  is  at  E, 


§  66]  Perspective  Ranges  of  Points  159 

When  the  revolving  ray  y  has  turned  beyond  the  position 
represented  by  the  straight  line  e,  the  point  Y  which  had 
just  vanished  at  one  end  E  of  the  straight  line  x  now  re- 
appears from  the  other  end  E,  proceeding  along  it  still  in 
the  same  sense  from  left  to  right.     Thus,  before  the  ray  y 
has  executed  a  complete  revolution,  the  point  Y  will  pass 
through  A,  and  at  this  moment,  AY  =  AA  =  0,  and 
aBmn     AC  AY    AC   AA 
(ABCY)=BC:BY  =  BC:BA=-°°; 
and  thus  we  see  that  as  the  point  Y  has  traversed  the  straight 
line  x  from  B  via  the  infinitely  distant  point  E  to  A,  the  double 
ratio  (ABCY)  has  assumed  all  negative  values  from  0  to  —  oo  . 
Finally,  as  the  ray  y  completes  its  revolution  by  turning  from 
the  position  a  to  its  initial  position  b,  the  point  Y  moves  from 
A  via  C  to  B.    When  Y  is  at  C,  AY  =  AC,  BY  =  BC,  and 

(ABCY)  =  |§:g  =  +  l; 

so  that  in  passing  along  x  from  A  to  C,  (ABCY)  assumes  all 
positive  values  comprised  between  +  oo  and  +1.  Between 
C  and  B,  it  has  all  positive  values  less  than  unity.  Thus, 
as  the  point  Y  traverses  the  straight  line  x  continually  in 
the  same  sense  until  it  has  returned  to  its  starting  point, 
the  double  ratio  (ABCY)  will  assume  all  possible  values 
both  positive  and  negative. 
In  general,  since 

(ABUD)     BC:BD     AD-AC     DA'DB     CB  '  CA  ' 
we  may  write: 

(ABCD)  =  (BADC)  =  (CDAB)  =  (DCBA). 
66.  Perspective  Ranges  of  Points. — If  A,  B,  C,  etc.,  desig- 
nate the  positions  of  the  points  of  a  point-range  x  (Fig.  92) 
these  points  are  said  to  be  " projected"  from  a  point  0  out- 
side of  x  by  the  straight  lines  or  "rays"  OA,  OB,  OC,  etc.; 
and  if  these  rays  intersect  another  straight  line  x'  (Fig.  93) 
in  the  points  A',  B',  C,  etc.,  the  two  point-ranges  x,  x'  are 
said  to  be  in  perspective  with  respect  to  the  point  0  as  center 


160 


Mirrors,  Prisms  and  Lenses 


[§66 


of  perspective.  The  points  A,  A';  B,  B';  C,  C;  etc.,  are 
called  pairs  of  corresponding  points  of  the  two  perspective 
point-ranges  x,  x' . 


b    f 


Fig.  93.— The  point-ranges  ABCD  and  A'B'C'D' 

are  in  perspective  relation  with  respect  to  the 
point  O  as  centre  of  perspective. 

If  A,  B,  C,  D  designate  the  positions  of  any  four  points  of 
x,  and  if  A',  B',  C,  D'  designate  the  corresponding  points 
on  x',  then 

(A'B'C'D')  =  (ABCD), 

as  we  shall  proceed  to  show. 


Fig.  94. — Straight  lines  x,  x'  are  bases  of  two  point-ranges  in 
perspective,  so  that  (ABCD)  =  (A'B'C'D'). 

Through  the  points  A,  B,  Ar  and  B'  (Fig.  94)  draw  four 
parallel  lines  AAC,  BBC,  A'AC'  and  B'B/  meeting  the  ray  OG 


§  67]  Harmonic  Range  161 

or  c  in  the  points  Ac,  Bc,  Ac'  and  Bc',  respectively;  and 
through  these  same  points  draw  four  other  parallel  lines 
AAd,  BBd,  A' Ad'  and  B'Bd'  meeting  the  ray  OD  or  d  in  the 
points  Ad,  Bd,  Ad'  and  Bd',  respectively.    Then,  evidently, 

AC  =  AAC   AD  _  AAd 

BC    BBC'  BD     BBd ' 
A'C     A'AC'   A'D'     A'Ad' 


hence, 


B'C    B'BC' 

'  B'D'     B'Bd'  ' 

AP 

(ABCD)  =  — 

BC 

AD_AAC    AAd 
'  BD     BBC '  BBd ' 

AT/ 

3'C'D')  =— , 

B'C 

A'D'A'Ae7   A'Ad' 
'  B'D'     B'BC' "  B'Bd' 

A  Ac     A  Ad 

B  Bc      B  Bd 

Now 


A'AC'    A'A/  B'Bc'     B'Bd' ' 
and,  consequently, 

(A,B'C'D')  =  (ABCD), 
as  was  to  be  proved. 

67.  The  Harmonic  Range. — The  special  case  when  the 
points  C  and  D  divide  the  line-segment  AB  internally  and 
externally  in  the  same  numerical  ratio,  so  that 

AC=_AD 

BC        BD' 
demands  attention,  particularly  because  it  is  a  case  that  we 
shall  meet  again  in  the  theory  of  the  reflection  of  paraxial  rays 
at  a  curved  mirror.     Under  these  circumstances,  the  value 
of  the  double  ratio  is 

(ABCD)=-1; 
and  then  we  say  that  the  segment  AB  is  divided  harmonically 
at  C  and  D,  or  also  the  segment  CD  is  divided  harmonically 
at  A  and  B.  For  example,  the  perpendicular  bisectors  of 
the  exterior  and  interior  angles  of  a  triangle  divide  the  op- 
posite side  of  the  triangle  harmonically  in  the  ratio  of  the 
other  two  sides. 


162 


Mirrors,  Prisms  and  Lenses 


67 


Four  harmonic  points  may  be  denned  not  merely  by  the 
metrical  relation  that  their  double  ratio  is  equal  to  - 1,  but 
also  by  a  geometrical  relation,  as  we  shall  now  show. 
Let  P,  Q,  R,  S  (Figs.  95  and  96)  designate  the  positions  of 
\z  four  points  lying  all  in 

one  plane,  no  three  of 
which  are  in  the  same 
straight  line.  These 
four  points  will  deter- 
mine six  straight  lines, 
viz.,  PQ,  PR,  PS,  QR, 

Fig.   95.— Complete    quadrilateral    PQRS;    QS,   and   RS,   which  are 
(ABCD)=-1.  called    the    sides    Qf    the 

complete  quadrilateral  whose  four  vertices  are  at  the  points 
P,  Q,  R,  and  S.  Any  two  of  these  lines  which  together  con- 
tain all  the  vertices  form  a  pair  of  opposite  sides  of  the 
quadrilateral.  Accordingly,  there  are  three  pairs  of  opposite 
sides,  viz.,   PQ  and  RS  . 

which  meet  in  a  point 
designated  by  A,  PS  and 
QR  which  meet  in  a 
point  designated  by  B, 
and  QS  and  PR  which 
meet  in  a  point  desig- 
nated by  O.     The  three 

points   A,    B    and    0    are    FlG-  96.— Complete   quadrilateral   PQRS; 

sometimes     called     the 

secondary  vertices  of  the  quadrilateral.    We  shall  explain  now 

what  connection  this  figure  has  with  a  harmonic  range  of 

points. 

The  secondary  vertices  A  and  B  are  determined  by  the 
two  pairs  of  opposite  sides  PQ,  RS  and  PS,  QR;  and  the 
points  C  and  D  where  the  third  pair  of  opposite  sides  QS 
and  PR  meet  the  straight  line  AB  divide  the  segment  AB 
harmonically.    For,  since  A,  B,  C,  D  and  P,  R,  O,  D  are  in 


§  67]  Harmonic  Range  163 

perspective  relation  with  respect  to  the  point  Q  as  center  of 
perspective  (§  66),  therefore 

(ABCD)  =  (PROD). 
But  P,  R,  0,  D  and  B,  A,  C,  D  are  also  in  perspective  to 
each  other  with  respect  to  the  point  S  as  center  of  perspective; 
consequently, 

(PROD)  =  (BACD). 
It  follows  therefore  that 

(ABCD)  =  (BACD). 
But  by  the  definition  of  the  double  ratio 

(BACD)  = . 

(ABCD) 

Accordingly,  here  we  must  have: 

(ABCD)= , 

(ABCD) 

or 

(ABCD)2=1. 

According   to   this   equation,    therefore,    the   double   ratio 

(ABCD)  must  be  equal  to  +1  or  —1.    But  we  saw  above 

(§  65)  that  the  double  ratio  of  four  points  A,  B,  C,  D  in  a 

straight  line  can  be  equal  to  +1  only  in  case  one  of  the 

points  A,  B  is  coincident  with  one  of  the  pair  C,  D;  which 

cannot  happen  in  case  of  the  four  points  A,  B,  C,  D  of  the 

quadrilateral  PQRS.    Therefore,  we  must  have  here: 

(ABCD)=-1; 

and  hence,  by  definition,  the  points  A,  B  are  harmonically 

separated  by  the  points  C,  D.     Similarly,  also,  the  points 

P,  R  are  harmonically  separated  by  the  points  O,  D. 

If  A,  B,  C,  D  is  a  harmonic  range  of  points,  then 


that  is, 


BC=DB        BA+AC_DA+AB 
AC    AD'  °r      AC  AD 


AC-ABAB-AD 
AC  AD 


164  Mirrors,  Prisms  and  Lenses  [§  68 

which  may  finally  be  written  in  the  form: 
11  2 


AC    AD    A  B' 

an  equation  that  is  characteristic  of  a  harmonic  range  of 
four  points  A,  B,  C,  D  (c/.  §  64). 

68.  Application  to  the  Case  of  the  Reflection  of  Paraxial 
Rays  at  a  Spherical  Mirror. — When  paraxial  rays  are  re- 
flected at  a  spherical  mirror  whose  center  is  at  C,  we  saw 
(§  64)  that  CM  :  AM  =  M'C  :  AM',  where  M,  M'  designate 
the  positions  of  a  pair  of  conjugate  points  lying  on  a  central 
ray  which  crosses  the  mirror  at  the  point  marked  A  (Fig.  90, 
a  and  b) ;  and  therefore 

Consequently,  the  four  points  C,  A,  M,  M'  are  a  harmonic 
range  of  points  lying  on  the  central  ray  AC,  and  we  may  say 
that  the  pair  of  conjugate  points  M,  M'  is  harmonically 
separated  by  the  center  of  the  mirror  C  and  the  point  A 
where  the  central  ray  meets  the  mirror.  Thus,  if  we  know 
the  positions  of  three  of  these  points,  we  can  construct  the 
position  of  the  fourth  point  by  the  aid  of  the  properties  of 
the  complete  quadrilateral  (§  67).  For  example,  the  image- 
point  M'  conjugate  to  a  given  point  M  with  respect  to  a 
spherical  mirror  may  be  constructed  as  follows: 

Draw  a  straight  line  x  (Fig.  97,  a  and  b)  to  represent  the 
-axis  of  the  mirror,  and  mark  on  it  the  positions  of  the  three 
given  points,  A,  C  and  M,  which  may  be  ranged  along  this 
line  in  any  sequence  whatever  depending  on  the  form  of 
the  mirror  and  on  whether  the  object-point  M  is  real  or 
virtual.  Through  M  draw  another  straight  line  in  any  con- 
venient direction,  and  mark  on  it  two  points  which  we  shall 
call  Q  and  S,  and  draw  the  straight  lines  AQ  and  CS  meeting 
in  a  point  R  and  the  straight  lines  AS  and  CQ  meeting  in  a 
point  P.  Then  the  straight  line  PR  will  intersect  the  straight 
line  x  in  the  point  M'  which  is  conjugate  to  M  with  respect 
to  a  spherical  mirror  whose  vertex  is  at  A  and  whose  center 


§  68]     Spherical  Mirror:  Conjugate  Axial  Points      165 

is  at  C.  It  will  be  remarked  that  in  performing  this  con- 
struction the  only  drawing  instrument  that  is  needed  is  a 
straight-edge. 


Fig.   97,  a. — Concave  Mirror:  Construction  of  point 
M'  conjugate  to  axial  point  M  in  front  of  the  mirror. 

If  the  mirror  is  concave,  the  possible  sequences  of  these 
four  points  on  the  axis  are  M,  C,  M',  A;  M',  C,  M,  A;  and 
C,  M,  A,  M',  when  the  object-point  M  is  real,  and  C,  M', 


Fig.    97,   b. — Convex    Mirror:    Construction    of   point    M' 
conjugate  to  virtual  object-point  M  on  axis  of  mirror. 

A,  M,  when  the  object-point  M  is  virtual.  In  the  case  of  a 
convex  mirror  the  points  may  occur  in  any  one  of  the  follow- 
ing arrangements:  M,  A,  M',  C,  when  the  object-point  M 


166 


Mirrors,  Prisms  and  Lenses 


69 


FlG. 


a. — Focal     point 
mirror  (AF  =  FC) 


of 


is  real,  and  M',  A,  M,  C;  A,  M,  C,  M'  and  A,  M',  C,  M, 
when  the  object-point  M  is  virtual.  The  student  should 
satisfy  himself  as  to  the  accuracy  of  these  statements  by 
drawing  a  diagram  for  each  of  these  eight  sequences  accord- 
ing to  the  directions  for  the  construction  as  given  above. 
Fig.  97,  a  shows  the  case  of  a  concave  mirror  with  the  points 
in  the  order  M,  C,  M',  A;  whereas  Fig.   97,  b  represents 

a  convex  mirror  with  a 
virtual  object-point  at  M, 
the  order  in  this  case  be- 
ing A,  M,  C,  M'. 

69.  Focal  Point  and 
Focal  Length  of  a  Spheri- 
cal Mirror. — I  n  the 
special  case  when  the  ob- 
ject-point M  coincides 
with  the  infinitely  distant 
point  E  of  the  x-axis,  the  conjugate  point  M'  will  lie  at  a 
point  F'  (Fig.  98,  a  and  b)  determined  by  the  relation: 

(CAEF')=-1, 
and  since  here  CE  =  AE=  oo ,  we  must  have: 

AF'=F'C. 
This  means  that  a  cyl- 
indrical bundle  of  inci- 
dent paraxial  rays  parallel 
to  the  axis  of  a  spherical 
mirror  will  be  transformed 
into  a  conical  bundle  of 
reflected  rays  with  its 
vertex  at  a  point  F'  which 
is  midway  between  the 
vertex  A  and  the  center  C. 

If,  on  the  other  hand,  the  image-point  M'  coincides  with 
the  infinitely  distant  point  E,  the  conjugate  object-point  M 
will  He  on  the  axis  at  a  point  F  determined  by  the  relation: 
(CAFE')  =  - 1, 


To  E  a*«° 

\s 

TeE  lltOO 

■*               -* 

TbE  «*«              / 

\a 

ToE  at  CO 

^° 

r/r 

aYs 

Fig.    98, 


b. — Focal    point     of 
mirror  (AF  =  FC) 


concave 


§  69]  Focal  Length  of  Mirror  167 

and  therefore  we  obtain  here  in  the  same  way  as  above: 

AF  =  FC. 
Accordingly,  a  conical  bundle  of  incident  rays  with  its 
vertex  at  a  point  F  midway  between  the  vertex  of  the  mirror 
and  its  center  will  be  transformed  into  a  cylindrical  bundle 
of  reflected  rays  parallel  to  the  axis  of  the  mirror.  The 
letters  F  and  F'  will  be  used  to  designate  the  positions  of 
the  so-called  focal  points  of  an  optical  system  which  is  sym- 
metric around  an  axis.  They  are  not  a  pair  of  conjugate 
points,  as  might  naturally  be  inferred  from  the  fact  that 
they  are  designated  by  the  same  letter.  In  the  case  of  a 
spherical  mirror  these  two  points,  as  we  have  seen,  are  coin- 
cident with  each  other,  which  is  a  consequence  of  the  identity 
of  object-space  and  image-space  to  which  reference  was  made 
at  the  conclusion  of  §  64.  The  focal  point  of  a  concave  mirror 
lies  in  front  of  the  mirror,  as  shown  in  Fig.  98,  b,  so  that 
paraxial  rays  parallel  to  the  axis  will  be  reflected  at  a  con- 
cave mirror  to  a  real  focus  at  F;  whereas  in  the  case  of  a 
convex  mirror  the  focal  point  F  lies  behind  the  mirror  (vir- 
tual focus),  as  shown  in  Fig.  98,  a. 

The  focal  length  f  of  a  spherical  mirror  may  be  defined  as 
the  abscissa  of  the  vertex  A  with  respect  to  the  focal  point 
F  as  origin;  that  is,  /=FA.  Hence,  according  as  the  mirror 
is  concave  or  convex,  the  focal  length  will  be  positive  or  negative, 
respectively.  It  may  be  remarked  that  the  signs  of  /  and 
r  are  always  opposite,  the  relation  between  these  magnitudes 
being  given  by  the  following  formula: 

/=-^orr=-2/. 

Hence,  also,  the  abscissa-relation  obtained  in  §  64  may  be 
written  in  terms  of  /  instead  of  r  as  follows : 

u    v!    f 
where,  however,  it  must  be  borne  in  mind  that,  whereas  the 
abscissae  u,  u'  are  measured  from  the  vertex  A  as  origin, 
the  focal  length  /  is  measured  from  the  focal  point  F. 


168 


Mirrors,  Prisms  and  Lenses 


[§70 


If  the  abscissae,  with  respect  to  the  focal  point  F,  of  the 
pair  of  conjugate  axial  points  M,  M'  are  denoted  by  x,  x't 
that  is,  if  FM  =  x,  FM'  =  x',  then,  since 

AM  =  AF-f-FM,        AM'  =  AF+FM', 
the  connection  between  the  w's  and  the  x's  is  given  by  the 
following  equations: 

u=x-f,        u'  =  x'-f; 
and  substituting  these  values  in  the  formula  above  and 
clearing  of    fractions,   we   derive  the   so-called   Newtonian 
formula,  viz.: 

x.x'=f; 
which  is  an  exceedingly  simple  and  convenient  form  of  the 
abscissa-relation  between  a  pair  of  conjugate  axial  points. 
The  right-hand  side  of  this  equation  is  essentially  positive, 
and  hence  the  abscissae  x,  xr  must  always  have  like  signs. 
Consequently,  in  a  spherical  mirror  the  conjugate  axial  points 
M,  M'  lie  always  both  on  the  same  side  of  the  focal  point  F. 

70.  Graphical  Method  of  exhibiting  the  Imagery  by 
Paraxial  Rays. — The  points  M,  M'  in  Fig.  99,  a  and  b  desig- 


Fig.  99,  a. — For  paraxial  rays  the  reflecting  (or 
refracting)  surface  must  be  represented  in  diagram 
by  the  straight  line  Ay,  not  by  the  curved  line  AZ. 


nate  the  positions  on  the  axis  of  a  spherical  mirror  of  a  pair 
of  conjugate  points  constructed  according  to  the  method 
explained  in  §  68.  On  the  reflecting  sphere  ZZ  take  a  point 
D,  and  draw  the  straight  lines  MD,  M'D  meeting  the  tan- 


70] 


Diagrams  for  Paraxial  Rays 


169 


gent  Ay  in  the  plane  of  these  lines  in  the  points  B,  G,  re- 
spectively. Also,  draw  the  straight  line  M'B.  Now  if  the 
point  D  were  very  close  to  the  vertex  A  of  the  mirror,  then 
the  straight  line  MD  would  represent  the  path  of  an  incident 


Fig.  99,  b. — For  paraxial  rays  the  reflecting  (or 
refracting)  surface  must  be  represented  in  diagram 
by  the  straight  line  Ay,  not  by  the  curved  line  AZ. 

paraxial  ray  crossing  the  axis  at  M,  and  the  path  of  the 
corresponding  reflected  ray  would  be  along  the  straight  line 
DM'.  But  under  these  circumstances,  the  three  points 
designated  here  by  the  letters  D,  B,  G  would  all  be  so  near 
together  that  even  when  we  cannot  regard  D  as  absolutely 
coincident  with  A,  we  may  consider  D,  B  and  G  as  all  coin- 
cident with  one  another.  Therefore,  when  the  ray  is  paraxial, 
we  may,  and,  in  fact,  in  the  diagram  we  must,  regard  the 
straight  line  BM'  as  showing  the  path  of  the  reflected  ray. 
It  is  quite  essential  that  this  point  which  is  seldom  clearly 
explained  should  be  rightly  apprehended  by  the  student.  In 
diagrams  showing  the  imagery  by  means  of  paraxial  ra3rs 
the  duty  of  the  straight  lines  that  are  drawn  is  not  primarily 
to  represent  the  actual  paths  of  the  rays  themselves  but  to 
locate  by  their  intersections  the  correct  positions  of  the  pairs 
of  corresponding  points  in  the  object-space  and  image-space. 
In  the  construction  of  such  diagrams,  a  practical  difficulty 


170  Mirrors,  Prisms  and  Lenses  [§  70 

is  encountered  due  to  the  fact  that,  whereas  in  reality  par- 
axial rays  are  comprised  within  the  very  narrow  cylindrical 
region  immediately  surrounding  the  axis  of  the  spherical 
surface  (§  63),  it  is  obviously  quite  impossible  to  show  them 
this  way  in  the  figure,  because  it  would  be  necessary  to  take 
the  dimensions  of  the  drawing  at  right  angles  to  the  axis 
so  small  that  magnitudes  of  the  second  order  of  smallness 
would  no  longer  be  perceptible  at  all;  thus,  for  example,  the 
points  B,  D,  G  in  Fig.  99  would  have  to  be  shown  as  one 
point.  On  the  other  hand,  if  the  lines  in  the  diagram  are 
not  all  drawn  close  to  the  axis,  the  relations  which  have  been 
found  above  will  cease  to  be  applicable,  so  that,  for  instance, 
the  rays  shown  in  such  a  drawing  would  not  intersect  in  the 
places  demanded  by  the  formulae. 

Accordingly,  in  order  to  overcome  this  difficulty,  a  method 
of  constructing  these  figures  has  been  very  generally  adopted, 
which,  although  it  is  confessedly  in  the  nature  of  a  com- 
promise, has  been  found  to  be  on  the  whole  quite  satisfactory, 
and  wherein  at  any  rate  the  geometrical  relations  are  in 
agreement  with  the  algebraic  conditions,  which  is  the  essen- 
tial requirement.  In  this  plan,  while  the  dimensions  parallel 
to  the  axis  remain  absolutely  unaltered,  the  dimensions  at 
right  angles  to  the  axis  are  all  prodigiously  magnified  in  the 
same  proportion.  Thus,  for  example,  if  the  incidence-height 
h  =  ~DB  (Fig.  89)  is  a  small  magnitude  of  the  order,  say,  of 
one-thousandth  of  the  unit  of  length,  it  will  be  shown  in 
the  figure  magnified  a  thousand  times;  whereas  another  or- 
dinate whose  height  was  only  one  one-millionth  of  the  unit 
of  length  and  which,  therefore,  would  be  of  the  second  order 
of  smallness  as  compared  with  h,  would  appear  even  in  the 
magnified  diagram  as  a  magnitude  of  the  first  order  of  small- 
ness. And  if  the  ordinate  denoted  by  h,  although  in  reality 
infinitely  small,  is  represented  in  the  drawing  by  a  line  of 
finite  length,  an  ordinate  of  the  second  order  of  smallness 
as  compared  with  h  will  be  entirely  unapparent  in  the 
magnified  diagram. 


§  71]  Extra-Axial  Conjugate  Points  171 

Of  course,  as  already  intimated,  one  effect  of  this  lateral 
enlargement  will  be  to  misrepresent  to  some  extent  the  rela- 
tions of  the  lines  and  angles  in  the  figure.  For  instance,  the 
circle  in  which  the  spherical  mirror  (or  refracting  surface) 
is  cut  by  the  plane  of  a  meridian  section  will  thereby  be 
transformed  into  an  infinitely  elongated  ellipse  with  its 
major  axis  perpendicular  to  the  axis  of  the  spherical  surface, 
and  this  ellipse  will  appear  in  the  diagram  as  a  straight  line 
Ay  tangent  to  the  circle  at  A.  The  minor  axis  of  the  ellipse 
remains  unchanged  and  equal  to  the  diameter  2r  of  the  circle, 
and  moreover  the  center  of  the  ellipse  remains  at  the  center  C 
of  the  circle.  But  the  most  apparent  change  will  be  in  the 
angular  magnitudes  which  will  be  completely  altered  and 
distorted.  For  example,  every  straight  line  drawn  through 
the  center  C  really  meets  the  circle  ZZ  (Fig.  89)  normally, 
but  in  the  distorted  figure  the  axis  of  symmetry  will  be  the 
only  one  of  such  lines  which  will  be  perpendicular  to  the 
straight  line  Ay  which  takes  the  place  of  the  circular  arc  ZZ. 
Angles  which  in  reality  are  equal  will  appear  unequal,  and 
vice  versa.  However — and  after  all  this  is  the  really  essential 
matter — the  absolute  dimensions  of  the  abscissa?  and  the  rela- 
tive dimensions  of  the  ordinates  will  not  be  changed  at  all;  and 
therefore  lines  which  are  really  straight  will  appear  as 
straight  lines  in  the  figure,  and  straight  lines  which  are 
parallel  will  be  shown  as  such.  The  abscissa  of  the  point  of 
intersection  of  a  pair  of  straight  lines  in  the  drawing  will  be 
the  true  abscissa  of  this  point. 

In  such  a  diagram,  therefore,  any  ray,  no  matter  what 
slope  it  may  have  nor  how  far  it  may  be  from  the  axis,  is  to 
be  considered  as  a  paraxial  ray.  The  meridian  section  of 
the  spherical  reflecting  or  refracting  surface  must  be  repre- 
sented in  the  figure  by  the  straight  fine  Ay  (?/-axis),  and  the 
position  of  the  center  C  with  respect  to  the  vertex  A  will 
show  whether  the  surface  is  convex  or  concave. 

71.  Extra-Axial  Conjugate  Points. — If  we  suppose  that  the 
axis  of  the  spherical  mirror  is  rotated  about  the  center  C 


172 


Mirrors,  Prisms  and  Lenses 


[§71 


through  a  small  angle  ACU,  so  that  the  vertex  A  moves 
along  the  mirror  to  a  neighboring  point  U,  the  conjugate 
axial  points  M,  M'  will  describe  also  small  arcs  MQ,  M'Q' 
of  concentric  circles;  and,  evidently,  the  points  Q,  Q'  will  be 


A  x 


Fig.  100. — Concave  mirror:  Object  is  a  small  line  MQ  perpendicular  to 
axis;  its  image  M'Q'  is  real  and  inverted. 


harmonically  separated  (§§67,  68)  by  the  points  C,  U,  so  that 
(CUQQO  =  (CAMM')  =  - 1.  Thus,  we  see  how  the  point  Q' 
is  the  image-point  conjugate  to  the  extra-axial  object-point  Q. 
In  the  diagram  (Fig.  100)  the  circular  arcs  AU,  MQ  and 
M'Q'  will  appear  as  straight  lines  perpendicular  to  the  axis,  as 
explained  in  §  70.  We  derive,  therefore,  without  difficulty 
the  following  conclusions: 

(1)  Th  image,  in  a  spherical  mirror,  of  a  plane  object  per- 
pendicular to  the  axis  is  likewise  a  plane  perpendicular  to  the 
axis;  (2)  A  straight  line  passing  through  the  center  of  the 
spherical  mirror  intersects  a  pair  of  such  conjugate  planes  in  a 
pair  of  conjugate  points;  and  (3)  To  a  homocentric  bundle  of 
incident  paraxial  rays  proceeding  from  a  point  Q  in  a  plane 
perpendicular  to  the  axis  of  a  spherical  mirror  there  corre- 


§  71]      Spherical  Mirror:  Construction  of  Image       173 

sponds  a  homocentric  bundle  of  reflected  rays  with  its  vertex  Q' 
lying  in  the  conjugate  image-plane. 

In  order  to  construct  the  image-point  Q'  of  the  extra-axial 
object-point  Q,  we  have  merely  to  find  the  point  of  inter- 
section after  reflection  at  the  spherical  mirror  of  any  two 


Fig.  101,  a. — Lateral    magnification   and    construction   of  image   in 
concave  mirror. 


rays  emanating  originally  from  Q.  The  diagrams  (Fig.  101, 
a  and  b) ,  which  are  drawn  according  to  the  method  explained 
in  §  70,  exhibit  this  construction  for  the  cases  when  the  mirror 
is  concave  and  convex.  Of  the  incident  rays  proceeding 
from  Q,  it  is  convenient  to  select  for  this  purpose  two  of  the 
following  three,  namely:  the  ray  QC  which  proceeding  to- 
wards the  center  C  meets  the  spherical  mirror  normally  at 
U,  whence  it  is  reflected  back  along  the  same  path;  the  ray 
QV  which  proceeding  parallel  to  the  axis  and  meeting  the 
mirror  in  the  point  designated  by  V  is  reflected  at  V  along 
the  straight  line  joining  V  with  the  focal  point  F;  and  the 
ray  QW  which  being  directed  towards  the  focal  point  F  is 
reflected  at  W  in  a  direction  parallel  to  the  axis.  The  point 
where    these    reflected    rays   intersect   will   be   the   image- 


174 


Mirrors,  Prisms  and  Lenses 


[§71 


point  Q'.  Moreover,  having  located  the  position  of  Q',  we 
can  draw  QM,  Q'M'  perpendicular  to  the  axis  at  M,  M',  re- 
spectively; and  then  M'Q'  will  be  the  image  of  the  small 
object-line  MQ.  In  Fig.  101,  a  the  image  M'Q'  is  real  and 
inverted,  whereas  in  Fig.   101,  b  it  is  virtual  and  erect. 


;\\f 


^  c 


ir 


Fig.  101,  b. — Lateral  magnification  and  construction  of  image  in  convex 

mirror. 


Whether  the  image  is  real  or  virtual  and  erect  or  inverted 
will  depend  both  on  the  position  of  the  object  and  on  the 
form  of  the  mirror. 

If  the  object-point  Q  is  supposed  to  move,  say,  from  left 
to  right  along  the  straight  line  QV  drawn  parallel  to  the 
axis  of  the  mirror,  the  corresponding  image-point  Q'  will 
traverse  the  straight  line  VF  continuously  in  the.  same 
direction.  Thus,  in  the  diagrams  (Fig.  102,  a  &  b)  the 
numerals  1,  2,  3,  etc.,  ranged  in  order  from  left  to  right  along 
a  straight  line  parallel  to  the  axis  of  the  mirror,  show  a 
number  of  successive  positions  of  the  object-point,  while  the 
primed  numbers  1',  2',  3',  etc.,  lying  along  the  straight  line 
VF,  show  the  corresponding  positions  of  the  image-point. 
The  straight  lines  11',  22',  33',  etc.,  all  meet  at  the  center 
C  of  the  mirror. 


§71] 


Imagery  in  Spherical  Mirror  175 


TO  4   AT  00 


^         ' 


Sf 


TO  E 
AT0O^-^VOBJE,CT  f       f      /    J     ) 


RAY 


/8         7 


TO    E  AT  CO 


CONCAVE    MIRROR 

(a) 


TO  5  AT  00 


IMAGE^ray 


TO  E  AT  OO 


OBJECT   RAY 


TO   E  ATCO 

3'  4-     5       6   7  8 

^WTr — r- 


<V 


\       c 

K 
X 


CONVEX    MIRROR 

lb) 


K 


% 

TO  S    AT  00 


Pig.   102,    a    and  b. — Imagery    in    (a)    concave    mirror,    (6)    convex 

mirror. 


176  Mirrors,  Prisms  and  Lenses  [§  73 

72.  The  Lateral  Magnification. — If  the  ordinates  of  the 
pair  of  extra-axial  conjugate  points  Q,  Q'  are  denoted  by 
y,  y',  respectively,  that  is,  if  in  Fig.  101,  a  and  b,  MQ  =  y, 
M'Q'  =  y',  the  ratio  y'jy  is  called  the  lateral  magnification  at 
the  axial  point  M.  This  ratio  will  be  denoted  by  y;  thus, 
y  =  y'jy.  The  sign  of  this  function  y  indicates  whether  the 
image  is  erect  or  inverted.  The  lateral  magnification  may 
have  any  value  positive  or  negative  depending  only  on  the 
position  of  the  object. 

In  the  similar  triangles  MCQ,  M'CQ' 

M'Q'  :MQ=M'C:MC; 
and  since 

M'C  =  r-u',        MC  =  r-u, 
where  u=AM,  u'  =  AM',  r  =  AC;  and  since  according  to  the 
abscissa-formula  (§  64) 

r-u'  _     v! 
r—u         u 
we  derive  the  following  formula  for  the  lateral  magnification 
in  the  case  of  a  spherical  mirror: 

V         u 
Also,  from  the  figure  we  see  that 

M'Q'  _  AW  _  FA  _  M'Q'  _  FM' , 
~MQ~MQ"FM      AV       FA' 
and  since  FM  =  z,  FM'  =  zr,  and  FA=/,  we  derive  also  an- 
other formula  for  the  lateral  magnification,  as  follows: 

y   y    x    f 

This  expression  shows  that  the  lateral  magnification  is  in- 
versely proportional  to  the  distance  of  the  object  from  the 
focal  plane. 

73.  Field  of  View  of  a  Spherical  Mirror.— When  the 
image  of  a  luminous  object  is  viewed  in  a  spherical  mirror, 
the  axis  of  the  mirror  is  determined  by  the  straight  line  O'C 


§73] 


Field  of  View  of  Spherical  Mirror 


177 


(Fig.  103,  a  and  b)  joining  the  center  O'  of  the  pupil  of  the 
observer's  eye  with  the  center  C  of  the  mirror;  and,  on  the 
assumption  that  the  image  is  formed  by  the  reflection  of 
paraxial  rays,  the  actual  portion  of  the  mirror  that  is  utilized 


Fig.  103,  a. — Field  of  view  for  eye  in  front  of  convex  mirror. 

consists  of  a  small  circular  zone  immediately  surrounding 
the  vertex  A  where  the  axis  meets  the  reflecting  surface.  Ac- 
cording to  the  method  of  drawing  these  diagrams  which  was 
described  in  §  70,  the  line-segment  GH  which  is  perpendicu- 
lar to  the  axis  at  A  and  which  is  bisected  at  A  will  represent 
a  meridian  section  of  this  zone  in  the  plane  of  the  figure,  so 
that  the  points  designated  by  G,  H  are  opposite  extremities 
of  a  diameter  of  the  effective  portion  of  the  mirror. 

All  the  reflected  rays  that  enter  the  eye  at  O'  must  neces- 
sarily lie  within  the  conical  region  determined  by  revolving 
the  isosceles  triangle  O'GH  around  the  axis  of  the  mirror. 


178 


Mirrors,  Prisms  and  Lenses 


[§73 


The  outermost  rays  that  can  possibly  be  reflected  into  the  eye 
at  O'  will  be  the  rays  that  are  reflected  along  the  straight 
lines  HO'  and  GO'.  In  order  to  see  a  real  image  in  a  concave 
mirror  (Fig.  103,  6),  the  eye  must  be  placed  in  front  of  the 


Fig.  103,  b. — Field  of  view  for  eye  in  front  of  concave  mirror. 


mirror  at  a  distance  greater  than  the  length  of  the  radius. 
The  incident  rays  corresponding  to  the  extreme  reflected 
rays  will  intersect  in  a  point  O  which  is  conjugate  to  0'; 
and  hence  the  field  of  view  (§  9)  within  which  all  object-points 
must  lie  in  order  that  their  images  in  the  mirror  may  be 
visible  to  an  eye  at  O'  will  be  limited  by  the  surface  of  a 
right  circular  cone  generated  by  the  revolution  of  the  isosceles 
triangle  OHG  around  the  axis  of  the  mirror.  Thus,  exactly 
as  in  the  case  of  the  corresponding  problem  in  connection 
with  the  field  of  view  of  a  plane  mirror  (§  16),  the  contour  of 
the  effective  portion  of  the  spherical  mirror  acts  also  as  a 
field-stop  for  the  imagery  produced  by  paraxial  rays. 

Through  Or  draw  B'J'  at  right  angles  to  the  axis  of  the 
mirror,  and  mark  the  points  B',  J'  at  equal  distances  from 
0'  on  opposite  sides  of  the  axis.  Then  B'J'  may  be  supposed 
to  represent  the  diameter  in  the  plane  of  the  diagram  of  the 
iris  opening  of  the  pupil  of  the  observer's  eye.  Construct 
by  the  method  described  in  §  71,  the  object-line  BJ  whose 
image  in  the  mirror  is  B'J'.    Evidently,  any  ray  which  after 


§  74]  Spherical  Refracting  Surface  179 

reflection  enters  the  pupil  of  the  eye  between  B'  and  J'  must 
before  reflection  have  passed,  really  or  virtually,  through 
the  conjugate  point  on  the  straight  line  between  B  and  J. 
In  fact,  the  circle  described  around  0  as  center  in  the  trans- 
versal plane  perpendicular  to  the  axis  at  0  with  radius  OB 
will  act  like  a  material  stop  to  limit  the  apertures  of  the 
bundles  of  incident  rays.  It  is  the  so-called  entrance-pupil 
of  the  system,  while  the  pupil  of  the  eye  plays  the  part  of 
the  exit-pupil  (see  §  16).  Thus,  for  example,  if  S  designates 
the  position  of  a  luminous  point  lying  anywhere  within  the 
field  of  view,  the  eye  at  O'  will  see  the  image  of  S  at  S'  by 
means  of  a  bundle  of  rays  which  are  drawn  from  S  to  all  points 
of  the  entrance-pupil  and  which  after  reflection  at  the 
mirror  are  comprised  within  the  cone  which  has  its  vertex 
at  S'  and  the  exit-pupil  as  base.  The  entrance-pupil  BJ  is 
the  aperture-stop  of  the  system  (§  11). 

74.  Refraction  of  Paraxial  Rays  at  a  Spherical  Surface. — 
In  the  accompanying  diagrams  Fig.  104,  a  and  b,  the  straight 
line  RB  represents  an  incident  ray  meeting  the  spherical 
refracting  surface  ZZ  at  B,  while  the  straight  line  BS  shows 


Fig.  104,  a. — Convex  spherical  refracting  surface  (n'>n). 

the  path  of  the  corresponding  refracted  ray.  If  the  position 
of  the  point  M  where  the  incident  ray  crosses  the  axis  is 
given,  the  problem  is  to  determine  the  position  of  the  point 
M'  where  the  refracted  ray  meets  the  axis.    The  angles  of 


180  Mirrors,  Prisms  and  Lenses  [§  74 

incidence  and  refraction  are  ZNBR=a,  ZN'BS=  a',  and 
by  the  law  of  refraction : 

n'.sina'  =  n.sina, 
where  n,  nf  denote  the  indices  of  refraction  of  the  first  and 
second  media,  respectively.    In  the  triangles  MBC,  M'BC, 
we  have: 

CM  :  BM  =  sina  :  sin0,        CM'  :  BM'  =  sina'  :  sin0, 


Fig.  104,  b. — Concave  spherical  refracting  surface  {nf  >  n) . 

where  $  =  ZBCA.    Dividing  one  of  these  equations  by  the 
other,  we  obtain: 

CM  .BM  _n' 

CM' :  BM'  n  ' 
Now  if  the  ray  RB  is  a  paraxial  ray,  the  incidence-point  B 
will  be  so  near  the  vertex  A  of  the  spherical  refracting  surface 
that  A  may  be  written  in  place  of  B,  according  to  the  def- 
inition of  a  paraxial  ray  as  given  in  §  63.  Therefore,  in  the 
case  of  the  refraction  of  paraxial  rays  at  a  spherical  surface 
the  four  points  C,  A,  M,  M'  on  the  axis  are  connected  by 
the  following  relation : 

CM  .AM  _n' 

CM,:AM'    n 


§  74]  Spherical  Refracting  Surface  181 

which  may  be  written  (§  65) : 

(CAMM')=-; 

n 

that  is,  the  double  ratio  of  the  four  axial  points  C,  A,  M,  M' 
is  constant  and  equal  to  the  relative  index  of  refraction  from 
the  first  medium  to  the  second. 

Thus,  for  a  given  spherical  surface  (that  is,  for  known 
positions  of  the  points  A  and  C) ,  separating  a  pair  of  media 
of  known  relative  index  of  refraction  (n'/ri),  the  point  M' 
on  the  axis  corresponding  to  a  given  position  of  the  axial 
point  M  has  a  perfectly  definite  position,  entirely  independent 
of  the  actual  slope  of  the  incident  paraxial  ray  RB;  whence 
it  may  be  inferred  that  M'  is  the  image  of  M,  so  that  to  a 
homocentric  bundle  of  incident  paraxial  rays  with  its  vertex 
lying  on  the  axis  of  the  spherical  refracting  surface  there  corre- 
sponds also  a  homocentric  bundle  of  refracted  rays  with  its 
vertex  on  the  axis. 

In  Fig.  104,  a  the  image  at  Mr  is  real,  whereas  in  Fig.  104,  6 
it  is  virtual.  Since  the  relative  index  of  refraction  is  never 
less  than  zero,  the  value  of  the  double  ratio  (CAMM')  in 
the  case  of  refraction  at  a  spherical  surface  is  necessarily 
positive;  consequently,  the  pair  of  conjugate  points  M,  M' 
is  not  "separated"  (§  65)  by  the  pair  of  points  A,  C,  as  was 
found  to  be  the  case  in  reflection  at  a  spherical  mirror  (§  68) . 
Thus,  if  M,  M'  designate  the  positions  of  a  pair  of  conjugate 
axial  points  with  respect  to  a  spherical  refracting  surface,  it 
is  always  possible  to  pass  from  M  to  M'  along  the  axis  one 
way  or  the  other  without  going  through  either  of  the  points 
A  or  C,  although  in  order  to  do  this  it  may  sometimes  be 
necessary  to  pass  through  the  infinitely  distant  point  of  the 
axis  (see  §  65) .  Accordingly,  depending  only  on  the  form 
of  the  surface  and  on  whether  n  is  greater  or  less  than  n', 
there  will  be  found  to  be  sixteen  possible  orders  of  arrange- 
ment of  these  four  points,  viz.: 

A,  C,  M,  M';  A,  C,  M',  M;  A,  M,  M',  C;  A,  M',  M,  C; 
M,  A,  C,  M';  M',  A,  C,  M;  M,  M',  A,  C;  M',  M,  A,  C; 


182  Mirrors,  Prisms  and  Lenses  [§75 

together  with  the  eight  other  arrangements  obtained  by  re- 
versing the  order  of  the  letters  in  each  of  these  combinations; 
in  other  words,  exactly  the  series  of  combinations  that  are 
not  possible  in  the  case  of  a  spherical  mirror  where  the  pair 
of  conjugate  axial  points  M,  M'  is  harmonically  separated 
by  the  pair  of  points  A,  C,  so  that  (CAMM')  =  -1  (§  68). 
The  student  should  draw  a  diagram  similar  to  Fig.  104  for 
each  of  the  possible  arrangements  of  the  four  points  above 
mentioned.  Fig.  104,  a  shows  the  case  M,  A,  C,  M'  and 
Fig.  104,  b  shows  the  case  M,  M',  C,  A. 

Moreover,  if  (CAMM')=w'/n,  then  also  (CAM'M)  = 
ft/ft',  as  follows  from  the  definition  of  the  double  ratio  (§  65). 
Consequently,  if  a  paraxial  ray  is  refracted  at  a  point  B  of 
a  spherical  surface  from  medium  n  to  medium  n'  along  the 
broken  line  RBS,  a  ray  directed  from  S  to  B  will  be  refracted 
from  medium  nf  to  medium  n  in  the  direction  BR;  which  is 
in  accordance  with  the  general  principle  of  the  reversibility 
of  the  light-path  (§  29).  If  therefore  M'  is  the  image  of  M 
when  the  light  is  refracted  across  the  spherical  surface  in  a 
given  sense,  then  also  M  will  be  the  image  of  M'  when  the 
refraction  takes  place  in  the  reverse  sense. 

75.  Reflection  Considered  as  a  Special  Case  of  Refrac- 
tion.— It  was  implied  above  that  if  it  were  possible  for  the 
ratio  n'/n  to  have  not  only  positive  values  but  also  the  unique 
negative  value  —1,  the  single  formula  (CAMM.')=n'ln 
would  express  the  relation  between  a  pair  of  conjugate  axial 
points  M,  M'  both  for  a  spherical  refracting  surface  and 
for  a  spherical  mirror.  The  question  naturally  arises,  there- 
fore, Is  there  a  general  rule  of  this  kind  applicable  also  to 
other  problems  in  optics  that  are  not  necessarily  concerned 
with  paraxial  rays  or  particular  conditions?  Returning  to 
fundamental  principles  and  recalling  the  laws  of  reflection 
and  refraction,  we  observe  that  while  the  angles  of  incidence 
and  refraction  always  have  like  signs,  the  angles  of  incidence 
and  reflection,  on  the  contrary,  have  opposite  signs.  In 
order,  therefore,  that  the  refraction-formula  nf .  sina'  =  n .  sina 


§  761       Construction  of  Conjugate  Axial  Points        183 

may  include  also  the  law  of  reflection  as  well,  the  values 
of  n  and  n'  in  the  latter  case  must  be  such  that  a'=  —  a 
is  a  solution  of  the  equation  in  question;  and  obviously  this 
solution  can  be  obtained  only  by  putting 

n =  —n,  or  —  =  —  1. 
n 

Accordingly,  the  rule  discovered  above  to  be  true  in  a  special 
case  is  found  to  be  entirely  general,  so  that,  at  least  from  a 
purely  mathematical  point  of  view,  the  reflection  of  light 
may  be  regarded  as  a  particular  case  of  refraction  back  again 
into  the  medium  of  the  incident  light,  provided  we  assign  to 
this  medium  two  equal  and  opposite  values  of  the  absolute 
index  of  refraction.  The  convenience  of  this  artifice  is  ap- 
parent, since  it  makes  it  quite  unnecessary  to  investigate  sep- 
arately and  independently  each  special  problem  of  reflection 
and  refraction;  for  when  in  any  given  case  the  relation  be- 
tween an  incident  ray  and  the  corresponding  refracted  ray 
has  been  ascertained,  it  will  be  necessary  merely  to  impose 
the  condition  n'=  —n  in  order  to  derive  immediately  the 
analogous  relation  between  the  incident  ray  and  the  corre- 
sponding reflected  ray.  Thus,  for  example,  any  formula 
hereafter  to  be  derived  concerning  the  refraction  of  paraxial 
rays  at  a  spherical  surface  may  be  converted  into  the  corre- 
sponding formula  for  the  case  of  a  spherical  mirror  by 
putting  n'=  —n. 

76.  Construction  of  the  Point  M'  Conjugate  to  the  Axial 
Point  M. — In  order  to  construct  the  point  M'  conjugate  to 
the  axial  point  M  with  respect  to  a  spherical  refracting  sur- 
face, we  may  proceed  as  follows : 

Through  the  vertex  A  (Fig.  105,  a,  b,  c  and  d)  and  the  center 
C  draw  a  pair  of  parallel  straight  lines  (preferably  but  not 
necessarily)  at  right  angles  to  the  axis;  and  on  the  line  going 
through  C  take  two  points  0  and  O'  such  that 

CO:CO'  =  n':n. 
Join  the  given  axial  point  M  by  a  straight  line  with  the  point 
O,  and  let  B  designate  the  point  where  this  straight  line, 


184 


Mirrors,  Prisms  and  Lenses 


76 


R     M 


Fig.   105. — Spherical  refracting  surface:  Construction  of  image-point   M' 
conjugate  to  axial  object-point  M;  construction  of  focal  points  F,  F\ 
(a)  Convex  surface,  n'>n;  order  MACM'. 
(6)  Concave  surface,  n'>n;  order  MM'CA. 

(c)  Convex  surface,  n' <n;  order  MM'AC. 

(d)  Concave  surface,  n'<n;  order  MCAM'. 


§  76]       Construction  of  Conjugate  Axial  Points       185 

produced  if  necessary,  meets  the  line  drawn  through  A 
parallel  to  CO;  then  the  required  point  M'  will  be  at  the 
place  where  the  straight  line  BO',  produced  if  necessary, 
intersects  the  axis. 

The  straight  line  Ay  drawn  perpendicular  to  the  axis  at  A 
will  be  tangent  to  the  spherical  surface  at  its  vertex ;  and  this 
line  will  represent  the  spherical  surface  in  the  diagram,  since 
we  are  concerned  here  only  with  paraxial  rays  (§  70) .  Thus, 
to  the  incident  ray  RB  crossing  the  axis  at  M  and  incident 
on  the  surface  at  B,  there  will  correspond  the  refracted  ray 
BS  crossing  the  axis  at  M'. 

The  proof  of  the  construction  consists  in  showing  that 

n' 
the  double  ratio  (CAMM')  is  equal  to  — ,  in  accordance  with 

n 

the  relation  which,  as  we  saw  above  (§  74),  connects  the  two 
conjugate  points  M,  M'. 
In  the  pair  of  similar  triangles  CMO,  AMB, 
CM:AM  =  CO:AB; 
and  in  the  pair  of  similar  triangles  CM'O',  AM'B, 

AM':CM'  =  AB:CO'. 
Multiplying  these  two  proportions,  we  obtain: 
CM     AM'     CO 


or 


and  hence 


CM'  *  AM      CO' ' 
CM     AM      n' 


CM'   AM'    n 


(CAMM')  =  -. 

n 

The  diagrams  illustrate  four  cases,  viz.,  the  cases  when 
the  points  A,  C,  M,  M'  are  ranged  along  the  axis  from  left 
to  right  in  the  orders  MACM',  MM'CA,  MM'AC  and 
MCAM'.  In  the  diagrams  Fig.  105,  a  and  6,  the  second 
medium  is  represented  as  more  highly  refracting  than  the 


186  Mirrors,  Prisms  and  Lenses  [§  77 

first  (n'>n),  whereas  in  the  two  other  diagrams  Fig.  105, 
c  and  d,  the  opposite  case  is  shown  (n'<ri);  in  a  and  c  the 
surface  is  convex,  and  in  b  and  d  it  is  concave. 

77.  The  Focal  Points  (F,  F')  of  a  Spherical  Refracting 
Surface. — The  object-point  F  which  is  conjugate  to  the  in- 
finitely distant  image-point  E  and  the  image-point  F'  which 
is  conjugate  to  the  infinitely  distant  object-point  E  of  the 
axis  are  the  so-called  focal  points  of  the  spherical  refracting 
surface.  A  conical  bundle  of  incident  paraxial  rays  with  its 
vertex  at  the  primary  focal  point  F  will  be  converted  into  a 
cylindrical  bundle  of  refracted  rays  all  parallel  to  the  axis 
and  meeting  therefore  in  the  infinitely  distant  point  E  of 
the  axis;  and,  similarly,  a  cylindrical  bundle  of  paraxial  rays 
proceeding  from  the  infinitely  distant  point  E  of  the  axis 
will  be  transformed  into  a  conical  bundle  of  refracted  rays 
with  its  vertex  at  the  secondary  focal  point  F'. 

According  to  the  method  explained  in  §  76,  the  focal  point 
F  may  be  constructed  by  drawing  the  straight  line  O'H 
(Fig.  105,  a,  b,  c  and  d)  through  0'  parallel  to  the  axis  meeting 
the  straight  line  AB  in  the  point  designated  by  H;  and  then 
the  straight  line  OH  will  intersect  the  axis  in  the  primary  focal 
point  F.  Similarly,  if  the  straight  line  OK  is  drawn  through 
O  parallel  to  the  axis  meeting  AB  in  a  point  K,  the  point  of 
intersection  of  the  straight  line  KO'  with  the  axis  will  de- 
termine the  position  of  the  secondary  focal  point  F'.  In 
brief,  the  diagonals  of  the  parallelogram  OO'HK  meet  the 
axis  in  the  focal  points  F,  F'.  The  spherical  refracting  surface 
is  said  to  be  convergent  or  divergent  according  as  the  focal 
point  F'  is  real  or  virtual,  respectively.  Thus,  in  the  dia- 
grams Fig.  105,  a  and  d,  incident  rays  parallel  to  the  axis  are 
brought  to  a  real  focus  at  F',  so  that  the  surface  is  convergent 
for  each  of  these  cases;  whereas  in  the  diagrams  Fig.  105, 
b  and  c,  incident  rays  parallel  to  the  axis  are  refracted  as  if 
they  proceeded  from  a  virtual  focus  at  F'. 

Moreover,  certain  characteristic  metric  relations  may  be 
derived  immediately  from  the  diagrams  Fig.  105,  a,  6,  c,  and  d. 


§  77]     Spherical  Refracting  Surface:  Focal  Points      187 

For  example,  in  the  two  pairs  of  similar  triangles  FAH,  HO'O 
and  F'CO',  O'HK,  we  obtain  the  proportions: 

FA :  HO'  =  AH :  O'O,  CF' :  HO7  =  CO' :  HK, 

and  since  CO'  =  AH,  HK  =  0'0,  we  find: 

FA  =  CF'; 
and  hence  also : 

F'A  =  CF. 

Accordingly,  concerning  the  positions  of  the  focal  points  of 
a  spherical  refracting  surface  we  have  the  following  rule: 

The  focal  points  of  a  spherical  refracting  surface  lie  on  the 
axis  at  such  places  that  the  step  from  one  of  them  to  the  center 
is  identical  with  the  step  from  the  vertex  to  the  other  focal  point. 
-  This  statement  should  be  verified  for  each  of  the  diagrams. 
Not  only  will  the  center  C  be  seen  to  be  at  the  same  distance 
from  the  primary  focal  point  as  the  secondary  focal  point  is 
from  the  vertex  A,  but  the  direction  from  F  to  C  will  always 
be  the  same  as  that  from  A  to  F'. 

This  relation  may  also  be  expressed  in  a  different  way; 
for,  since 

FA  =  CF'  =  CA+AF', 
we  have  the  following  equation : 

FA-f-F'A  =  CA;  or  AC  =  AF+AF'; 
which  may  be  put  in  words  by  saying  that  the  step  from  the 
vertex  to  the  center  of  a  spherical  refracting  surface  is  equal  to 
the  sum  of  the  steps  from  the  vertex  to  the  two  focal  points. 

And,  finally,  since  in  the  pair  of  similar  triangles  FAH, 
FCO,  we  have: 

FC:FA=CO:AH=CO:CO'=n':w, 
and  since  FC=  —  CF=  —  F'A,  we  obtain  also  another  useful 
and  important  relation,  viz. : 

F'A__n' 
FA        n ' 
and,  consequently:  The  two  focal  points  F,  F'  of  a  spherical 
refracting  surface  lie  on  opposite  sides  of  the  vertex  A,  and  at 
distances  from  it  which  are  in  the  ratio  of  n  to  nf.    If,  there- 
fore, we  are  given  the  positions  of  one  of  the  two  focal  points, 


188 


Mirrors,  Prisms  and  Lenses 


[§77 


AIR         /    GLASS 


w 


Fig.  106,  a,  b,  c  and  d. — Focal  points  of  spherical  refracting  surface  sep- 
arating air,  of  index  1,  and  glass,  of  index  1.5. 
(a)  Refraction  from  air  to  glass  at  convex  surface. 
(6)  "      "   "       "     "    concave     " 

(c)  "     glass  to  air  "   convex       " 

(d)  "       "       "  "  "   concave     " 


§  77]     Spherical  Refracting  Surface:  Focal  Points      189 

F  or  F',  as  well  as  the  positions  of  the  points  A,  C  which  de- 
termine the  size  and  form  of  the  spherical  surface,  we  have 
all  the  data  necessary  to  enable  us  to  locate  the  point  M' 
conjugate  to  a  given  axial  object-point  M.  For  we  can 
locate  the  position  of  the  other  focal  point  and  thus  determine 
the  value  of  the  ratio  n' :  n. 

Whether  the  secondary  focal  point  will  lie  on  one  side  or 
the  other  of  the  spherical  refracting  surface,  that  is,  whether 
the  surface  will  be  convergent  or  divergent,  will  depend  on 
each  of  two  things,  viz.:  (1)  Whether  the  surface  is  convex 
or  concave,  and  (2)  Whether  n'  is  greater  or  less  than  n.  For 
example,  if  the  rays  are  refracted  from  air  to  glass  (n'/n  = 
3/2),  according  to  the  above  relations  we  find  that  AF= 2  CA, 
AF'  =  3  AC ;  so  that  starting  at  the  vertex  A  and  taking  the 
step  CA  twice  we  can  locate  the  primary  focal  point  F;  and 
returning  to  the  vertex  A  and  taking  the  step  AC  three 
times,  we  arrive  at  the  secondary  focal  point  F\  The  dia- 
grams Fig.  106,  a  and  b,  show  the  positions  of  the  focal  points 
for  refraction  from  air  to  glass  for  a  convex  surface  and  for 
a  concave  surface.  In  this  case  the  convex  surface  is  con- 
vergent and  the  concave  surface  is  divergent.  On  the  other 
hand,  when  the  light  is  refracted  from  glass  to  air  (n'/n= 
2/3),  we  find  AF  =  3  AC,  AF'  =  2  CA  (Fig.  106,  c  and  d),  and 
in  this  case  the  concave  surface  is  convergent  and  the  convex 
surface  is  divergent. 

In  conclusion,  it  may  be  added  that  the  constructions  and 
rules  which  have  been  given  above  for  the  case  of  a  spherical 
refracting  surface  are  entirely  applicable  also  to  a  spherical 
mirror.  In  fact,  here  we  have  an  excellent  illustration  of 
the  method  of  treating  reflection  as  a  special  case  of  refrac- 
tion, which  was  explained  in  §  75.  For  if  we  take  n'=  -n, 
the  two  points  O,  0'  (Fig.  107,  a  and  b)  will  lie  on  a  straight 
line  passing  through  the  center  C  of  the  mirror  at  equal  dis- 
tances from  C  in  opposite  directions.  The  point  M'  con- 
jugate to  the  axial  object-point  M  and  the  focal  points  F, 
F'  will  be  found  precisely  according  to  the  directions  for 


190 


Mirrors,  Prisms  and  Lenses 


l§78 


S\ 

O 

\\^) 

O 

-"-"M 

A 

/  d»    \ 

0 

o' 

Fig.  107,  a  and  b. — Reflection  at  spherical  mirror:  Con- 
struction of  image-point  M'  conjugate  to  axial  object- 
point  M ;  construction  of  focal  point. 

(a)   concave  mirror,   (b)  convex  mirror. 

drawing  'the  diagrams  of  Fig.  105.  Obviously,  the  focal 
points  of  a  spherical  mirror  will  coincide  with  each  other 
at  a  point  midway  between  the  vertex  and  center  (§  69). 

78.  Abscissa-Equation  referred  to  the  Vertex  of  the 
Spherical  Refracting  Surface  as  Origin. — If  the  vertex  A  of 
the  spherical  refracting  surface  is  taken  as  the  origin  (§  63) 
from  which  distances  or  steps  along  the  axis  are  reckoned, 
and  if  the  symbols  r,  u  and  v!  are  employed  as  in  the  case 
of  a  spherical  mirror  (§  64)  to  denote  the  abscissa?  of  the 


§  79]  Spherical  Refracting  Surface  191 

center  C  and  the  pair  of  conjugate  axial  points  M,  M',  that 
is,  if  AC  =  r,  AM  =  u,  AM'  =  u',  then 

CM  =  CA+ AM  =  u-r,        CM'  =  CA+ AM'  =  u'-  r; 

n' 
and   since   the   formula    (CAMM/)  =  —   may  evidently  be 

written  as  follows: 

,  CM'_     CM 
nAM'    nAM' 


we  obtain 


u  —r_     u—r 


u  u 

Dividing  both  sides  by  r,  we  derive  the  so-called  invariant 
relation  in  the  case  of  refraction  of  paraxial  rays  at  a  spherical 
surface,  in  the  following  form: 

.f.-iy-.p-!). 

\r    ul        \r     ul 
Usually,  however,  this  equation  is  written  as  follows: 


-,  =  -+ 
u      u        r 

which  is  to  be  regarded  as  one  of  the  fundamental  formulae  of 
geometrical  optics.  If  the  two  constants  r  and  n'/n  are  known, 
the  abscissa  u'  corresponding  to  any  given  value  of  u  may 
easily  be  determined.  Putting  n'  =  —  n  (§  75),  we  obtain  the 
abscissa-formula  for  reflection  of  paraxial  rays  at  a  spherical 
mirror  (§  64) ;  and  if  we  put  r  =  oo ,  we  derive  the  formula 

— ,  =  -  for  the  refraction  of  paraxial  rays  at  a  plane  surface 

(§41).  It  is  because  this  linear  equation  connecting  the 
abscissae  of  a  pair  of  conjugate  axial  points  includes  these 
other  cases  also  that  some  writers  have  proposed  that  the 
formula  above  should  be  called  the  characteristic  equation  of 
paraxial  imagery. 

79.  The  Focal  Lengths  f,  f  of  a  Spherical  Refracting 
Surface. — The  steps  from  the  focal  points  F  and  F'  to  the  vertex 


192  Mirrors,  Prisms  and  Lenses  [§  79 

A  are  called  the  focal  lengths  of  the  spherical  refracting  surface; 
the  primary  focal  le?igth,  denoted  by  f,  is  the  abscissa  of  A  with 
respect  to  F  (/=FA),  and  the  secondary  focal  length,  denoted 
by  f,  is  the  abscissa  of  A  with  respect  to  F'(/' =  F'A). 

Since  FA+F'A  =  CA  (§  77),  and  since  CA=  -r,  the  focal 
lengths  and  the  radius  of  the  surface  are  connected  by  the 
following  relation: 

f+f'+r=0, 

and  hence  if  two  of  these  magnitudes  are  known,  the  value 
of  the  third  may  always  be  determined  from  the  fact  that  their 
algebraic  sum  is  equal  to  zero.  For  example,  starting  at  any 
point  on  the  axis  and  taking  in  succession  in  any  order  the 
three  steps  denoted  by  /,  /'  and  r,  one  will  find  himself  at 
the  end  of  the  last  step  back  again  at  the  starting  point. 

Moreover,  the  focal  lengths  are  connected  with  the  indices 
of  refraction  by  the  following  relation  (§  77) : 

V        n' 

J-=--orn.f'+n'.f=0; 
f         n 

and,  hence,  the  focal  lengths  of  a  spherical  refracting  surface 
are  opposite  in  sign  and  in  the  same  numerical  ratio  as  that  of 
the  indices  of  refraction.  This  formula,  as  we  shall  see  (§  122), 
represents  a  general  law  of  fundamental  importance  in  geo- 
metrical optics. 

Expressions  for  the  focal  lengths  in  terms  of  the  radius 
r  and  the  relative  index  of  refraction  {n!  :  n)  may  be  derived 
immediately  from  the  pair  of  simultaneous  equations  above 
by  solving  them  for  /  and  /'.  The  same  expressions  may 
likewise  be  easily  obtained  by  substituting  in  succession  in 
the  abscissa-formula  (§  78)  the  two  pairs  of  corresponding 
values,  viz.,  u=  — /,  u'=  go  and  w=oo,  u'=—f.  And, 
finally,  they  may  also  be  obtained  geometrically  from  one  of 
the  diagrams  of  Fig.  107  by  observing  that,  since  by  con- 
struction CO  :  CO'  —  n'\n,  it  follows  that 

CO':  0'0  =  n:  {n'-n),        CO:  0'0  =  n':  (n'-n). 


§  80]  Spherical  Refracting  Surface  193 

Now  from  the  two  pairs  of  similar  triangles  FAH,  HO'O  and 
F'AK,  O'HK  we  obtain  the  two  proportions: 

FA:  HO'  =  AH:  O'O,        F'A:  0'H  =  AK:  HK; 
and  since 

FA=/,  HO'  =  AC  =  r,  AH  =  CO',  F'A=/',  AK  =  CO,  and 
HK  =  0'0, 

we  have,  finally: 

,_     n  „  n' 

J     n'-nTl      J  n'-n'T' 

which  are  exceedingly  useful  forms  of  the  expressions  for  the 
focal  lengths. 
Since 

n    n' —  n        nf 


f        r  f" 

the  abscissa-relation  connecting  u  and  v!  may  be  expressed 
in  terms  of  one  of  the  focal  lengths  instead  of  in  terms  of  the 
radius  r,  for  example,  in  terms  of  the  focal  length/,  as  follows: 

n'  _n.n 
u      u     f 

80.  Extra-Axial  Conjugate  Points ;  Conjugate  Planes  of  a 
Spherical  Refracting  Surface. — If  the  axis  AC  of  a  spherical 
refracting  surface  is  revolved  in  a  meridian  plane  through 
a  very  small  angle  about  an  axis  perpendicular  to  this  plane 
at  the  center  C,  so  that  the  vertex  of  the  surface  is  displaced 
a  little  to  one  side  of  its  former  position  A  to  a  point  U  on 
the  surface,  the  pair  of  conjugate  points  M,  M'  will  likewise 
undergo  slight  displacements  into  the  new  positions  Q,  Q'; 
and,  evidently,  the  same  relation  will  connect  the  four  points 
C,  U,  Q,  Q'  on  the  central  line  UC  as  exists  between  the  four 
points  C,  A,  M,  M'  on  the  axis  AC,  and  accordingly  (§  76) 
we  may  write: 

(CUQQ')=£'; 


194  Mirrors,  Prisms  and  Lenses  [§  81 

and  hence  it  is  obvious  that  the  points  Q,  Q'  are  a  pair  of 
extra-axial  conjugate  points  with  respect  to  the  spherical 
refracting  surface.  Thus,  if  the  points  belonging  to  an  ob- 
ject are  all  congregated  in  the  immediate  vicinity  of  the  axis 
on  an  element  of  a  spherical  surface  which  is  concentric  with 
the  refracting  sphere,  the  corresponding  image-points  will 
all  be  assembled  on  an  element  of  another  concentric  spherical 
surface,  and  any  straight  line  going  through  C  will  determine 
by  its  intersections  with  this  pair  of  concentric  surfaces  two 
conjugate  points  Q,  Q'.  In  order  that  the  rays  concerned 
may  all  be  incident  near  the  vertex  A,  it  is  necessary 
to  assume  that  ZUCA  is  very  small,  which  means  that 
the  little  elements  of  the  surfaces  described  around  C  may 
in  fact  be  regarded  as  plane  surfaces  perpendicular  to  the 
axis  AC.  Accordingly,  the  imagery  produced  by  the  re- 
fraction of  paraxial  rays  at  a  spherical  surface  may  be  de- 
scribed by  the  following  statements: 

(1)  The  image  of  a  plane  object  perpendicular  to  the  axis 
of  a  spherical  refracting  surface  is  similar  to  the  object,  and 
will  lie  likewise  in  a  plane  perpendicular  to  the  axis;  (2)  A 
straight  line  drawn  through  the  center  C  will  intersect  a  pair  of 
conjugate  planes  in  a  pair  of  conjugate  points  Q,  Q';  and  (3) 
Incident  rays  which  interesct  in  Q  will  be  transformed  into 
refracted  rays  which  intersect  in  Q'. 

Diagrams  showing  the  refraction  of  paraxial  rays  at  a 
spherical  surface  should  be  drawn  therefore  according  to  the 
plan  explained  in  §  70,  as  has  been  already  stated.  The 
spherical  refracting  surface  must  be  represented  in  the  figure 
by  the  plane  tangent  to  the  surface  at  its  vertex  A,  whose 
trace  in  the  meridian  plane  of  the  drawing  is  the  straight 
line  Ay  which  is  taken  as  the  y-axis  of  the  system  of  rect- 
angular coordinates  whose  origin  is  at  A  (§  63). 

81.  Construction  of  the  Point  Q'  which  with  Respect  to  a 
Spherical  Refracting  Surface  is  Conjugate  to  the  Extra- 
axial  Point  Q. — The  point  Q'  conjugate  to  the  extra-axial 
point  Q  is  easily  constructed.    Having  first  located  the  focal 


81] 


Spherical  Refracting  Surface 


195 


points  F,  F'  (§  77),  we  draw  through  Q  (Figs.  108  and  109)  a 
straight  line  parallel  to  the  z-axis  meeting  the  y-axis  in  the 
point  designated  by  V;  then  the  point  of  intersection  of  the 


Fig.    108. — Spherical  refracting  surface:  Lateral  magnification  and 
construction  of  image.    Convex  surface,  n'  >  n. 

straight  lines  VF'  and  QC  will  be  the  required  point  Q'. 
A  third  line  may  also  be  drawn  through  Q,  viz.,  the  straight 
line  QF  meeting  the  y-axis  in  the  point  marked  W;  and  if  a 


Fig.    109. — Spherical    refracting    surface:   Lateral    magnification    and 
construction  of  image.    Concave  surface,  n'>n. 

straight  line  is  drawn  through  W  parallel  to  the  x-axis,  it 
will  likewise  pass  through  Q'. 

If  M,  M'  designate  the  feet  of  the  perpendiculars  let  fall 


196  Mirrors,  Prisms  and  Lenses  [§  82 

from  Q,  Q'  respectively,  on  the  z-axis,  then  M'Q'  will  be 
the  image  of  the  small  object-line  MQ.  In  Fig.  108,  which 
represents  the  case  of  a  convex  refracting  surface,  the  image 
is  real  and  inverted,  whereas  in  Fig.  109  the  surface  is 
concave  and  the  image  is  virtual  and  erect.  Both  diagrams 
are  drawn  for  the  case  when  n'>n. 

If  the  object-point  Q  coincides  with  the  point  marked  V, 
the  image-point  Q'  will  also  be  at  V,  and  image  and  object 
will  be  congruent.  The  pair  of  conjugate  planes  of  an  optical 
system  for  which  this  is  the  case  are  called  the  principal 
planes  (see  §  119);  and  hence  the  principal  planes  of  a  spher- 
ical infracting  surface  coincide  with  each  other  and  are  identical 
with  the  tangent-plane  at  the  vertex. 

82.  Lateral  Magnification  for  case  of  Spherical  Refract- 
ing Surface.— The  ratio  M'Q':  MQ  (Figs.  108  and  109)  is  the 
so-called  lateral  magnification  of  the  spherical  refracting  sur- 
face with  respect  to  the  pair  of  conjugate  axial  points  M,  M'. 
Since 


M'Q':MQ  =  CM':CM 

and  since  (§  74) 

CM'    n'    AM' 

CM     n  '  AM  ' 

we  find: 

M'Q'n     AM' 

MQ      n'    AM 

If  y,  yr  denote  the  heights  of  object  and  image,  that  is,  if 
2/  =  MQ,  y'  =  M'Q',  and  if  we  put  the  lateral  magnification 
equal  to  y,  as  in  §  72,  then,  evidently: 


_y  _n     u 

y     n' "  u} 

where  u  =  AM,  v!  =  AM'.  The  lateral  magnification  depends, 
therefore,  on  the  position  of  the  object,  and  the  image  is 
erect  or  inverted  according  as  this  ratio  is  positive  or  nega- 
tive. 


§  83]     Spherical  Refracting  Surface:  Focal  Planes     197 

83.  The  Focal  Planes  of  a  Spherical  Refracting  Surface. 
— The  focal  planes  are  the  pair  of  planes  which  are  perpendic- 
ular to  the  axis  at  the  focal  points  F,  F'.  "The  infinitely 
distant  plane  of  space/'  which,  according  to  the  notions  of 
the  modern  geometry,  is  to  be  regarded  as  the  locus  of  the 
infinitely  distant  points  (§  65)  of  space,  is  the  image-plane 
conjugate  to  the  primary  focal  plane,  which  is  the  plane 
perpendicular  to  the  axis  at  F.  On  the  other  hand,  re- 
garded as  belonging  to  the  object-space,  the  infinitely  dis- 
tant plane  is  imaged  by  the  secondary  focal  plane  perpendicu- 
lar to  the  axis  at  F'. 

The  rays  proceeding  from  an  infinitely  distant  object-point 
I  (Fig.  110)  constitute  a  cylindrical  bundle  of  parallel  in- 


Toj'at  <*> 

Fig.  110. — Focal  planes  and  focal  lengths  of  spherical  refracting  surface. 


cident  rays.  Since  I  lies  in  the  infinitely  distant  plane  of 
space,  its  image  V  will  be  formed  in  the  secondary  focal 
plane,  and  the  position  of  V  in  this  plane  may  be  located  by 
drawing  through  the  center  C  of  the  spherical  refracting 
surface  a  straight  line  parallel  to  the  system  of  parallel 
rays  which  meet  in  the  infinitely  distant  point  I.  Thus,  for 
example,  the  image  of  a  star  which  may  be  regarded  as  a 
point  infinitely  far  away  will  be  formed  in  the  secondary 
focal  plane ;  and  if  the  apparent  place  of  the  star  in  the  firma- 
ment is  in  the  direction  CI,  the  star's  image  will  be  at  the 


198  Mirrors,  Prisms  and  Lenses  [§  83 

point  I'  where  the  straight  line  CI  meets  the  secondary  focal 
plane. 

Similarly,  if  J  designates  the  position  of  an  object-point 
lying  in  the  primary  focal  plane,  its  image  J'  will  be  the  in- 
finitely distant  point  of  the  straight  line  JC.  Thus,  to  a 
homocentric  bundle  of  incident  paraxial  rays  with  its  vertex  in 
the  primary  focal  plane,  there  corresponds  a  cylindrical  bundle 
of  refracted  rays;  and  to  a  cylindrical  bundle  of  incident  paraxial 
rays  there  corresponds  a  homocentric  bundle  of  refracted  rays 
with  its  vertex  in  the  secondary  focal  plane. 

The  directions  of  the  infinitely  distant  points  I  and  J'  are 
given  by  assigning  the  values  of  the  slope-angles 

0  =  ZFCI  =  ZF'CP,  0'  =  ZFCJ  =  ZF'CJ'; 
and  the  points  I'  and  J  conjugate  to  them  will  lie  in  the  sec- 
ondary and  primary  focal  planes  on  straight  lines  passing 
through  the  center  C  and  inclined  to  the  axis  at  the  angles 
6  and  0',  respectively.  The  angle  0,  which  is  the  measure 
of  the  angular  distance  from  the  axis  of  the  infinitely  distant 
object-point  I,  determines  the  apparent  size  of  an  object 
in  the  infinitely  distant  plane  of  the  object-space;  and,  sim- 
ilarly, the  angle  0'  is  the  measure  of  the  apparent  size  of  the 
infinitely  distant  image  of  the  object  FJ. 

Draw  the  straight  lines  JG  and  FK  paralled  to  the  optical 
axis  and  meeting  the  ?/-axis  in  the  points  designated  by  G 
and  K,  respectively;  then  the  straight  lines  FK  and  CP  will 
be  parallel  to  each  other,  and  the  same  will  be  true  with 
respect  to  the  straight  lines  GF'  and  JC.  Hence, 
ZAFK=  0,  ZAF'G  =  0';  and  since  AK=FT  and  AG  =  FJ, 
we  find: 

|l=tan0,!^  =  tan0'. 

Putting  FA=/  and  F'A=/'  (§79),  we  obtain  the  following 
expressions  for  the  focal  lengths: 

FT  FJ 

/•=__,    /'  =  . 


tan  0  tan  6' 


§  84]  Spherical  Refracting  Surface  199 

and  since  the  tangents  of  the  small  angles  8,  8'  are  indis- 
tinguishable from  the  angles  themselves  (see  §  63) ,  we  obtain 
new  definitions  of  the  focal  lengths,  as  follows: 

The  primary  focal  length  is  the  ratio  of  the  height  of  the  image, 
in  the  secondary  focal  plane,  of  an  infinitely  distant  object  to 
the  apparent  size  of  the  object;  and  the  secondary  focal  length 
is  the  ratio  of  the  height  of  an  object  in  the  primary  focal  plane 
to  the  apparent  size  of  the  infinitely  distant  image. 

The  ratio  of  the  apparent  size  of  the  infinitely  distant 
image  to  the  height  of  an  object  in  the  primary  focal  plane 
is  a  measure  of  the  magnifying  power  of  the  optical  system 
(see  §  158),  and  in  this  sense  we  may  say  that  the  magnifying 
power  of  a  spherical  refracting  surface  is  equal  to  the  reciprocal 
of  the  secondary  focal  length. 

84.  Construction  of  Paraxial  Ray  Refracted  at  a  Spherical 
Surface. — The  refracted  ray  corresponding  to  a  paraxial  ray 
IB  (Fig.  110)  incident  on  a  spherical  refracting  surface  at 
the  point  B  may  easily  be  constructed,  for  example,  in  one 
of  the  following  ways: 

(a)  Through  the  primary  focal  point  draw  the  straight 
line  FK  parallel  to  IB  meeting  the  y-axis  in  the  point  K ;  and 
through  K  draw  a  straight  line  parallel  to  the  z-axis  meeting 
the  secondary  focal  plane  in  the  point  I';  the  path  of  the 
refracted  ray  will  lie  along  the  straight  line  BI'. 

(b)  Through  the  center  C  draw  a  straight  line  CI'  parallel 
to  the  given  incident  ray  meeting  the  secondary  focal  plane 
in  the  point  I';  the  path  of  the  corresponding  refracted  ray 
will  be  along  the  straight  line  BI'. 

(c)  Let  J  designate  the  point  where  the  given  incident 
ray  crosses  the  primary  focal  plane,  and  draw  the  straight 
line  JG  parallel  to  the  z-axis  meeting  the  y-axis  in  the 
point  designated  by  G;  then  the  path  of  the  required 
refracted  ray  will  lie  along  the  straight  line  BI'  drawn 
through  the  incidence-point  B  parallel  to  the  straight  line 
GF',  where  F'  designates  the  position  of  the  secondary  focal 
point. 


200  Mirrors,  Prisms  and  Lenses  [§  85 

(d)  Finally,  the  required  refracted  ray  will  be  along  the 
straight  line  BP  drawn  parallel  to  the  straight  line  JC. 

85.  The  Image-Equations  in  the  case  of  Refraction  of 
Paraxial  Rays  at  a  Spherical  Surface. — The  rectangular  co- 
ordinates of  the  image-point  Q'  may  easily  be  expressed  in 
terms  of  the  coordinates  of  the  object-point  Q.  But  the 
forms  of  these  expressions  will  depend  partly  on  the  particu- 
lar pair  of  constants  (n'/n,  r  and  /,  /')  which  define  the  sur- 
face and  partly  on  the  system  of  axes  to  which  the  coordinates 
are  referred.  The  axis  of  the  spherical  surface  will  always 
represent  the  axis  of  abscissae  (x-axis),  and  the  ?/-axis  will 
be  at  right  angles  to  it;  but  the  origin  may  be  taken  at  any 
place  along  the  x-axis.  If  the  vertex  A  is  taken  as  the  origin 
(§  63),  the  coordinates  of  Q,  Q'  will  be  (u,  y)  and  (u',  y'); 
that  is,  u  =  AM,  u'  =  AM',  £/  =  MQ,  2/'=M'Q';  and  since 
(§§  78  and  82) 

n'  _  n.n'  —  n       yr  _nu' 
v!    u        r    '      y     n'u  ' 
we  obtain  by  solving  for  u'  and  y'; 

,_         n'ru  ,_         nry 

(n' '— n)  u+nr'  (n'-n)  u+nr' 

In  terms  of  the  same  coordinates,  but  with  a  different 
pair  of  constants,  viz.,  /,  /',  instead  of  nf:n,  r,  the  image- 
equations  may  be  put  also  in  other  forms,  as  follows: 

It  will  be  recalled  that  in  §  79  the  abscissa-formula  was 
written : 

n' _n  .  n 
u     u     f 
and  since  (§  79)  n'/n  =  —f'/f,  n  and  n'  may  be  eliminated 
and  the  image-equations  will  become: 

/+  /'.  1=0  yf-  i  J'+u'-  /< 

tt-rtt,i-i    u,  y    f+u       ff  f,u, 

which  are  also  frequently  employed.     These  formulae  may 
also   be   easily   derived  from  the  geometrical   relations  in 
Figs.  108  and  109,  since  we  have  the  proportions: 
FM:  AM  =  VA:  VW  =  AF':  AM'. 


86] 


Smith-Helmholtz  Equation 


201 


Instead  of  a  single  system  of  rectangular  coordinates,  we 
may  have  two  systems,  one  for  the  object-space  and  the 
other  for  the  image-space.  For  example,  if  the  focal  points 
F,  F'  are  selected  as  the  origins  of  two  such  systems,  and 
if  the  abscissae  of  the  pair  of  conjugate  axial  points  M,  M' 
are  denoted  by  x,  x',  that  is,  if  #  =  FM,  z^F'M',  then,  since 
u  =  AM  =  AF+  FM  =  x  -/,  u'  =  AM'  =  AF'+  F'M'  =  x'  -/', 
the  abscissae,  u,  v!  may  be  eliminated  from  the  equations 
above,  and  the  image-equations  will  be  obtained  finally  in 
their  simplest  forms,  as  follows: 

y    *   /'" 

These  relations  may  be  derived  directly  from  the  two  pairs 
of  similar  triangles  FMQ,  FAW  and  F'M'Q',  F'AV  in 
Figs.  108  and  109.    The  abscissa-relation 

x.x'=f.f 
is  the  so-called  Newtonian  formula  (see  §  69) .    If  the  x's  are 
plotted  as  abscissae  and  the  x"s  as  ordinates,  this  equation 
will  represent  a  rectangular  hyperbola. 

86.  The^o-called  Smith-Helmholtz  Formula. — In  Fig.  Ill 
if  M'Q'  =  2/'  represents  the  image  in  a  spherical  refracting 


Fig.  111. — Spherical  refracting  surface:  Smith-Helmholtz  law. 

surface  Ay  of  a  small  object-line  MQ,  =  y  perpendicular  to 
the  axis  at  M,  and  if  B  designates  the  incidence-point  of  a 


202  Mirrors,  Prisms  and  Lenses  [§  86 

paraxial  ray  which  crosses  the  axis  before  and  after  refrac- 
tion at  M  and  M',  respectively,  then  in  the  triangle  MBM' 

sin0:sin0'  =  BM':BM, 
where    0  =  ZAMB,    0'  =  ZAM'B  denote  the  slopes  of  the 
incident  ray  MB  and  the  corresponding  refracted  ray  BM'. 
Since  the  ray  is  paraxial,  we  may  put  0  =  sin  0,   0'  =  sin  0' 
and  also  BM  =  AM  =  u,  BM'  =  AM'  =  v!  (§  63) .    Hence, 


^  =  -,  or    u'.  0'  =  u.  0. 
6'    u' 


But  (§  82) 


ri .y'  _n.ym 


and,  therefore,  by  multiplying  these  two  equations  so  as  to 
eliminate  u  and  uf,  we  obtain  the  important  invariant- 
relation  in  the  case  of  refraction  of  paraxial  rays  at  a 
spherical  surface,  viz.: 

ri.y'.  6'  =  n.y.  0. 
This  formula  states  that  the  function  obtained  by  the  con- 
tinued product  of  the  three  factors  n,  y,  0  has  the  same  value 
after  refraction  at  a  spherical  surface  as  it  had  before  re- 
fraction. It  is  a  special  case  of  a  general  law  which  is  found 
to  apply  to  a  centered  system  of  spherical  refracting  sur- 
faces (§  118)  and  which  is  usually  known  as  Lagrange's  law; 
but  undoubtedly  Robert  Smith  who  announced  the  law  for 
the  case  of  a  system  of  thin  lenses  as  early  as  1738  is  entitled 
to  the  credit  of  it.  The  importance  of  the  relation  was 
recognized  by  Helmholtz(  182 1-1894),  and  the  form  in  which 
it  is  written  above  is  due  to  him.  On  the  whole  it  seems 
proper  to  adopt  the  suggestion  of  P.  Culmann  and  to  refer 
to  this  equation  as  the  Smith-Helmholtz  formula. 


Ch.  VI]  Problems  203 

PROBLEMS 

1.  If  A  designates  the  vertex  and  C  the  center  of  a  spher- 
ical mirror,  and  if  M,  M'  designate  the  points  where  a 
paraxial  ray  crosses  the  straight  line  AC  before  and  after 
reflection,  respectively,  show  that 

1+1=2 

u     u      r 
where  r  =  AC,  w  =  AM,  u'  =  AM'. 

2.  The  radius  of  a  concave  mirror  is  30  cm.  Paraxial  rays 
proceed  from  a  point  60  cm.  in  front  of  it;  find  where  they 
are  focused  after  reflection. 

Ans.  At  a  point  20  cm.  in  front  of  the  mirror. 

3.  The  radius  of  a  concave  mirror  is  60  cm.  A  luminous 
point  is  placed  in  front  of  the  mirror  at  a  distance  of  (a)  120 
cm.,  (b)  60  cm.,  (c)  30  cm.,  and  (d)  20  cm.  Find  the  position 
of  the  image-point  for  each  of  these  positions  of  the  object. 

Ans.  (a)  40  cm.  in  front  of  mirror;  (b)  60  cm.  in  front  of 
mirror;  (c)  at  infinity;  and  (d)  60  cm.  behind  mirror. 

4.  A  candle  is  placed  in  front  of  a  concave  spherical  mir- 
ror, whose  radius  is  1  foot,  at  a  distance  of  5  inches  from 
the  mirror.    Where  will  the  image  be  formed? 

Ans.  30  inches  behind  the  mirror. 

5.  An  object  is  24  inches  in  front  of  a  concave  mirror  of 
radius  1  foot;  where  will  its  image  be  formed?  If  the  object 
is  displaced  through  a  small  distance  z,  through  what  dis- 
tance will  the  image  move? 

Ans.  Image  is  8  inches  in  front  of  mirror;  distance  through 
which  image  moves  will  be  2z/(z— 18). 

6.  An  object  is  placed  1  foot  from  a  concave  mirror  of 
radius  4  feet.  If  the  object  is  moved  1  inch  nearer  the  mirror, 
what  will  be  the  corresponding  displacement  of  the  image? 

Ans.  The  image  moves  3.7  inches  nearer  the  mirror. 

7.  An  object-point  is  10  cm.  in  front  of  a  convex  mirror  of 
radius  60  cm.    Find  the  position  of  the  image-point. 

Ans.  7.5  cm.  behind  the  mirror. 


204  Mirrors,  Prisms  and  Lenses  [Ch.  VI 

8.  Given  the  positions  on  the  axis  of  a  spherical  mirror 
of  the  vertex  A,  the  center  C  and  an  object-point  M;  show- 
how  to  construct  the  position  of  the  image-point  M'.  There 
are  eight  possible  arrangements  of  these  four  points;  draw 
a  diagram  for  each  one  of  them. 

9.  If  x,  x'  denote  the  abscissae,  with  respect  to  the  focal 
point  F  as  origin,  of  a  pair  of  conjugate  points  on  the  axis 
of  a  spherical  mirror,  show  that 

x.x'=}\ 
where  /  denotes  the  focal  length  of  the  mirror.     How  are 
object  and  image  situated  with  respect  to  the  focal  plane? 

10.  An  object  is  placed  at  a  distance  of  60  cm.  in  front  of 
a  spherical  mirror,  and  the  image  is  found  to  be  on  the  same 
side  of  the  mirror  at  a  distance  of  20  cm.  What  is  the  focal 
length  of  the  mirror,  and  is  it  concave  or  convex? 

Ans.  Concave  mirror  of  focal  length  15  cm. 

11.  How  far  from  a  concave  mirror  of  focal  length  18 
inches  must  an  object  be  placed  in  order  that  the  image 
shall  be  magnified  three  times? 

Ans.  1  ft.  or  2  ft.  from  the  mirror,  according  as  image  is 
erect  or  inverted. 

12.  A  candle-flame  one  inch  high  is  18  inches  in  front  of 
a  concave  mirror  of  focal  length  15  inches.  Find  the  position 
and  size  of  the  image. 

Ans.  The  image  will  be  real  and  inverted,  90  inches  from 
the  mirror,  and  5  inches  long. 

13.  A  small  object  is  placed  at  right  angles  to  the  axis  of 
a  spherical  mirror;  show  how  to  construct  the  image,  and 
derive  the  magnification-formula: 

y       u' 

14.  A  luminous  point  moves  from  left  to  right  along  a 
straight  line  parallel  to  the  axis  of  a  spherical  mirror.  Show 
by  diagrams  for  both  concave  and  convex  mirrors  how  the 
conjugate  image-point  moves. 

15.  The  center  of  a  spherical  mirror  is  at  C,  and  the 


Ch.  VI]  Problems  205 

straight  line  QQ'  joining  a  pair  of  conjugate  points  meets 
the  mirror  in  a  point  U.  If  P  designates  the  position  of  a 
point  which  is  not  on  the  straight  line  QQ',  and  if  a  straight 
line  is  drawn  cutting  the  straight  lines  PU,  PQ,  PC  and  PQ' 
in  the  points  V,  R,  Z  and  R',  respectively;  show  that  R,  R' 
are  a  pair  of  conjugate  points  with  respect  to  another  spher- 
ical mirror  whose  center  is  at  Z  and  whose  radius  is  equal  to 
VZ. 

16.  Show  by  geometrical  construction  that  the  focal  point 
of  a  spherical  mirror  lies  midway  between  the  center  and  the 
vertex. 

17.  An  object  is  placed  5  inches  from  a  spherical  mirror  of 
focal  length  6  inches.  Assuming  that  the  object  is  real, 
where  will  the  image  be  formed,  and  what  will  be  the  mag- 
nification? Draw  diagrams  for  both  convex  and  concave 
mirrors. 

Ans.  For  concave  mirror,  image  is  30  in.  behind  the  mirror, 
magnification  =+6;  for  convex  mirror,  image  is  2T8T  inches 
behind  the  mirror,  magnification  =  +  tV 

18.  How  far  from  a  concave  mirror  must  a  real  object  be 
placed  in  order  that  the  image  shall  be  (a)  real  and  four 
times  the  size  of  the  object,  (b)  virtual  and  four  times  the 
size  of  the  object,  and  (c)  real  and  one-fourth  the  size  of  the 
object?  Draw  diagrams  showing  the  construction  for  each 
of  these  three  cases. 

Ans.  Distance  of  mirror  from  the  object  is  equal  to  (a) 
5//4,  (b)  3//4,  and  (c)  5/,  where  /  denotes  the  focal  length. 

19.  What  kind  of  image  is  produced  in  a  concave  mirror 
by  a  virtual  object?  Illustrate  and  explain  by  means  of  a 
diagram. 

Ans.  Image  is  real  and  erect  and  smaller  than  object. 

20.  Determine  the  position  and  magnification  of  the  image 
of  a  virtual  object  lying  midway  between  the  vertex  and 
focal  point  of  a  convex  mirror.  Draw  diagram  showing 
construction. 

Ans.  The  vertex  of  the  mirror  will  be  midway  between  the 


206  Mirrors,  Prisms  and  Lenses  [Ch.  VI 

axial  point  of  the  image  and  the  focal  point  of  the  mirror,  and 
the  image  will  be  real  and  erect  and  twice  as  large  as  object. 

21.  Show  that  when  an  object  is  placed  midway  between 
the  focal  point  and  the  vertex  of  a  concave  mirror  the  image 
will  be  virtual  and  erect  and  twice  as  large  as  the  object. 

22.  An  object  3  inches  high  is  placed  10  inches  in  front  of 
a  convex  mirror  of  30  inches  focal  length.  Find  the  position 
and  size  of  the  image. 

Ans.  Virtual  image  7.5  inches  from  the  mirror  and  234 
inches  high. 

23.  An  object  is  placed  in  front  of  a  concave  mirror  at  a 
distance  of  one  foot.  If  the  image  is  real  and  three  times  as 
large  as  the  object,  what  is  the  focal  length  of  the  mirror? 

Ans.  9  inches. 

24.  The  radius  of  a  concave  mirror  is  23  cm.  An  object, 
2  cm.  high,  is  placed  in  front  of  the  mirror  at  a  distance  of 
one  meter.    Find  the  position  and  size  of  the  image. 

Ans.  A  real  image,  0.26  cm.  high,  13  cm.  from  the  mirror. 

25.  Find  the  position  and  size  of  the  image  of  a  disk  3 
inches  in  diameter  placed  at  right  angles  to  the  axis  of  a 
spherical  mirror  of  radius  6  feet,  when  the  distance  from  the 
object  to  the  mirror  is  (a)  1  ft.,  (6)  3  ft.,  and  (c)  9  ft. 

Ans.  For  a  concave  mirror:  (a)  Virtual  image,  4.5  inches 
in  diameter,  18  inches  from  mirror;  (6)  Image  at  infinity; 
(c)  Real  inverted  image,  1.5  inches  in  diameter,  4.5  feet  from 
the  mirror. 

26.  Assuming  that  the  apparent  diameter  of  the  sun  is 
30',  calculate  the  approximate  diameter  of  the  sun's  image 
in  a  concave  mirror  of  focal  length  1  foot. 

Ans.  A  little  more  than  one-tenth  of  an  inch. 

27.  A  gas-flame  is  8  ft.  from  a  wall,  and  it  is  required  to 
throw  on  the  wall  a  real  image  of  the  flame  which  shall  be  mag- 
nified three  times.  Determine  the  position  and  focal  length 
of  a  concave  mirror  which  would  give  the  required  image. 

Ans.  The  mirror  must  have  a  focal  length  of  3  ft.  and  must 
be  placed  at  a  distance  of  4  ft.  from  the  object. 


Ch.  VI]  Problems  207 

28.  It  is  desired  to  throw  on  a  wall  an  image  of  an  object 
magnified  12  times,  the  distance  of  the  object  from,  the 
wall  being  11  feet.  Find  the  focal  length  of  a  concave 
mirror  which  will  do  this,  and  state  where  it  must  be 
placed. 

Ans.  The  focal  length  of  the  mirror  must  be  f|  ft.,  and 
it  must  be  placed  1  ft.  from  the  object. 

29.  Assuming  that  the  eye  is  placed  on  the  axis  of  a  spher- 
ical mirror,  and  that  the  rays  are  paraxial,  explain  how  the 
field  of  view  is  determined.  Draw  accurate  diagrams  for 
concave  and  convex  mirrors. 

30.  A  man  holds,  halfway  between  his  eye  and  a  convex 
mirror  3  feet  from  his  eye,  two  fine  parallel  wires,  so  that 
they  are  seen  directly  and  also  by  reflection  in  the  mirror. 
Show  that  if  the  apparent  distance  between  the  wires  as 
seen  directly  is  5  times  that  as  seen  by  reflection,  the  radius 
of  the  mirror  is  3  feet. 

31.  A  scale  etched  on  a  thin  sheet  of  transparent  glass  is 
placed  between  the  eye  of  an  observer  and  a  convex  mirror 
of  focal  length  one  foot.  When  the  distance  between  the 
eye  and  the  scale  is  three  feet,  one  of  the  scale  divisions 
appears  to  cover  three  divisions  of  the  image  in  the  mirror. 
Find  the  position  of  the  mirror. 

Ans.  The  mirror  is  one  foot  from  the  scale. 

32.  A  scale  etched  on  a  thin  sheet  of  transparent  glass  is 
interposed  between  the  eye  of  an  observer  and  a  convex 
mirror  of  focal  length  /.  When  the  distance  of  the  scale  from 
the  eye  is  b  feet,  one  division  of  the  scale  appears  to  cover 
m  divisions  of  its  image  in  the  mirror.  If  now  the  scale  is 
displaced  through  a  distance  c  in  the  direction  of  the  axis 
of  the  mirror,  it  is  found  that  one  division  of  the  scale  ap- 
pears to  cover  k  divisions  in  the  mirror.  Find  an  expression 
for  /  in  terms  of  m,  k,  b  and  c. 

Ans. 

(k—m)  (b—c)  be 


f= 


{b{k-m)-(k-l)c\  {6(fc-m)-(fc+l)c 


208  Mirrors,  Prisms  and  Lenses  [Ch.  VI 

33.  A  concave  and  a  convex  mirror,  each  of  radius  20  cm., 
are  placed  opposite  to  each  other  and  40  cm.  apart  on  the 
same  axis.  An  object  3  cm.  high  is  placed  midway  between 
them.  Find  the  position  and  size  of  the  image  formed  by 
reflection,  first,  at  the  convex,  and  then  at  the  concave  mirror. 
Draw  accurate  diagram,  and  trace  the  path  of  a  ray  from 
a  point  in  the  object  to  the  corresponding  point  in  the  image. 

Ans.  The  image  is  12i8r  cm.  from  the  concave  mirror, 
real  and  inverted,  and  i\  cm.  high. 

34.  Same  problem  as  No.  33,  except  that  in  this  case  the 
image  is  formed  by  rays  which  have  been  reflected  first  from 
the  concave  mirror  and  then  from  the  convex  mirror. 

Ans.  The  image  is  6f  cm.  behind  the  convex  mirror, 
virtual  and  inverted,  and  1  cm.  high. 

35.  Two  concave  mirrors,  of  focal  lengths  20  and  40  cm., 
are  turned  towards  each  other,  the  distance  between  their 
vertices  being  one  meter.  An  object  1  cm.  high  is  placed 
between  the  mirrors  at  a  distance  of  10  cm.  from  the  mirror 
whose  focal  length  is  20  cm.  Find  the  position  and  size  of 
the  image  produced  by  rays  which  are  reflected  first  from 
the  nearer  mirror  and  then  from  the  farther  mirror. 

Ans.  A  real  inverted  image,  1  cm.  long,  at  a  distance  of 
60  cm.  from  the  mirror  that  is  farther  from  the  object. 

36.  The  distance  between  the  vertices  Ai  and  A2  of  two 
spherical  mirrors  which  face  each  other  is  denoted  by  d, 
that  is,  <2  =  A2Ai.  The  focal  points  of  the  mirrors  are  at  Fi 
and  F2,  and  the  focal  lengths  are  /i  =  FiAi  and  /2  =  F2A2. 
An  object  is  placed  between  the  mirrors  at  a  distance  u\ 
from  Ai.  Rays  proceeding  from  the  object  are  reflected, 
first,  from  the  mirror  Ai  and  then  from  the  mirror  A2;  show 
that  the  distance  of  the  final  image  from  the  mirror  A2  is 

</i.m~(/i+m)  Ah. 

and  that  the  magnification  is 

fi.f* 
(/i+«o  (/2+d)-/i.«r 


Ch.  VI]  Problems  209 

37.  If  the  rays  fall  first  on  the  mirror  A2  and  then  on  Ai, 
these  letters  having  exactly  the  same  meanings  as  in  No.  36, 
then  the  distance  of  the  image  from  mirror  Ai  will  be 

fi\(f2+d)    (ttrH)+/2rfl 
B(/i7d)/2+(t*i+d)  Ui-h-dY 
and  the  magnification  will  be 

Uh 

(A-d)  f2+(ui+d)  Ui~h-d)m 

38.  If  the  mirror  Ax  in  Nos.  36  and  37  is  a  plane  mirror, 
show  that  when  the  light  is  reflected  froni  the  plane  mirror 
first  the  distance  of  the  image  from  the  curved  mirror  is 

(ui—d)f2 


and  that  the  magnification  is 


h+d—ui 


fr\-d-ui 

and  that  when  the  light  is  reflected  from  the  curved  mirror 
first,  the  distance  of  the  image  from  the  plane  mirror  is 

ih+d)  (U!+d)+f2d 

\  h+ui+d 
and  that  the  magnification  is 

h 

f2+Ui+d  ' 

If  both  the  mirrors  are  plane,  the  magnification  will  be 
unity,  and  the  image  after  two  reflections,  first  at  Ai  and 
then  at  A2,  will  be  formed  at  a  distance  of  (u\—d)  from 
A2;  whereas  if  the  light  falls  first  on  mirror  A2,  the  distance 
of  the  image  from  the  other  mirror  will  be  (uy\-2d). 

39.  If  M,  M'  are  a  pair  of  conjugate  points  on  the  axis 
of  a  spherical  refracting  surface  which  divides  two  media 
of  indices  n  and  nf,  show  that 

(CAMM')=-, 

n 

where  A  and  C  designate  the  vertex  and  the  center  of  the 
spherical  surface. 

40.  Show  how  to  construct  the  position  of  the  point  M' 


210  Mirrors,  Prisms  and  Lenses  [Ch.  VI 

conjugate  to  a  given  point  M  on  the  axis  of  a  spherical  re- 
fracting surface;  and  draw  diagrams  for  all  the  possible  ar- 
rangements of  the  four  points  A,  C,  M,  M'.  Prove  the  con- 
struction, and  derive  the  formula  n'/u'  =  n/u-\-(n'—ri)/r, 
where  n,  n'  denote  the  indices  of  refraction,  and  w  =  AM, 
u'  =  AM',r  =  AC. 

41.  Show  how  the  formula  in  No.  40  includes  as  special 
cases  the  case  of  refraction  of  paraxial  rays  at  a  plane  sur- 
face and  the  case  of  reflection  at  a  spherical  mirror. 

42.  From  the  formula  in  No.  40  derive  expressions  for 
the  focal  lengths  /,  /'  of  a  spherical  refracting  surface,  and 
show  that 

f+f+r  =  0,       ra./'+rc'./=0. 

43.  Does  the  construction  found  in  No.  40  apply  to  the 
case  of  a  spherical  mirror?    Explain  with  diagrams. 

44.  Apply  the  construction  employed  in  No.  40  to  de- 
termine the  positions  of  the  focal  points  F,  F'  of  a  spherical 
refracting  surface,  and  show  that 

FA  =  CF',     F'A  =  CF,       F'A:  FA=  -»':  n. 

45.  Where  are  the  focal  points  of  a  plane  refracting  sur- 
face?   Explain  clearly. 

46.  Explain  how  the  results  of  No.  44  are  applicable  to 
a  spherical  mirror. 

47.  Air  and  glass  are  separated  by  a  spherical  refracting 
surface  of  radius  7'  =  AC.  Find  the  positions  of  the  focal 
points  F,  F'  for  the  cases  when  the  refraction  is  from  air  to 
glass  and  from  glass  to  air  and  when  the  surface  is  convex 
and  concave;  illustrating  your  answers  by  four  accurately 
drawn  diagrams.  (Take  indices  of  refraction  of  air  and 
glass  equal  to  1  and  1.5,  respectively.) 

48.  From  the  figures  used  in  No.  44  for  constructing  the 
positions  of  the  focal  points  F,  F',  derive  the  formulae  for 
the  focal  lengths  which  were  obtained  in  No.  42. 

49.  Light  falling  on  a  concave  surface  separating  water 
(n=1.33)  from  glass  (n'  =  1.55)  is  convergent  towards  a 
point  10  cm.  beyond  the  vertex.    The  radius  of  the  surface 


Ch.  VI]  Problems  211 

is  20  cm.    Find  the  point  where  the  refracted  rays  cross  the 
axis. 

Ans.  13.19  cm.  beyond  the  vertex  of  the  sphere  in  the 
glass  medium. 

50.  Light  is  refracted  from  air  to  glass  (nr:  n  =  3:  2)  at  a 
spherical  surface.  If  the  vertex  of  the  bundle  of  incident 
rays  is  in  the  glass  and  20  cm.  from  the  vertex  of  the  re- 
fracting surface,  and  if  the  refracted  rays  are  converged  to 
a  point  in  the  glass  and  5  cm.  from  the  vertex,  determine 
the  form  and  size  of  the  surface. 

Ans.  Convex  surface  of  radius  2  cm. 

51.  A  small  air-bubble  in  a  glass  sphere,  4  inches  in  di- 
ameter, viewed  so  that  the  speck  and  the  center  of  the  sphere 
are  in  line  with  the  eye,  appears  to  be  one  inch  from  the 
point  of  the  surface  nearest  the  eye.  What  is  its  actual  dis- 
tance, assuming  that  the  index  of  refraction  of  glass  is  1.5? 

Ans.  1.2  inches. 

52.  The  radius  of  a  concave  refracting  surface  is  20  cm. 
A  virtual  image  of  a  real  object  is  formed  at  a  distance  of 
40  cm.  from  the  vertex,  and  the  distance  from  the  object 
to  the  image  is  60  cm.  The  first  medium  is  air  (n  =  1).  Find 
the  index  of  refraction  of  the  second  medium. 

Ans.  n'  =  1.6. 

53.  Light  diverging  from  a  point  M  in  air  is  converged 
by  a  spherical  refracting  surface  to  a  point  M'  in  glass  of 
index  1.5.  The  distance  MM' =18  cm.,  and  the  point  M 
is  twice  as  far  from  the  surface  as  the  point  M'.  Find  the 
radius  of  the  surface.  Ans.  1.5  cm. 

54.  Find  the  positions  of  the  focal  points  F,  F'  of  a  con- 
cave spherical  refracting  surface  separating  air  from  a  me- 
dium of  index  1.6,  having  found  that  the  image  of  a  luminous 
point  30  cm.  in  front  of  the  surface  is  midway  between  the 
luminous  point  and  the  surface. 

Ans.  AF  = +  13.63  cm.;       AF'=  -21.81  cm. 

55.  A  convergent  bundle  of  rays  is  incident  on  a  spherical 
refracting  surface  of  radius  10  cm.     The  relative  index  of 


212  Mirrors,  Prisms  and  Lenses  [Ch.  VI 

refraction  from  the  first  medium  to  the  second  medium  is 
equal  to  2  (nr:  n  =  2:l).  If  the  incident  rays-  cross  the  axis 
at  M  and  the  refracted  rays  at  M',  and  if  M'M  =  +60  cm., 
determine  the  positions  of  the  points  M,  M'. 

Ans.  If  the  surface  is  convex,  AM  =  +77.72  cm.,  AM' 
=  +  17.72  cm.  If  the  surface  is  concave,  then  either  AM  = 
+30  cm.,  AM' =  —30  cm.  or  AM  =  +20  cm.,  AM'  =  -40  cm. 

56.  A  beam  of  parallel  rays  passing  through  water  (n  = 
1.3)  is  refracted  at  a  concave  surface  into  glass  (n'  =  1.5). 
If  the  radius  of  the  surface  is  20  cm.,  where  will  the  light  be 
focused?  Ans.  Virtual  focus,  150  cm.  from  the  surface. 

57.  A  small  air-bubble  is  imbedded  in  a  glass  sphere  at 
a  distance  of  5.98  cm.  from  the  nearest  point  of  the  surface. 
What  will  be  the  apparent  depth  of  the  bubble,  viewed  from 
this  side  of  the  sphere,  if  the  radius  of  the  sphere  is  7.03  cm., 
and  the  index  of  refraction  from  air  to  glass  is  1.42? 

Ans.  5.63  cm. 

58.  Assuming  that  the  cornea  of  the  eye  is  a  spherical 
refracting  surface  of  radius  8  mm.  separating  the  outside  air 
from  the  aqueous  humor  (of  index  f),  find  the  distance 
of  the  pupil  of  the  eye  from  the  vertex  of  the  cornea,  if  its 
apparent  distance  is  found  to  be  3.04  mm.  Also,  if  the  ap- 
parent diameter  of  the  pupil  is  4.5  mm.,  what  is  its  real 
diameter?  Ans.  3.6  mm.;  4  mm. 

59.  Construct  the  image  M'Q'  of  a  small  object  MQ  per- 
pendicular at  M  to  the  axis  of  a  spherical  refracting  surface, 
and  derive  the  magnification-formula  in  terms  of  the  dis- 
tances of  M  and  M'  from  the  vertex  of  the  surface.  Draw 
two  diagrams,  one  for  convex,  and  one  for  concave  surface. 

60.  Derive  the  image-equations  of  a  spherical  refracting 
surface  referred  to  the  focal  points  as  origins. 

61.  Derive  the  image  equations  of  a  spherical  refracting 
surface  in  the  forms 

f/u+f'/u'+ 1  =  0,       y'/y  =f/(f+u)  =  (f'+u')/f. 

62.  Show  that  there  are  two  positions  on  the  axis  of  a 
spherical  refracting  surface  where  image  and  object  coincide. 


Ch.  VI]  Problems  213 

63.  Locate  the  two  pairs  of  conjugate  planes  of  a  spheri- 
cal refracting  surface  for  which  image  and  object  have  the 
same  size. 

64.  A  real  object,  1  cm.  high,  is  placed  12  cm.  from  a  con- 
vex spherical  refracting  surface,  of  radius  30  cm.,  which 
separates  air  (n  =  l)  from  glass  (n'  =  1.5).  Find  the  position 
and  size  of  the  image. 

Ans.  Image  is  virtual  and  erect,  1.25  cm.  high,  22.5  cm. 
from  vertex. 

65.  In  the  preceding  example,  suppose  that  the  object 
is  a  virtual  object  at  the  same  distance  from  the  spherical 
refracting  surface.  Find  the  position  and  size  of  the  image 
in  this  case. 

Ans.  Image  is  real  and  erect,  I  cm.  high,  and  15  cm. 
from  vertex. 

66.  Solve  Nos.  64  and  65  for  the  case  when  the  surface 
is  concave;  and  draw  diagrams  showing  construction  of  the 
image  in  all  four  cases. 

67.  Solve  No.  64  on  the  supposition  that  the  first  medium 
is  glass  and  the  second  medium  air. 

Ans.  Image  will  be  virtual  and  erect,  if  cm.  high,  and 
f  ?  cm.  from  vertex. 

68.  (a)  The  human  eye  from  which  the  crystalline  lens 
has  been  removed  (so-called  "aphakic  eye")  may  be  re- 
garded as  consisting  of  a  single  spherical  refracting  surface, 
namely,  the  anterior  surface  of  the  cornea.  If  the  radius 
of  this  surface  is  taken  as  8  mm.,  and  if  the  index  of  refrac- 
tion of  the  eye-medium  (both  the  aqueous  and  vitreous 
humors)  is  put  equal  to  |,  what  will  be  the  focal  lengths 
of  the  aphakic  eye?  (b)  Assuming  that  the  length  of  the 
eye-ball  of  an  aphakic  eye  is  22  mm.,  where  will  an  object 
have  to  be  placed  to  be  imaged  distinctly  on  the  retina  at 
the  back  of  the  eye? 

Ans.  (a)/=+24mm.,/'=-32mm.;  (b)  ^  =  +52.8  mm., 
which  means  that  the  object  must  be  virtual  and  lie  behind 
the  eye. 


214  Mirrors,  Prisms  and  Lenses  [Ch.  VI 

69.  Listing's  " reduced  eye"  is  composed  of -a  single 
convex  spherical  refracting  surface  of  radius  5.2  mm.  sep- 
arating air  (n  =  l)  from  the  vitreous  humor  (n'  =  1.332). 
Calculate  the  focal  lengths. 

Ans.  /=  +  15.68  mm.,  /'=  -20.90  mm. 

70.  In  Donder's  "reduced  eye"  the  focal  lengths  are 
assumed  to  be  +15  and  —20  mm.  Calculate  the  radius  of 
the  equivalent  spherical  refracting  surface  and  the  index  of 
refraction  of  the  vitreous  humor  for  these  values  of  the  focal 
lengths.  Ans.  r  =  +5mm.;  n'  =  i. 

71.  The  angular  distance  of  a  star  from  the  axis  of  a 
spherical  refracting  surface  which  separates  air  (n  =  l)  from 
glass  (n'  =  1.5)  is  10°.  The  surface  is  convex  and  of  radius 
10  cm.    Find  the  position  of  the  star's  image. 

Ans.  A  real  image  will  be  formed  in  the  secondary  focal 
plane  about  3.5  cm.  from  the  axis. 

72.  What  is  the  size  of  the  image  on  the  retina  of  List- 
ing's " reduced  eye"  (No.  69)  if  the  apparent  size  of  the 
distant  object  is  5°?  Ans.  1.36  mm. 

73.  A  hemispherical  lens,  the  curved  surface  of  which  has 
a  radius  of  3  inches,  is  made  of  glass  of  index  1.5.  Show 
that  rays  of  light  proceeding  from  a  point  on  its  axis  4  inches 
in  front  of  its  plane  surface  will  emerge  parallel  to  the  axis. 

74.  A  paraxial  ray  parallel  to  the  axis  of  a  solid  refracting 
sphere  of  index  n'  is  refracted  into  the  sphere  at  first  towards 
a  point  X  on  the  axis,  and  after  the  second  refraction  crosses 
the  axis  at  a  point  F'.  If  the  first  and  last  media  are  the 
same  and  of  index  n,  show  that  the  point  F'  lies  midway  be- 
tween the  second  vertex  of  the  sphere  and  the  point  X. 

75.  A  small  object  of  height  y  is  placed  at  the  center  of 
a  spherical  refracting  surface  in  a  plane  at  right  angles  to 
the  axis.  Determine  the  position  and  size  of  the  image. 
Show  how  the  Smith-Helmholtz  formula  (§  86)  is  appli- 
cable to  a  part  of  this  problem. 

Ans.  Image  is  in  same  plane  as  object,  erect,  and  of  size 
y'  =  n.y/n'. 


Ch.  VI]  Problems  215 

76.  A  plane  object  is  placed  parallel  to  a  plane  refracting 
surface.  Show  that  its  image  formed  by  paraxial  rays  is 
erect  and  of  same  size  as  object.  Is  the  Smith-Helmholtz 
formula  (§  86)  applicable  to  a  plane  refracting  surface?  Is 
it  applicable  to  a  spherical  mirror?    Explain  clearly. 

77.  In  a  convex  spherical  refracting  surface  of  radius 
0.75,  which  separates  air  (n  =  l)  from  water  (n'  =  -|),  the 
image  is  real,  inverted  and  one-third  the  size  of  the  object. 
Find  the  positions  of  object  and  image.  If  a  ray  pro- 
ceeding from  the  axial  point  of  the  object  is  inclined  to  the 
axis  at  an  angle  of  3°,  what  will  be  the  slope  of  the  correspond- 
ing refracted  ray? 

Ans.  Object  is  in  air  and  image  is  in  water,  their  distances 
from  the  surface  being  9  and  4,  respectively;  slope  of  re- 
fracted ray  is  —4.5°. 

78.  In  a  spherical  refracting  surface 

a=6+<p,  a'=d'+<p, 
where  a,  a/  denote  the  angles  of  incidence  and  refraction, 
6,  6'  denote  the  inclinations  of  the  ray  to  the  axis  before 
and  after  refraction,  and  <p  denotes  the  so-called  central 
angle  (ZBCA).  For  a  paraxial  ray  the  law  of  refraction 
may  be  written 

n'.a'  =  n.a. 
From  these  formulae  deduce  the  abscissa-relation  in  the  form 

n' _n  ,  n'  —  n 

uf     u         r 

79.  The  curved  surface  of  a  glass  hemisphere  is  silvered. 
Rays  coming  from  a  luminous  point  at  a  distance  u  from 
the  plane  surface  are  refracted  into  the  glass,  reflected  from 
the  concave  spherical  surface,  and  refracted  at  the  plane 
surface  back  into  the  air.  If  r  denotes  the  radius  of  the 
spherical  surface  and  n  the  index  of  refraction  of  the  glass, 
show  that 

u    u       r 


216  Mirrors,  Prisms  and  Lenses  [Ch.  VI 

where  v!  denotes  the  distance  of  the  image  from  the  plane 
surface. 

80.  A  plane  object  of  height  one  inch  is  placed  at  right 
angles  to  the  axis  of  a  spherical  mirror.  The  slope  of  the  re- 
flected ray  corresponding  to  an  incident  paraxial  ray  which 
emanates  from  the  axial  point  of  the  object  at  a  slope  of  +5° 
is  +10°.  Is  the  image  erect  or  inverted,  and  what  is  its  size? 
Ans.  Inverted  image,  one-half  inch  high. 


CHAPTER  VII 

REFRACTION  OF  PARAXIAL  RAYS  THROUGH  AN  INFINITELY 
THIN  LENS 

87.  Forms  of  Lenses. — In  optics  the  word  lens  is  used 
to  denote  a  portion  of  a  transparent  substance,  usually 
isotropic,  comprised  between  two  smooth  polished  surfaces, 
one  of  which  may  be  plane.  These  surfaces  are  called  the 
faces  of  the  lens.  The  curved  faces  are  generally  spherical, 
and  this  may  always  be  considered  as  implied  unless  the 
contrary  is  expressly  stated.  Lenses  with  spherical  faces 
are  sometimes  called  " spherical  lenses"  to  distinguish  them 
from  cylindrical,  sphero-cylindrical  and  other  forms  of 
lenses  which  are  also  quite  common,  especially  in  modern 
spectacle  glasses.  A  plane  face  may  be  regarded  as  a  spher- 
ical or  cylindrical  surface  of  infinite  radius. 

The  axis  of  a  lens  is  the  straight  line  which  is  normal  to 
both  faces,  and,  consequently,  a  ray  whose  path  lies  along 
the  axis  (the  so-called  axial  ray)  will  pass  through  the  lens 
without  being  deflected  from  this  line.  The  axis  of  a  spher- 
ical lens  is  the  straight  line  joining  the  centers  d,  C2  of 
the  two  spherical  faces,  and  since  a  lens  of  this  kind  is  sym- 
metric around  the  axis,  it  may  be  represented  in  a  plane 
figure  by  a  meridian  section  showing  the  arcs  of  the  two 
great  circles  in  which  this  plane  intersects  the  spherical 
faces.  Depending  on  the  lengths  of  the  radii  in  comparison 
with  the  length  of  the  line-segment  CiC2,  these  arcs  inter- 
sect in  two  points  equidistant  from  the  axis  or  else  they  do 
not  intersect  each  other  at  all. 

(a)  If  they  intersect,  then  CiC2  is  less  than  the  arith- 
metical sum  but  greater  than  the  arithmetical  difference 
of  the  radii,  and   the  lens   may  be  a  double   convex  lens 

217 


218  Mirrors,  Prisms  and  Lenses  [§  87 


Fig.  112,  a. — Double  convex  lens. 


V 


>TO     n    AT  00 

2       1 


Fig.  112,  &. — Plano-convex  lens. 


§  87]  Forms  of  Lenses  219 


Fig.  112,  c. — Convex  meniscus. 

(Fig.  112,  a)  or  a  convex  meniscus  (Fig.  112,  c).  A  particular 
case  of  a  double  convex  lens  is  a  plano-convex  lens  (Fig.  112,  b). 

(b)  If  they  do  not  intersect,  then  either  one  circle  lies 
wholly  outside  the  other,  the  distance  between  the  centers 
being,  therefore,  greater  than  the  arithmetical  sum  of  the 
radii,  so  that  the  lens  is  a  double  concave  lens  (Fig.  113,  a), 
or,  in  case  one  of  the  surfaces  is  plane,  a  plano-concave  lens 
(Fig.  113,  b);  or  else  one  circle  lies  wholly  inside  the  other, 
so  that  the  distance  between  the  centers  is  less  than  the 
arithmetical  difference  of  the  radii,  and  then  the  lens  has 
the  form  of  a  concave  meniscus  (Fig.  113,  c). 

The  first  face  of  a  lens  is  the  side  turned  towards  the  in- 
cident light.  The  points  where  the  axis  meets  the  two  faces 
are  called  the  vertices,  and  the  distance  from  the  vertex  Ai  of 
the  first  face  to  the  vertex  A2  of  the  second  face,  which  is 
denoted  by  d,  is  called  the  thickness  of  the  lens;  thus,  d  — 
AiA2.  Since  the  direction  which  the  light  takes  in  going 
across  the  lens  from  Ai  to  A2  is  the  positive  direction  along 
the  axis  (see  §  63),  the  thickness  d  is  essentially  a  positive 
magnitude. 


220  Mirrors,  Prisms  and  Lenses  [§  87 


s 


\ 


A I  IA„ 

1/  12. 


\ 

\ 


\ 


/ 


Fig.  113,  a. — Double  concave  lens. 


TO     C,     AT  oo 

* fc 


N 


Fig.  113,  b. — Plano-concave  lens. 


\ 


\  \ 


A\      \Az  Cz  Ci  ' 


/ 


Fig.  113,  c. — Concave  meniscus. 


87] 


Forms  of  Lenses 


221 


The  radii  of  the  surfaces,  denoted  by  rh  r2,  are  the  ab- 
scissa? of  the  centers  Ci,  C2  with  respect  to  the  vertices 
Ai,  A2,  respectively;  thus,  ri  =  AiCi,  r2  =  A2C2. 

Certain  special  forms  of  spherical  lenses  may  be  mentioned 
here,  viz. : 

(a)  Symmetric  Lenses,  which  are  double  convex  or  double 
concave  lenses  whose  surfaces  have  equal  but  opposite 
curvatures  (ri-f-r2  =  0).  A  particular  case  of  double  convex 
symmetric  lens  is  one  whose  two  faces  are  portions  of  the 
same  spherical  surface;  a  lens  of  this  kind  being  sometimes 
called  a  solid  sphere  (d  =  ri  —  r2  =  2r1) . 

(b)  Concentric  Lenses,  whose  two  faces  have  the  same 
center  of  curvature  (CiC2  =  0).    A  concentric  lens  may  be 


Fig.  114. — Concentric  concave  meniscus. 


a  double  convex  lens  characterized  by  the  relation  d  =  r\— r2, 
of  which  a  "solid  sphere"  is  a  special  case;  or  it  may  have 
the  form  of  a  concave  meniscus  for  which  either  ri>r2>0 
and  d  =  ri  —  r2  (Fig.  114)  or  ri<r2<0  and  d  =  r2-  r\. 

(c)  Lenses  of  Zero  Curvature,  in  which  the  axial  thickness 


222  Mirrors,  Prisms  and  Lenses  [§  87 


Fig.  115. — Lens  of  zero  curvature  (ri  =  r2). 

of  the  lens  is  equal  to  the  distance  between  the  centers  (d  = 
AiA2  =  CiC2).  This  lens  is  a  convex  meniscus  characterized 
by  the  condition  that  r\—  r2  =  0  (Fig.  115). 

Lenses  may  also  be  conveniently  classified  in  two  main 
groups,  viz.,  convex  lenses  and  concave  lenses,  depending  on 
the  relative  thickness  of  the  lens  along  the  axis  as  compared 
with  its  thickness  at  the  edges.  The  thickness  of  a  convex 
lens  is  greater  along  the  axis  than  it  is  out  towards  the  edge, 
whereas  a  concave  lens  is  thinnest  in  the  middle.  Each  of 
these  two  main  divisions  includes  three  special  forms  which 
have  already  been  mentioned.  Thus,  the  three  types  of  con- 
vex lenses  are  the  double  convex,  the  plano-convex  and  the 
convex  or  "  crescent-shaped "  meniscus,  as  shown  in  Fig.  112; 
and,  similarly,  the  types  of  concave  lenses  are  the  double 
concave,  plano-concave  and  the  concave  or  " canoe-shaped" 
meniscus  (Fig.  113). 

A  convex  glass  lens  of  moderate  thickness  held  in  air  with 
its  axis  towards  the  sun  has  the  property  of  a  burning  glass 
and  converges  the  rays  to  a  real  focus  on  the  other  side  of 


§  88]  Optical  Center  of  Lens  223 

the  lens.  A  convex  lens  is  called  therefore  also  a  convergent 
lens  or  a  positive  lens.  On  the  other  hand,  under  the  same 
circumstances,  a  concave  lens  will  render  a  beam  of  sun- 
light divergent,  and,  accordingly,  a  concave  lens  is  called 
also  a  divergent  lens  or  a  negative  lens.  The  explanation  of 
the  terms  " positive"  and  " negative"  as  applied  to  lenses 
will  be  apparent  when  we  come  to  speak  of  the  positions  of 
the  focal  points  of  a  lens  (§  90). 

Finally,  if  the  curvatures  of  the  two  faces  of  the  lens  are 
opposite  in  sign,  the  lens  is  double  convex  or  double  con- 
cave; if  the  curvatures  have  the  same  sign,  the  lens  is  a 
meniscus;  and  if  the  curvature  of  one  face  is  zero,  the  lens 
is  plano-convex  or  plano-concave. 

88.  The  Optical  Center  0  of  a  Lens  surrounded  by  the 
same  medium  on  both  sides. — When  a  ray  of  light  emerges 
at  the  second  face  of  a  lens  into  the  surrounding  medium 
in  the  same  direction  as  it  had  when  it  met  the  first  face, 
the  path  of  the  ray  inside  the  lens  lies  along  a  straight  line 
which  crosses  the  axis  at  a  remarkable  point  O  called  the 
optical  center  of  the  lens,  which  is  indeed  the  (internal  or 
external)  " center  of  similitude"  of  the  two  circles  whose 
arcs  are  the  traces  of  the  spherical  faces  of  the  lens  in  the 
meridian  plane  which  contains  the  ray. 

In  order  to  prove  this,  draw  a  pair  of  parallel  radii  CiBi  and 
C2B2  (Fig.  116),  and  suppose  that  a  ray  enters  the- lens  at 
Bi  and  leaves  it  at  B2,  so  that  the  straight  line  BiB2  repre- 
sents the  path  of  the  ray  through  the  lens.  If  the  straight 
line  RBi  represents  the  path  of  the  incident  ray,  a  straight 
line  B2S  drawn  through  B2  parallel  to  RBi  will  represent 
the  path  of  the  emergent  ray;  because,  since  the  tangents 
to  the  circular  arcs  at  Bi,  B2  are  parallel  to  each  other,  the 
lens  behaves  towards  this  ray  which  enters  it  at  Bi  exactly 
like  a  slab  of  the  same  material  with  plane  parallel  sides 
(§44).  Consequently,  the  position  of  the  point  O  where 
the  straight  line  BiB2,  produced  if  necessary,  crosses  the 
axis   of  the  lens  is  seen  to  be  entirely  dependent  on  the 


224 


Mirrors,  Prisms  and  Lenses 


[§88 


Fig.  116. — Optical  center  of  lens. 

geometrical  form  of  the  lens.  In  particular,  the  position 
of  this  point  will  not  depend  on  the  direction  of  the  inci- 
dent ray,  as  will  be  shown  by  the  following  investigation. 
From  the  similar  triangles  OCiB!  and  OC2B2,  we  derive 
the  proportion : 

OCi:  OC2  =  BiCi:  B2C2=AiCi:  A2C2. 
Accordingly,  we  may  write: 

OAi+AiC1  =  AiCi 
OA2+A2C2    A2CV 
and,  consequently: 

AiO_n 

A20    r2 ' 

Now  A20  =  A2Ai+AiO  =  AiO— d;  so  that  we  obtain  finally: 


AiO 


d. 


ri-r2 

The  function  on  the  right-hand  side  of  this  equation  depends 
only  on  the  form  of  the  lens,  so  that  the  position  of  the 


88] 


Optical  Center  of  Lens 


225 


point  0  with  respect  to  the  vertex  of  the  first  face  of  the 
lens  may  be  found  immediately  as  soon  as  we  know  the 
magnitudes  denoted  by  rh  r2  and  d. 


TO  CA  AT  oo  * 


Fig.  117. — Optical  center  of  lens  with  one  plane  face  is  at  the  vertex 
of  curved  face. 

If  the  lens  is  double  convex  or  double  concave,  the  optical 
center  O  will  lie  inside  the  lens  between  the  vertices  Ai  and 


Fig.  118. — Optical  center  of  meniscus  lies  outside  lens. 

A2;  if  one  face  of  the  lens  is  plane  (Fig.  117),  the  optical 
center  will  coincide  with  the  vertex. of  the  curved  face;  and, 


226  Mirrors,  Prisms  and  Lenses  [§  89 

finally,  if  the  lens  is  a  meniscus  (Fig.  118),  the  optical  center 
will  lie  outside  the  lens  entirely. 

In  general,  the  positions  of  the  points  designated  in  the 
diagrams  by  the  letters  N,  N'  will  vary  for  different  ray- 
paths  Bi  B2  within  the  lens;  but  if  the  rays  are  paraxial, 
the  positions  of  N,  N'  are  fixed.  In  fact,  if  the  ray  RBi  B2  S 
is  a  paraxial  ray,  the  points  N,  N'  are  the  so-called  nodal 
points  of  the  lens  (see  §  119). 

89.  The  Abscissa-Formula  of  a  Thin  Lens,  referred  to 
the  axial  point  of  the  lens  as  origin. — Ordinarily,  the  axial 
thickness  of  a  lens  is  much  smaller  than  either  of  the  radii 
of  curvature,  so  that  in  many  lens-problems  this  dimension 
is  negligible  in  comparison  with  the  other  linear  dimensions 
that  are  involved.  Moreover,  the  lens-formulae  are  greatly 
simplified  by  ignoring  the  thickness  of  the  lens.  However, 
in  using  these  formulae  one  must  be  duly  cautious  about 
taking  too  literally  results  that  are  strictly  applicable  only 
to  an  infinitely  thin  lens,  whose  vertices  are  regarded  as 
coincident,  that  is,  Ai  A2  =  d  =  0.  The  approximate  formulae 
that  are  obtained  for  lenses  of  zero-thickness  are  often  of 
very  great  practical  utility,  especially  in  the  preliminary 
design  of  an  optical  instrument  composed,  it  may  be,  of 
several  lenses  whose  thicknesses  are  by  no  means  negligible. 

The  optical  center  O  of  an  infinitely  thin  lens  coincides 
with  the  two  vertices  Ai,  A2,  and  hereafter  these  three  co- 
incident points  in  which  the  axis  meets  an  infinitely  thin  lens 
will  be  designated  by  the  simple  letter  A.  An  infinitely  thin 
lens  is  represented  in  a  diagram  by  the  segment  of  a  straight 
fine  which  is  bisected  at  right  angles  by  the  axis  of  the  lens; 
the  actual  form  of  the  lens  being  indicated  by  assigning  the 
positions  of  the  centers  Ci,  C2  of  the  two  faces.  In  order  to 
tell  at  a  glance  the  character  of  a  lens,  the  form  of  it  at  the 
edges  may  be  indicated,  as  shown  in  Fig.  119.  Fig.  119,a 
is  a  conventional  representation  of  an  infinitely  thin  con- 
vex lens,  and  Fig.  119,  b  is  a  similar  diagram  for  an  infinitely 
thin  concave  lens. 


89] 


Abscissa-Formula  of  Thin  Lens 


227 


Let  us  assume  that  the  lens  is  surrounded  by  the  same 
medium  on  both  sides;  and  let  n  denote  the  index  of  refrac- 


Fig.  119,  a. — Infinitely  thin  convex  lens;  M,  M'  conjugate  points  on  axis. 

tion  of  this  medium,  while  n'  denotes  the  index  of  refraction 
of  the  lens-substance  itself. 

The  broken  line  RBS  (Fig.  119)  represents  the  path  of 
a  paraxial  ray  which  enters  and  leaves  the  infinitely  thin 


Fig.  119,  fr. — Infinitely  thin  concave  lens;  M,  M'  conjugate  points 
on  axis. 

lens  at  the  point  marked  B.  The  points  where  the  ray 
crosses  the  axis  before  and  after  passing  through  the  lens 
will  be  designated  by  M,  M',  respectively.     The  straight 


228  Mirrors,  Prisms  and  Lenses  [§  89 

line  BM/  which  intersects  the  axis  at  the  point  marked  Mi' 
shows  the  path  the  ray  takes  after  being  refracted  at  the  first 
face  of  the  lens.  Obviously,  the  points  M,  M/  are  a  pair  of 
conjugate  axial  points  with  respect  to  the  first  surface  of  the 
lens,  and,  similarly,  the  points  MY,  M'  are  a  pair  of  con- 
jugate axial  points  with  respect  to  the  second  surface  of 
the  lens,  and,  therefore,  M,  M'  are  a  pair  of  conjugate  axial 
points  with  respect  to  the  lens  as  a  whole,  so  that  M'  will 
be  the  image  in  the  lens  of  an  axial  object-point  M.  The 
abscissae  of  these  points  with  respect  to  the  axial  point  A 
as  origin  will  be  denoted  by  u,  u';  thus,  w  =  AM,  w'  =  AM'. 
Also,  put  wi/  =  AMi/.  The  radii  of  curvature  of  the  two 
faces  are  ri  =  ACi,  r2  =  AC2- 

Accordingly,  in  order  to  obtain  the  formulae  connecting 
u  and  u',  we  have  merely  to  apply  the  fundamental  equa- 
tion (§  78)  for  the  refraction  of  paraxial  rays  at  a  spherical 
surface  to  each  face  of  the  lens  in  succession,  bearing  in 
mind  that  the  first  refraction  is  from  medium  n  to  medium 
n',  while  the  second  refraction  is  from  medium  n'  to  me- 
dium n.    Thus,  we  obtain : 

n'      n     n'—n        n      n'  n'—n 


U\      u        ri  u'     U\  r<t 

Eliminating  U\  by  adding  these  equations,  and  dividing 
through  by  n,  we  derive  the  abscissa-formula  for  the  refrac- 
tion of  paraxial  rays  through  an  infinitely  thin  lens,  in  the 
following  form : 


1      ln'-n/l      1\ 

u'     u        n     \ri     rj' 


The  expression  on  the  right-hand  side  of  this  equation,  in- 
volving only  the  lens-constants  r±f  r2  and  n'/n,  has  for  a 
given  lens  a  perfectly  definite  value,  which  may  be  com- 
puted once  for  all.    And  so  if  we  put 

l_n'^n/l     1\ 

f        n     \ri     r2/' 
where  the  magnitude  denoted  by  /  is  a  constant  of  the  lens 


90] 


Focal  Points  of  Thin  Lens 


229 


(which  we  shall  afterwards  see  is  the  focal  length  of  the 
lens),  the  formula  above  may  be  written: 

1  _1  =  1 
u'    u    /' 
which  is  the  form  of  the  lens-formula  that  is  perhaps  most 
common.    For  a  given  value  of  u  we  find  u'  =f.u/(f-\-u). 

Incidentally,  it  may  be  observed  that  the  equation  above 
is  symmetrical  with  respect  to  u  and  -vl ;  that  is,  the  equa- 
tion will  remain  unaltered  if —it  is  written  in  place  of  u'  and 
— u'  in  place  of  u.  Accordingly,  if  the  positions  of  a  pair  of 
conjugate  points   on  the   axis   are  designated   by  M,  M' 


Fig.  120.— Infinitely  thin  lens:  AP  =  M'A  =  BM,  AP'  =  MA. 

(Fig.  120),  the  pair  of  axial  points  designated  by  P,  P'  will 
likewise  be  conjugate,  provided  AP  =  M'A  and  AP'  =  MA;  so 
that  the  thin  lens  at  A  bisects  the  two  segments  PM'  and 
P'M.  Another  and  more  striking  way  of  exhibiting  this 
characteristic  property  of  an  infinitely  thin  lens  consists 
in  saying,  that  if  M'  is  the  image  of  an  axial  object-point 
at  M,  and  if  then  the  lens  is  shifted  from  its  first  position 
at  A  to  a  point  B  such  that  MB  =  AM',  the  object-point  M 
will  again  be  imaged  at  M'. 

90.  The  Focal  Points  of  an  Infinitely  Thin  Lens.— If 
the  object-point  M  is  at  the  infinitely  distant  point  on  the 
axis  of  the  lens,  its  image  will  be  formed  at  a  point  F'  whose 
position  on  the  axis  may  be  found  by  putting  u=  oo  ,u'  =  AF' 


230 


Mirrors,  Prisms  and  Lenses 


90 


in  the  formula  l/u'—l/u=lff;  thus,  we  find  AF'=/.  Sim- 
ilarly, the  object-point  F  conjugate  to  the  infinitely  dis- 
tant point  of  the  axis  is  found  by  substituting  in  the  same 
equation  the  pair  of  values  u  =  AF,  ur  —  oo  J  whence  we  ob- 
tain AF  =  — /.  These  points  F,  F'  are  the  primary  and  sec- 
ondary focal  points,  respectively,  and,  accordingly,  it  is 
evident  that  the  focal  points  of  an  infinitely  thin  lens  are  equi- 
distant from  the  lens  and  on  opposite  sides  of  it. 

The  character  of  the  imagery  in  the  case  of  an  infinitely 
thin  lens  is  completely  determined  as  soon  as  we  know  the 
positions  of  the  two  focal  points  F,  F';  and  since  the  point  A 
where  the  axis  meets  the  lens  lies  midway  between  F  and 
F',  it  is  obvious  that  the  natural  division  of  lenses  is  into 
two  classes  depending  on  the  order  in  which  the  three  points 
above  mentioned  are  ranged  along  the  axis. 

(1)  7/  the  primary  focal  point  is  in  front  of  the  lens  (Fig. 
121,  a),  that  is,  if  the  order  of  the  points  named  in  the  se- 


Fig.  121,  a. — Focal  points  (F,  F'),of  infinitely  thin  lens 
(FA  =  AF'=/).  In  a  positive  (or  convex  or  conver- 
gent) lens  the  first  focal  point  (F)  lies  on  same  side 
of  lens  as  incident  light  (real  focus) . 

quence  in  which  they  are  reached  by  light  traversing  the 
axis  of  the  lens  is  F,  A,  F',  then  incident  rays  parallel  to 


90] 


Focal  Points  of  Thin  Lens 


231 


the  axis  will  be  converged  to  a  real  focus  at  F'  on  the  other 
side  of  the  lens,  and  the  lens  is  a  convergent  lens  (§  87).  It 
is  also  called  a  positive  lens,  because  the  lens-constant  (or 
primary  focal  length)  /=FA  =  AF'  is  measured  along  the 
axis  in  the  positive  sense.  If  it  is  assumed  that  n'>n  (as, 
for  example,  in  the  case  of  a  glass  lens  in  air),  the  sign  of 
this  constant  /,  according  to  the  formula  above  which  de- 
fines 1//,  will  be  the  same  as  that  of  the  term  (l/ri— l/r2), 
which  is  the  algebraic  expression  of  the  difference  of  curva- 
tures (§  99)  between  the  two  faces  of  the  lens.  If  the  lens 
is  double  convex,  plano-convex  or  a  crescent-shaped  me- 
niscus— that  is,  in  all  forms  of  lenses  that  are  thicker  in 
the  middle  than  out  towards  the  edges — the  difference  of 
curvatures  (l/ri — l/r2)  will  be  found  to  be  positive.  And 
hence,  as  already  stated  (§  87),  thin  lenses  of  this  descrip- 
tion are  convergent  if  n'>n. 

(2)  //  the  secondary  focal  point  is  in  front  of  the  lens  (Fig. 
121,  b),  that  is,  if  the  points  F',  A,  F  are  ranged  along  the 


W 


F' 


Fig.  121,  b—  Focal  points  (F,F')  of  infinitely  thin 
lens  (FA  =  AF'=/).  In  a  negative  (or  concave  or 
divergent)  lens  the  first  focal  point  (F)  lies  on 
the  other  side  of  the  lens  from  the  incident  light 
(virtual  focus) . 


232  Mirrors,  Prisms  and  Lenses  [§91 

axis  in  the  order  named,  incident  parallel  rays  will  be  made 
to  diverge  from  a  virtual  focus  at  F',  and  in  this  case  the 
lens  is  said  to  be  a  divergent  or  negative  lens,  since  now  the 
lens-constant  /=FA=AF'  is  measured  along  the  axis  in 
the  negative  sense.  For  lenses  which  are  thinner  in  the 
middle  than  at  the  edges,  that  is,  for  double  concave,  plano- 
concave and  canoe-shaped  meniscus  lenses  the  difference  of 
curvatures  (1/ri—  l/r2)  will  be  found  to  be  negative;  and 
hence  for  such  lenses  the  constant  /  will  be  negative  if  nr>n. 

A  case  of  rather  more  theoretical  than  practical  interest  is 
afforded  by  an  infinitely  thin  concentric  lens  (§  87)  for  which 
r2  =  rh  and  which  is  therefore  uniformly  thick  in  a  direc- 
tion parallel  to  the  axis,  so  that  according  to  the  above 
classification  it  should  be  neither  convergent  nor  divergent. 
In  fact,  the  value  of  the  lens-constant  /  for  this  lens  is  in- 
finity, and  hence  u\=u,  so  that  object-point  M  and  image- 
point  M'  are  coincident  always.  A  bundle  of  parallel  rays 
traversing  an  infinitely  thin  concentric  lens  will  emerge 
from  the  lens  just  as  though  the  lens  had  not  been  inter- 
posed in  the  path  of  the  rays. 

91.  Construction  of  the  Point  M'  Conjugate  to  the  Axial 
Point  M  with  respect  to  an  Infinitely  Thin  Lens. — The 
planes  which  are  perpendicular  to  the  axis  of  the  lens  at  the 
focal  points  F,  F'  are  called  the  primary  and  secondary  focal 
planes,  respectively. 

The  point  M'  conjugate  to  a  point  M  on  the  axis  of  an 
infinitely  thin  lens  surrounded  by  the  same  medium  on  both 
sides  may  be  constructed  as  follows : 

Through  the  given  point  M  (Fig.  122,  a  and  b)  draw  a 
straight  line  MB  meeting  the  lens  at  B,  and  through  the 
axial  point  (A)  of  the  lens  draw  a  straight  line  AI'  parallel 
to  MB  and  meeting  the  secondary  focal  plane  in  the  point 
V;  then  the  point  where  the  straight  line  BF,  produced  if 
necessary,  crosses  the  axis  will  be  the  required  point  M' 
conjugate  to  M. 

The  point  M'  may  also  be  constructed  in  another  way, 


91] 


Thin  Lens:  Conjugate  Axial  Points 


233 


as  follows:  Let  J  designate  the  point  where  the  straight 
line  MB  crosses  the  primary  focal  plane,  and  through  B 
draw  a  straight  line  parallel  to  the  straight  line  JA,  which 


Fig.  122,  a  and  6. — Infinitely  thin  lens:  Construction  of  point 
M'  conjugate  to  axial  object-point  M.  (a)  Convex, 
(6)  Concave  lens. 

will  intersect  the  axis  of  the  lens  in  the  required  point  M'. 
Fig.  122,  a  shows  the  construction  in  the  case  of  a  convex 
lens  and  Fig.  122,  b  shows  it  for  a  concave  lens. 

The  proof  is  obvious.  From  the  two  pairs  of  similar 
triangles  MAB,  AFT  and  MM'B,  AMT,  we  obtain  the 
proportions: 

MA=MB=MM'. 
AF'     AI'     AM'  ' 


234 


Mirrors,  Prisms  and  Lenses 


92 


and  if  we  introduce  the  symbols  w  =  AM,  w'  =  AM',  /=AF', 
we  get: 

—  u  _u'— u 

which  is  the  same  as  the  abscissa-relation  found  in  §  89. 
92.  Extra-Axial  Conjugate  Points  Q,  Q';  Conjugate 
Planes. — Since  the  axial  point  A  of  an  infinitely  thin  lens 
is  also  the  optical  center  of  the  lens  (§  89),  a  straight  line 
drawn  through  A  will  represent  the  path  of  a  ray  both  be- 
fore and  after  passing  through  the  lens  at  this  point.  If 
the  axis  of  the  lens  is  rotated  in  a  meridian  plane  through 


Fig.   123. — Infinitely  thin  lens:  Image-point  Q'  conjugate  to  extra-axial 
object-point  Q. 


a  very  small  angle  FAJ  (Fig.  123)  around  the  point  A  as 
vertex,  the  focal  points  F,  F'  will  describe  the  small  arcs 
FJ,  FT  and  the  straight  line  JI'  will  represent  the  path  of 
a  paraxial  ray  traversing  the  lens  at  A.  The  points  Q,  Q' 
at  the  ends  of  the  arcs  MQ,  M'Q'  traced  out  in  this  angular 
movement  of  the  axis  by  a  pair  of  conjugate  axial  points 
M,  M'  will  evidently  occupy  the  same  relation  to  each 
other  on  the  straight  line  JI'  as  M,  M'  have  to  each  other 


92J 


Image  in  Infinitely  Thin  Lens 


235 


on  the  straight  line  FF',  and  therefore  Q,  Q'  are  a  pair  of 
extra-axial  conjugate  points. 

Accordingly,  if  the  points  of  an  object  lie  in  the  vicinity 
of  the  axis  on  an  element  of  a  spherical  surface  described 


Fig.  124,  a  and  b. — Infinitely  thin  lens:  Lateral  magnification 
and  construction  of  image  M'Q'  conjugate  to  short  object- 
line  MQ  perpendicular  to  axis,     (a)  Convex,  (6)  Concave  lens. 

around  the  vertex  A  of  the  infinitely  thin  lens  as  center, 
the  corresponding  points  of  the  image  will  be  assembled 
on  a  concentric  spherical  surface;  and  since,  within  the 
region  of  paraxial  rays,  these  spherical  elements  may  be 
regarded  as  plane,  it  follows  that  a  small  plane  object  at 


236  Mirrors,  Prisms  and  Lenses  [§  93 

right  angles  to  the  axis  will  be  reproduced  by  a  similar 
plane  image  also  at  right  angles  to  the  axis. 

Conjugate  planes  are  pairs  of  parallel  planes  perpendicular 
to  the  axis  of  the  lens;  and  any  straight  line  drawn  through 
the  center  of  an  infinitely  thin  lens  will  pierce  a  pair  of  conju- 
gate planes  in  a  pair  of  conjugate  points. 

In  particular,  the  planes  conjugate  to  the  focal  planes 
are  the  infinitely  distant  planes  of  the  image-space  and 
object-space,  according  as  the  infinitely  distant  plane  is 
regarded  as  belonging  to  one  or  the  other  of  these 
regions. 

The  construction  of  the  point  Q'  conjugate  to  an  extra- 
axial  object-point  Q  (Fig.  124,  a  and  b),  with  respect  to 
an  infinitely  thin  lens,  is  made  by  a  method  precisely  sim- 
ilar to  that  employed  in  the  corresponding  problem  in  the 
cases  both  of  a  spherical  mirror  (§71)  and  of  a  spherical 
refracting  surface  (§  81) ;  the  only  difference  in  this  case 
being  that  the  center  of  the  lens  takes  the  place  of  the  center 
of  the  spherical  surface  and  that  the  focal  points  of  the 
lens  are  at  equal  distances  on  opposite  sides  of  it. 

93.  Lateral  Magnification  in  case  of  Infinitely  Thin  Lens. 
— The  lateral  magnification  in  the  case  of  an  infinitely  thin 
lens,  defined,  as  in  §§  72  and  82,  as  the  ratio  of  the  height 
of  the  image  (y'  =  M'Q')  to  the  height  of  the  object  (y  =  MQ), 
may  be  obtained  from  the  diagram  (Fig.  124)  and  is  evi- 
dently given  by  the  following  formula: 

y'    W 

y=-=-; 

y     u 

so  that  the  linear  dimensions  of  object  and  image  are  in  the 
same  ratio  as  their  distances  from  the  thin  lens.  Moreover, 
it  appears  that  the  image  is  erect  or  inverted  according  as 
object  and  image  lie  on  the  same  side  or  on  opposite  sides  of 
the  lens. 

Another  expression  for  the  lateral  magnification  may 
be  derived  by  considering  the  two  pairs  of  similar  right 


§  94]  Imagery  in  Thin  Lens  237 

triangles  FMQ,  FAW  and  F'M'Q',  F'AV,  from  which  we 
obtain  the  proportions: 

AW_  FA         M'Q'_F'M'. 

MQ      FM'        AV        F'A  ; 
and  since 

AW  =  M'Q'  =  ?/,       AV  =  MQ  =  y,       FA  =  AF'=/, 
we  find: 

y'    f      x' 
y    *      f 

where  z  =  FM,  x'  =  F'M'  denote  the  abscissae  of  M,  M' 
with  respect  to  the  focal  points  F,  F',  respectively,  as  ori- 
gins. Accordingly,  the  lateral  magnification  varies  inversely 
as  the  distance  of  the  object  from  the  primary  focal  plane,  and 
directly  as  the  distance  of  the  image  from  the  secondary  focal 
plane. 

94.  Character  of  the  Imagery  in  a  Thin  Lens. — The 
Newtonian  form  of  the  abscissa-relation  (c/.  §  85)  for  an 
infinitely  thin  lens  surrounded  by  air  is : 

x.x'=~f, 
which  shows  that  object  and  image  lie  on  opposite  sides  of 
the  focal  planes;  so  that  if  M  is  a  point  on  the  axis  to  the 
right  of  the  primary  focal  point  F,  the  conjugate  point  M' 
will  be  found  on  the  axis  at  the  left  of  the  secondary  focal 
point  F',  and  vice  versa. 

The  character  of  the  imagery  produced  by  the  refraction 
of  paraxial  rays  through  an  infinitely  thin  lens  is  exhibited 
in  the  diagrams  Fig.  125,  a  and  b.  The  numerals  1,  2,  3, 
etc.,  mark  the  successive  positions  of  an  object-point  which 
is  supposed  to  traverse  a  straight  line  parallel  to  the  axis 
(so-called  " object-ray")  from  an  infinite  distance  in  front 
of  the  lens  to  an  infinite  distance  on  the  other  side  of  it. 
Until  it  reaches  the  lens  at  the  point  marked  V  the  object 
is  real,  thereafter  it  is  virtual.  The  corresponding  numerals 
with  primes,  viz.,  1',  2',  3',  etc.,  ranged  along  the  straight 
line  VF'  (called  the  " image-ray")  mark  the  successive 
positions    of   the    image-point,    which,    starting,    from   the 


238 


Mirrors,  Prisms  and  Lenses 


[§94 


secondary  focal  point  F',  moves  along  this  line  always  in 
the  same  direction  out  to  infinity  and  back  again  to  its 
starting  point.     The  straight  lines  11',  22',  33',  etc.,  con- 


T0  5ATW 


N«' 


1 

2                     J           4 

V*  g>               6                          7 

OBJECT 

RAY      ^ 

\     \ 

1 

? 

A 
\ 

"\IMAGE 

/„i                      \ray 

TO  3  ATOO 


^ 


'(//TO  5' AT  00 

Fig.  125,    a   and   b. — Character    of    imagery    in    infinitely    thin    lens, 
(a)  Convex,   (b)  Concave  lens. 

necting  corresponding  positions  of  object-point  and  image- 
point  form  a  pencil  of  rays  all  passing  through  the  optical 
center  A  of  the  lens,  which  is  the  center  of  perspective  of 


[§  94]  Imagery  in  Thin  Lens  239 

object-space  and  image-space.  At  the  point  V  object  and 
image  coincide  with  each  other  in  the  lens  itself,  and  here 
object  and  image  are  congruent.  The  so-called  principal 
planes  (§  119)  of  an  infinitely  thin  lens  coincide  with  each 
other  in  the  plane  perpendicular  to  the  axis  of  the  lens  at  Us 
optical  center  A.  The  fact  that  object-point  and  image- 
point  coincide  with  each  other  at  V  is  expressed  geometri- 
cally by  saying  the  y-axis  is  the  base  of  a  range  of  self  conju- 
gate points. 

In  a  convex  lens  (Fig.  125,  a)  the  image  of  a  real  object 
is  seen  to  be  real  and  inverted  as  long  as  the  object  lies  in 
front  of  the  lens  beyond  the  primary  focal  plane;  whereas 
the  image  is  virtual  and  erect  if  the  object  is  placed  between 
the  primary  focal  plane  and  the  lens.  The  image  of  a  vir- 
tual object  in  a  convex  lens  is  formed  between  the  lens  and 
the  secondary  focal  plane  and  is  real  and  erect. 

In  a  concave  lens  (Fig.  125,  b)  the  image  of  a  real  object 
lies  between  the  lens  and  the  secondary  focal  plane,  and  it 
is  virtual  and  erect.  If  the  object  is  virtual,  its  image  in  a 
concave  lens  will  be  real  and  erect  if  the  object  lies  between 
the  lens  and  the  primary  focal  plane,  but  it  will  be  virtual 
and  inverted  if  the  object  lies  beyond  the  primary  focal 
plane. 

If  2  =  MM'  denotes  the  distance  between  a  pair  of  con- 
jugate axial  points  M,  M',  then  u'  =  u-\-z,  where  u  —  AM, 
u'  =  AM'.  Substituting  this  value  of  u  in  the  formula 
l/u'—l/u  =  l/f,  we  obtain  a  quadratic  in  u,  which  implies, 
therefore,  that  for  a  given  value  of  the  interval  z  between 
object  and  image,  there  are  always  two  positions  of  the 
object-point  M  with  respect  to  the  lens  (§  89).  But  under 
some  circumstances  the  assigned  value  of  the  interval  z 
may  be  such  that  the  roots  of  the  quadratic  prove  to  be 
imaginary,  and  then  it  will  be  quite  impossible  with  the 
given  lens  to  produce  an  image  at  the  given  distance  z  from 
the  object.  For  example,  if  the  object  lies  in  front  of  a 
convex  lens  (/>  0)  at  a  distance  greater  than  the  focal  length, 


240 


Mirrors,  Prisms  and  Lenses 


[§95 


then  u<0  and  z>0.  Put  a  =  MA  =  -w,  so  that  the  magni- 
tudes denoted  by  /,  z  and  a  are  all  positive.  Eliminating 
v!  from  the  abscissa-formula,  we  obtain  a  quadratic  in  a 
whose  roots  are  given  by  the  following  expression: 

s±Vz(s~4/). 


which  will  be  imaginary  if  (2—  4/)<  0.  Hence,  the  distance 
(z)  between  a  real  object  and  its  real  image  in  a  convex  lens 
cannot  be  less  than  four  times  the  focal  length  f. 

95.  The  Focal  Lengths  f,  V  of  an  Infinitely  Thin  Lens. — 
The  focal  lengths  of  a  thin  lens  are  defined  exactly  in  the 
same  way  as  the  focal  lengths  of  a  spherical  refracting  sur- 


I 

^B 

K  ^"^\ 

l' 

// 

H 

TO    E    ATOO             y 

4 

^^-v^^     TO  B  AT00 

/M/ 

F 

A 

F~*^-^^ 

^^*"^^^^  TO    j'ATOO 

//TO   I 

AT  00 

1 

' 

Fig.  126. 


-Focal  planes  and  focal  lengths  of  infinitely  thin  lens 
(/=FA  =  — f'  =  KF'). 


face  (§  83).  Thus,  the  primary  focal  length  of  a  lens  is  the 
ratio  of  the  height  of  the  image,  in  the  secondary  focal  plane, 
to  the  apparent  size  of  the  infinitely  distant  object.  In  Fig.  126 
FT'  is  the  height  of  the  image  of  the  infinitely  distant  ob- 
ject EI  which  is  seen  under  the  angle  6  =  Z  EFI  =  Z  AFK, 
and  the  primary  focal  length  is,  therefore,  FT/tan  6  = 
AK/tan  6  =  FA  =/;  and  hence,  as  already  observed,  the 
primary  focal  length  is  identical  with  the  lens-constant 
denoted  by/,  which,  as  we  have  seen  (§  90),  is  the  abscissa 
of  the  axial  point  A  of  a  thin  lens  with  respect  to  its  primary 
focal  point  F.     Similarly,  the  secondary  focal  length  (/')  is 


§  95]  Infinitely  Thin  Lens:  Focal  Lengths  241 

the  ratio  of  the  height  of  an  object  in  the  primary  focal  plane  of 
the  lens  to  the  apparent  size  of  its  infinitely  distant  image. 
For  example,  in  the  diagram  the  image  of  the  object  FJ 
lying  in  the  primary  focal  plane  is  EM',  which  lying  in  the 
infinitely  distant  plane  of  the  image-space,  subtends  the 
angle  0'  =  ZEF'J'  =  Z  AF'H;  and  hence  /'  =  FJ/tan  6'  = 
AH/tan  0'  =  F'A;  so  that  the  secondary  focal  length  may 
also  be  defined  as  the  abscissa  of  the  axial  point  A  of  an  in- 
finitely thin  lens  with  respect  to  the  secondary  focal  point  F'. 
And  since  F'A  = -AF' = -FA,  evidently: 

Accordingly,  the  focal  lengths  (/,  /')  of  a  lens  surrounded  by 
the  same  medium  on  both  sides  are  equal  in  magnitude  and 
opposite  in  sign. 

If  the  lens  is  reversed  by  turning  it  through  180°  about 
an  axis  perpendicular  to  the  axis  of  the  lens,  that  is,  if  the 
light  is  made  to  traverse  the  lens  in  a  sense  exactly  opposite 
to  that  which  it  had  at  first,  the  focal  lengths  /,  /'  will  not 
be  altered.  This  is  evident  from  the  fact  that  the  expres- 
sion for  the  focal  length/,  viz., 

J        J      (n'-n)  (r2~n)\ 

remains  the  same  when  -rh  -r2  are  substituted  in  place  of 

7*1,  r2,  respectively.     Thus,  the  character  of  the  lens  (§  90) 

and  its  action  are  not  changed  by  presenting  the  opposite 

face  to  the  incident  rays. 

The  focal  length  of  an  infinitely  thin  symmetric   lens 

n  r 
(§87),   for  which  ri=~r2  =  r  (say)  is  f=^—f — r*  and  if 

71  =  1,  w'=1.5,  we  find  f=r.  Accordingly,  the  focal  length 
of  an  infinitely  thin  symmetric  glass  lens  surrounded  by  air 
(n=l,  n' =1.5)  is  equal  to  the  radius  of  the  first  face.  Spec- 
tacle glasses  were  at  first  symmetric  lenses,  and  in  the  old 
inch  system  of  designation  a  No.  10  spectacle  glass,  for  ex- 
ample, was  a  lens  whose  radius  of  curvature  on  each  surface 
was  10  inches  and  whose  focal  length  was  10  inches. 


242  Mirrors,  Prisms  and  Lenses  [§  96 

If  one  face  of  the  lens  is  plane,  for  example,  if  r\=  go  , 

71  V  71  T 

r2  =  r,  we  find/=  ——^—  ;  or  if  i\  =  r,  r2=  oo  ,  then/= 


n—n  n—n 

where  in  each  case  r  denotes  the  radius  of  the  curved  sur- 
face. Comparing  this  with  the  value  of  /  obtained  in  the 
preceding  case,  we  see  that  if  one  of  the  faces  of  a  symmetric 
lens  be  ground  off  plane,  the  focal  length  of  the  lens  will  thereby 
be  doubled. 

96.  Central  Collineation  of  Object-Space  and  Image- 
Space. — Comparing  the  methods  and  results  of  this  chap- 
ter with  those  obtained  in  the  preceding  chapter,  the  serious 
student  cannot  have  failed  to  remark  a  striking  parallelism 
that  exists  between  the  imagery  by  paraxial  rays  in  a  spher- 
ical refracting  surface  and  the  imagery  under  the  same  con- 
ditions in  an  infinitely  thin  lens.  In  some  instances  the 
formulae  are  actually  identical,  and  a  closer  examination 
will  show  that  this  similarity  extends  even  to  comparative 
details.  For,  example,  the  focal  points  lie  on  opposite  sides 
of  a  lens  just  as  they  were  found  to  do  in  the  case  of  a  spher- 
ical refracting  surface,  and  the  resemblance  goes  still  far- 
ther. For  in  a  spherical  refracting  surface  the  connection 
between  the  focal  lengths  (/,  /')  and  the  indices  of  refraction 
(n,  n')  is  expressed  by  the  formula  n'.f-\-n.f'  =  0  (§79); 
and  if  in  this  formula  we  put  7i'  =  n,  we  obtain  the  relation 
/-f//  =  0,  which  is  the  algebraic  statement  of  the  fact  that 
the  focal  lengths  of  a  lens  surrounded  by  the  same  medium 
on  both  sides  are  equal  and  opposite  (§  95) . 

It  has  already  been  pointed  out  that  the  imagery  in  a 
spherical  mirror  may  be  regarded  as  a  special  case  of  refrac- 
tion at  a  spherical  surface  (§§  75,  77  and  78) ;  and  now  it 
is  proposed  to  advance  a  step  farther  in  this  generalization 
process  and  to  show  that  all  these  types  of  imagery  which 
have  been  investigated  separately  and  independently  are 
in  reality  embraced  in  a  concept  of  geometry  known  as 
collinear  correspondence  between  one  space  and  another 
(called  in  the  theory  of  optical  imagery   " object-space" 


§  96]  Central  Collineation  243 

and  "image-space")-  Moreover,  these  types  of  imagery 
belong  to  a  particularly  simple  kind  of  collinear  correspond- 
ence to  which  the  name  central  collineation  has  been  given. 

A  lens  or  an  optical  instrument  is  said  to  divide  the  sur- 
rounding space  into  two  parts,  viz.,  the  object-space  and 
the  image-space;  but  these  are  not  to  be  thought  of  as  sep- 
arate and  distinct  regions  but  as  interpenetrating  and  in- 
cluding each  other;  so  that  a  point  or  ray  may  be  regarded 
at  one  time  as  belonging  to  the  object-space  and  at  another 
time  as  belonging  to  the  image-space,  depending  merely 
on  the  point  of  view.  Thus,  for  example,  the  infinitely 
distant  plane  of  space  may  be  viewed  as  the  image  of  the 
primary  focal  plane  of  a  lens,  and  then  it  is  a  part  of  the 
image-space;  but  if  the  secondary  focal  plane  is  regarded 
as  the  image  of  the  infinitely  distant  plane,  the  latter  is  a 
part  of  the  object-space. 

Now  the  distinguishing  characteristics  of  the  optical 
imagery  which  is  produced  by  the  refraction  of  paraxial 
rays  at  a  single  spherical  surface  or  through  an  infinitely 
thin  lens  may  be  summarized  in  the  two  following  state- 
ments: 

(a)  All  straight  lines  joining  pairs  of  conjugate  points  in- 
tersect in  one  point,  viz.,  the  center  (C)  of  the  spherical  re- 
fracting surface  or  the  optical  center  (A)  of  the  thin  lens. 
This  point  which  is  the  center  of  perspective  of  object- 
space  and  image-space  is  called  the  center  of  collineation, 
and  will  be  referred  to  here  as  the  point  C. 

(b)  Any  pair  of  corresponding  incident  and  refracted  rays 
lying  in  a  meridian  plane  meet  in  a  straight  line  Ay  called  the 
axis  of  collineation  (or  the  y-eixis)  which  is  perpendicular  at 
A  to  the  optical  axis  (or  the  rc-axis) . 

Any  straight  line  going  through  the  center  of  collinea- 
tion is  called  a  central  ray.  Every  central  ray  is  a  self -cor- 
responding ray;  that  is,  image-ray  and  corresponding  object- 
ray  lie  along  one  and  the  same  straight  line.  Moreover, 
any  point  lying  on  the  axis  of  collineation  is  a  self-conjugate 


244 


Mirrors,  Prisms  and  Lenses 


[§97 


point;  that  is,  along  this  line  object-point  and  image-point 
are  coincident  with  each  other.  The  center  of  collineation 
is  also  a  self-conjugate  point,  and  hence,  in  general,  there  are 
two  self-conjugate  points  on  a  central  ray,  viz.,  the  center 
of  collineation  and  the  point  where  the  ray  meets  the  axis 
of  collineation.  Only  in  case  the  center  of  collineation  lies 
on  the  axis  of  collineation  will  there  be  only  one  self-conju- 
gate or  so-called  double  point  on  a  central  ray. 

97.  Central  Collineation  (cont'd).  Geometrical  Con- 
structions.— Starting  from  these  simple  propositions,  we 
can  easily  develop  a  complete  theory  of  optical  imagery 
for  the  simple  cases  mentioned  above.    Thus,  for  example, 


Fig.  127. — Central  collineation:  Construction  of  pairs  of  conjugate  points 
M,  M';  P,  P';  Q,  Q';  R,  R';  S,  S';  T,  T';  and  U,  U'.  Axis  of  collineation 
Ay;  center  of  collineation  C. 

being  given  the  axis  of  collineation  (Ay)  and  the  center  of  col- 
lineation (C),  together  with  the  positions  of  a  pair  of  conjugate 
points  P,  P',  we  can  construct  the  position  of  a  point  Q'  con- 
jugate to  a  given  point  Q,  as  follows: 

(a)  In  general,  the  straight  line  PQ  (Fig.  127)  will  not 
pass  through  the  center  of  collineation.  Let  the  self- 
conjugate  point  in  which  the  straight  line  PQ  meets  the 
axis  of  collineation  be  designated  by  T;  the  image-ray  cor- 


97] 


Central  Collineation 


245 


responding  to  the  object-ray  PT  will  lie  along  the  straight 
line  TP',  and  since  this  ray  must  pass  likewise  through  the 
point  Q'  conjugate  to  Q,  the  required  point  will  be  at  the 
intersection  of  the  straight  lines  TP',  QC. 

(b)  But  in  the  special  case  when  the  straight  line  PQ  is 
a  central  ray  (Fig.  128)  the  construction  which  has  just 
been  given  fails,  and  we  must  resort  to  a  different  procedure, 


V 

*a 

j 

T 

>^WL^L 

^                                    X 

F 

A                 /Cl^A 

^ 

Fig.  128. — Central  collineation:  Straight  line  PQ  passes  through  center 
of  collineation  (C) .  Diagram  shows  case  when  C  does  not  lie  on  axis 
of  collineation  Ay;  as  in  spherical  refracting  surface  (c>  1). 

as  follows:  Through  the  points  P  and  C  draw  a  pair  of 
straight  lines  PO,  CO  meeting  in  a  point  O,  and  let  the 
point  where  the  straight  line  PO  meets  the  axis  of  collinea- 
tion be  designated  by  T.  Also,  let  O'  designate  the  point 
of  intersection  of  the  straight  lines  TP'  and  CO.  Then  if 
the  point  where  the  straight  line  QO  meets  the  axis  of  col- 
lineation is  designated  by  U,  the  required  point  Q'  will  be 
the  point  of  intersection  of  the  straight  lines  UO'  and  QC. 
The  image-point  I'  conjugate  to  the  infinitely  distant 
object-point  I  of  the  pencil  of  parallel  rays  whose  central 
ray  is  PP'  may  be  constructed  exactly  as  described  above 
in  (b),  provided  we  have  the  same  data.    The  straight  line 


246 


Mirrors,  Prisms  and  Lenses 


[§97 


OG  is  drawn  parallel  to  PP'  meeting  the  axis  of  collineation 
in  G,  and  the  required  point  I'  is  the  point  of  intersection 
of  the  straight  lines  GO'  and  PP'. 

Similarly,  the  position  of  the  object-point  J  conjugate 
to  the  infinitely  distant  image-point  J'  of  the  central  ray 
PP'  is  found  by  drawing  the  straight  line  O'H  parallel  to 
PP'  meeting  the  axis  of  collineation  in  H;  then  the  point 
of  intersection  of  the  straight  lines  OH,  PP'  will  be  the  re- 
quired point  J. 

u 


Fig.  129. — Central  collineation:  Straight  line  PQ  passes  through  center 
of  collineation  (C).  Diagram  shows  case  when  C  lies  on  axis  of  col- 
lineation Ay,  as  in  infinitely  thin  lens  (c—  1). 

The  focal  points  F,  F'  on  the  optical  axis  are  constructed 
in  precisely  the  same  way  as  the  two  points  J,  I'  on  the 
central  ray  PP'. 

The  special  case  when  the  center  of  collineation  (C)  lies  on 
the  axis  of  collineation,  that  is,  when  the  two  points  A  and  C 
are  coincident,  is  shown  in  Fig.  129,  which  evidently  cor- 
responds to  the  case  of  an  infinitely  thin  lens  surrounded 
by  the  same  medium  on  both  sides. 


§  98]  Field  of  View  of  Thin  Lens  247 

It  would  be  easy  to  show  by  the  methods  of  projective 
geometry  that  the  straight  lines  FJ,  FT  are  parallel  to 
the  axis  of  collineation  and  that  we  have  the  following  re- 
lations between  the  points  J,  I'  and  the  two  self -con  jugate 
points  B,  C  on  the  central  ray  JF: 

JB  =  CI',     I'B  =  CJ,     ^?  =  c, 

where  c  denotes  a  constant  called  the  invariant  of  central 
collineation,  which  has  the  value  n'\  n  for  a  spherical  re- 
fracting surface  and  the  value  +1  for  a  thin  lens  surrounded 
by  the  same  medium  on  both  sides.  For  a  spherical  mirror, 
c  =  —  1.  For  the  axial  ray  the  above  relations  may  be  written : 

FA  =  CF',     F'A  =  CF,     ^  =  c. 
AF 

The  reader  who  wishes  to  pursue  this  subject  will  find  a 
complete  discussion  at  the  end  of  Chapter  V  of  the  author's 
Principles  and  Methods  of  Geometrical  Optics  published  by 
The  Macmillan  Company  of  New  York. 

98.  Field  of  View  of  an  Infinitely  Thin  Lens. — If  it  is 
assumed  that  there  are  no  artificial  stops  present  except 
in  the  plane  of  the  lens,  and  that  the  imagery  is  produced 
by  means  of  paraxial  rays  only,  the  field  of  view  in  the  case 
of  an  observer  looking  through  the  lens  along  its  axis  is 
easily  determined  by  drawing  the  straight  lines  O'G,  O'H 
(Fig.  130,  a  and  b)  in  a  meridian  plane  of  the  lens  from  the 
center  0'  of  the  eye-pupil  to  the  ends  G,  H  of  the  diameter 
of  the  lens-opening.  For  the  lens-opening  acts  here  just 
like  a  round  window  or  port-hole  in  an  opaque  wall  to  limit 
the  field  of  view  in  the  image-space  of  the  lens.  If  O  desig- 
nates the  position  of  the  axial  object-point  which  is  repro- 
duced by  the  image-point  O',  then  the  straight  lines  OG,  OH 
determine  the  limits  in  the  meridian  plane  of  the  diagram  of 
the  field  of  view  of  the  object-space.  Let  the  straight  line 
B'C  bisected  at  right  angles  at  O'  by  the  axis  of  the  lens 
represent  the  diameter  of  the  pupil  in  the  meridian  plane  of 
the  lens;  and  construct  the  line  BOC  whose  image  in  the  lens 


248 


Mirrors,  Prisms  and  Lenses 


[§98 


is  the  diameter  B'O'C  of  the  pupil  of  the  eye.  Then  the 
image  S'  of  the  luminous  point  S  lying  within  the  object-side 
field  of  view  may  be  constructed  by  drawing  through  S  the 
straight  lines  SB,  SC  to  meet  the  lens  in  two  points  which 


ENTRANCE 
PUPIL 


PUPIL    OF    EYE 


S^% 

^ST; 

A 

/* 

B><^ 

lB' 

ENTRANCE 

PUPIL        EXIT 

PUPIL 

Fig.  130,  a  and  b. — Field  of  view  of  infinitely  thin  lens  for  given  position  of 
eye  on  axis  of  lens,    (a)  Convex,   (6)  Concave  lens. 

must  be  joined  with  B',  C,  respectively;  and  the  point  of 
intersection  of  these  latter  fines  will  be  the  required  point 
S'  conjugate  to  S.  In  brief,  the  circular  opening  whose  di- 
ameter is  BC  is  the  common  base  of  all  the  cones  of  effective 
rays  in  the  object-space  of  the  lens,  just  as  the  pupil  of  the 
eye  itself  is  the  common  base  of  the  cones  of  effective  rays 


Ch.  VII]  Problems  249 

in  the  image-space.  Assuming  that  the  lens-opening  is 
large  enough  to  permit  the  entire  pupil  of  the  eye  to  be  filled 
with  rays  emanating  from  an  axial  object-point,  the  lens- 
opening  GH  acts  as  field-stop  and  the  pupil  of  the  eye  as 
aperture-stop  (Chapter  XII). 

PROBLEMS 

1.  Show  how  to  construct  the  optical  center  of  a  lens. 
Draw  diagrams  for  the  various  forms  of  convex  and  con- 
cave lenses;  and  prove  that  the  distance  of  the  optical 
center  from  the  vertex  of  the  first  face  is  equal  to  rid/(ri  —  r2), 
where  rh  r2  denote  the  radii  of  the  two  surfaces  and  d  de- 
notes the  axial  thickness  of  the  lens. 

2.  In  each  of  the  following  lenses  the  axial  thickness  is 
2  cm.  Find  the  position  of  the  optical  center,  and  draw 
a  diagram  for  each  lens  showing  the  position  of  this  point. 

(a)  Double  convex  lens  of  radii  10  and  16  cm.;  (6)  Double 
concave  lens  of  radii  10  and  16  cm.;  (c)  Plano-convex  lens; 
(d)  Positive  meniscus  of  radii  10  and  16  cm.;  (e)  Negative 
meniscus  of  radii  20  and  16  cm.;  (/)  Lens  of  zero  curvature. 

3.  Rays  of  light  diverging  from  a  point  one  foot  in  front 
of  a  thin  lens  are  brought  to  a  focus  4  inches  beyond  it. 
Find  the  focal  length.  Ans.  /=-f-3  inches. 

4.  An  object  is  placed  one  foot  in  front  of  a  thin  convex 
lens  of  focal  length  9  inches.    Where  is  the  image  formed? 

Ans.  3  feet  from  the  lens  on  the  other  side. 

5.  Rays  coming  from  a  point  6  inches  in  front  of  a  thin 
lens  are  converged  to  a  point  18  inches  on  the  other  side  of 
the  lens.    Find  the  focal  length.  Ans.  /=  +4.5  inches. 

6.  An  object  is  placed  in  front  of  a  thin  lens  at  a  distance 
of  30  cm.  from  it.  The  image  is  virtual  and  10  cm.  from 
the  lens.    Find  the  focal  length.  Ans. /=— 15  cm. 

7.  The  radius  of  the  first  face  of  a  thin  double  convex 
lens  made  of  glass  of  index  1.5  is  20  cm.  If  the  focal  length 
of  the  lens  is  30  cm.,  what  must  be  the  radius  of  the  second 
face?  Ans.  60  cm. 


250  Mirrors,  Prisms  and  Lenses  [Ch.  VII 

8.  A  thin  convex  lens  made  of  glass  of  index  1.5  has  a 
focal  length  of  12.5  cm.  If  the  radius  of  the  second  face  is 
+  17.5  cm.,  what  is  the  radius  of  the  first  face?  And  if  the 
lens  is  concave,  and  the  radius  of  the  first  face  is  +17.5  cm., 
what  is  the  radius  of  the  second  face? 

Ans.  In  both  cases  the  radius  is  +4.6  cm. 

9.  The  focal  length  of  a  double  convex  lens  was  found 
to  be  30.6  cm.,  and  its  radii  30.4  and  34.5  cm.  Find  the 
index  of  refraction  of  the  glass.  Ans.  1.528. 

10.  The  focal  length  of  a  glass  lens  in  air  is  5  inches. 
What  will  be  the  focal  length  of  the  lens  in  water,  assuming 
that  the  indices  of  refraction  of  air,  glass  and  water  are  1, 
|  and  I-,  respectively?  Ans.  20  inches. 

11.  Show  that  any  thin  lens  which  is  thicker  in  the  middle 
than  out  towards  the  edges  is  convergent,  provided  the 
lens-medium  is  more  highly  refracting  than  the  surrounding 
medium. 

12.  Show  that  the  focal  length  of  a  thin  plano-convex 
lens  is  twice  that  of  a  double  convex  lens,  if  the  curvatures 
of  the  curved  surfaces  are  all  equal  in  magnitude. 

13.  Find  the  focal  length  of  a  thin  double  convex  diamond 
lens,  of  index  2.4875,  the  radius  of  each  surface  being  4  cm. 

Ans.  13.4  mm. 

14.  The  curved  surface  of  a  thin  plano-convex  lens  of  glass 
of  index  1.5  has  a  radius  of  12  inches.  Find  its  focal  length. 
What  must  be  the  radii  of  a  symmetric  double  convex  lens 
of  same  material  which  has  same  focal  length? 

Ans.  /=24  inches;  r  =  24  inches. 

15.  The  radii  of  a  thin  double  convex  lens  are  9  cm.  and 
12  cm.  The  lens  is  made  of  glass  of  index  1.5.  If  light  di- 
verges from  a  point  18  cm.  in  front  of  the  lens,  where  will 
it  be  focused?  Ans.  Real  image,  24  cm.  from  lens. 

16.  A  thin  lens  is  made  of  glass  of  index  n.  If  the  focal 
length  of  the  lens  in  air  is  a,  and  if  its  focal  length  in  a  liquid 
is  6,  show  that  the  index  of  refraction  of  the  liquid  is 

bn 
6+a(n~l)' 


Ch.  VII]  Problems  251 

17.  Draw  figures,  approximately  to  scale,  showing  the 
paths  of  the  rays  of  light,  and  the  positions  of  the  images 
formed  when  a  luminous  object  is  placed  at  a  distance  of 
(a)  1  inch,  (6)  6  inches  from  a  convex  lens  of  focal  length 
2  inches. 

18.  An  object  is  placed  8  inches  from  a  thin  convex  lens, 
and  its  image  is  formed  24  inches  on  the  other  side  of  the  lens. 
If  the  object  were  moved  nearer  the  lens  until  its  distance 
was  4  inches,  where  would  the  image  be? 

Ans.  Virtual  image,  1  foot  from  lens. 

19.  A  virtual  image  of  an  object  30  cm.  from  a  thin  lens  is 
formed  on  the  same  side  of  the  lens  at  a  distance  of  10  cm. 
from  it.    Find  the  focal  length  of  the  lens. 

Ans.  /=  ~  15  cm. 

20.  Light  converging  towards  a  point  M  on  the  axis  of 
a  lens  is  intercepted  and  focused  at  a  point  M'  on  the  same 
side  of  the  lens  as  M.  The  distances  of  M  and  M'  from  the 
lens  are  5  cm.  and  10  cm.,  respectively.  Find  the  focal 
length  of  the  lens.  Ans.  /=  —  10  cm. 

21.  A  far-sighted  person  can  see  distinctly  only  at  a  dis- 
tance of  40  cm.  or  more.  How  much  will  his  range  of  dis- 
tinct vision  be  increased  by  using  spectacles  of  focal  length 
+32  cm.? 

Ans.  The  spectacles  will  enable  him  to  see  distinctly 
objects  as  near  to  his  eye  as  17.78  cm.,  so  that  his  range  of 
distinct  vision  will  be  increased  by  22.22  cm. 

22.  The  projection  lens  of  a  lantern  has  a  focal  length  of 
one  foot.  If  the  screen  is  1024  feet  away,  how  far  back  of 
the  lens  must  the  glass  slide  be  placed?        Ans.  1024/1023  ft. 

23.  An  engraver  uses  a  magnifying  glass  of  focal  length 
+4  inches,  holding  it  close  to  the  eye.  At  what  distance 
must  the  lens  be  from  the  work  so  that  the  magnification 
may  be  fourfold?  Ans.  3  inches. 

24.  Assuming  that  the  optical  system  of  the  eye  is  equiva- 
lent to  a  thin  convex  lens  of  focal  length  15  mm.,  what  will 


252  Mirrors,  Prisms  and  Lenses  [Ch.  VII 

be  the  size  of  the  retinal  image  of  a  child  1  meter  high  at  a 
distance  of  15  meters  from  the  eye?  Ans.  1  mm. 

25.  A  millimeter  scale  is  placed  at  a  distance  of  84  cm. 
in  front  of  a  convex  lens,  and  it  was  found  that  10  mm.  of 
the  scale  corresponded  to  29  mm.  of  its  real  inverted  image. 
Find  the  focal  length  of  the  lens.  Ans.  /=  +62.5  cm. 

26.  If  X,  X'  and  Y,  Y'  are  two  pairs  of  conjugate  points 
on  the  axis  of  an  infinitely  thin  lens,  and  if  the  lens  is  mid- 
way between  X  and  Y',  show  that  it  is  also  midway  be- 
tween X'  and  Y. 

27.  M  and  M'  are  a  pair  of  conjugate  axial  points  with 
respect  to  an  infinitely  thin  lens  whose  optical  center  is  at 
a  point  designated  by  A.  Show  that  when  the  lens  is  shifted 
from  A  to  a  point  B  such  that  MB=AM',  the  points  M 
and  M'  will  be  conjugate  to  each  other  with  respect  to  the 
lens  in  this  new  position. 

28.  Given  the  positions  of  the  focal  points  F,  F'  of  an 
infinitely  thin  lens,  show  how  to  construct  the  image-point 
M'  conjugate  to  an  axial  object-point  M.  Draw  diagrams 
for  convex  and  concave  lenses. 

29.  At  the  optical  center  (A)  of  a  thin  lens  erect  a  per- 
pendicular to  the  axis  of  the  lens,  and  take  a  point  L  on 
this  perpendicular  such  that  AL=/,  where  /  denotes  the 
primary  focal  length.  Through  A  draw  a  line  AP  in  such 
a  direction  that  ZF'AP  =  45°,  where  F'  designates  the  sec- 
ondary focal  point  of  the  lens.  Take  a  point  M  on  the  axis 
of  the  lens,  and  draw  the  straight  line  ML  meeting  the 
straight  line  AP  in  a  point  S.  If  M'  designates  the  foot  of 
the  perpendicular  let  fall  from  S  to  the  axis  of  the  lens, 
show  that  M,  M'  are  a  pair  of  conjugate  axial  points.  Draw 
two  diagrams,  one  for  a  convex  and  the  other  for  a  concave 
lens. 

30.  Derive  the  image-equations  in  the  case  of  an  infinitely 
thin  lens  in  the  form :  l/u'  =  \ju-\- 1//,  y'/y  =  u'\u. 

31.  Show  that  the  focal  points  of  an  infinitely  thin  lens 
are  at  equal  distances  on  opposite  sides  of  the  lens. 


Ch.  VII]  Problems  253 

32.  A  candle  is  placed  at  a  distance  of  2  meters  from  a 
wall,  and  when  a  lens  is  placed  between  the  candle  and  the 
wall  at  a  distance  of  50  cm.  from  the  candle,  a  distinct  image 
of  the  latter  is  cast  upon  the  wall.  Find  the  focal  length  of 
the  lens  and  the  magnification  of  the  image. 

Ans.  /=37.5  cm.;  image  is  3  times  as  large  as  object. 

33.  The  distance  between  a  real  object  and  its  real  image 
in  an  infinitely  thin  lens  is  32  inches.  If  the  image  is  3  times 
as  large  as  the  object,  find  the  position  and  focal  length  of 
the  lens. 

Ans.  The  lens  is  a  convex  lens  of  focal  length  6  inches 
placed  between  object  and  image  at  a  distance  of  8  inches 
from  the  object. 

34.  When  an  object  is  held  at  a  distance  of  6  cm.  from 
one  face  of  a  thin  lens,  the  image  of  the  object  formed  by 
reflection  in  this  face  is  found  to  lie  in  the  same  plane  as  the 
object.  If  the  object  is  placed  at  a  distance  of  20  cm.  from 
the  lens,  the  image  produced  by  the  lens  is  inverted  and  of 
the  same  size  as  the  object.  The  lens  is  made  of  glass  of 
index  1.5.    Find  the  radii  of  the  two  surfaces. 

Ans.  The  lens  is  a  convex  meniscus  of  radii  6  and  |4  cm. 

35.  In  a  magic  lantern  the  image  of  the  slide  is  thrown 
upon  a  screen  by  means  of  a  thin  convex  lens.  Show  that 
the  adjustment  for  focusing  is  always  possible  provided 
that  the  distance  from  the  slide  to  the  screen  is  not  less 
than  4  times  the  focal  length  of  the  lens,  and  provided  that 
the  lens  can  move  in  its  tube  to  a  distance  from  the  slide 
equal  to  twice  the  focal  length. 

36.  A  person  holds  a  lens  in  front  of  his  eye  and  ob- 
serves that  by  reflection  at  the  nearer  surface  an  object 
which  is  6  feet  from  the  lens  appears  upright  and  diminished 
to  one-twentieth  of  its  height.  Looking  through  the  lens 
at  an  object  on  the  other  side  6  feet  from  the  lens,  its  image 
is  inverted  and  diminished  in  height  to  one-tenth.  The 
lens  is  a  glass  lens  of  index  1.5.  Find  the  radii  of  its  sur- 
faces. Ans.  A  double  convex  lens  of  radii  ||  and  44  ft. 


254  Mirrors,  Prisms  and  Lenses  [Ch.  VII 

37.  How  far  from  a  lens  must  an  object  be  placed  so 
that  its  image  will  be  erect  and  half  as  high  as  the  object? 

Ans.  The  object  must  be  in  the  second  focal  plane  of  the 
lens.  (Draw  diagram  showing  construction  of  image  for 
convex  lens  and  also  a  diagram  for  concave  lens.) 

38.  How  far  from  a  thin  lens  must  an  object  be  placed 
so  that  its  image  will  be  inverted  and  half  as  high  as  the 
object?  Draw  two  diagrams,  showing  construction  of  image 
for  convex  lens  and  for  concave  lens. 

Ans.  If  the  optical  center  of  the  lens  and  the  primary 
focal  point  are  designated  by  A  and  F,  respectively,  and  if 
the  axial  point  of  the  object  is  designated  by  M,  then 
AM  =  3AF. 

39.  An  object  is  to  be  placed  in  front  of  a  convex  lens  of 
focal  length  18  inches  in  such  a  position  that  its  image  is 
magnified  3  times.  Find  the  two  possible  positions,  and 
draw  diagram  for  each  position  showing  the  construction 
of  the  image. 

Ans.  If  image  is  inverted,  object  must  be  2  ft.  from  lens; 
if  it  is  erect,  object  must  be  1  ft.  from  lens. 

40.  In  the  preceding  example  if  the  lens  were  concave, 
where  would  the  object  have  to  be? 

Ans.  The  object  would  be  virtual,  at  a  distance  of  1  ft. 
from  the  lens  for  an  erect  image,  and  at  a  distance  of  2  ft. 
for  an  inverted  image. 

41.  A  person  can  see  distinctly  at  a  distance  of  1  foot, 
and  he  finds  that  when  he  holds  a  certain  lens  close  to  his 
eye  small  objects  are  seen  distinctly  and  magnified  6  times. 
Find  the  focal  length  of  the  lens.  Ans.  /=  +2.4  inches. 

42.  Derive  the  Newtonian  formula  x.x' =  — f2  for  a  lens. 

43.  A  convex  lens  is  used  to  produce  an  image  of  a  fixed 
object  on  a  fixed  screen.  Show  that,  in  general,  there  will 
be  two  possible  positions  of  the  lens,  and  prove  that  the 
height  of  the  object  is  the  geometrical  mean  between  the 
heights  of  the  two  images. 

44.  A  copper  cent  is  19  mm.  in  diameter  and  a  silver 


Ch.  VII  Problems  255 

half  dollar  is  30.4  mm.  in  diameter.  How  far  from  a  con- 
vex lens  of  focal  length  10  cm.  must  the  smaller  coin  be 
placed  so  that  its  image  in  the  lens  will  be  just  the  size  of 
the  larger  one? 

Ans.  It  must  be  placed  in  front  of  the  lens  at  a  distance 
of  either  16.25  cm.  or  3.75  cm. 

45.  What  must  be  the  radius  of  the  curved  surface  of  a 
thin  plano-convex  lens  made  of  glass  of  index  1.5  which 
will  give  a  real  image  of  an  object  placed  2  cm.  in  front  of 
the  lens  and  magnified  3  times?  Ans.  9  mm. 

46.  Find  the  magnification  of  a  convex  lens  of  focal 
length  0.2  inch  for  an  eye  whose  distance  of  most  distinct 
vision  is  14  inches.  Ans.  71  times. 

47.  An  object  is  placed  in  front  of  a  convex  lens  at  a  dis- 
tance from  it  equal  to  1.5  times  the  focal  length.  Find  the 
linear  magnification.  If  the  object  is  removed  to  twice  this 
distance,  what  will  be  the  magnification?  Ans.   -  2;  —  |. 

48.  An  object  5  cm.  high  is  placed  12  cm.  in  front  of  a 
thin  lens  of  focal  length  8  cm.  Find  the  position,  size  and 
nature  of  the  image  (a)  for  a  convex  lens,  and  (6)  for  a 
concave  lens;  and  draw  accurate  diagram  for  each  case. 

Ans.  (a)  Real,  inverted  image,  10  cm.  high,  24  cm.  from 
lens;  (6)  Virtual,  erect  image,  2  cm.  high,  4.8  cm.  from 
lens. 

49.  When  an  object  is  placed  at  a  point  R  on  the  axis  of 
a  thin  lens  of  focal  length  /,  the  image  is  erect,  and  when 
the  object  is  moved  to  a  point  S  the  image  is  the  same  size 
as  before  but  inverted;  show  that 

m 
where  m  is  a  positive  number  denoting  the  value  of  the 
ratio  of  the  size  of  the  image  to  that  of  the  object. 

50.  A  screen,  placed  at  right  angles  to  the  axis  of  a  thin 
lens  of  focal  length  /,  receives  the  image  of  a  small  object. 
If  the  image  is  20  times  as  large  as  the  object,  show  that 
the  distance  of  the  screen  from  the  lens  is  equal  to  21/. 


256  Mirrors,  Prisms  and  Lenses  [Ch.  VII 

51.  Given  a  convex  lens,  a  concave  lens,  a  concave  mirror 
and  a  convex  mirror,  each  of  focal  length  20  cm.  An  object 
is  placed  in  front  of  each  in  turn  at  distances  of  40,  20  and 
10  cm.  Draw  diagrams  showing  the  construction  of  the 
image  for  each  lens  and  each  mirror  and  for  each  of  the 
three  given  positions  of  the  object;  and  find  the  position 
and  character  of  the  image  in  each  case. 

52.  A  plane  mirror  is  placed  anywhere  behind  a  convex 
lens  with  its  plane  at  right  angles  to  the  axis  of  the  lens. 
A  needle  is  set  up  perpendicular  to  the  axis  in  the  primary 
focal  plane  of  the  lens.  Show  that  the  image  of  the  needle 
produced  by  rays  that  have  passed  twice  through  the  lens 
will  lie  also  in  the  primary  focal  plane  and  will  be  of  the 
same  size  as  the  object  but  inverted. 

53.  An  object  is  placed  in  front  of  a  thin  convex  lens  at 
a  distance  a  from  it  not  greater  than  twice  its  focal  length  /; 
and  a  plane  mirror  is  adjusted  in  the  secondary  focal  plane 
of  the  lens.  Show  that  a  real  image  formed  by  rays  which 
have  passed  twice  through  the  lens  will  be  formed  at  a  dis- 
tance b  in  front  of  the  lens;  and  that  f=(a-\-b)/2.  Show 
also  that  the  image  is  of  the  same  size  as  the  object  but  in- 
verted. Draw  a  diagram  showing  the  construction  of  the 
image. 

54.  A  convex  lens  of  focal  length  10  cm.  is  placed  at  a 
distance  of  2  cm.  in  front  of  a  plane  mirror  which  is  per- 
pendicular to  the  axis  of  the  lens.  Where  must  an  eye  be 
placed  in  front  of  the  lens  so  that  it  may  see  its  own  image 
by  means  of  rays  which,  after  having  traversed  the  lens 
twice,  return  into  the  eye  as  bundles  of  parallel  rays? 

Ans.  3.75  cm.  from  the  lens. 

55.  A  thin  convex  lens  of  focal  length  10  inches  is  placed 
in  front  of  a  concave  mirror  of  focal  length  5  inches.  The 
distance  between  the  lens  and  the  mirror  is  10  inches.  An 
object  is  placed  in  front  of  the  lens  at  any  distance  from  it. 
Show  that  its  image  formed  by  rays  which  have  passed 
twice  through  the  lens  will  lie  at  an  equal  distance  from  the 


Ch.  VII]  Problems  257 

lens  on  the  other  side  of  it,  and  that  it  will  be  of  the  same 
size  as  the  object  but  inverted. 

56.  A  thin  convex  lens  of  focal  length  12  inches  is  placed 
12  inches  in  front  of  a  concave  mirror  of  focal  length  8  inches. 
An  object  is  placed  3  inches  in  front  of  the  lens.  Show  that 
its  image  formed  by  rays  which  have  passed  twice  through 
the  lens  is  in  the  same  plane  as  the  object  and  of  the  same 
size,  but  inverted. 

57.  The  focal  length  of  a  thin  symmetric  double  concave 
lens  made  of  glass  of  index  1.5  is  five  inches.  A  luminous 
point  lies  on  the  axis  so  far  away  that  it  may  be  considered 
as  being  at  infinity.  Prove  that  its  image  formed  by  rays 
which  are  reflected  at  the  first  surface  is  2.5  inches  in  front 
of  the  lens;  the  image  formed  by  rays  which  are  refracted 
twice  at  the  first  surface  and  reflected  once  at  the  second 
surface  is  on  the  other  side  of  the  lens  at  a  distance  of  1.25 
inches  from  it;  and,  finally,  the  image  formed  by  rays  which 
after  being  reflected  twice  at  the  second  surface  have  emerged 
again  into  the  surrounding  air  is  0.5  inch  from  the  lens  on 
the  side  away  from  the  source. 

58.  A  concave  mirror,  of  radius  r,  has  its  center  at  the 
optical  center  of  a  thin  lens,  of  focal  length  /,  and  the  axes 
of  lens  and  mirror  are  in  the  same  straight  line.  Rays  com- 
ing from  an  axial  object  point  at  a  distance  u  from  the  lens 
traverse  the  lens  and  after  being  reflected  at  the  mirror 
pass  through  the  lens  again  and  emerge  from  it  as  a  bundle 
of  rays  parallel  to  the  axis.    Prove  that 

W=o. 

u    r    f 


CHAPTER  VIII 

CHANGE    OF    CURVATURE    OF    THE    WAVE-FRONT    IN    REFLEC- 
TION AND  REFRACTION.      DIOPTRY  SYSTEM 

99.  Concerning  Curvature  and  its  Measure. — Since  the 
rays  or  lines  of  advance  of  the  light-waves  are  always  at 
right  angles  to  the  wave-surface  (§  7),  one  way  of  investi- 
gating the  procedure  of  light  is  to  study  the  form  of  the 
wave-surface;  for,  in  general,  the  effect  of  reflection  and  re- 
fraction will  be  to  produce  an  abrupt  change  of  curvature 
of  the  wave-front.  In  this  method  attention  is  concen- 
trated primarily  on  the  wave-surface  rather  than  on  the 
rays  themselves;  but  in  reality  the  only  difference  between 
it  and  the  ray-method  consists  in  a  new  point  of  view,  which 
may,  however,  be  serviceable.  Thus,  when  a  plane  wave 
is  incident  on  a  lens,  the  wave-front  on  emergence  will  no 
longer  be  plane  but  curved  in  such  fashion  that  the  light- 
waves either  converge  to  or  diverge  from  a  point  in  the  second 
focal  plane  of  the  lens.  The  effect  of  the  lens  or  optical 
system  is  to  imprint  a  new  curvature  on  the  wave-front, 
and  if  the  change  of  curvature  which  is  thus  produced  can 
be  ascertained,  the  final  form  of  the  wave  can  be  determined 
by  mere  algebraic  addition  of  the  initial  and  impressed 
curvatures.  It  will  be  necessary,  however,  to  explain  pre- 
cisely what  is  meant  by  this  term  curvature  and  how  it  is 
measured. 

In  passing  along  an  arc  of  a  plane  curve  from  a  point  A 
(Fig.  131)  to  a  point  B,  the  total  curvature  of  the  arc  AB  is 
the  change  of  direction  of  the  curve  between  A  and  B,  which 
is  evidently  measured  by  the  angle  between  the  tangents 
to  the  curve  at  these  two  places.  This  angle  is  equal  to 
the  angle  at  O  between  the  normals  AO  and  BO  which  are 

258 


99] 


Curvature  of  an  Arc 


259 


perpendicular  to  the  tangents  at  A  and  B.  The  mean  curva- 
ture between  A  and  B  is  the  change  of  this  angle  per  unit 
length  of  the  arc  AB.  If,  therefore,  the  length  of  the  arc 
AB  is  denoted  by  a  and  the  magnitude  of  the  angle  BOA 


Fig.  131. — Mean  curvature  of  arc  AB  measured  by 
<P/a,  where  a-denotes  length  of  arc  and  <f>  denotes 
angle  between  the  normals  AO  and  BO. 

by  <p,  the  mean  curvature  between  A  and  B  is  equal  to  (pja. 
And  the  limiting  value  of  this  quotient  when  the  point  B  is 
infinitely  near  to  A  is  the  measure  of  the  actual  curvature 
at  the  point  A  or,  as  we  say,  the  curvature  at  A.  If  the  curva- 
ture at  A  is  denoted  by  the  capital  letter  R,  then  R  is  equal 
to  the  Umiting  value  of  <p/a  when  the  arc  a  is  indefinitely 
small. 

In  Fig.  132  the  point  B  is  supposed  to  be  infinitely  near 
to  A;  and  the  point  of  intersection  C  of  the  normals  drawn 


260 


Mirrors,  Prisms  and  Lenses 


[§99 


to  the  two  contiguous  points  A,  B  on  the  curve  passing 
through  these  two  points  is  called  the  center  of  curvature 
of  the  curve  at  the  point  A;  the  circle  described  in  the 
plane  of  the  curve  around  this  point  C  as  center  with  radius 


Fig.  132,  a  and  b. — Curvature  of  arc  BAB  at  point  A  midway  between 
B  and  B  is  measured  by  the  sagitta  AD.    (a)  Convex,  (b)  Concave  arc. 

r= AC,  which  will  coincide  with  the  given  curve  throughout 
the  infinitely  small  arc  AB,  is  called  the  circle  of  curvature 
and  its  radius  r  is  called  the  radius  of  curvature  at  the  point 
A.  Now  since  by  definition  the  angle  <p  is  equal  to  the  arc 
BA  divided  by  the  radius  r,  that  is,  since  <p  =a/r,  the 
curvature  at  A  is  equal  to  l/r;  that  is,  the  curvature  at  any 
point  on  a  curve  is  equal  to  the  reciprocal  of  the  radius  of  curva- 
ture at  that  point,  or 

r 
The  sign  of  the  curvature  is  the  same  as  that  of  the  radius 
of  curvature.     Accordingly,  if  the  surface  is  convex  with  re- 
spect to  the  incident  light,  the  curvature  is  to  be  counted  as  posi- 
tive, in  accordance  with  our  previous  usage  in  this  respect. 


§  99]  Measure  of  Curvature  261 

Thus,  for  example,  when  spherical  waves  spread  out  from 
a  point-source,  the  wave-front  at  any  instant  is  concave 
and  its  curvature  is  reckoned,  therefore,  as  negative.  If  a 
convex  lens  is  interposed  at  a  distance  from  the  point-source 
greater  than  its  focal  length,  the  light-waves  will  thereby 
be  converged  to  a  focus  on  the  other  side  of  the  lens  whence 
they  will  ultimately  diverge  again.  While  the  wave-front 
is  advancing  from  the  lens  to  the  focus,  its  curvature  is  pos- 
itive; at  the  focus  itself  the  wave-front  collapses  into  a  point, 
the  curvature  of  the  wave  at  this  place  being  infinite;  and 
beyond  the  focus  the  curvature  becomes  negative.  As  long 
as  the  wave  does  not  undergo  any  reflection  or  refraction, 
its  curvature  varies  continuously;  whereas  a  sudden  change 
of  curvature  is  imprinted  on  the  wave  when  there  is  a  transi- 
tion from  one  medium  to  another. 

Another  method  of  measuring  the  curvature  of  a  small 
arc  BB  (Fig.  132)  is  in  terms  of  its  bulge  AD,  where  the 
points  designated  by  A  and  D  are  the  middle  points  of  the 
arc  and  its  chord.  If  the  points  A  and  B  are  so  close  to- 
gether that  they  may  be  regarded  as  lying  on  the  circle  of 
curvature  corresponding  to  the  point  A,  the  ordinate  DB  =  h 
will  be  a  mean  proportional  between  the  two  segments  into 
which  the  diameter  of  the  circle  is  divided  by  the  point  D, 
so  that  we  have  the  proportion : 

XD:h  =  h:(2r~AD). 
Since  the  segment  AD  is  always  very  small  in  comparison 
with  the  diameter  of  the  circle  of  curvature,  only  a  vanish- 
ingly  small  error  will  be  introduced  by  writing  2r  in  place 
of  (2r— AD)  in  the  above  proportion.     Thus,  we  obtain: 

h2 


or  since  R  =  1/r, 


AD=2r' 


h2 

ad- |  a 


If  the  arc  BB  is  not  infinitely  small,  this  equation  contains 
a  certain  error  which  is  more  and  more  negligible  in  pro- 


262 


Mirrors,  Prisms  and  Lenses 


99 


portion  as  the  arc  is  taken  smaller  and  smaller.  For  a  small 
arc,  therefore,  we  may  say  that  the  segment  AD  is  propor- 
tional to  the  curvature  (R)  at  the  point  A,  and  hence  it 
may  be  said  to  measure  the  curvature  at  this  place.  This 
segment  AD  was  called  by  Kepler  the  sagitta  of  the  arc 
BB  because  of  its  resemblance  to  an  "arrow"  on  a  bow. 


Fig.  133. — Curvatures  of  arcs 
BAB  and  BKB  are  in  same 
ratio  as  their  sagittae  AD 
and  KD. 


Fig.  134. — Curvatures  of 
arcs  AP  and  AQ  in 
same  ratio  as  their 
sagittae  VP  and  VQ. 


Obviously,  it  does  measure  the  bulge  or  "  sag  "  of  the  curve 
at  A.  In  Fig.  133,  where  the  straight  line  BDB  is  the  com- 
mon chord  of  the  small  arcs  BAB  and  BKB,  the  curvatures 
at  A  and  K  are  evidently  in  the  ratio  of  AD  to  KD.  Or, 
again,  consider  Fig.  134,  where  the  two  arcs  AP  and  AQ 
have  a  common  tangent  at  A.  If  on  this  tangent  a  point  V 
is  taken  very  close  to  A,  and  if  through  V  a  straight  line  is 
drawn  perpendicular  to  AV  intersecting  the  two  arcs  in 


99] 


Spherometer  and  Lens-Gauge 


263 


the  points  designated  by  P  and  Q,  the  curvatures  at  A  will 
be  in  the  ratio  of  VP  to  VQ. 

In  many  optical  problems  (as  has  been  explained  in  the 
last  two  chapters)  we  are  concerned  only  with  a  very  small 
portion  of  the  reflecting  and 
refracting  surface  (case  of 
paraxial  rays),  and  under 
such  circumstances  it  is 
especially  convenient  and 
simple  to  measure  the  curv- 
atures of  the  wave-fronts 
before  and  after  refraction 
or  reflection  and  the  curva- 
tures of  the  mirrors  or 
lenses  by  means  of  their 
sagittae.  In  fact,  the  ordi- 
nary method  of  determining 
the  curvature  of  an  optical 
surface  with  an  instru- 
ment called  a  spherometer 
(Fig.  135)  consists  essen- 
tially in  employing  a  mi- 
crometer screw  to  measure  the  sagitta  of  the  arc  whose 
chord  is  equal  to  the  diameter  of  the  circle  circumscribed 
about  the  equilateral  triangle  formed  by  the  conical  points 
of  the  tripod  which  supports  the  instrument  on  the  curved 
surface  to  be  measured.  The  simple  lens-gauge  (Fig.  136) 
used  by  opticians  to  measure  the  power  of  a  spectacle  lens 
is  based  on  the  same  principle.  In  size  and  external  ap- 
pearance it  resembles  a  watch,  except  that  on  its  lower  side 
it  has  three  metallic  pins  projecting  from  it  in  parallel  lines 
which  all  lie  in  a  plane  parallel  to  the  face  of  the  gauge.  The 
two  outer  pins  are  stationary  and  symmetrically  placed  so  that 
when  the  instrument  is  held  in  a  vertical  plane  with  the  pins 
pointing  downwards,  the  straight  line  BB  (Fig.  132)  joining 
the  conical  points  of  the  outer  pins  is  horizontal;  whereas 


Fig.  135. — Spherometer. 


264 


Mirrors,  Prisms  and  Lenses 


[§99 


the  other  pin  which  is  midway  between  the  two  outer  ones 
is  capable  of  being  pushed  upwards  by  a  slight  pressure  so 
that  its  tip  A  which  left  to  itself  falls  a  little  below  the 
straight  line  BB  can  be  made  to  ascend  a  little  above  this 

line.  The  vertical  dis- 
placement of  the  tip  A  of 
the  middle  pin  above  or 
below  the  level  of  the 
chord  BB,  which  is  equal 
to  the  sagitta  of  the  arc 
BAB  whose  curvature  is 
to  be  measured,  is  regis- 
tered on  the  dial  (see 
§  108)  by  the  angular 
movement  of  a  light  hand 
or  pointer  with  which  the 
movable  pin  is  connected. 
If  the  circle  is  drawn 
which  passes  through  the 
end-points  of  the  three 
pins  B,  A  and  B,  the 
diameter  drawn  through  A  will  bisect  the  chord  BB  at  a 
point  D;  and  since  the  products  of  the  segments  of  two 
intersecting  chords  of  a  circle  are  equal,  we  obtain  imme- 
diately : 

AD  (2r-AD)=h2, 
where  r  denotes  the  radius  of  the  circle  and  2h  =  chord  BB. 
Hence,  exactly  as  above,  we  obtain  here  also: 

h2 
AD=  —  ,  approximately; 

thus  proving  again  that  the  sagitta  AD  is  proportional  to 
the  curvature  l/r  =  R.  In  using  the  lens-gauge  care  must 
be  taken  to  see  that  the  plane  of  the  instrument  is  not  tilted 
out  of  the  vertical,  and  this  is  one  reason  why  a  spherometer 
is  more  accurate.  On  the  other  hand,  the  lens-gauge,  be- 
sides being  more  handy  and   convenient,   possesses  a  de- 


Fig.  136. — Lens-gauge. 


100] 


Plane  Refracting  Surface 


265 


cided  advantage  over  a  spherometer  supported  on  a  tripod 
by  reason  of  the  fact  that  it  can  be  used  to  measure  the 
curvatures  in  different  meridians  of  a  non-spherical  surface 
of  revolution,  for  example,  the  curvatures  of  the  normal 
sections  (§  111)  of  a  cylindrical  or  of  a  toric  surface  (§  112). 
How  the  lens-gauge  is  graduated  will  be  explained  presently 
(§  108). 

100.  Refraction  of  a  Spherical  Wave  at  a  Plane  Surface. 
— The  whole  duty  of  an  optical  system,  therefore,  whether 


z 

:-•'.'■'-        ■'  ^^^ 

B 

Pr<T*.    •   '  • 

AIR            ^</ 

%    ■ :    *  GLASS 

a\\  ;  ;  • 

>':"}  V: A       *  ."    .         -       . 

i-     \.\               -       ' 

:■■•'  \  -\  '  ' 
■■■'■: 'A  \    '   ■  • 

■;'■•'    I    >      •    * 

M'\         M\                         A 

".  K"  ,'j  .       ,. 

•*               \ 
\ 

:■'--/    i  ' v  •  . 

&  •'■•/•  '         •   •      ' 

r-. :  ■//*•' 

■    J  i     .  ' 
■'•"•  /  /  ■       '  • 
:••:/  /.  ..  •  , 

'■'/ ' '  • '  • «     ■    ' 

B 

fe>sv 

Z 

3fc 


Fig.   137,  a. — Divergent  spherical    waves    refracted    at    plane 
surface  from  air  to  glass. 

it  be  a  single  lens  or  mirror  or  a  combination  of  such  parts 
is  to  imprint  a  certain  curvature  on  the  surface  of  the  in- 
cident wave;  and  if  we  consider  only  such  portions  of  the 
wave-fronts  as  lie  very  close  to  the  axis  of  symmetry  of 
the  instrument,  it  is  evident  that  this  method  of  investi- 
gating the  change  of  curvature  that  is  produced  in  the 
wave-front  at  the  point  where  the  axis  meets  it  should  lead 
to  precisely  the  same  results  as  have  been  found  already 


266 


Mirrors,  Prisms  and  Lenses 


100 


in  the  corresponding  problems  concerning  the  reflection 
and  refraction  of  paraxial  rays.  In  fact,  according  to  this 
method,  these  results  should  be  found  to  apply  not  merely 
to  the  case  when  the  reflecting  and  refracting  surfaces  are 
plane  or  spherical,  but  equally  also  to  the  more  general 
case  when  these  surfaces  have  any  form  whatever,  provided 
they  are  symmetrical  around  the  optical  axis. 


Z 

\1: 

AIR 

/! 
/ 1 

•V^^              GLASS 

■■'■■  ■  \  \ 

ft 

;:•.••    •  \   \ 

i 

/  i 

i 
i 
i 

v      '  ■  .           N    "     ^V^ 

•    •  -  ;           n           -A. 

■■:•'-.     •               \           ^ 

J 

A  •  •        .              /M 

^V 

\  \ 

\  \ 

:•...••    •         /       ^s^' 

\  \ 

•   s      ^-^ 

\\ 

■■■.-•    /  ^y^ 

\\ 

>■  '• .  /  ^y^  •     -  - 

\\ 

!•:  -.  s^s^ 

M 

B:  -;•/.:,    ' 

z 

Fig.  137,  b. 


-Convergent  spherical  waves  refracted  at  plane  surface  from 
air  to  glass. 


We  shall  begin  by  investigating  the  simple  case  of  the 
refraction  of  a  spherical  wave  at  a  plane  surface. 

In  the  diagrams  (Fig.  137,  a,  b,  c,  and  d)  the  straight  line  ZZ 
represents  the  trace  in  the  plane  of  the  paper  of  a  plane  re- 
fracting surface  separating  two  media  of  indices  n,  n' . 
Around  the  point  M  as  center  spherical  waves  are  supposed 
to  be  advancing  in  the  first  medium  (n)  towards  the  refract- 
ing surface,  and  at  a  certain  instant  when  the  disturbance 


100] 


Plane  Refracting  Surface 


267 


has  begun  to  affect  a  point  B  on  this  surface  the  incident 
wave  will  be  represented  in  the  plane  of  the  figure  by  the 
circular  arc  BJB  described  around  M  as  center  with  radius 
equal  to  BM;  the  point  designated  by  J  lying  on  the  arc 
midway  between  its  two  ends  B,  B,  so  that  the  straight 
line  MJ  is  the  perpendicular  bisector  at  A  of  the  chord  BB. 
The  two  points  M,  J  will  be  found  to  lie  always  on  opposite 


GLASS 


Fig.   137,  c. — Divergent  spherical  waves  refracted  at  plane 
surface  from  glass  to  air. 

sides  of  the  refracting  plane.  In  Fig.  137,  a  and  c,  where  the 
point  M  is  shown  as  lying  in  front  of  the  surface  ZZ,  the 
arc  BJB  is  indicated  by  a  dotted  line,  because  it  marks  the 
position  which  the  incident  wave-front  would  have  had 
if  the  refracting  surface  had  not  been  interposed.  But 
the  waves  travel  faster  in  the  rarer  medium  (air)  than  in 
the  denser  medium  (glass);  and,  consequently,  the  vertex 
of  the  refracted  wave-front  instead  of  being  at  the  point  J 


268 


Mirrors,  Prisms  and  Lenses 


[§100 


on  the  axis  will  be  at  a  point  K  on  this  line,  and  therefore 
the  position  of  the  refracted  wave-front  at  the  moment 
when  the  disturbance  arrives  at  B  will  be  represented  by 
the  arc  BKB  of  a  circle  whose  center  is  at  a  point  M' 
on  the  axis.  If,  for  example,  the  waves  are  refracted  from 
air  to  glass,  that  is,  if  n'>nt  the  velocity  v  in  the  first  me- 


z 

GLASS     •' 

/ 
*    '     .    .''1. 
■   '•/ 
".■.  *       •  / 

"        .  1  • 

■A 
if}. 

1  !■'■ 
1 "' ' 

B 

V.                   AIR 

\  \ 

\       ^ 

«!■■ 

■.»■  ..»v 

J  : 

v\'J 

\\- 
■  x\ 

A                         /M'     ^"M 

'    J^$ 

B 

* 

Z 

3C 


Fig.  137,  d. — Convergent  spherical  waves  refracted  at  plane  surface  from 

glass  to  air. 

dium  will  be  greater  than  the  velocity  v'  in  the  second  me- 
dium, so  that  for  this  case  AK  will  be  shorter  than  AJ,  and 
the  effect  of  the  retardation  will  be  to  flatten  the  wave- 
front,  as  shown  in  Fig.  137,  a  and  b.  On  the  other  hand,  if 
n'<n,  then  v'>v,  so  that  now  AK  will  be  longer  than  AJ, 
and  the  effect  of  the  refraction  will  be  to  increase  the  curva- 
ture or  bulge  of  the  wave-front,  as  shown  in  Fig.  137,  c  and  d. 
Since  (see  §  31) 

AJ:  AK  =  v:  v'  =  n':n, 


101] 


Spherical  Refracting  Surface 


269 


it  follows  that 


.KA  =  n.JA. 


Now  JA  and  KA  are  the  sagittce  (§  99)  of  the  small  arcs 
BJB  and  BKB,  respectively,  and  hence  they  are  propor- 
tional to  the  curvatures  of  these  arcs,  that  is,  to  l/JM  and 
1/KM'.  If  the  point  B  is  infinitely  near  to  A,  we  may  put 
JM  =  AM  =  u,  KM'  =  AM'  =  <-  and  thus  we  obtain: 


u      u 
which  will  be  recognized  as  the  relation  which  we  found 
for  the  refraction  of  paraxial  rays  at  a  plane  surface  (§41). 
101.  Refraction  of  a  Spherical  Wave  at  a  Spherical  Sur- 
face.— Here  the  same  method  is  employed  as  in  the  preced- 


Fig.  138,  a. 


-Divergent  spherical  waves  refracted  at  convex  surface 
from  air  to  glass. 


ing  section.  In  each  of  the  diagrams  (Fig.  138,  a,  b,  c,  d, 
e,  f,  g,  and  h)  the  circular  arc  ZZ  represents  the  trace  in  the 
plane  of  the  paper  of  a  meridian  section  of  the  spherical 
refracting  surface  with  its  vertex  at  A  and  center  at  C.  The 
surface  is  convex  in  Fig.  138,  a,  b,  c,  and  d  and  concave  in 
Fig.  138,  e,  f,  g,  and  h.  The  point  M  on  the  axis  is  the  center 
of  a  system  of  spherical  waves  which  are  advancing  in  the 
first  medium,  of  index  n,  towards  the  refracting  surface. 
In  Fig.  138,  a,  c,  e,  and  g  the  point  M  lies  in  front  of  the  re- 


270 


Mirrors,  Prisms  and  Lenses 


101 


fracting  surface,  whereas  in  Fig.  138,  b,  d,  f,  and  h  this  point 
is  situated  on  the  other  side  of  the  surface.     The  points 


y^T 

>b^l    '  ' 

AIR 

jt/'l' 
f'/r 

■    />£* 

^        GLASS 

/'/  • 
/•■/• 

.  /   ' 

1  • 

a|;K 

•  I  • 

•  \  ■  ■■ 

D 

\\ 

/•          s' 

V  \  - 

'•\\.\  • 

s-  * 

^\\ 

/     ■"" 

' 

NA\ 

Fig.  138,  b. — Convergent  spherical  waves  refracted  at  convex 
surface  from  air  to  glass. 

marked  B,  B  are  two  points  on  the  arc  ZZ  very  close  to- 
gether but  at  equal  distances  on  opposite  sides  of  the  op- 


glass 


Fig.  138,  c. — Divergent  spherical  waves  refracted  at  convex 
surface  from  glass  to  air. 

tical  axis,  so  that  the  arc  BJB  described  around  M  as  center 
with  radius  equal  to  BM  shows  the  position  of  the  wave- 
front  of  the  incident  waves  at  the  instant  when  the  disturb- 


101] 


Spherical  Refracting  Surface 


271 


ance  begins  to  affect  the  points  B,  B;  the  point  where  this 
arc  crosses  the  optical  axis  being  designated  by  J. 


NB^^Z 

.  •  •   •  ■  ::]yi 

*5^^ 

//] 

^^^^^ 

•'•'-;/// 

^^^n,,^ 

I  glass  .  :■ 

•/   // 

X^\.               AIR 

/    /  ' 

VN                   ^^^\ 

/ 

l 

• ..  a| 

J  J 

\ 

K 

D                    C              /'M' 

^^>M 

■  •  :'% 

\ 

s                             ^r^^ 

\     v  \ 

''                   J^f 

•  •  '* :." 

\    ^  \ 

**                *^*^ 

;  • 

•:\    \\ 

''          ^"'^^ 

•-:\    M 

,'      ^**^ 

:%  \\ 

,'      -*^^ 

x\\ 

^^^ 

Fig.  138,  d. — Convergent  spherical  waves  refracted  at  convex 
surface  from  glass  to  air. 

When  the  waves  enter  the  second  medium,  of  index  n', 
they  will  proceed  with  augmented  or  diminished  speed  ac- 


GLASS 


Fig.  138,  e. — Divergent  spherical  waves  refracted  at 
concave  surface  from  air  to  glass. 

cording  as  n  is  greater  or  less  than  n'.  In  the  diagrams 
Fig.  138,  a,  b,  e,  /,  the  case  is  represented  where  n'  >n;  and 
in  the  diagrams  Fig.  138,  c,  d,  g,  h  the  second  medium  is 


272 


Mirrors,  Prisms  and  Lenses 


[§101 


supposed  to  be  less  highly  refracting  than  the  first  {n'<n). 
The  center  of  curvature  of  the  refracted  waves  will  lie  at 


GLASS 


Fig.  138,  /. — Convergent  spherical  waves  refracted  at  concave 
surface  from  air  to  glass. 

a  point  M'  on  the  axis,  so  that  the  wave-front  in  the  second 
medium  which  passes  through  B,  B  will  be  represented  by 


.  ."V'-.^N 

3^- 

GLASS 

^"''"■y^.- 

AIR 

^*-«* 

""*                j<        .• 

„-*"" 

-""*'''                   y^ 

M'~^-^                        M\^ 

C                .      D 

K!  :• 

J 

A 

***"•**»                      ^\ 

"****»„ 

^v          v  -'■ 

*•/ 

** 

^**^JnJ 

Fig.  138,  g. — Divergent  spherical  waves  refracted  at  con- 
cave surface  from  glass  to  air. 

the  arc  BKB  of  a  circle  described  around  M'  as  center  with 
radius  equal  to  BM';  the  point  where  this  arc  crosses  the 
axis  being  designated  by  K. 

In  each  of  the  diagrams  of  Fig.  138  one  of  the  two  arcs 


101] 


Spherical  Refracting  Surface 


273 


BJB  and  BKB  is  shown  by  a  dotted  line,  because,  on  ac- 
count of  the  interposition  of  the  refracting  surface  ZZ,  the 
part  of  one  or  the  other  of  these  wave-fronts  which  is  com- 
prised between  B,  B  does  not  actually  materialize;  but  this 
circumstance  does  not  in  the  least  affect  the  geometrical 
relations. 

Thus,  during  the  time  the  light  takes  to  go  in  the  first 
medium  from  J  to  A  (or  from  A  to  J),  it  will  travel  in  the 


Fig.  138,  h. — Convergent  spherical  waves  refracted  at 
concave  surface  from  glass  to  air. 


second  medium  from  K  to  A  (or  from  A  to  K).  In  other 
words,  the  optical  lengths  (§  39)  of  the  axial  line-segments 
AJ  and  AK  are  equal,  and  therefore : 

n.AJ  =  n.AK. 

This  shows  how  the  position  of  the  point  IVT  may  be  found, 
for  we  have  only  to  lay  off  on  the  axis  a  piece 

AK=-,AJ, 

n 

and  to  locate  the  point  M'  at  the  place  where  the  perpendic- 
ular bisector  of  the  chord  BK  intersects  the  optical  axis. 
Draw  the  chord  BDB  crossing  the  optical  axis  at  right 


274 


Mirrors,  Prisms  and  Lenses 


[§102 


angles  at  the  point  D;  then,  evidently,  since 

AJ  =  AD+DJ  =  AD-JD,  AE>AD+DK  =  AD-KD, 

we  have:  n(AD-JD)  =  n'(AD-KD). 

Now  recalling  the  fact  that  the  points  B,  B  were  assumed 
to  be  very  close  to  the  vertex  A  of  the  spherical  refracting 
surface,  we  remark  that  the  arcs  whose  summits  are  at  A,  J 
and  K  are  all  very  small;  and  hence  the  segments  AD,  JD 
and  KD  may  be  regarded  as  the  sagittce  of  these  arcs  and 
proportional  to  their  curvatures  (§  99),  viz.,  l/r,  1/u  and 
1/u',  respectively,  where  r  =  AC,  w  =  AM  =  JM,  u'  =  AM' 
=  KM',  approximately.  Introducing  these  values  in  the 
equation  above,  we  obtain  the  characteristic  invariant  re- 
lation for  the  case  of  the  refraction  of  paraxial  rays  at  a 
spherical  surface,  viz., 


(i_lW(I_l,) 

\r      u)         \r     u  / 


in  the  same  form  as  was  found  in  §  78. 

102.  Reflection  of  a  Spherical  Wave  at  a  Spherical  Mir- 
ror.— The  problem  of  reflection  at  a  spherical  mirror  may 


Fig.  139,  a. — Divergent  spherical  waves  reflected  at  convex  mirror. 

be  investigated  in  the  same  way.  In  Fig.  139,  a  and  b, 
the  arcs  BAB,  BJB  and  BKB  represent  the  traces  of  the 
mirror  and  of  the  wave-fronts  of  the  incident  and  reflected 


102] 


Spherical  Mirror 


275 


waves,  respectively.    In  the  case  of  reflection  the  condition 
evidently  is : 

KA  =  AJ, 
because  while  the  incident  wave  advances  along  the  optical 
axis  through  the  distance  A  J  or  J  A,  the  reflected  wave  will 
travel  in  the  opposite  direction  through  an  equal  distance 


Fig.  139,  b. — Divergent  spherical  waves  reflected  at  concave  mirror. 

KA  or  AK.  Therefore  the  center  M'  of  the  reflected  wave 
may  be  found  by  laying  off  AK  =  JA  and  locating  the  point 
where  the  perpendicular  bisector  of  the  chord  KB  inter- 
sects the  axis. 

Here  also  the  segments  AD,  JD  and  KD  are  to  be  re- 
garded as  the  measures  of  the  curvatures  of  the  small  arcs 
BAB,  BJB  and  BKB,  respectively,  and  proportional,  there- 
fore, to  the  reciprocals  of  the  radii  of  curvature,  viz.,  l/r, 
1/m  and  1/V,  where  r  =  AC,  u=AM=JM,  u'=AM'  =  KM' 
in  the  limit  when  the  arcs  are  infinitely  small.  Now 
KD  =  KA+ AD  =  AJ+ AD  =  AD+DJ+ AD, 


276  Mirrors,  Prisms  and  Lenses  [§  103 

that  is, 

JD+KD  =  2AD; 

hence,  substituting  the  symbols  u,  v!  and  r,  we  derive  the 
abscissa-formula  for  the  reflection  of  paraxial  rays  at  a 
spherical  mirror  (§  64),  viz., 

2+1  =  2  . 

u    u      r 
which  may  be  expressed  in  words  by  saying  that  the  curva- 
ture of  the  mirror  is  the  arithmetical  mean  of  the  curvatures 
of  the  incident  and  reflected  waves  at  the  vertex  of  the  mirror; 
that  is, 

R 2-' 

where  U=l/u,  U'  =  l/u'  denote  the  curvatures  of  the  in- 
cident and  reflected  waves,  and  R  =  l/r  denotes  the  curva- 
ture of  the  mirror.  Thus,  for  example,  if  an  incident  plane 
wave  (U  =  0)  is  advancing  parallel  to  the  axis  of  the  mirror, 
the  curvature  of  the  reflected  wave  will  be  twice  that  of 
the  mirror,  and  consequently,  the  center  F  of  the  reflected 
wave-front  will  lie  midway  between  the  vertex  A  and  the 
center  C  of  the  mirror  (§  69). 

Of  course,  the  condition  KA  =  AJ  might  have  been  de- 
rived at  once  from  the  condition  n.AJ  =  n'.XK,  which  was 
found  in  §  101,  by  putting  in  this  equation  n'=—n,  in  ac- 
cordance with  the  general  rule  given  in  §  75. 

103.  Refraction  of  a  Spherical  Wave  through  an  In- 
finitely Thin  Lens. — Since,  as  has  been  shown  (§  89),  a 
homocentric  bundle  of  incident  paraxial  rays  with  its  ver- 
tex at  a  point  M  on  the  axis  of  a  thin  lens  is  transformed 
into  a  homocentric  bundle  of  emergent  rays  with  its  vertex 
at  the  conjugate  point  M',  we  know  that  if  the  waves  are 
spherical  before  traversing  the  lens,  they  will  issue  from  it 
as  spherical  waves,  at  least  in  the  neighborhood  of  the  axis. 

Each  of  the  diagrams  (Fig.  140,  a  and  b)  represents  a 
meridian  section  of  the  lens  which  is  convex  in  one  figure 
and  concave  in  the  other.     As  a  matter  of  fact  the  lens  is 


§103] 


Infinitely  Thin  Lens 


277 


assumed  to  be  infinitely  thin,  and  perhaps  it  is  well  to  call 
particular  attention  to  this  fundamental  consideration,  be- 
cause in  the  diagrams,  in  order  to  exhibit  the  relations  by 
means  of  the  sagittce,  the  lens-thickness  is  shown  very  much 
exaggerated. 


Fig.  140,  a. — Divergent  spherical  waves  refracted  through  thin  convex  lens. 

Take  a  point  Bi  on  the  first  surface  of  the  lens  not  very 
far  from  the  vertex  Ai  of  this  surface,  and  around  the  axial 
object-point  M  as  center  with  radius  equal  to  BiM  describe 
the  circular  arc  BiJBi  which  is  bisected  by  the  axis  of  the 
lens  in  the  point  designated  by  J;  evidently,  this  arc  will 
represent  the  trace  in  the  plane  of  the  diagram  of  the  wave- 
front  of  the  incident  waves  at  the  moment  when  the  dis- 
turbance reaches  Bi.  Now  the  disturbance  which  is  propa- 
gated onwards  from  Bi  will  proceed  across  the  lens  to  a 
point  B2  on  the  second  face  of  the  lens,  and  since  the  lens 
is  supposed  to  be  infinitely  thin,  the  distances  of  Bi,  B2 
from  the  axis  are  to  be  regarded  as  equal,  that  is,  DiBi  = 
D2B2,  where  Di,  D2,  designate  the  feet  of  the  perpendicu- 
lars let  fall  from  Bi,  B2,  respectively,  to  the  axis  of  the 
lens.  If,  therefore,  around  the  point  M'  conjugate  to  M 
an  arc  B2KB2  is  described  with  radius  equal  to  B2M',  which 
is  bisected  by  the  axis  at  the  point  designated  by  K,  this 
arc  will  represent  the  trace  in  the  plane  of  the  diagram  of 
the  wave-front  of  the  emergent  waves  at  the  same  instant 


278 


Mirrors,  Prisms  and  Lenses 


[§103 


that  the  arc  B1JB1  shows  the  wave-front  of  the  incident 
waves. 

With  M,  M'  as  centers  and  with  any  convenient  radii 
describe  also  the  arcs  GH,  SL  intersecting  the  axis  of  the 


Fig.  140,  b. — Divergent  spherical  waves  refracted  through  thin  concave  lens. 

lens  at  G,  S  and  meeting  the  straight  lines  BiM,  B2M',  in 
H,  L,  respectively;  so  that  these  arcs  represent,  therefore, 
successive  positions  of  the  wave-front  before  and  after 
transmission  through  the  lens.  Now  the  optical  length  of 
the  light-path  from  H  to  L  is  equal  to  that  along  the  axis 
of  the  lens  from  G  to  S  (§  39);  and,  hence,  if  n,  n'  denote 
the  indices  of  refraction  of  the  two  media  concerned,  we 
may  write : 

n.HBi+n,.B1B2+n.B2L  =  n.GAi+n,.AiA2+n.A2S; 
and  since 

n(MH+LM')  =n(MG+SM0, 
we  obtain  by  addition  of  these  two  equations : 

n(MB1-fB2M,)+n,.B1B2  =  n(MAi+A2M,)+n,.AiA2. 
Now  MBi  =  MJ,  B2M'  =  KM',  B1B2  =  D1D2; 

and  therefore: 

w(M  J- M  Ai+ KM  -  A2M')  =  n'(Af  A2  -  DiD2) . 


§  104]  Reduced  Distance  279 

Substituting  in  this  equation  the  following  expressions,  viz. : 
MJ-MAi  =  AiM+MJ  =  AJ  =  AiDd-DiJ  =  A1D1-JD1, 
KM'-A2M'  =  KM'+M'A2  =  KA2  =  KD2+D2A2 
=  KD2-A2D2, 
AiA2  =  AiDi+DiD2+D2A2  =  AiDi+DiD2-A2D2, 
we  obtain : 

n(AiDi-JD1+KD2-A2D2)=n,(AiDi-A2D2); 
which  may  be  put  finally  in  the  following  form: 
w(KD2-JDi)  =  (n'-n)  (AiDi-A2D2). 
It  has  been  assumed  here  that  the  lens  is  surrounded  by 
the  same  medium  (n)  on  both  sides,  but  the  same  method 
would  lead  to  a  more  general  formula  for  which  the  initial 
and  final  media  were  different. 

Evidently,  since  the  points  Bi,  B2  are  very  near  the  verti- 
ces Ai,  A2,  the  segments  A1D1,  JDi,  A2D2,  KD2  may  be  re- 
garded as  the  sagittce  of  the  small  arcs  B1A1B1,  B1JB1, 
B2A2B2,  B2KB2,  respectively;  and  since  these  arcs  all  have 
equal  chords,  the  reciprocals  of  the  radii  of  curvature  may  be 
substituted  in  the  equation  above  in  place  of  the  sagittce. 
Accordingly,  if  the  radii  of  the  lens-surfaces  are  denoted 
by  n,  r2,  and  if  we  put  AM  =  JM  =  w,  A2M/  =  KM'=w'J  as 
is  permissible  in  this  case,  we  derive  immediately  the  fa- 
miliar lens-formula  for  the  refraction  of  paraxial  rays  (§  89), 
viz.: 

W      ul  \ri      r2/     f 

where  /  denotes  the  primary  focal  length  of  the  lens. 

104.  Reduced  Distance. — If  P,  Q  designate  the  positions 
of  two  points  lying  both  in  the  same  medium  of  refractive 
index  n,  el  distinction  has  already  been  pointed  out  (see  §  39) 
between  the  actual  or  absolute  distance  of  these  points  from 
each  other  and  the  so-called  " optical  length"  of  the  seg- 
ment PQ  of  the  straight  line  joining  these  points,  which 
is  obtained  by  multiplying  the  absolute  length  by  the  index 
of  refraction  of  the  medium,  and  which  is  equal  therefore 
to  n.PQ.     A  further  distinction,  due  originally  to  Gauss, 


280  Mirrors,  Prisms  and  Lenses  [§  104 

is  to  be  made  now  by  employing  the  term  reduced  distance 
between  P  and  Q  to  mean,  not  the  product,  but  the  quo- 
tient of  the  distance  PQ  by  the  index  of  refraction  of  the  medium 
in  which  the  two  points  P  and  Q  lie;  that  is,  the  reduced  dis- 

PQ 

tance  from  P  to  Q  is  equal  to  — — .     Thus,  for  example,  if 

the  medium  is  glass  of  index  1.5,  and  if  the  distance  PQ  = 
12  inches,  the  optical  distance  or  equivalent  light-path  in 
air  will  be  18  inches,  whereas  the  reduced  distance  will  be 

8  inches.     The  reduced  thickness  of  a  lens  is  c  =  -,  where 

n 

d  =  AiA2  denotes  the  distance  of  the  second  vertex  A2  of 
the  lens  from  the  first  vertex  Ai  and  n  denotes  the  index  of 
refraction  of  the  lens-substance.  The  optical  distance  is 
never  less,  and  the  reduced  distance  is  never  greater,  than 
the  actual  distance.  If  the  medium  is  air  (n=l),  the  op- 
tical distance  and  the  reduced  distance  are  both  equal  to 
the  absolute  distance.  Apparently,  the  first  use  of  the  term 
"reduced  distance"  in  this  sense  in  English  occurs  in 
Pendlebury's  Lenses  and  systems  of  lenses,  treated  after 
the  manner  of  Gauss  (Cambridge,  1884).  A  distinct  ad- 
vantage in  the  direction  of  simplification  is  usually  gained 
in  mathematical  formulation  by  denoting  a  more  or  less 
complex  function  by  a  single  symbol;  and  modern  optical 
writers,  notably  Gullstrand  and  his  disciples  in  Germany, 
have  recognized  the  convenience  of  this  idea  of  "  reduced 
distance"  and  utilized  it  to  express  the  relations  between 
object  and  image  in  their  simplest  forms;  as  we  shall  show 
presently  by  several  examples. 

In  this  connection  the  attention  of  the  student  needs  to 
be  called  to  a  point  which  has  been  alluded  to  before  (see 
§  8),  but  which  is  not  always  clearly  understood.  Although 
two  points  P,  Q  may  be  situated  physically  in  different 
media,  they  may  be  regarded  as  optically  in  the  same  me- 
dium. Thus,  any  point  which  is  on  the  prolongation,  in 
either  direction,  of  the  line-segment  which  represents  the 


§  105]  Refracting  Power  281 

actual  path  of  a  ray  of  light  through  a  certain  medium,  may, 
and  in  fact  generally  must,  be  regarded  as  a  point  belonging 
to  the  medium  in  question,  no  matter  what  may  be  its  ac- 
tual physical  environment.  No  better  illustration  of  this 
notion  can  be  given  than  is  afforded  by  considering  the 
focal  points  on  the  axis  of  a  spherical  refracting  surface. 
The  points  F  and  F'  lie  always  on  opposite  sides  of  the  ver- 
tex A,  but  no  matter  whether  the  first  focal  point  F  is 
on  one  side  of  A  or  on  the  other,  it  is  to  be  considered 
always  as  a  point  in  the  first  medium;  and,  similarly, 
the  second  focal  point  F'  is  to  be  considered  always  as 
a  point  in  the  second  medium,  so  that  the  reduced  dis- 
tance between  F  and  F'  is  FA/n+AF'/n'  both  for  a  conver- 
gent and  for  a  divergent  system.    The  reduced  focal  lengths 

f  f 

of  a  spherical  refracting  surface  are  -  and  — .;  so  that  the 
^  n  n 

f     f 
reduced  distance  of  F'  from  F  is  equal  to  -  -  — ,  . 

n    n 

The  boundary  between  two  optical  media  is  a  "twilight 
zone,"  so  to  speak,  which  cannot  be  said  properly  to  be- 
long to  either  medium;  and  hence  linear  magnitudes  which 
refer  specifically  to  the  interface  or  surface  of  separation 
cannot  be  definitely  assigned  to  one  medium  or  the  other. 
This  applies,  for  example,  to  the  radius  of  curvature  of  a 
mirror  or  of  a  refracting  surface.  Whether  a  surface  which 
separates  air  from  glass  is  convex  or  concave,  we  have  no 
right  to  say  that  the  radius  of  curvature  lies  in  the  air  or 
in  the  glass;  and  thus  we  never  speak  of  the  "reduced  ra- 
dius" of  a  reflecting  or  refracting  surface. 

105.  The  Refracting  Power. — In  the  w-form  of  the  ab- 
scissa-equation which  gives  the  relation  between  a  pair  of  con- 
jugate points  on  the  axis,  we  are  concerned  not  so  much  with 
the  linear  magnitudes  themselves,  that  is,  with  the  abscissa, 
as  with  the  reciprocals  of  these  magnitudes,  which,  as  we 
have  seen,  represent  the  curvatures  of  the  surfaces  of  which 
these  abscissa?  are  the  radii.     It  is  partly  for  this  reason 


282  Mirrors,  Prisms  and  Lenses  [§  105 

that  many  teachers  of  geometrical  optics  regard  the  so- 
called  " curvature  method"  of  studying  these  problems  as 
both  more  natural  and  more  direct  than  the  "ray  method." 
There  is  certainly  much  to  be  said  in  its  favor,  but  the  truth 
is,  both  methods  have  their  advantages,  and  neither  is  to 
be  preferred  to  the  other.  The  student  who  desires  to  have 
more  than  a  mere  elementary  knowledge  of  optics  will  find 
it  necessary  to  be  acquainted  with  both  points  of  view;  and 
when  he  has  attained  this  position,  he  will  realize  that  the 
two  methods  are  perfectly  equivalent  and  that  the  distinc- 
tion between  them  is  more  or  less  artificial. 

But  whether  we  have  the  so-called  "curvature  method" 
in  mind  or  not,  it  will  evidently  be  a  step  in  the  direction  of 
simplifying  the  abscissae-formula  if  we  introduce  symbols 
for  the  reciprocals  of  the  abscissae,  and  thereby  get  rid  of 
the  fractional  forms.  Thus,  instead  of  employing  the  re- 
duced focal  length,  it  will  be  better  to  introduce  a  term  for 
the  reciprocal  of  this  magnitude.  Accordingly,  the  refract- 
ing power  of  an  optical  system  is  defined  to  be  the  reciprocal 
of  the  reduced  primary  focal  length.  These  reciprocal  mag- 
nitudes will  be  denoted  by  capital  italic  letters.  For  ex- 
ample, the  refracting  power  of  an  optical  system  will  be  de- 
noted by  F;  that  is,  according  to  the  above  definition: 

The  refracting  power  of  a  spherical  refracting  surface  (see 
§79)  is: 

F  =  -f=~j,  =  (n'-n)R, 

where  R  =  -  denotes  the  curvature  of  the  surface.     If  the 
r 

first  medium  is  air  (n  =  l),  then  F=j.    The  refracting  power 

of  a  spherical  refracting  surface  is  directly  proportional  to 
the  curvature  of  the  surface. 


§  105]  Refracting  Power  283 

The  reflecting  power  of  a  spherical  mirror  {nf  =  —  n,  /'=/) 
is  defined  in  the  same  way,  viz., 

F  =  -  =  -2n.R, 

where  n  denotes  the  index  of  refraction  of  the  medium  in 
front  of  the  mirror.  Thus,  although  the  position  of  the 
focal  point  (F)  and  the  magnitude  of  the  focal  length  (/)  of 
a  curved  mirror  will  not  be  altered  by  changing  the  medium 
in  front  of  the  mirror,  its  reflecting  power  will  be  affected; 
and  this  will  be  the  case  whether  the  mirror  is  concave  or 
convex.  If  the  focal  length  of  a  mirror  is  8,  its  reflecting 
power  will  be  one-eighth  when  the  mirror  is  in  contact  with 
air  (n=l),  but  it  will  be  raised  to  one-sixth  if  the  medium 
in  front  of  the  mirror  is  water  (n  =  4) . 

The  refracting  power  of  a  lens  surrounded  by  the  same 
medium  (ri)  on  both  sides  is 

F  =  ^=-^ 
f        /'' 
If  the  curvatures  of  the  two  faces  of  an  infinitely  thin  lens 
are  denoted  by  Ri  and  R2)  that  is,  if 

Ri  =  —  ,  R2  =  —  , 

ri  r2 

then 

F=(n'-n)  (Ri-R2), 
where  n'  denotes  the  index  of  refraction  of  the  lens-substance 
and  n  denotes  the  index  of  refraction  of  the  surrounding 
medium.  If  either  one  of  these  media  is  changed,  other 
things  remaining  the  same,  the  refracting  power  of  the  lens 
will  be  altered. 

If  F\,  F2  denote  the  refracting  powers  of  the  two  surfaces 
of  a  lens,  then 

F^tn'-^Ri,         F2  =  (n-n')R2, 
and  in  place  of  the  preceding  equation  we  may  write : 

F  =  F1+F2; 
and  thus  it  appears  that  the  refracting  power  of  an  infinitely 


284  Mirrors,  Prisms  and  Lenses  [§  106 

thin  lens  is  equal  to  the  algebraic  sum  of  the  refracting  powers 
of  the  lens-surfaces. 

The  refracting  power  of  a  lens  depends,  therefore,  on 
the  curvatures  of  both  faces,  but  evidently  a  lens  of  given 
material  and  of  prescribed  refracting  power  may  have  very 
different  forms.  One  of  the  minor  problems  of  optical 
construction  is  to  "bend"  a  lens,  as  the  technicab  phrase 
is,  that  is,  being  given  the  curvature  of  one  face  of  the  lens, 
to  find  the  curvature  of  the  other  face  so  that  the  refracting 
power  of  the  lens  may  have  a  given  value.  If,  for  example 
the  magnitudes  denoted  by  n,  n' ,  R2  and  F  are  assigned, 
the  curvature  of  the  first  face  must  be : 

F 

Ri  =  R2-\-    , 

n  —n 

If  the  media  are  different  on  the  two  sides  of  the  lens,  and  if 
the  indices  of  refraction  of  the  three  media  in  the  order  in 
which  they  are  traversed  by  the  light  are  denoted  by  n\, 
n<t  and  n3,  we  find  easily  the  following  formula  for  the  re- 
fracting power  of  an  infinitely  thin  lens : 

F=  J=  -  J,  =(n2-n1)R1+(n3-n2)R2  =  Fl+F2, 

where  the  symbols  have  precisely  the  same  meanings  as 
before. 

It  will  be  seen  from  these  examples  that  one  effect  of  in- 
troducing the  term  refracting  power  is  a  simplification  in 
consequence  of  the  fact  that  the  two  magnitudes  denoted 
by  /  and  /'  are  now  expressed  in  terms  of  a  single  magni- 
tude F. 

106.  Reduced  Abscissa  and  Reduced  "  Vergence  ". — The 
reduced  abscissae  of  a  pair  of  conjugate  axial  points  M,  M' 
are  defined  in  exactly  the  same  way  as  the  reduced  focal 
lengths.  The  point  designated  by  M  is  to  be  regarded  al- 
ways as  lying  in  the  first  medium  of  the  system,  and,  sim- 
ilarly, the  point  designated  by  M'  is  to  be  regarded  as  lying 
in  the  last  medium,  entirely  irrespective  of  the  question  as 


§  106]  Reduced  Abscissa  and  Reduced  "Vergence"   285 

to  whether  either  of  these  points  is  "real"  or  "virtual," 
as  explained  in  §  104. 

By  way  of  illustration,  suppose  that  the  optical  system 
consists  of  a  single  spherical  refracting  surface  separating 
two  media  ef  indices  n  and  n' .  If  the  origin  of  abscissae 
is  taken  at  the  vertex  A,  so  that  w  =  AM,  w'  =  AM',  then 

the  reduced  abscissae  will  be  -,  — .    The  reciprocals  of  these 

n'  n' 

magnitudes,  denoted  by  U,  U'  are  called  the  reduced  "ver- 

gences,"  with  respect  to  the  point  A;  thus, 

U=-,       U'=-,. 
u  u 

These  functions  U,  U'  are  the  measures  of  the  convergence 
or  divergence  of  the  bundles  of  object-rays  and  image-rays; 
and  in  this  illustration  these  magnitudes  are  evidently  pro- 
portional to  the  curvatures  of  the  incident  and  refracted 
wave-fronts  at  the  instant  when  the  disturbance  arrives 
at  the  surface  of  separation  of  the  two  media. 

Since  (§  79)  the  abscissa-formula  for  a  spherical  refracting 
surface  may  be  written  in  the  form : 

nf _n  ,  n 
u~u~f' 
this  relation  may  now  be  expressed  in  the  elegant  and  con- 
venient form: 

U'=U+F. 

This  same  formula  holds  in  the  case  of  a  spherical  mirror, 
in  which  case  Uf  =  —  n/u' ',  where  n  denotes  the  index  of  re- 
fraction of  the  medium  in  front  of  the  mirror. 

Moreover,  the  same  formula  U'=U-\-F  is  found  to  be 
applicable  to  the  case  of  an  infinitely  thin  lens.  If  the  lens 
is  surrounded  by  the  same  medium  (n)  on  both  sides,  then 
we  must  put  U  =  n/u,  U'  =  n/u'  and  F  =  n/f,  where  n'  de- 
notes the  index  of  refraction  of  the  lens-substance.  Or  in 
case  the  last  medium  (w3)  is  different  from  the  first  medium 
(ni),  then  U  =  ni/u,  U'  =  ns/u',  and  F  =  n\jf.  In  both  cases 
the  formula  will  be  found  to  be  identical  in  form  with  that 


286  Mirrors,  Prisms  and  Lenses  [§  107 

given  above.  In  fact,  as  we  shall  see  in  Chapter  X,  the 
formula  U'=U-\-F  is  perfectly  general  and  applicable  to 
any  optical  instrument  which  is  symmetrical  about  an  axis. 
The  advantage  of  a  single  formula  which  has  such  wide  ap- 
plicability is  obvious.  It  is  easy  to  remember*  that  the  re- 
duced vergence  (W)  on  the  image-side  of  the  instrument  is 
equal  to  the  algebraic  sum  of  the  reduced  vergence  (U)  on  the 
object-side  and  the  refracting  power  (F) . 

If  the  abscissas  are  measured  from  the  focal  points  F,  F', 
that  is,  if  we  put  z  =  FM,  x/--=F/Mr,  the  magnitudes 

X  =  -,      X'  =  -' 
x  x 

are  called  the  reduced  focal  point  vergences;  and  the  relation 

between  X,  X'  is  expressed  by  the  equation : 

X-X'=-F\ 

107.  The  Dioptry  as  Unit  of  Curvature. — Obviously,  the 

magnitudes   which   have   been   denoted   above   by   capital 

italic  letters,   since  they  are  all  equal  or  proportional  to 

the  reciprocals  of  certain  linear  magnitudes,  are  essentially 

measures  of  curvature,  and  hence  they  must  be  described  or 

expressed  in  terms  of  some  unit  of  curvature,  which  will  itself 

be  dependent  on  the  unit  of  length.     Opticians  guided  by 

purely  practical  considerations  were  the  first  to  recognize 

the  need  of  a  suitable  optical  unit  for  this  purpose.     The 

unit  of  curvature  which  is  now  almost  universally  used  in 

spectacle  optics  and  which  is  coming  to  be  employed  more 

and  more  in  all  other  branches  of  optics  is  the  curvature 

of  an  arc  whose  radius  of  curvature  is  one  meter.     To  this 

unit  the    name  dioptry*  has  been    given.      Originally,  the 

*  The  name  "dioptrie"  was  first  suggested  by  Monoyer  of  France 
in  1872  (see  Annates  d'oculistique,  LXV1II,  111),  being  derived  from 
the  Greek  tcl  o\o7rrpi/<a,  whence  came  also  the  term  "dioptrics" 
which  was  formerly  much  used  by  scientific  writers  as  applying  to  the 
phenomena  of  refraction,  especially  through  lenses.  The  word  is 
generally  written  dioptre  in  French  and  Dioptrie  in  German.  Etymo- 
logically,  the  correct  English  form  would  appear  to  be  dioptry,  and 
this  spelling  has  been  adopted  by  the  American  translators  of  both 


§  107]  Dioptry  287 

dioptry  was  defined  as  the  refracting  power  of  a  lens  in  air 
of  focal  length  one  meter.  Consequently,  a  lens  whose  focal 
length  was  50  cm.  or  half  a  meter  would  have  a  refracting 
power  of  2  dptr.,  whereas  another  lens  of  focal  length  2  meters 
would  have  a  refracting  power  of  \  dptr.  In  general,  if 
the  focal  length  of  a  lens  surrounded  by  air  is  /  centimeters, 
its  refracting  power  will  be  100//  dptr.  But  according  to 
the  definition  which  we  have  given,  the  dioptry  is  a  unit 
not  of  refracting  power  only  but  of  any  similar  magnitude 
of  the  nature  of  a  curvature.  Thus,  for  example,  if  the 
radius  of  a  mirror  or  of  a  spherical  refracting  surface  is  half 
a  meter,  its  curvature  is  2  dptr.  If  the  distances  denoted 
by  f>  r,  u>  x)  etc.,  are  expressed  in  meters,  the  magnitudes 
denoted  by  the  corresponding  capital  letters  F,  R,  U,  X, 
etc.,  will  be  in  dioptries.  Dr.  Drysdale  has  suggested 
that  we  introduce  also  the  convenient  terms  millidioptry 
(  =  0.001  dptr.),  Hectodioptry  (  =  100  dptr.)  and  Kilodioptry 
(  =  1000  dptr.)  corresponding,  respectively,  to  the  Kilo- 
meter, centimeter  and  millimeter  as  units  of  length.  Thus, 
the  refracting  power  of  a  lens  of  focal  length  10  cm.  might 
be  variously  described  as  equal  to  100  millidioptries,  to 
10  dioptries,  to  0.1  Hectodioptry  or  to  0.01  Kilodioptry. 
But  these  terms  have  not  come  into  general  use. 

If  the  focal  length  of  a  lens  in  water  (n  =  1.3)  is  13  cm., 
its  refracting  power  will  be  the  same  as  that  of  a  lens  in 
air  (n  =  l)  of  focal  length  10  cm.,  viz.,  10  dptr.  If  the  pri- 
mary focal  point  of  a  spherical  refracting  surface  is  situated 

Landolt's  and  Tscherning's  books  on  physiological  optics;  notwith- 
standing the  fact  that  the  word  is  usually  spelled  and  pronounced 
dioptre  in  England  and  diopter  in  America.  Dr.  Crew  in  his  well  known 
text-book  of  physics  writes  dioptric.  The  author  has  concluded  that 
on  the  whole  it  is  best  to  adopt  the  spelling  used  in  the  text. 

The  usual  abbreviation  of  dioptry  is  a  capital  D.;  but  as  this  letter 
is  liable  to  be  confused  with  the  symbols  of  magnitude  employed  in  the 
formulae,  it  seems  preferable  to  follow  the  usage  of  Von  Rohr  and 
other  modern  writers  on  optics  who  have  adopted  the  abbreviation 
dptr.,  although  doubtless  many  will  object  to  this  long  form. 


288  Mirrors,  and  Prisms  Lenses  [§  108 

(optically)  in  air  (n=l)  at  a  distance  of  1  meter  from  the 
vertex,  the  refracting  power  of  the  surface  will  be  1  dptr. 
and  the  radius  of  the  surface  will  be  equal  to  (nf  —  1)  meters, 
where  ,nf  denotes  the  index  of  refraction  of  the  second  me- 
dium. If  the  radius  of  curvature  of  a  mirror  is  50  cm.,  its 
reflecting  power  will  be  4  dptr.  if  the  reflecting  surface  is 
in  contact  with  air  (n  =  l),  but  it  will  be  5^  dptr.  if  the 
surface  is  in  contact  with  water  (w =-§-)•  These  examples 
are  given  merely  to  illustrate  how  the  term  dioptry  is  used. 

108.  Lens-Gauge — The  dial  of  the  opticians'  lens-gauge 
described  in  §  99  is  usually  graduated  so  as  to  give  in  di- 
optries  the  refracting  power  of  the  surface  which  is  measured. 
The  refracting  power  of  a  spherical  refracting  surface  is 
proportional  to  its  curvature,  as  we  have  seen  (§  105),  but 
it  is  dependent  also  on  the  indices  of  refraction  of  the  two 
media.  If  the  first  medium  is  air  and  if  the  index  of  re- 
fraction of  the  second  medium  is  denoted  by  n,  then  F  = 
(n—l)R.  The  gauge  actually  measures  the  curvature  R,  and 
the  readings  on  the  dial  correspond  to  the  values  of  R  mul- 
tiplied by  the  factor  (n—  1).  Direct  readings  of  the  refract- 
ing power  (F)  imply,  therefore,  that  the  maker  has  assumed 
a  certain  value  of  the  index  of  refraction  n;  and  if  the  actual 
value  of  n  is  different  from  this  assumed  value,  the  readings 
will  be  erroneous.  The  value  of  n  assumed  by  the  maker  is  a 
constant  of  the  instrument,  which  should  be  marked  on  it, 
although  it  may  easily  be  determined  empirically  by  com- 
paring the  readings  with  the  determination  of  the  curvature 
as  obtained  with  an  ordinary  spherometer. 

Suppose  that  this  constant  is  denoted  by  c,  and  that  we 
wish  to  use  the  gauge  to  measure  the  refracting  power  (F) 
of  a  lens  of  negligible  thickness  made  of  glass  of  index  n. 
If  the  refracting  powers  of  the  two  surfaces  of  the  lens  are 
denoted  by  F\  and  F2  and  the  curvatures  by  Rx  and  R2, 
then  F  =  Fi+F2  where  Fi  =  (n-l)Rh  F2  =  —  {n-l)R2}  the 
minus  sign  in  front  of  the  last  expression  being  necessary 
because  the  refraction  in  this  case  takes  place  from  glass 


§  109]        Lens-System  of  Negligible  Thickness  289 

to  air.    But  if  the  constant  c  has  a  value  different  from  n, 
the  readings  of  the  instrument  for  the  two  faces  of  the  lens 
will  not  give  the  correct  values  Fh  F2  of  the  refracting  powers. 
Suppose  the  readings  are  denoted  by  Fi,  F2,  so  that 
iY=  {c-\)Ri,F2'=-{c-l)R2.    Then  evidently 


and  hence 


t\  — 7^1>  ^2 7^2, 

c-  1  c  —  1 


F=~  (Fi'+ftO. 

c—  1 


The  gauge-readings  must  be  multiplied  therefore  by  the 
factor 

n-1 

c.-l 

in  order  to  obtain  the  correct  values  of  the  refracting  powers. 

Suppose,  for  example,  that  the  graduations  on  the  dial  cor- 
respond to  a  value  c=1.54  and  that  the  index  of  the  lens 
to  be  measured  is  n=1.52.  Then  the  value  of  the  factor 
is  0.963;  so  that  if  the  lens-gauge  gives  for  the  refracting 
power  F  the  value  6.25  dptr.,  the  correct  value  is  obtained 
by  multiplying  this  value  by  0.963,  that  is,  the  correct 
value  will  be  6.02  dptr. 

109.  Refraction  of  Paraxial  Rays  through  a  Thin  Lens- 
System. — Let  Mi'  designate  the  position  of  a  point  conju- 
gate to  an  axial  object-point  with  respect  to  an  infinitely 
thin  lens  of  refracting  power  F\,  and  let  the  point  where 
the  axis  crosses  the  lens  be  designated  by  Ai.  If  the  lens 
is  surrounded  by  air,  and  if  we  put  Wi  =  AiMi,  wi'  =  AiMi', 
tfi  =  l/wi,  tfi'  =  l/wi', 
then 

£/i'=£7i+Fi. 
If  now  at  a  point  A2  on  the  axis  of  the  lens  beyond  Ai  (such 
that  the  distance  d  =  AiA2  is  measured  in  the  direction  in 
which  the  light  is  going)  another  infinitely  thin  lens  is  set 
up  with  its  axis  in  the  same  straight  line  with  that  of  the 
first  lens,  then  Mi'  may  be  regarded  as  an  axial  object-point 


290  Mirrors,  Prisms  and  Lenses  [§  109 

M2  with  respect  to  the  second  lens;  and  if  M2'  designates 
the  position  of  the  point  conjugate  to  M2  (or  Mi')  with  re- 
spect to  this  lens,  then  also  (supposing  that  the  second  lens 
is  surrounded  by  air  and  that  its  refracting  power  is  denoted 

by  F2), 

U2'  =  U2+F2, 

where  £72  =  l/w2,  UJ  =  1/1*2',  w2  =  A2M2  =  A2M/,  w'2  =  A2M2'. 
Obviously,  the  point  M2'  is  the  image-point  conjugate  to 
the  axial  object-point  Mi  with  respect  to  the  two  lenses; 
so  that  regarding  the  system  as  a  whole,  we  may  write  M,  M' 
in  place  of  Mi,  M2'  and  U,  U'  in  place  of  Ui,  U2,  respectively. 
Now  let  us  impose  the  condition  that  the  two  thin  lenses 
are  in  contact  with  each  other  or  that  they  are  as  close  together 
as  possible;  in  other  words,  that  the  axial  distance  d  between 
the  lenses  is  negligible.  If  this  is  the  case,  the  points  Ai,  A2 
are  to  be  regarded  as  a  pair  of  coincident  points,  and  hence 

w=u2] 

and,  therefore,  we  may  write  now : 

UJ  =  U+FU      U'=Ui'+F2. 
Eliminating  Ui',  we  obtain : 

U^U+^+Ft); 
and  if  we  put 

we  have  finally: 

U'=U+F. 
Since  this  formula  is  seen  to  be  identical  in  both  form  and 
meaning  with  the  formula  for  a  single  thin  lens,  it  appears 
therefore  that  a  combination  of  two  thin  coaxial  lenses  in 
contact  is  equivalent  to  a  single  lens  of  refracting  power  F  equal 
to-  the  algebraic  sum  of  the  refracting  powers  F\  and  F2  of  the 
component  lenses. 

Theoretically,  this  rule  can  be  applied  to  a  centered  system 
of  any  number  of  thin  lenses  in  contact.  Thus,  the  total  re- 
fracting power  of  a  thin  lens-system  will  be 

F=Fl+F2+  .  .  .  +Fm, 


F=F1+F2, 


§  110]  Prismatic  Power  of  Thin  Lens  291 

where  the  total  number  of  lenses  is  denoted  by  m.     This 
formula  may  be  written : 

i=m 

F=  2  Pit 

where  F\  denotes  the  refracting  power  of  the  ith.  lens. 

In  the  case  of  actual  lenses  placed  together  in  this  fashion 
it  will  always  be  a  question,  How  far  are  we  justified  in 
neglecting  the  total  thickness  of  the  system?  Two  adjacent 
lenses  may  be  placed  in  actual  contact,  but  a  third  lens  can- 
not be  in  contact  with  the  first.  Moreover,  even  when 
there  are  only  two  lenses,  their  outward  forms  may  be  such 
that  it  will  not  be  possible  to  place  them  in  tangential  con- 
tact at  their  vertices,  although  they  can  always  be  made 
to  touch  at  two  points  symmetrically  situated  with  respect 
to  their  common  axis.  Attention  is  directed  to  this  ques- 
tion chiefly  in  connection  with  the  method  of  neutraliza- 
tion of  lenses  which  is  practiced  extensively  in  the  fitting 
of  spectacle  glasses.  Two  infinitely  thin  lenses  of  equal  and 
opposite  refracting  powers  are  said  to  "neutralize"  each 
other,  because  when  they  are  placed  in  contact  their  total 
refracting  power  (F1+F2)  is  equal  to  zero.  Strictly  speak- 
ing, the  neutralization  of  a  negative  glass  by  a  positive  glass 
implies  not  only  that  the  focal  lengths  are  equal  in  magni- 
tude but  also  that  the  primary  focal  point  of  one  lens  shall 
coincide  with  the  secondary  focal  point  of  the  other.  Both 
of  these  conditions  are  realized  in  a  combination  of  a  plano- 
concave with  a  plano-convex  lens  fitted  together  so  as  to  form 
a  slab  with  plane  parallel  sides.  But  even  with  the  relatively 
thin  lenses  employed  in  spectacles  sensible  errors  may  be 
introduced  by  assuming,  as  is  usually  done,  that  the  con- 
dition Fi+F2  =  0  is  the -sole  or  even  the  main  consideration 
for  neutralization. 

110.  Prismatic  Power  of  a  Thin  Lens. — Only  such  rays 
as  go  through  the  optical  center  (§  88)  emerge  from  a  lens 
without  being  deviated  from  their  original  directions.  The 
prismatic  power  of  a  thin  lens,  which,  like  the  power  of  a 


292 


Mirrors,  Prisms  and  Lenses 


[§110 


Fig.  141,  a  and  b. — Prismatic  power  of  infinitely  thin  lens,     (a)   Convex, 
(b)   Concave  lens. 

thin  prism  (§  70),  is  measured  by  the  deviation  of  a  ray  in 
passing  through  it,  depends  not  only  on  the  refracting  power 
of  the  lens  but  also  on  the  place  where  the  ray  enters  the 
lens.     In  the  accompanying  diagram  (Fig.   141,  a  and  b) 


§  110]  Prismatic  Power  of  Thin  Lens  293 

the  point  A  designates  the  axial  point  of  a  thin  lens  of  re- 
fracting power  F.  A  ray  RB  incident  on  the  lens  at  B  passes 
out  in  the  direction  BS.  If  M,  M'  designate  the  points 
where  the  incident  and  emergent  rays  cross  the  axis,  then 
ZM'BM=  e  is  the  angle  of  deviation;  and  if  0  =  Z  AMB  de- 
notes the  slope  of  the  incident  ray  and  0'  =  Z  AM'B  denotes 
the  slope  of  the  emergent  ray,  evidently  we  have  the  rela- 
tion: 

e=  0-  $'. 
The  distance  /i  =  AB  of  the  incidence-point  B  from  the  axis 
of  the  lens  or  the  incidence-height  of  the  ray  is  called  by 
the  spectacle-makers  the  decentration  of  the  lens.  Since 
the  decentration  of  an  ophthalmic  lens  is  always  compara- 
tively small,  the  ray  RB  may  be  regarded  as  a  paraxial 
ray,  and  hence  we  can  put  6  and  6'  in  place  of  tan0  and 
tan#'  and  write: 

6=--=-h.U,  $'=--  = -h.U', 

u  u 

where  u  =  AM,  m'  =  AM',   U=l/u,  U'  =  1/u',  since  the  lens 

is  supposed  to  be  surrounded  by  air  (n=l).     Accordingly, 

e  =  h(U'-U), 
the  deviation-angle  e  being  expressed  in  radians  if  h,  u  and 
v!  are  all  expressed  in  terms  of  the  same  linear  unit.    But 

U'-U=F; 
and  hence 

e  =  h.F  radians. 
In  this  formula  the  decentration  h  must  be  expressed  in 
meters  if  the  refracting  power  F  is  given  in  dioptries.  The 
above  relation  may  be  derived  immediately  also  from 
Fig.  142,  where  the  incident  ray  RB  is  drawn  parallel  to 
the  axis  of  the  lens,  so  that  in  this  case  6'  +  e  =  0;  and 

since  tan  6'  =  6'  =  =rr-r  =  -tt,  =  —  t  =  —  h.F.  we  obtain,  as  above, 
F'A    /'        / 

e  —  h.F.  If  a  screen  is  placed  perpendicular  to  the  incident 
light  coming  in  the  direction  RB,  a  spot  of  light  will  be  pro- 
duced on  the  screen  at  the  point  N  where  the  straight  line 


294 


Mirrors,  Prisms  and  Lenses 


[§110 


RB  meets  the  screen;  and  if  now  a  lens  is  interposed  at  a 
certain  known  distance  from  the  screen,  the  deviation  e  can 
easily  be  determined  by  measuring  the  distance  NL  through 
which  the  spot  of  light  is  deflected. 

However,  both  the  radian  and  the  meter  are  inconven- 
iently large  units  for  expressing  the  values  of  the  small  mag- 

£ 


Fig.   142. — Prismatic  power  of  infinitely  thin  lens;  incident  ray  parallel 

to  axis. 

nitudes  denoted  by  e  and  h.  Opticians  measure  the  devia- 
tion in  terms  of  the  centrad  or  in  terms  of  the  prism-dioptry, 
which  in  the  case  of  small  angles,  as  we  have  seen  (§  70), 
is  practically  the  same  unit  as  the  centrad.  If  the  angle 
of  deviation  expressed  in  centrads  or  prism-dioptries  is  de- 
noted by  p,  while  e  denotes  the  value  of  this  angle  in  ra- 
dians, then 

p  =  100  €. 
Moreover,  if  the  decentration  h  is  given  in  centimeters  in- 
stead of  in  meters,  we  obtain  the  following  formula: 

p  =  h.F; 
that  is,  the  deviation  (p)  in  prism-dioptries  (or  centrads)  pro- 
duced by  a  thin  lens  in  any  zone  is  equal  to  the  product  of  the 
refracting  power  (F)  of  the  lens  in  dioptries  by  the  radius  (h) 
of  the  zone  in  centimeters;  or  as  the  opticians  usually  express 
it,  the  prismatic  power  of  a  thin  lens  in  prism-dioptries  is 


Ch.  VIII]  Problems  295 

equal  to  the  product  of  the  refracting  power  of  the  lens  in 
dioptries  by  the  decentration  in  centimeters.  For  example, 
a  spectacle  glass  of  refracting  power  5  dptr.  must  be  de- 
centered  about  0.4  cm.  or  4  mm.  in  order  to  have  a  pris- 
matic power  of  2  prism-dptr. 

If  in  Fig.  142  the  distance  AP  of  the  screen  from  the  lens 
is  1  meter,  the  deflection  LN  in  centimeters  of  the  spot  of 
light  will  be  equal  to  the  prismatic  power  of  a  lens  of  focal 
length  /=  AF'  decentered  by  the  amount  /i  =  AB. 

PROBLEMS 

1.  How  is  the  curvature  of  a  wave  affected  by  reflection 
at  a  plane  mirror?  How  is  the  curvature  of  a  plane  wave 
affected  by  reflection  at  a  spherical  mirror? 

2.  The  distance  between  a  luminous  point  and  the  eye 
of  an  observer  is  50  cm.  A  plate  of  glass  (n=1.5),  10  cm. 
thick,  is  interposed  midway  between  the  point  and  the  eye 
with  its  two  parallel  faces  perpendicular  to  the  line  of  vision. 
Spherical  waves  spreading  out  from  the  luminous  point 
are  refracted  through  the  plate  and  into  the  eye.  Find  the 
curvature  of  the  wave-front:  (a)  just  before  it  enters  the 
glass,  (6)  immediately  after  entering  the  glass,  (c)  im- 
mediately after  leaving  the  glass,  and  (d)  when  it  reaches 
the  eye. 

Ans.  (a) -5  dptr.;  (6)  —  3|  dptr.;  (c)-3|  dptr.;  (d) 
-2|  dptr. 

3.  What  is  the  refracting  power  of  a  spherical  refracting 
surface  of  radius  20  cm.  separating  air  (n  =  l)  from  glass 
(n' =  1.5)? 

Ans.  +2.5  dptr.  or  —2.5  dptr.,  according  as  the  surface 
is  convex  or  concave,  respectively. 

4.  If  the  cornea  of  the  eye  is  regarded  as  a  single  spheri- 
cal refracting  surface  of  radius  7.7  mm.  separating  air  (n=l) 
from  the  aqueous  humor  (n' =  1.336),  what  is  its  refracting 
power?  Ans.  43.6  dptr. 


296  Mirrors,  Prisms  and  Lenses  [Ch.  VIII 

5.  Using  the  data  of  the  preceding  problem,  find  the  re- 
fracting power  of  the  cornea  when  the  eye  is  under  water 
(n  =  1.33).  Ans.  Nearly  0.78  dptr. 

6.  What  is  the  reflecting  power  of  a  concave  mirror  of 
radius  20  cm.  when  the  reflecting  surface  is  in  contact  with 
(a)  air  (n=l)  and  (6)  water  (n=-J)? 

Ans.  (a)  10  dptr.;  (b)  13.33  dptr. 

7.  A  convex  spherical  surface  of  radius  25  cm.  separates 
air  (n=l)  from  glass  (V  =  1.5).  Find  the  refracting  power 
and  the  reflecting  power  of  the  surface. 

Ans.  Refracting  power  is  +2  dptr.;  reflecting  power  is 
-  8  dptr. 

8.  The  reflecting  power  of  a  spherical  mirror  in  contact 
with  air  is  +2  dptr.    Determine  the  form  of  the  mirror. 

Ans.  A  concave  mirror  of  radius  1  meter. 

9.  A  spherical  mirror  is  in  contact  with  a  liquid  of  re- 
fractive index  n.  If  the  reflecting  power  of  the  mirror  is 
+2  dptr.,  show  that  the  mirror  is  a  concave  mirror  of  radius 
n  meters. 

10.  The  index  of  refraction  of  carbon  bisulphide  is  1.629. 
What  is  the  reflecting  power  of  a  concave  mirror  of  radius 
25  cm.  in  contact  with  this  liquid?  Ans.  +13.032  dptr. 

11.  What  is  the  refracting  power  of  a  thin  symmetric 
convex  lens  made  of  glass  of  index  1.5,  if  the  radius  of  cur- 
vature of  each  surface  is  5  cm.?  Ans.  +20  dptr. 

12.  The  refracting  power  of  a  thin  plano-convex  lens 
made  of  glass  of  index  1.5  is  20  dptr.  Find  the  radius  of 
the  curved  surface.  Ans.  2.5  cm.  or  nearly  1  inch. 

13.  A  thin  convex  meniscus  lens  is  made  of  glass  of  in- 
dex 1.5.  The  radius  of  the  first  surface  is  10  and  that  of  the 
second  surface  is  25  cm.  Assuming  that  the  lens  is  sur- 
rounded by  air  (n=  1),  find  its  refracting  power. 

Ans.  +3  dptr. 

14.  If  the  lens  in  the  preceding  example  were  made  of 
water  of  index  ~,  what  will  be  its  refracting  power? 

Ans.  +2  dptr. 


Ch.  VIII]  Problems  297 

15.  If  the  first  surface  of  the  lens  in  No.  13  were  in  con- 
tact with  water  (fti  =  |)  and  the  second  surface  in  contact 
with  air  (n3=  1),  what  will  be  the  refracting  power? 

Ans.  —  ■§-  dptr. 

16.  If  the  first  surface  of  the  lens  in  No.  13  were  in  con- 
tact with  air  (fti=l)  and  the  second  surface  in  contact  with 
water  (n3  =  4),  what  will  be  the  refracting  power? 

Ans.  +4^  dptr. 

17.  In  examples  13,  14,  15  and  16  suppose  the  lens  were 
reversed  so  that  the  opposite  face  was  turned  to  the  inci- 
dent light.  What  would  be  the  answers  to  these  problems 
then? 

Ans.  The  same  answers  would  be  obtained  for  Nos.  13 
and  14;  but  the  answers  for  Nos.  15  and  16  would  be  inter- 
changed. 

18.  Show  that  the  lateral  magnification  in  a  spherical 
mirror,  a  spherical  refracting  surface  or  an  infinitely  thin  lens 
is  equal  to  the  ratio  of  the  reduced  "  vergences  "  U  and  U'. 

19.  Describe  the  spherometer  and  the  lens-gauge  and 
explain  their  principles. 

20.  Show  how  a  plane  wave  is  refracted  through  a  thin 
lens,  and  derive  from  a  diagram  for  this  case  the  formula 
for  the  refracting  power. 

21.  Show  how  a  plane  wave  is  refracted  through  a  thin 
prism,  and  derive  the  formula  for  the  deviation  in  terms  of 
the  refracting  angle  of  the  prism  and  the  relative  index  of 
refraction. 

22.  The  refracting  power  of  a  thin  lens  is  +6  dptr.  It 
is  made  of  glass  of  index  1.5  and  surrounded  by  air  (w=l). 
If  the  radius  of  the  first  surface  is  +10  cm.,  what  is  the 
radius  of  the  second  surface?  Ans.  r2  =  —  50  cm. 

23.  A  convex  lens  produces  on  a  screen  14.4  cm.  from 
the  lens  an  image  which  is  three  times  as  large  as  the  object. 
Find  the  refracting  power  of  the  lens.  Ans.  27.78  dptr. 

24.  A  lens-gauge  graduated  in  dioptries  for  glass  of  in- 
dex 1.5  is  used  to  measure  a  thin  double  convex  lens  made 


298  Mirrors,  Prisms  and  Lenses  [Ch.  VIII 

of  glass  of  index  1.6.  The  readings  on  the  dial  give  +4  for 
both  surfaces.  Find  the  refracting  power  of  the  lens,  assum- 
ing that  its  thickness  is  negligible.  Ans.  +9.6  dptr. 

25.  Modern  spectacle  glasses  are  meniscus  lenses  with 
the  concave  surface  worn  next  the  eye.  If  the  glass  is  to 
give  the  proper  correction,  it  is  very  important  for  it  to  be 
adjusted  at  a  certain  measured  distance  from  the  eye.  In 
determining  this  distance  it  is  necessary  to  ascertain  the 
" vertex  depth"  of  the  concave  surface,  that  is,  the  perpen- 
dicular distance  (t)  of  the  vertex  from  the  plane  of  the  edge 
or  contour  of  the  surface.  If  the  diameter  of  this  contour 
expressed  in  millimeters  is  denoted  by  2h,  and  if  the  refract- 
ing power  of  the  surface  next  the  eye,  expressed  in  dioptries, 
is  denoted  by  F2,  and,  finally,  if  the  index  of  refraction  of 
the  glass  is  denoted  by  n,  show  that  the  vertex  depth  of  the 
surface  is  approximately: 

t  =  _  0.0005  -^4  millimeters. 
n—1 

26.  What  is  the  refracting  power  of  a  lem  which  is  equiva- 
lent to  two  thin  convex  lenses  of  focal  lengths  15  and  30  cm., 
placed  in  contact?  Ans.  10  dptr. 

27.  A  concave  lens  of  focal  length  12  cm.  is  placed  in 
contact  with  a  convex  lens  of  focal  length  7.5  cm.  Find 
the  refracting  power  of  the  combination.         Ans.  5  dptr. 

28.  The  refracting  power  of  a  thin  concave  lens  is  5  times 
that  of  a  thin  convex  lens  in  contact  with  it.  If  the  focal 
length  of  the  combination  is  8  cm.,  find  the  refracting  power 
of  each  of  the  components.  Ans.   — 15  §  and  +3  §  dptr. 

29.  Two  thin  lenses,  made  of  glass  of  indices  1.5  and  1.6, 
are  fitted  together  with  the  second  surface  of  the  first  lens 
coincident  with  the  first  surface  of  the  second  lens  (rz  =  r2). 
The  radii  of  the  surfaces  are  all  positive  and  equal  to  4,  11 
and  6  cm.  taken  in  the  order  named.  Find  the  refracting 
power  of  the  combination.  Ans.  12.5  dptr. 

30.  What  is  the  prismatic  effect  of  a  lens  of  power  +4  dptr. 
decentered  0.75  cm.?  Ans.  3  prism-dioptries. 


Ch.  VIII]  Problems  299 

31.  Two  thin  convex  lenses  have  each  a  focal  length  of 
1  inch.  Find  the  position  of  the  second  focal  point  of  the 
combination  of  these  two  lenses  when  they  are  placed  with 
their  axes  in  the  same  straight  line:  (a)  when  they  are  in 
contact,  (b)  when  they  are  separated  by  1.5  inches,  and 
(c)  when  they  are  separated  by  3  inches.  Draw  a  diagram 
for  each  case  showing  the  path  of  a  beam  of  light  coming 
from  a  distant  axial  object-point. 

Ans.  (a)  Half  an  inch  beyond  the  combination;  (b)  be- 
tween the  lenses  and  1  inch  from  second  lens;  (c)  2  inches 
beyond  second  lens. 

32.  A  convex  lens  of  focal  length  20  cm.  and  a  concave 
lens  of  focal  length  5  cm.  are  placed  16  cm.  apart.  Find  the 
positions  of  the  focal  points  of  the  combination. 

Ans.  One  of  the  focal  points  is  420  cm.  from  the  convex 
lens  and  436  cm.  from  the  concave  lens;  and  the  other  focal 
point  is  36  cm.  from  the  convex  lens  and  20  cm.  from  the 
concave  lens. 

33.  How  much  must  a  lens  of  5  dptr.  be  decentered  in 
order  to  produce  a  deviation  of  3°  307  Ans.  1.22  cm. 

34.  The  radius  of  a  spherical  surface  is  measured  by  a 
spherometer  and  found  to  be  14.857  cm.  Measured  by  a 
lens-gauge  the  reading  is  3.5  dptr.  What  is  the  index  of  re- 
fraction of  the  glass  for  which  the  readings  on  the  dial  of  the 
gauge  have  been  calculated?  Ans.  1.52. 

35.  The  radii  of  each  surface  of  a  thin  symmetric  double 
convex  glass  lens  is  6  inches.  The  lens  is  supported  with 
its  lower  face  in  contact  with  the  horizontal  surface  of  still 
water.  Assuming  that  the  sun  is  in  the  zenith  vertically 
above  the  lens,  and  that  its  apparent  diameter  is  30',  find 
the  position  and  size  of  the  sun's  image.  (Take  the  indices 
of  refraction  of  air,  glass  and  water  equal  to  1,  f  and  f, 
respectively.) 

Ans.  A  real  image  12  inches  below  the  surface  of  the  water, 
0.0785  inch  in  diameter. 


CHAPTER  IX 

ASTIGMATIC  LENSES 

111.  Curvature  and  Refracting  Power  of  a  Normal  Sec- 
tion of  a  Curved  Refracting  Surface. — The  refracting  power 
(F)  of  a  spherical  surface  is  proportional  to  the  curvature 
(R)  of  the  surface,  that  is,  F=(?i'—ri)R,  where  n  and  n' 
denote  the  indices  of  refraction  of  the  media  on  opposite 
sides  of  the  surface  (§  105).  A  spherical  surface  has  the 
same  curvature  in  every  meridian,  and  hence  also  its  re- 
fracting power  is  uniform,  so  that  the  refracted  rays  in 
one  meridian  plane  are  brought  to  the  same  focus  as  those 
in  another  meridian  plane.  But  the  surfaces  of  a  lens  are 
not  always  spherical  (§  87),  and  therefore,  in  order  to  ascer- 
tain what  happens  when  a  narrow  bundle  of  rays  is  inci- 
dent perpendicularly  on  a  curved  reflecting  or  refracting 
surface  of  any  form,  we  must  investigate  the  reflecting  or 
refracting  power  in  different  sections  of  the  surface;  and 
this  means  that  we  must  investigate  the  curvature  of  these 
sections.  In  general,  this  is  a  problem  of  some  difficulty 
and  involves  a  more  or  less  extensive  knowledge  of  the 
theory  of  curved  surfaces  and  the  methods  of  infinitesimal 
geometry.  No  attempt  can  be  made  to  explain  this  theory 
here,  but  for  the  student  who  is  not  already  familiar  with 
it,  certain  general  definitions  and  propositions  of  geometry 
which  have  a  direct  bearing  on  the  optical  problems  to  be 
treated  in  this  chapter  will  be  stated  as  succinctly  as  pos- 
sible. 

The  normal  to  a  curved  surface  at  any  point  is  a  straight 
line  drawn  perpendicular  to  the  tangent  plane  at  that  point. 
The  curved  line  which  is  traced  on  the  surface  by  a  plane 
containing  the  normal  at  a  point  A  of  the  surface  is  called 

300 


§111] 


Normal  Sections  of  Curved  Surface 


301 


a  normal  section  through  this  point.  The  normal  sections 
of  a  sphere,  like  the  meridians  of  longitude  of  the  earth  (as- 
sumed to  be  a  perfect  sphere),  are  all  great  circles  of  the 
sphere,   and  their  curvatures  are  equal.     But,   generally, 


Fig.  143. — Normal  sections  of  curved  surface:  xAy  and  xAz  planes  of 
principal  sections;  xAP  plane  of  oblique  normal  section. 

the  curvatures  of  the  normal  sections  through  a  point  on 
a  curved  surface  will  vary  from  one  section  to  the  next;  so 
that  if  we  imagine  a  plane  containing  the  normal  to  be  turned 
around  this  line  as  axis,  we  shall  find  that  for  one  special 
azimuth  of  this  revolving  plane  the  curved  line  which  it 
carves  out  on  the  surface  will  have  the  greatest  curvature, 
and  that  then  as  the  plane  continues  to  revolve  the  curva- 
ture of  the  section  decreases  and  reaches  its  least  value  for 
an  azimuth  which  is  exactly  90°  from  that  for  which  the 
curvature  was  greatest.  Thus,  for  example,  in  a  cylindri- 
cal surface  the  curvature  at  any  point  is  least  and  equal  to 
zero  in  a  normal  section  whose  plane  is  parallel  to  the  axis 
of  the  cylinder,  and  it  is  greatest  in  a  normal  section  made 


302  Mirrors,  Prisms  and  Lenses  [§111 

by  a  plane  perpendicular  to  the  axis.  At  each  point  A  of  a 
curved  surface  the  normal  sections  of  greatest  and  least  curva- 
tures lie  always  in  two  perpendicular  planes,  which  are  called 
the  planes  of  the  principal  sections  of  the  surface  at  A.  The 
lines  of  intersection  of  these  planes  with  each  other  and 
with  the  tangent  plane  at  A  may  be  chosen  as  the  axes 
of  reference  of  a  system  of  rectangular  coordinates  x,  y,  z 
whose  x-axis  is  the  normal  Ax  (Fig.  143).  The  centers  of 
curvature  of  the  principal  sections  made  by  the  xy-plsme 
and  the  £2-plane  will  be  designated  by  Cy  and  Cz,  respec- 
tively; and  the  curvatures  of  the  principal  sections  will  be 
denoted  by  Ry  and  Rz,  so  that  if  ry  =  ACy  and  rz  =  ACz 
denote  the  principal  radii  of  curvature  of  the  surface  at 
the   point  A,  we  must  have  here  (§  99)  Ry  =  l/ry  and  R2 

Now  there  is  a  remarkable  geometrical  relation  between 
the  curvature  of  any  normal  section  at  A  and  the  curvatures 
of  the  principal  sections  of  the  surface  at  this  point  which 
will  be  stated  also  without  giving  the  proof.  Let  a  plane 
containing  the  normal  Ax  intersect  the  tangent  plane  (or 
2/2-plane)  in  the  straight  line  AP  (Fig.  143)  and  put  ZyAP 
=  0.  The  center  of  curvature  of  the  normal  section  made 
by  this  plane  lies  also  on  the  normal  Ax  at  a  point  which 
may  be  designated  as  Gg,  so  that  the  radius  of  curvature 
is  ACo  =  re,  and  the  curvature  itself  is  He=lfro.  The  con- 
nection between  Re  and  the  principal  curvatures  Ry  and 
Rz  is  expressed  by  the  following  formula: 
Re  =  Ry.cos2  d+Rz.sm2  0, 
where  0  denotes  the  angle  which  the  normal  section  makes 
with  the  xy-pl&ne. 

In  a  normal  section  at  right  angles  to  the  first  we  should 
have,  therefore, 

Re+w0  =  Ry.cos2(  0+9O°)+#z.sin2(  (9+90°), 
or,  since  cos(  (9+90°)  =  -  sin  0,    sin(  0+90°)  =cos  0, 

#0+9O°  =  -Ry.sin2  0+#zcos2  0. 


§  ill]        Normal  Sections  of  Curved  Surface  303 

Adding  the  curvatures  Re  and  Rd+<d0°,  we  obtain  the  rela- 
tion: 

Rd+Re+9o°=Ry+Rz; 
that  is,  the  algebraic  sum  of  the  curvatures  of  any  two  normal 
sections  intersecting  each  other  at  right  angles  at  a  point  on 
a  curved  surface  has  a  constant  value,  which  is  equal  to  the 
algebraic  sum  of  the  principal  curvatures  at  this  point. 

These  theorems  concerning  the  curvatures  of  the  normal 
sections  at  a  point  of  a  curved  surface  are  due  to  the  great 
mathematician  Euler  (1707-1783),  who  made  notable  con- 
tributions also  to  the  theory  of  optics. 

Since,  therefore,  the  curvature  of  a  surface  at  the  point  A 
varies  from  one  azimuth  to  another  as  has  just  been  ex- 
plained, the  power  of  a  refracting  surface  will  vary  in 
exactly  the  same  way.  Accordingly,  the  principal  sections 
for  which  the  curvature  of  a  refracting  surface  has  its  great- 
est and  least  values  (Ry,  Rz)  are  also  the  sections  at  this 
place  of  greatest  and  least  refracting  powers  (Fy,Fz),  because 

Fy  =  (n'-n)Ry,       Fy=(n'-ri)Rz. 
The  refracting  power  at  this  place  in  an  oblique  normal 
section  which  is  inclined  to  the  xy-plsaie  at  an  angle  6  will  be : 

Fd={nf~n)Re; 
and  the  relation  between  Fe  and  Fy,  Fz  is  given  by  the 
formula: 

F0  =  Fy.cos20+Fz.sin20; 
and  moreover: 

Fe+Fe+90o=Fy+Fz; 
that  is,  the  algebraic  sum  of  the  refracting  powers  in  any  two 
normal  sections  through  a  point  on  a  curved  refracting  sur- 
face is  constant  and  equal  to  the  algebraic  sum  of  the  princi- 
pal refracting  powers. 

For  example,  in  Fig.  144,  let  A  designate  a  point  of  a 
curved  refracting  surface,  and  let  the  normal  at  this  point 
be  represented  by  the  straight  line  Ax,  which  in  accordance 
with  the  preceding  discussion  is  to  be  taken  as  the  z-axis 
of  a  system  of  rectangular  coordinates  with  its  origin  at  A. 


304 


Mirrors,  Prisms  and  Lenses 


[§111 


The  y-Sixis  is  represented  by  a  straight  line  drawn  in  the 
plane  of  the  paper  perpendicular  to  Ax.  The  plane  of  the 
paper  represents  the  plane  of  one  of  the  principal  sections, 
whereas  the  £2-plane  at  right  angles  to  this  plane  represents 


Fig.  144. — Chief  ray  of  narrow  bundle  of  rays  normal  to  curved  re- 
fracting surface:  Principal  sections  xAy,  xkz;  tangent  plane  ykz. 

the  plane  of  the  other  principal  section.  The  tangent-plane 
at  A  is  represented  by  the  2/2-plane  perpendicular  to  the 
normal.  Consider  now  a  narrow  bundle  of  rays  which  pro- 
ceeding from  a  point  M  on  the  normal  are  incident  on  the 
curved  refracting  surface  at  points  which  are  all  very  close 
to  A.  This  point  M  may  be  designated  also  by  My  or  by 
Mz  according  as  it  is  regarded  as  lying  in  the  one  or  the  other 
of  the  two  principal  sections;  or  it  may  be  designated  also 
by  Me  if  it  is  to  be  considered  as  lying  in  an  oblique  normal 
section  which  is  inclined  to  the  ^-plane  at  an  angle  6.  The 
chief  ray  of  the  bundle  is  the  ray  which  coincides  with  the 
normal  to  the  surface  at  A  and  which  proceeds  therefore 
into  the  second  medium  without  being  deviated.  A  plane 
containing  this  chief  ray  will  cut  out  from  the  bundle  a  pen- 
cil of  rays  which  will  be  refracted  at  points  of  the  surface 
which  lie  in  a  normal  section.  The  pencil  of  rays  proceed- 
ing from  My  in  the  xy-pleme  will  be  refracted  to  a  point  My', 
while  the  pencil  of  rays  proceeding  from  Mz  will  be  refracted 
to  a  point  Mz';  and,  in  general,  these  points  My'  and  M/  will 
be  two  different  points  on  the  normal  Ax.  Now  if  Uyt  Uy' 
denote  the  reduced  "vergences"  (§  106)  of  the  pair  of  conju- 


§  112]  Surfaces  of  Revolution  305 

gate  points  My,  My'  in  one  principal  section;  and,  similarly, 
if  Uz,  Uz  denote  the  reduced  '' vergences"  of  the  pair  of  con- 
jugate points  Mz,  Mz'  in  the  other  principal  section,  evi- 
dently we  shall  have  the  following  relations: 

Uy'=Uy+Fy,  UZ'=UZ+FZ. 

Similarly,  also,  a  pencil  of  rays  proceeding  from  Me  and 
meeting  the  refracting  surface  at  points  in  an  oblique  nor- 
mal section  will  be  refracted  to  a  point  M0'  which  will  lie 
on  Ax  between  M/  and  M/,  so  that 
U0'=Ue+Fe. 

If  the  bundle  of  incident  rays  is  homocentric,  that  is,  if 
the  points  designated  by  My,  Mz  and  M0  are  all  coincident, 
then  Uy=Uz=Ud=U.  The  peculiarity  of  the  imagery 
consists  in  the  fact  that  instead  of  obtaining  a  single  image- 
point  M'  corresponding  to  an  object-point  M,  as  in  the  case 
of  a  spherical  refracting  surface,  we  find  here  a  whole  se- 
ries of  such  points  lying  on  the  segment  My'Mz'  of  the  nor- 
mal Ax.    This  will  be  explained  more  fully  in  §  113. 

112.  Surfaces  of  Revolution.  Cylindrical  and  Toric 
Surfaces. — The  curved  reflecting  and  refracting  surfaces 
of  optical  mirrors  and  lenses  are  almost  without  exception 
surfaces  of  revolution,  that  is,  surfaces  generated  by  the  revo- 
lution of  the  arc  of  a  plane  curve  around  an  axis  in  its  plane. 
Accordingly,  it  is  desirable  to  call  attention  to  some  of  the 
special  properties  of  these  surfaces.  The  curve  traced  on 
a  surface  of  revolution  by  a  plane  containing  the  axis  of 
revolution  is  called  a  meridian  section.  The  normals  to  the 
generating  curve  are  also  normals  to  the  surface;  and  since 
the  normal  at  any  point  of  the  surface  lies  in  the  meridian 
section  which  passes  through  that  point,  it  follows  that  the 
normals  to  a  surface  of  revolution  all  intersect  the  axis  of 
revolution. 

The  two  principal  sections  at  any  point  of  a  surface  of 
revolution  are  the  meridian  section  which  passes  through 
that  point  and  the  normal  section  which  is  perpendicular 
to  the  meridian  section.     The  center  of  curvature  of  the 


306 


Mirrors,  Prisms  and  Lenses 


112 


latter  principal  section  lies  on  the  axis  of  revolution  at  the 

point  where  the  normal  crosses  it. 

Not  only  are  the  surfaces  of  mirrors  and  lenses  generally 

surfaces  of  revolution,  but  usually  they  are  very  simple  types 

of  such  surfaces.  A  spher- 
ical surface  may  be  consid- 
ered as  generated  by  the. 
revolution  of  a  circle 
around  one  of  its  diame- 
ters. The  other  chief 
forms  of  reflecting  and  re- 
fracting surfaces  are  cyl- 
indrical and  toric  surfaces, 
which  are  also  compara- 
tively easy  to  grind. 

A  cylindrical  surface  of 
revolution  is  generated  by 
the  revolution  of  a  straight 
line  about  a  parallel  straight 
line  as  axis,  called  the  axis 
of  the  cylinder.  A  meridian 
section  of  a  cylinder  at  a 
point  A  on  the  surface 
(Fig.  145)  will  be  a  straight 
line  of  zero  curvature, 
whereas  the  other  principal 
section  at  right  angles  to 

Fig     145 -Refracting    power   of  cylin-  th    axig     f  th     cylinder  ^U 
drical     surface:    Principal     sections  ° 

made  by  planes  Ay  and  Kz;  oblique  be     the     arc     of     a      Circle 

section  AP.  whose  curvature  is  R  =  l/r, 

where  r  denotes  the  radius  of  the  cylinder.  If  the  i/-axis 
is  drawn  parallel  to  the  cylinder-axis,  then  Ry  =  0,  RZ  =  R; 
and  hence  according  to  Euler's  formula  given  in  §111, 
the  curvature  in  an  oblique  normal  section  AP  inclined  to 
the  axis  of  the  cylinder  at  an  angle  6  will  be 
Re=R.sm26. 


112] 


Cylindrical  Refracting  Surface 


307 


This  result  may  be  obtained  also  independently  by  observ- 
ing that  although  the  arcs  Az  and  AP  in  Fig.  145  have  the 
same  sagitta  (§  99),  their  chords  denoted  by  2h  and  2he  are 
unequal  in  length,  because  h  =  hd.smd.  Now  the  curva- 
tures of  two  arcs  having  the  same  sagitta  are  inversely  pro- 
portional to  the  squares  of  their  chords;  consequently, 

R     he*' 
and  hence 

Re  =  R.sin2d, 
exactly  as  above.     Moreover,  in  a  normal  section  perpen- 


Fig.  146. — Principal  sections  of  toric  surface. 


dicular  to  the  section  AP,  we  find,  by  writing  (0+90°)  in 
place  of  6, 

R0+CjQ°  =  R.cos2  6; 
and  therefore 

Re-\-Re+V0o=R' 
Accordingly,  in  the  case  of  a  cylindrical  refracting  sur- 
face, if  the  maximum  refracting  power  is  denoted  by  F> 
the  refracting  power  in  an  oblique  section  inclined  to  the 


308 


Mirrors,  Prisms  and  Lenses 


[§112 


axis  at  an  angle  6  will  be  F.sin2  6,  and  in  a  section  at  right 
angles  to  this  F.cos2  6.  The  refracting  power  F  of  a  cylin- 
drical refracting  surface  may,  therefore,  be  considered  as 
in  a  certain  sense  capable  of  resolution  into  a  refracting 
power  F.sin2  6  in  one  oblique  section  and  a  refracting  power 


Fig.  147,  a  and  b. — Toric  surfaces  (reproduced  from  Prentice's  Ophthalmic  Lenses 
and  Prisms  by  permission  of  the  author) . 

F.cos2  6  in  a  section  at  right  angles  to  the  first;  and  since 

F0+Fd+9O°=F, 
we  can  say  that  the  algebraic  sum  of  the  refracting  powers  in 
any  two  mutually  perpendicular  sections  of  a  cylindrical  re- 
fracting surface  is  constant  and  equal  to  the  maximum  refract- 
ing power. 

A  toric  or  toroidal  surface  (so-called  from  the  architect- 
ural term  torus  applied  to  the  molding  at  the  base  of  an 
Ionic  column)  is  a  surface  shaped  like  an  anchor-ring  which 


§  112]  Toric  Surfaces  and  Lenses  309 

is  generated  by  the  revolution  of  a  conic  section  around  an 
axis  which  lies  in  the  plane  of  the  generating  curve  but  does 
not  pass  through  its  center.  The  surface  of  an  automobile 
tyre  is  a  toric  surface,  being  generated  by  the  revolution 
of  the  circular  cross-section  of  the  tyre  around  an  axis  per- 


Fig.    148,   a  and    b. — Principal    sections    of    toric  lenses    (reproduced    from 
Prentice's  Ophthalmic  Lenses  and  Prisms  by  permission  of  the  author). 

pendicular  to  the  plane  of  the  wheel  at  its  center.  Toric 
refracting  surfaces  are  generated  always  by  the  revolution 
of  the  arc  of  a  circle  (Fig.  146).  The  arcs  of  the  two  prin- 
cipal sections  of  a  toric  surface  of  a  lens  bisect  each  other 
at  the  vertex  A  of  the  surface,  so  that  the  normal  Ax  is  an 
axis  of  symmetry.  If  the  axis  of  revolution  is  parallel  to 
the  2/-axis  of  the  system  of  rectangular  coordinates,  the 
center  of  the  meridian  section  through  A  is  at  the  center 
Cy  of  the  generating  circle,  whereas  the  center  of  the  other 
principal  section  at  A  is  at  the  point  of  intersection  Cz  of 
the  normal  Ax  with  the  axis  of  revolution. 

The  diagrams,  Fig.  147,  a  and  b  (which  are  copied  from 


310 


Mirrors,  Prisms  and  Lenses 


[§H3 


the  beautiful  drawings  of  Mr.  Prentice  in  his  valuable 
and  original  essay  on  " Ophthalmic  Lenses  and  Prisms' '  in 
the  American  Encyclopaedia  of  Opthhalmology)  show  the  two 
principal  forms  of  toric  surfaces.  The  principal  sections  of 
some  types  of  toric  lenses  are  indicated  in  Fig.  148,  a  and  b. 

A  cylindrical  surface  of  revolution  may  be  considered  as 
a  special  form  of  toric  surface  by  regarding  the  segment  of 
the  generating  straight  line  as  the  arc  of  a  circle  with  an 
infinite  radius. 

113.  Refraction  of  a  Narrow  Bundle  of  Rays  incident 
Normally  on  a  Cylindrical  Refracting  Surface.  Sturm's 
Conoid. — In  order  to  obtain  a  clear  idea  of  the  character 
of  a  bundle  of  rays  refracted  at  a  cylindrical  surface  or 
through  a  thin  cylindrical  lens,  suppose,  by  way  of  illustra- 


Fig.  149. — Chief  ray  of  narrow  bundle  meets  cylindrical  refracting  surface 
normally;  astigmatic  bundle  of  refracted  rays.  Principal  sections  xAy 
and  xAz. 


tion,  that  we  consider  a  special  case  of  the  problem  which  we 
had  in  §  111  in  connection  with  Fig.  144,  namely,  the  case  in 
which  a  narrow  homocentric  bundle  of  incident  rays,  origi- 
nally converging  towards  a  point  M,  is  intercepted  before  it 
reaches  this  point  by  being  received  on  a  cylindrical  refract- 
ing surface  which  is  placed  so  that  the  chief  ray  of  the  bundle 
meets  the  surface  normally  at  a  point  A  and  proceeds,  there- 
fore, along  the  normal  Ax  (Fig.  149)  without  being  deflected. 


§113]  Astigmatic  Bundle  of  Rays  311 

For  convenience  of  delineation,  the  cylindrical  surface  is 
represented  in  the  figure  as  the  first  surface  of  an  infinitely 
thin  piano-cylindrical  lens,  but  the  explanation  is  not  es- 
sentially affected  by  the  fact  that  it  applies  to  a  bundle  of 
rays  which  have  undergone  also  a  second  refraction  at  the 
plane  face  of  the  lens.  The  bundle  of  incident  rays  is  not 
represented  in  the  figure.  The  point  where  the  chief  ray 
meets  the  lens  is  designated  by  A.  In  the  drawing  this  point 
A  is  marked  on  the  second  or  plane  face  of  the  lens,  but  since 
the  lens  is  supposed  to  be  infinitely  thin,  this  point  may  be 
regarded  also  as  lying  on  the  first  face.  The  plane  of  the 
paper  represents  the  meridian  section  of  the  cylindrical  sur- 
face through  the  vertex  A,  and  hence  the  axis  of  the  cylinder 
is  in  this  plane  and  parallel  to  the  straight  line  Ay  perpendic- 
ular to  Ax  in  the  meridian  or  xy-pl&ne.  This  meridian  plane 
is  one  of  the  principal  sections  at  the  vertex  A  of  the  cylin- 
drical surface;  whereas  the  other  principal  section  is  the 
zz-plane  at  right  angles  to  the  plane  of  the  paper.  The 
bundle  of  rays  is  cut  by  these  principal  sections  in  a  pencil 
of  meridian  rays  lying  in  the  meridian  xy-pl&ne  and  a  pencil 
of  sagittal  rays  (named  by  analogy  with  the  so-called  "  sagittal 
suture"  in  anatomy)  lying  in  the  xz-pl&ne;  the  chief  ray  of 
the  bundle  being  common  to  both  of  these  pencils,  since  it 
is  the  line  of  intersection  of  the  two  principal  sections  of  the 
bundle.  Now  the  meridian  rays  traversing  the  infinitely 
thin  cylindrical  lens  in  a  section  containing  the  axis  of  the 
cylinder  will  be  entirely  unaffected  in  transit  and  will  pro- 
ceed therefore  to  the  point  M  just  as  though  the  thin  piece 
of  glass  had  not  been  interposed  in  the  way;  so  that  this 
point  regarded  now  as  the  point  of  rendezvous,  so  to  speak, 
of  the  meridian  rays  after  they  have  passed  through  the 
lens  may  also  be  designated  by  My',  as  in  fact  it  is  marked 
in  the  diagram.  On  the  other  hand,  the  rays  of  the  sagittal 
pencil  meet  the  surface  in  points  lying  on  the  arc  of  the  sec- 
tion made  by  the  zz-plane,  and  the  rays  in  this  plane  are 
refracted  just  as  they  would  be  through  a  piano-spherical 


312  Mirrors,  Prisms  and  Lenses  [§  113 

lens  of  the  same  curvature  as  that  of  the  cylinder;  and  ac- 
cordingly after  passing  through  the  lens  they  will  be  brought 
to  a  focus  at  a  point  Mz'  on  the  chief  ray  Ax,  which  in  the 
case  here  supposed  will  be  between  the  lens  and  the  point 
My',  as  represented  in  the  figure. 

The  bundle  of  rays  after  refraction  is  no  longer  homocentric, 
so  that  an  object-point  is  not  reproduced  in  a  cylindrical 
lens  by  a  single  image-point  or  even  by  a  pair  of  image-points, 
since  only  the  meridian  and  sagittal  image-rays  intersect 
in  the  so-called  image-points  My'  and  M/,  respectively. 
Under  such  circumstances,  the  bundle  of  image-rays  is  said 
to  be  astigmatic  (or  without  focus),  which,  in  fact,  is  the 
general  character  of  a  bundle  of  optical  rays,  as  will  be 
further  explained  in  Chapter  XV. 

Rays  which  are  incident  on  the  cylindrical  surface  in  an 
oblique  section  made  by  a  plane  containing  the  normal  Ax 
will  be  brought  to  a  focus  at  a  point  lying  between  My'  and 
Mz',  as  explained  in  §  111.  But  the  two  points  My'  and  Mz' 
have  a  superior  right  to  be  regarded  as  the  image-points  of 
the  astigmatic  bundle  of  rays,  not  only  because  they  are 
the  image-points  of  the  two  principal  pencils  of  the  bundle, 
but  also  because  the  so-called  image-lines  of  the  astigmatic 
bundle  of  rays  are  located  at  these  places,  as  we  shall  pro- 
ceed to  show. 

Imagine  a  straight  line  drawn  on  the  surface  of  the  cylin- 
der parallel  to  the  i/-axis  and  at  a  short  distance  from  the 
zy-plane,  and  consider  the  pencil  of  rays  which  meet  the 
surface  in  points  lying  along  this  line;  these  rays  after  pass- 
ing through  the  lens  will  meet  in  a  point  in  the  zz-plane  a 
little  to  one  side  of  the  image-point  My';  and  the  assemblage 
of  these  image-points  will  form  a  very  short  image-line  per- 
pendicular to  the  meridian  section  of  the  bundle  of  rays  at 
the  point  M/;  just  as  though  the  pencil  of  meridian  rays 
had  been  rotated  through  a  very  small  angle  around  an 
axis  parallel  to  the  y-a,xis  and  passing  through  Mz.'  And, 
similarly,  if  the  pencil  of  sagittal  rays  is  rotated  slightly 


113] 


Sturm's  Conoid 


313 


on  both  sides  of  the  zz-plane  around  an  axis  parallel  to  the 
2-axis  and  passing  through  the  image-point  M/,  the  image- 
point  Mz'  will  trace  out  a  little  image-line  perpendicular  to 
the  sagittal  section  of  the  astigmatic  bundle  of  rays.  Thus, 
instead  of  a  point-like  image  of  a  point-like  object  or  point- 
to-point  correspondence  between  object  and  image,  that  is, 
instead  of  the  so-called  punctual  imagery  which  we  have 
when  paraxial  rays  are  reflected  or  refracted  at  a  spherical 
surface,  we  obtain  here  something  essentially  different;  for 
in  this  case  each  point  of  the  object  is  reproduced  by  two 


A 

l2 ^"~-~---«^^^ 

x 

y 

o 

Z               3           A         5""^ 

-    ooQ 

_         7 

0 

Fig.  150. — Sturm's  conoid. 

tiny  image-lines,  each  perpendicular  to  the  chief  ray  of  the 
bundle,  one  in  one  principal  sectio?i  and  the  other  in  the  other 
principal  section;  so  that  if  one  of  the  image-lines  is  vertical, 
the  other  will  be  horizontal.  The  image-line  which  passes 
through  the  image-point  of  the  meridian  rays  lies  in  the 
plane  of  the  sagittal  section,  and  vice-versa. 

The  case  in  which  an  object-point  is  reproduced  by  two 
short  image-lines  is  the  simplest  form  of  astigmatism,  and 
it  is  only  under  exceptionally  favorable  circumstances  that 
it  can  be  actually  realized  as  described  above.  The  astig- 
matic bundle  of  rays  represented  in  Fig.  150,  which  is  com- 
pletely symmetrical  in  the  two  principal  sections  is  known 
as  Sturm's  conoid  after  the  celebrated  mathematician  who 
appears  to  have  been  the  first  to  make  a  systematic  investi- 
gation (1838)  of  the  characteristics  of  a  narrow  bundle  of 


314 


Mirrors,  Prisms  and  Lenses 


[§114 


optical  rays.  If  the  lens-opening  is  determined  by  a  small 
circular  stop  in  a  plane  at  right  angles  to  the  optical  axis 
(or  rr-axis)  and  with  its  center  on  this  axis,  the  transverse 
sections  of  the  astigmatic  bundle  of  refracted  rays  made  by 
planes  perpendicular  to  the  chief  ray  (that  is,  parallel  to 
the  2/2-plane)  will  be  ellipses  with  their  major  axes  parallel 
to  the  i/-axis  in  one  part  of  the  bundle  and  parallel  to  the 
2-axis  in  the  other  part.  These  elliptical  sections  become 
narrower  and  narrower  as  they  approach  either  of  the  image- 
lines,  at  both  of  which  places  the  elliptical  section  collapses 
into  the  major-axis  of  the  ellipse.  At  some  intermediate 
point  between  the  two  image-lines  the  section  of  the  bundle 
will  be  a  circle  (the  so-called  " circle  of  least  confusion"). 
114.  Thin  Cylindrical  and  Toric  Lenses. — Optical  lenses 
may  now  be  classified  in  two  principal  groups,  namely, 
anastigmatic  (or  simply  stigmatic)  lenses  and  astigmatic  lenses, 
according  as  the  imagery  produced  by  the  refraction  of  par- 


es, Concave. 


b,  Convex. 
Fig.  151,    a  and  b. — Piano-cylindrical  lenses. 

axial  rays  through  the  lens  is  punctual  imagery  or  not  (§  113). 
Anastigmatic  lenses  are  single  focus  lenses,  whereas  astig- 
matic lenses  may  be  said  to  be  double  focus  lenses.  The 
essential  requirement  is  that  the  optical  axis  of  the  lens, 


114] 


Cylindrical  Lenses 


315 


which  is  generally  an  axis  of  symmetry,  shall  meet  both 
faces  normally  (§  87) ;  and  another  condition  that  must 
always  be  fulfilled  in  an  actual  lens  is  that  the  planes  of  the 
principal  sections  at  the  vertex  of  the  first  surface  shall  also 
be  the  planes  of  the  principal  sections  at  the  vertex  of  the 
second  surface.  Astigmatic  lenses  are  generally  cylindrical 
or  toric. 

Cylindrical  lenses  are  made  in  three  forms,  namely,  piano- 
cylindrical    (one   surface   cylindrical   and   the   other   plane, 


Fig.  152. — Sphero-cylindrical  lens. 


Fig.  153. — Sphero-cylindrical  lens. 

Fig.  151,  a  and  6),  cross-cylindrical  (both  surfaces  cylindrical, 
the  axes  of  the  cylinders  being  at  right  angles),  and  sphero- 
cylindrical (one  surface  cylindrical  and  the  other  spherical, 
Figs.  152  and  153).  All  of  these  forms  are  quite  common  in 
modern  spectacle  glasses,  but  prior  to  1860  cylindrical  lenses 
were  hardly  employed  at  all.  The  first  scientific  use  of  a 
cylindrical  lens  seems  to  have  been  made  by  Fresnel 
(1788-1827)  in  1819  for  the  purpose  of  obtaining  a  luminous 
line.  In  1825  Sir  George  Airy  (1801-1892),  afterwards  the 
distinguished  astronomer-royal  at  Greenwich,  employed  a  con- 


316  Mirrors,  Prisms  and  Lenses  [§  114 

cave  sphero-cylindrical  glass  to  correct  the  myopic  astigma- 
tism of  one  of  his  eyes.  But  it  was  not  until  Donders 
(1818-1889)  published  his  treatise  on  astigmatism  and  cyl- 
indrical glasses  in  1862  that  their  importance  began  to  be 
recognized  by  ophthalmologists  all  over  the  world. 

In  a  toric  lens  usually  only  one  of  the  surfaces  is  toric 
(§  112),  while  the  other  is  plane  or  spherical.  The  diagrams, 
Fig.  147,  a  and  b,  and  Fig.  148  show  the  principal  types  of 
toric  lenses. 

Let  Fyti,  Fyt2  and  FZii,  FZy2,  denote  the  refracting  powers 
of  the  two  surfaces  of  an  astigmatic  lens  in  the  xy-plsaie  and 
zz-plane,  respectively,  which  are  the  planes  of  the  principal 
sections  of  the  thin  lens  with  respect  to  its  optical  center  A. 
Now  the  total  refracting  power  (F)  of  a  thin  lens  was  found 
(§  105)  to  be  equal  to  the  algebraic  sum  (F1+F2)  of  the 
powers  of  the  two  surfaces  of  the  lens;  so  that  applying  this 
formula  to  an  astigmatic  lens,  we  obtain  for  the  refracting 
power  in  the  two  principal  sections : 

Fy  =  Fy<i-\-Fyt2,         Fz  =  FZt\-\-Fz^ 

In  each  of  the  following  special  cases  the  lens  is  supposed 
to  be  surrounded  by  the  same  medium  (n)  on  both  sides, 
while  the  index  of  refraction  of  the  lens  itself  is  denoted  by  n'. 

(1)  Consider,  first,  the  case  of  a  piano-cylindrical  lens, 
which  in  a  principal  section  containing  the  axis  of  the  cylin- 
der acts,  as  was  remarked  (§  113),  like  a  slab  of  the  same 
material  with  plane  parallel  faces;  whereas  in  the  other  prin- 
cipal section  the  effect  is  the  same  as  that  of  a  piano-spherical 
lens  of  the  same  radius  (r)  as  that  of  the  cylinder.  If  the 
axis  of  the  cylinder  is  parallel  to  the  y-axis,  and  if  the  plane 
surface  is  supposed  to  be  the  second  surface,  we  shall  have 
in  this  case : 

Fy,i  =  Fy<2  =  Fz<2  =  0, 
and,  consequently: 

Fy  =  0,        Fz  =  FZtl  =  F=(n'-n)R, 
where  F  denotes  the  maximum  refracting  power  of  the  cylin- 
drical surface,  and  R  =  l/r  denotes  its  curvature. 


§  114]  Cylindrical  and  Toric  Lenses  317 

If  M  designates  the  position  of  an  object-point  lying  on 
the  optical  axis  (z-axis)  of  a  thin  piano-cylindrical  lens,  and 
if  M#'  designates  the  position  of  the  corresponding  image- 
point  produced  by  the  refraction  through  the  lens  of  the 
rays  which  lie  in  the  plane  of  a  normal  section  inclined  at 
an  angle  6  to  the  axis  of  the  cylinder;  and  if  we  put 

AM  =  u,        AMe'  =  u',        U  =  n/u,        Ud'  =  n/ue', 
then 

Ue'=  U+Fe,  where  Fe  =  F. sin2  6; 
and  for  the  two  principal  sections: 

Uy'=U,        UZ'=U+F. 

(2)  In  a  cross-cylindrical  lens  the  axes  of  y  and  z  are  par- 
allel to  the  axes  of  the  cylinders.  Assuming  that  the  cylin- 
drical axis  of  the  first  surface  of  the  lens  is  parallel  to  the 
2/-axis,  we  have  for  a  thin  lens  of  this  form : 

Fy  =  Fy,2=-(nf-n)R2,        Fz  =  FZil=(n' '-n)Rh 
F=(n'-n)(R1.sm2d  -ft.cos80); 
where  R\,  R2  denote  the  maximum  curvatures  of  the  cylin- 
ders and  Fe  denotes  the  refracting  power  in  a  section  in- 
clined at  an  angle  6  to  the  axis  of  the  first  surface. 

(3)  In  a  thin  sphero-cylindrical  lens,  if  we  suppose,  for 
example,  that  the  axis  of  the  cylindrical  surface  is  parallel 
to  the  ?/-axis  and  that  this  surface  is  also  the  first  surface 
of  the  lens,  then 

Fy,i  =  0,  Fy,2=Fz,2  =  F2, 

Fy  =  Fy,2=-(nf-n)R2, 
Fz  =  Fz,1+Fy  =  (nf-n)(Rl-R2), 
Fe=(n'-n)(R1.sm2d-R2); 
where  R\,  R2  denote  the  maximum  curvatures  of  the  cylin- 
drical and  spherical  faces,  respectively,  and  Fe  denotes  the 
refracting  power  of  the  combination  in  a  plane  inclined  at 
an  angle  6  to  the  axis  of  the  cylinder. 

(4)  Consider,  finally,  a  thin  toric  lens,  whose  second  face 
may  be  supposed  to  be  spherical,  so  that  if  r2  denotes 
the   radius    of   this   surface,   its    refracting    power  will  be 


318  Mirrors,  Prisms  and  Lenses  [§  115 

'Ft—  —  {n'  —  n)R2}  where  R2  =  lfr2.  Then  if  Ryyh  Rz,i  denote 
the  principal  curvatures  of  the  toric  surface,  the  refracting 
powers  of  the  lens  will  be 

Fy=(n'-n)  (Ry,1-R2))  Fz  =  {n'-n)  (Rz,i-Rt), 
Fe=  (n'-n)  (#y,i.cos2  d-\-Rz,i.sin2  0-R2). 
115.  Transposing  of  Cylindrical  Lenses. — The  orientation 
of  a  cylindrical  refracting  surface  is  described  by  assigning 
the  value  of  the  angle  <p  which  the  axis  of  the  cylinder  makes 
with  a  fixed  line  of  reference.  In  a  cylindrical  spectacle 
glass  this  line  of  reference  is  a  horizontal  line  usually  imag- 
ined as  drawn  from  a  point  opposite  the  center  of  the  pa- 
tient's eye  either  towards  his  temple  or  towards  his  nose; 


18CT x      \  y 10'  isoj 

TEMPLE  NOSE        NOSE  TEM.P.LE 

Fig.  154. — Mode  of  reckoning  axis  of  cylindrical  eye-glass. 

and  the  angle  through  which  this  line  has  to  be  rotated  in 
a  vertical  plane  in  order  for  it  to  be  parallel  to  the  axis  of 
the  cylinder  is  the  angle  denoted  by  <p.  In  England  and 
America  it  is  customary  to  imagine  the  horizontal  line  of 
reference  as  drawn  from  the  center  of  the  glass  towards  that 
temple  of  the  patient  which  is  on  the  right-hand  side  of  an  ob- 
server supposed  to  be  adjusting  the  glass  on  the  patient's 
eye;  so  that  for  a  glass  in  front  of  either  eye  the  radius  vector 
is  supposed  to  rotate  in  a  counter-clockwise  sense  from  0° 
to  180°,  as  represented  in  Fig.  154.  A  different  plan  was 
recommended  by  the  international  ophthalmological  con- 
gress which  met  in  Naples  in  1909,  whereby  the  angle  cp  was 
to  be  reckoned  from  an  initial  position  of  the  radius  vector 
drawn  horizontally  from  a  point  opposite  the  center  of  the 
eye  towards  the  nose.     According  to  this  plan,  the  sense  of 


§  115]  Transposing  of  Cylindrical  Lenses 


319 


rotation  will  be  clockwise  for  one  eye  and  counter-clockwise 
for  the  other  eye,  as  represented  in  Fig.  155. 

A  sphero-cylindrical  glass  is  described  in  an  ophthalmo- 
logical  prescription  by  giving  the  refracting  power  P  of 
the  cylindrical  component  and  the  refracting  power  Q  of 
the  spherical  component,  together  with  the  slope  <p  of  the 
axis  of  the  cylinder,  in  a  formula  which  is  usually  written 
as  follows: 

Q  sph.  3  P  cyl.,  slx.<p, 
where  the  symbol  O  means  "combined  with." 

Opticians  speak  of  transposing  a  lens  when  they  substi- 
tute a  glass  of  one  form  for  an  equivalent  glass  of  another 


180 


180° 


TEMPLE  NOSE      NOSE  TEMPLE 

Fig.  155. — Mode  of  reckoning  axis  of  cylindrical  eye-glass. 


form.  All  that  is  necessary  for  this  purpose  is  to  see  that 
the  powers  of  the  lens  in  the  two  principal  sections  remain 
the  same  as  before.  The  following  rules  for  transposing 
cylindrical  lenses  may  be  useful : 

(1)  To  transpose  a  sphero-cylindrical  lens  into  another 
sphero-cylindrical  lens  or  into  a  cross-cylindrical  lens: 

A  lens  given  by  the  formula  Q  sph.  O  P  cyl.,  ax.  <p  is 
equivalent  to  either  of  the  following  combinations : 

a.  Sphero-cylinder:  (P+Q)  sph.  C  -P  cyl.,  ax.  (<p  ±90°) 

b.  Cross-cylinder :  (P+  Q)  cyl. ,  ax.  <p  O  Q  cyl. ,  ax.  ( <p  ±  90°) . 
The  power  of  the  spherical  component  in  the  original  com- 
bination is  Q  dptr.  in  both  principal  sections,  and  the  power 
of  the  cylindrical  component  is  P  dptr.  in  the  section  which  is 
inclined  to  the  line  of  reference  at  an  angle  (<p  =*=  90°) ;  so  that 
the  combined  power  in  this  latter  section  is  (P+Q)  dptr. 


320  Mirrors,  Prisms  and  Lenses  [§  116 

Accordingly,  a  spherical  surface  of  power  (P+Q)  dptr. 
must  be  combined  with  a  cylindrical  surface  of  power 
— P  dptr.  and  of  axis-slope  (^>=*=90°).  With  respect  to  the 
double  sign  in  the  expression  v  <£>=•=  90°),  the  rule  is  to  select 
always  that  one  of  the  two  signs  which  will  make  the  slope 
of  the  cylinder-axis  positive  ard  less  than  180°.  Thus,  for 
example,  +8  dptr.  sph.  O  +2  dptr.  cyl.',  ax.  20°  is  equiva- 
lent to  +10  dptr.  sph.  O  —  2  dptr.  cyl.,  ax.  110°  or  to 
+  10  dptr.  cyl.,  ax.  20°  C  +8  dptr.  cyl.,  ax.  110°. 

(2)  To  transpose  a  cross-cylindrical  lens  into  a  sphero- 
cylindrical lens: 

The  combination  P  cyl.,  ax.  <p  O  R  cyl.,  ax.  (<£>=*=  90°) 
is  equivalent  to  either  of  the  following: 

a.  Sphero-cylinder:  P  sph.  O  (R—P)  cyl.,  ax.  (<p  =*=90°),  or 

b.  Sphero-cylinder:  R  sph.  O  (P—R)  cyl.,  ax.  <p. 

Thus,  +2  cyl.,  ax.  80°  C  +3  cyl.,  ax.  170°  may  be  replaced 
by  either  +2  sph.  C  +1  cyl.,  ax.  170°  or  +3  sph.  O 
-1  cyl.,  ax.  80°. 

(3)  To  transpose  a  spherical  lens  into  a  cross-cylinder: 

Q  sph.  is  equivalent  to  Q  cyl.,  ax.  <p  O  Q  cyl.,  ax.  (<p  =*=  90°), 
where  the  angle  (p  may  have  any  value  between  0°  and  180°. 
For  example,  +5  sph.  is  equivalent  to  +5  cyl.,  ax.  10°  O 
+5  cyl.,  ax.  100°. 

(4)  The  refracting  powers  of  a  toric  surface  in  the  prin- 
cipal sections  are  Fy=(nf  —  n)/ry  and  Fz  =  (n'  —  n)/rz.  Let 
us  suppose  that  the  axis  of  revolution  is  parallel  to  the 
2/-axis.  The  toric  refracting  surface  may  be  replaced  by  a 
sphero-cylindrical  lens  in  either  of  two  ways,  as  follows: 

.a.  Fz  sph.  O  (Fy—Fz)  cyl.,  axis  parallel  to  y-sads. 

b.  Fy  sph.  O  (Fz  —  Py)  cyl.,  axis  parallel  to  2-axis. 

116.  Obliquely  Crossed  Cylinders. — Oculists  and  optom- 
etrists sometimes  prescribe  a  bi-cylindrical  spectacle-glass 
with  the  axes  of  the  cylinders  crossed,  not  at  right  angles 
(as  in  the  so-called  cross-cylinder) ,  but  at  an  acute  or  obtuse 
angle  7;  and  as  it  is  not  easy  to  grind  a  lens  of  this  form, 
the  optician  prefers  to  make  an  equivalent  sphero-cylinder 


116] 


Obliquely  Crossed  Cylinders 


321 


or  a  cross-cylinder,  which  will  have  precisely  the  same  op- 
tical effect  as  the  prescribed  combination  of  obliquely  crossed 
cylinders.    His  problem  may  be  stated  thus : 

Being  given  the  refracting  powers  Fi,  F2  of  the  two  sur- 
faces of  the  bi-cylindrical  lens,  and  the  angle  y  between 
the  directions  of  the  axes  of  the  cylinders,  it  is  required  to 
calculate  the  refracting  powers  P  and  Q  of  the  cylindrical 
and  spherical  components,  respectively,  of  the  equivalent 
sphero-cylindrical  combination,  together  with  the  direction 
of  the  axis  of  the  cylinder;  that  is,  it  is  required  to  transpose 

Fi  cyl.,  ax.<p  C  F2  cyl.,  ax.  (<p+y) 
into 

Q  sph.  O  P  cyl.,  ax.  (<p-\-  a). 

Simple  working  formula?  for  converting  one  of  these  lenses 
into  the  other  were  developed 
first  by  Mr.  Charles  F. 
Prentice.  The  following 
method  is  based  on  an  ar- 
ticle "  On  obliquely  crossed 
cylinders"  by  Professor  S.  P. 
Thompson  published  in  the 
Philosophical  Magazine  (se- 
ries 5,  xlix.,  1900,  pp.  316- 
324). 

In  Fig.  156  the  straight 
lines  OA  and  OB  are  drawn 
parallel  to  the  cylindrical 
axes  of  the  bi-cylindrical  lens, 
sothatZAOB  =  7.  Through 
O  draw  another  straight  line 
OC,  and  let  ZAOC  be  de- 
noted by  6.  In  the  sec- 
tion of  the  lens  at  right 
angles  to  OC  the  total  r 
§112): 

Fi.cos20+^2.cos2(y 


Fig. 


156. — Axes  of  obliquely  crossed 
cylinders. 


efracting    power    will    be    (see 


ey, 


322  Mirrors,  Prisms  and  Lenses  [§  116 

and  in  the  section  containing  OC : 

Fi.sin20+^2.sin2(7-0). 

The  sum  of  these  two  expressions  is  equal  to  (F1+F2);  and 
according  to  the  theory  of  curved  surfaces  (§  111),  this  sum 
must  also  be  equal  to  the  sum  of  the  maximum  and  mini- 
mum refracting  powers  of  the  equivalent  sphero-cylindrical 
lens.  Now,  obviously,  (P+0)  will  be  the  maximum  (or 
minimum)  refracting  power  in  a  section  of  the  latter  lens 
at  right  angles  to  the  axis  of  the  cylinder,  whereas  Q  will  be 
the  minimum  (or  maximum)  refracting  power  in  the  sec- 
tion containing  the  axis  of  the  cy Under;  accordingly,  first 
of  all,  we  find  that  we  must  have: 

2Q+P  =  F1+F2. 

Now  there  is  a  certain  value  of  the  angle  6,  say,  6  =  a, 
for  which  the  first  of  the  two  expressions  above  will  be  a 
maximum  (or  minimum)  and  the  second  a  minimum  (or 
maximum) ;  and  if  we  can  determine  this  angle  a,  the  prob- 
lem will  practically  be  solved,  because  then  we  shall  have: 
P+0  =  ^i.cos2a+F2.cos2(7-  a), 
Q  =  Fi.sin2a+F2.sin2(7-  a); 

where  (on  the  assumption  that  Q  is  the  minimum  refracting 
power  in  the  section  containing  the  axis  of  the  cylinder)  a 
denotes  the  angle  between  the  cylindrical  axis  of  the  sphero- 
cylinder  and  the  cylindrical  axis  of  the  cylinder  whose  refract- 
ing power  is  denoted  by  F\.  Now  in  order  to  ascertain  this 
angle  a,  all  we  have  to  do  (as  will  be  obvious  to  any  one 
who  is  familiar  with  the  elements  of  the  differential  calculus) 
is,  first,  to  differentiate  the  expression 

Fi.cos20+F2.cos2(7-0) 

with  respect  to  6,  and  then,  after  writing  a  in  place  of  0, 
to  put  the  resultant  expression  equal  to  zero.  Thus  we  ob- 
tain the  following  equation  for  finding  the  angle  a  in  terms 
of  the  known  magnitudes  Fh  F2  and  7 : 

—  2i^i.sina  .cosa+2F2.sin(7—  a).cos(7—  a)=0; 


§116] 


Obliquely  Crossed  Cylinders 


323 


which  may  also  be  put  in  the  following  form: 
F\         _    F2 
sin2(7—  a)     sin2a' 

Moreover,  since  P  =  (P*+Q)  —  Q,  we  find: 

P  =  Fi(cos2a-sin2a)+F2  {cos2(7~  a)—  sin2(y-  a)} 
=  Fi.cos2a+F2.cos2(7—  a); 

and  if  in  this  formula  we  substitute  the  value 

sin2a 


F,  =  -. 


sin2(7  — a) 


Fi, 


we  shall  find : 


sin27 
sin2(7~  a) 


v*y 


*V 


a* 

Fig.  157. — Graphical  mode  of  finding  cylindrical  component  (P)  of 
sphero-cylinder  equivalent  to  two  obliquely  crossed  cylinders  of  powers 
F\   and  F2. 

Hence, 


Fl 


sin2(7~  a)     sin2a     sin27* 


324  Mirrors,  Prisms  and  Lenses  [§  116 

which  at  once  suggests  an  elegant  and  simple  graphical 
solution  of  the  problem.  For,  evidently,  according  to  the 
above  relations,  the  magnitudes  denoted  by  Fh  F2  and  P 
may  be  represented  in  a  diagram  (Fig.  157)  by  the  sides  of 
a  triangle  whose  opposite  angles  are  2(7— a),  2a  and 
(180°— 27),  respectively.  Hence  the  rule  is  as  follows: 
On  any  straight  line  lay  off  a  segment  AB  to  represent,  ac- 
cording to  a  certain  scale,  the  magnitude  of  the  refracting 
power  Fi;  and  let  X  designate  the  position  of  a  point  on  AB 
produced  beyond  B.  Construct  the  ZXBC  equal  to  twice 
the  angle  between  the  axes  of  the  two  given  cylindrical  com- 
ponents (ZXBC  =  2 7);  and  along  the  side  BC  of  this  angle 
lay  off  the  length  BC  to  represent  the  magnitude  of  the  re- 
fracting power  F2.  Then  the  straight  line  AC  will  repre- 
sent on  the  same  scale  the  magnitude  of  the  refracting  power 
P  of  the  cylindrical  member  of  the  equivalent  sphero- 
cylindrical lens,  and  the  Z  B AC  =  2  a  will  be  equal  to  twice 
the  angle  between  the  cylindrical  axes  of  the  surfaces  whose 
powers  are  denoted  by  F\  and  P.  For  calculating  the  values 
of  P,  Q  and  a,  we  have  by  trigonometry  the  following  sys- 
tem of  formulae :  

P  =  +  \/F21+F22+2Fi.F2.cos2y, 
Q=FM-P 


tan2a 


2 

F2.sm2y 


^1+F2.cos27' 

which  will  be  found  to  be  applicable  in  all  cases,  whether 
the  signs  of  Fh  F2  are  like  or  unlike. 

There  is,  to  be  sure,  another  solution  also,  in  which  the 
cylindrical  axis  of  the  sphero-cylindrical  lens  is  inclined  to 
the  cylindrical  axis  of  the  cylinder  of  power  Fi  at  the  angle 
(90°+  a).  For  if  the  refracting  power  Q  of  the  spherical 
member  is  assumed  to  be  the  maximum  (instead  of  the 
minimum)  refracting  power  of  the  sphero-cylindrical  com- 
bination, then  (P-f-Q)  will  be  the  minimum  power  in  a  sec- 
tion at  right  angles  to  the  axis  of  the  cylinder;  and  in  this 


§  116]  Obliquely  Crossed  Cylinders  325 

case  the  refracting  power  of  the  cylindrical  component  will 

be  represented  by  the  dotted  line  AC  in  Fig.  157  which  is 

equal  to  AC  in  length  but  opposite  to  it  in  direction.     In 

fact,  in  this  case  the  formulae  for  P  and  Q  will  be  as  follows: 

P  =  -  VFl-]-Fi+2Fi.F2.GOs2y, 

n_Fi±IW> 

Q  2         * 

This  result  could  have  been  obtained  from  the  first  result  by 

transposing;  for,  according  to  §  115,  Q  sph.  O  P  cyl.,  ax.  <f> 

is  equivalent  to  (P+Q)  sph.  O  —  P  cyl.  ax.  (<£=*=  90°),  where 

the  symbols  P  and  Q  denote  here  the  powers  of  the  first 

combination. 

Moreover,  since  Q  sph.  O  P  cyl.,  ax.<£  is  equivalent  also 
to  (P+Q)  cyl.,  ax.</>  O  Q  cyl.,  ax.  (<£=*=  90°),  two  obliquely 
crossed  cylinders  may  be  replaced  by  a  cross-cylinder  of 
powers  (P+Q)  and  Q.    In  fact,  since 

(P+Q)+Q=Pi+P2, 

(P+Q)~Q  =  VPi+Pi+2Pi.P2.cos2  y, 
it  follows  that : 

(P+Q)Q=F1.F2.sin'2y; 
so  that  this  formula  will  give  us  the  product  of  the  powers 
of  the  equivalent  cross-cylinder,  and  since  their  sum  P+2Q 
=  Fi+F2,  the  values  of  (P+Q)  and  Q  may  be  obtained  in- 
dependently, without  first  finding  the  value  of  P. 

The  following  numerical  example  will  serve  to  illustrate 
the  use  of  the  formula; : 

Given  a  combination  of  obliquely  crossed  cylinders  as 
follows : 

+4  cyl.,  ax.  20°  C  -2.75  cyl.,  ax.  65°; 
let  it  be  required  to  find  the  equivalent  sphero-cylinder 
and  also  the  equivalent  cross-cylinder. 

We  must  put  Pi  =  +4,  because  Pi  denotes  the  power  of 
the  cylinder  whose  axis-slope  is  the  smaller  of  the  two.  Then 
F2=  -2.75  and  y  =(65° -20°)  =45°.  Substituting  these 
values,  we  find : 

P=+4.86,       Q=-1.8,        a=-17°16'. 


326  Mirrors,  Prisms  and  Lenses  [Ch.  IX 

Accordingly,  the  given  combination  is  equivalent  to  one  of 
the  three  following: 

+4.85  cyl.,  ax.    2°  44'  O- 1.8  sph. ; 
-4.85  cyl.,  ax.  92°  44'  C  +3.05  sph.; 
+3.05  cyl.,  ax.    2°  44'  C  - 1.8  cyl.,  ax.  92°  44'. 
If   7  =  90o,     then    P  =  F1-F2,     Q  =  F2    and     a  =  0°,    or 
P  =  F2-Fi,  Q=F\  and  a  =  0°;  so  that  we  can  write: 

Fi  cyl.,  ax.  <£  C  F2  cyl.,  ax.  (<£±90°) 
is  equivalent  to 

Fi  sph.  C  (F2-Fi)  cyl.,  ax.  (<£±90°) 
or 

F2  sph.  C  (F1-F2)  cyl.,  ax.  cj>; 
exactly  as  found  in  §  115. 

PROBLEMS 

1.  The  radius  of  a  convex  cylindrical  refracting  surface 
separating  air  from  glass  (n  =  1.5)  is  8|-  cm.  What  is  its 
refracting  power  in  a  normal  section  inclined  to  the  axis  of 
the  cylinder  at  an  angle  of  60°?  Ans.  +4.5  dptr. 

2.  A  curved  refracting  surface  separates  air  and  glass 
(n':n  =  3:  2),  and  the  radii  of  greatest  and  least  curvature 
at  a  point  A  on  the  surface  are  ry  =  +  10  cm.  and  rz  = 
+5  cm.  Find  the  interval  between  the  two  principal  image- 
points  corresponding  to  an  object-point  lying  on  the  normal 
to  the  surface  at  A  in  front  of  the  surface  and  at  a  distance 
of  30  cm.  from  it.  Ans.  67.5  cm. 

3.  The  principal  refracting  powers  of  a  thin  astigmatic 
lens  surrounded  by  air  are  denoted  by  Fy  and  Fz.  The  prin- 
cipal image-points  corresponding  to  an  axial  object-point 
M  are  designated  by  My  and  Mz.  If  the  optical  center  of 
the  lens  is  designated  by  A,  and  if  we  put  U=l/u,  where 
u= AM,  then 

M'  M'  = Fy~Fz . 


Ch.  IX]  Problems  327 

4.  The  refracting  powers  of  a  thin  astigmatic  lens  in  the 
two  principal  sections  are  +3  and  +5  dptr.  The  lens  is 
made  of  glass  of  index  1.5.  Find  the  radii  of  the  two  sur- 
faces for  each  of  the  following  forms:  (a)  Cross-cylinder; 
(6)  Sphero-cylinder;  c)  Plano-toric. 

Ans.  (a)  Double  convex  cross-cylinder,  radii  10  and 
16  -|  cm.;  (6)  Double  convex  sphero-cylinder,  radius  of 
sphere  16  f  cm.,  radius  of  cylinder  25  cm.;  or  convex  me- 
niscus sphero-cylinder,  radius  of  sphere  10  cm.,  radius  of 
cylinder  25  cm. ;  (c)  Radii  of  toric  surface  10  and  16  J  cm. 

5  The  principal  refracting  powers  of  a  thin  lens  are  +4 
and  —5  dptr.  If  the  refracting  power  in  an  oblique  normal 
section  is  +2  dptr.,  what  will  be  its  refracting  power  in  a 
normal  section  at  right  angles  to  the  first?  and  what  is  the 
angle  of  inclination  of  the  +2  section  to  the  +4  section? 

Ans.  -3  dptr.;  28°  7' 32". 

6.  Two  cylinders  each  of  power  +1.18  dptr.  are  com- 
bined with  their  axes  inclined  to  each  other  at  an  angle  of 
32°  3'  50".  Show  that  the  combination  is  equivalent  to 
+0.18  sph.  O  +2  cyl.,  axis  midway  between  the  axes  of 
the  two  given  cylinders. 

7.  Show  that 

+2  cyl.,  ax.  0°  C  -3  cyl.,  ax.  53°  26'  14" 
is  equivalent  to 

-2.53  sph.  C  +4.06  cyl.,  ax.  -22°  30'. 

8.  Transpose 

-1.25  cyl.,  ax.  20°  C  +3.25  cyl.,  ax.  53°  41'  24.25" 
into  the  equivalent  sphero-cylinder. 

,  Ans.   —0.5  sph.  O  +  3  cyl.,  ax.  65°, 
or  +  2.5  sph.  O  —  3  cyl.,  ax.  155°. 

9.  Transpose 

+9.5  cyl.,  ax.  0°  C  +10  cyl.,  ax.  57°  40'  45" 
into  the  equivalent  sphero-cylinder. 

Ans.  +4.53  sph.  C  +10.43  cyl.,  ax.  30°, 
or  +14.96  sph.  C  - 10.43  cyl.,  ax.  120°. 


328  Mirrors,  Prisms  and  Lenses  [Ch.  IX 

10.  Find  the  sphero-cylindrical  equivalent  of 

+2  cyl,  ax.  20°  C  +3  cyl.,  ax.  70°. 

Ans.  +0.85  sph.  C  +  3.3  cyl.,  ax.  51°  42', 
or  +  4.15  sph.  C  -3.3  cyl.,  ax.  141°  42'. 

11.  Transpose 

-1.75  cyl.,  ax.  120°  C  +1.25  cyl.,  ax.  135° 
into  the  equivalent  cross-cylinder. 

Ans.  +0.207  cyl.,  ax.  98°  30'  C  -0.707  cyl.,  ax.  8°  30'. 

12.  Transpose  +4  cyl.,  ax.  80°  C  -2  cyl.,  ax.  120°  into 
the  equivalent  cross-cylinder. 

Ans.  +3.075  cyl.,  ax.  65°  50'  C  -1.075  cyl.,  ax.  155°  50'. 


/ 


CHAPTER  X 

GEOMETRICAL      THEORY       OF       THE       SYMMETRICAL       OPTICAL 
INSTRUMENT 

117.  Graphical  Method  of  tracing  the  Path  of  a  Paraxial 
Ray  through  a  Centered  System  of  Spherical  Refracting 
Surfaces. — Nearly  all  optical  instruments  consist  of  a  com- 
bination of  transparent,  isotropic  media,  each  separated 
from  the  next  by  a  spherical  (or  plane)  surface;  the  centers 
of  these  surfaces  lying  all  on  one  and  the  same  straight  line 
called  the  optical  axis  of  the  centered  system  of  spherical 
surfaces,  which  is  an  axis  of  symmetry.  In  a  symmetrical 
optical  instrument  of  this  kind  it  is  sufficient  to  investigate 
the  procedure  of  paraxial  rays  in  any  meridian  plane  con- 
taining the  axis. 

The  indices  of  refraction  of  the  media  will  be  denoted  by 
Tii,  712,  etc.,  named  in  the  order  in  which  they  are  traversed 
by  the  light;  so  that  if  m  denotes  the  number  of  refracting 
surfaces,  the  index  of  refraction  of  the  last  medium  into 
which  the  rays  emerge  after  refraction  at  the  mth  surface 
will  be  nm+1.  The  indices  of  refraction  of  the  two  media 
which  are  separated  by  the  A:th  surface  (where  k  denotes 
any  integer  between  1  and  m,  inclusive)  will  be  nk  and  nk+1. 
The  vertex  and  center  of  the  kth  surface  will  be  designated 
by  Ak  and  Ck,  respectively;  and  the  radius  of  this  surface 
will  be  denoted  by  rk  =  AkCk.  Moreover,  if  Mk,  Mk+i 
designate  the  positions  of  the  points  where  a  paraxial  ray 
crosses  the  axis  before  and  after  refraction,  respectively,  at 
the  kth  surface,  these  points  will  be  a  pair  of  conjugate  axial 
points  with  respect  to  this  surface;  and  the  points  Mi,  Mm+i 
will,  therefore,  be  a  pair  of  conjugate  axial  points  with  respect 

329 


330 


Mirrors,  Prisms  and  Lenses 


[§H7 


to  the  entire  centered  system  of  m  spherical  refracting  sur- 
faces. 

The  accompanying  diagram  (Fig.  158)  represents  a  merid- 
ian section  of  an  optical  system  of  this  kind.  The  straight 
line  MiBi  represents  the  path  of  a  paraxial  ray  in  the  first 
medium  (wi)  which  crossing  the  axis  at  Mi  meets  the  first 
surface r  (i/i)  in  the  point  marked  Bi.     Similarly,  the  path 


Fig.  158. — Path  of  paraxial  ray  through  centered  system  of  spherical  re- 
fracting surfaces. 


of  the  ray  from  the  first  surface  to  the  second  surface  is 
shown  by  the  straight  line  BiB2  which  crosses  the  axis  at  M2. 
Thus,  the  entire  course  of  the  ray  is  shown  by  the  broken 
line  M1B1B2B3M4  which  is  bent  in  succession  at  each  of  the 
incidence-points  Bi,  B2,  B3  (supposing  that  m  =  3,  as  repre- 
sented in  the  diagram) . 

The  figure  shows  also  the  path  of  another  paraxial  ray, 
emanating  from  an  object-point  Qi  near  the  optical  axis  but 
not  on  it  and  represented  here  as  lying  perpendicularly 
above  Mi.  This  ray  is  the  ray  which  leaves  Qi  along  a  straight 
line  which  passes  through  the  center  Ci  of  the  first  refracting 
surface  and  also  through  the  point  Q2  which  is  conjugate  to 
Qi  with  respect  to  this  surface.  This  point  Q2  can  be  lo- 
cated by  determining  the  point  of  intersection  of  the  straight 
line  Q1C1  with  the  straight  line  M2Q2  drawn  perpendicu- 
lar to  the  axis  at  M2.  Similarly,  the  point  Q3  conjugate  to 
Q2  with  respect  to  the  second  refracting  surface  will  be  at 


§  117]      Centered  System  of  Spherical  Surfaces      331 

the  point  of  intersection  of  the  straight  line  Q2C2  with  the 
straight  line  drawn  perpendicular  to  the  axis  at  M3;  and 
so  on  from  one  surface  to  the  next.  Provided,  therefore,  we 
know  the  path  of  one  paraxial  ray  through  the  system,  it 
is  easy  to  construct  the  path  of  a  second  ray. 

But  the  best  graphical  method  of  tracing  the  path  of  a 
paraxial  ray  through  a  centered  system  of  spherical  refract- 
ing surfaces  consists  in  applying  the  construction  described 


Fig.  159. — Graphical  method  of  tracing  path  of  paraxial  ray  through  cen-V 
tered  system  of  spherical  refracting  surfaces. 

in  §  76,  as  follows:  If  the  straight  line  M1B1  (Fig.  159)  rep- 
resenting the  path  of  the  ray  in  the  first  medium  meets  the 
perpendicular  erected  to  the  optical  axis  at  the  center  Ci  in 
the  point  Xi,  and  if  on  this  perpendicular  a  second  point  X/ 
is  taken  such  that  C1X1  :  CiXi'  =  n2  :  nh  then  the  straight 
line  BiXi'  will  determine  the  path  BiB2  of  the  ray  in  the 
second  medium.  Draw  C2Y2  parallel  to  C1X1,  and  let  Y2 
designate  the  point  of  intersection  of  the  straight  lines 
BiB2  and  C2Y2;  and  on  C2Y2  take  a  point  Y2'  such  that 
C2Y2:  C2Y2'  =  n3:  n2,  and  draw  the  straight  line  Y2'B2  meeting 
the  third  refracting  surface  in  B3  and  intersecting  in  Z3  the 
straight  line  drawn  through  C3  parallel  to  C2Y2.  If  on  C3Z3 
a  point  Z/  is  taken  such  that  C3Z3  :  C3Z3'  =  n4  :  n3,  then  the 
straight  line  B3Z3'  will  determine  the  path  of  the  ray  after 
refraction  at  the  third  surface.  This  process  is  to  be  re- 
peated until  the  ray  has  been  traced  into  the  last  medium. 


332  Mirrors,  Prisms  and  Lenses  [§  118 

118.  Calculation  of  the  Path  of  a  Paraxial  Ray  through 
a  Centered  System  of  Spherical  Refracting  Surfaces. — Ob- 
viously, just  as  in  the  case  of  a  single  spherical  refracting 
surface  (§  80),  any  figure  lying  in  a  plane  in  the  object-space 
perpendicular  to  the  optical  axis  of  a  centered  system  of 
spherical  refracting  surfaces  will  be  reproduced  by  means 
of  paraxial  rays  by  a  similar  figure  in  the  image-space  also 
lying  in  a  plane  perpendicular  to  the  optical  axis. 

Moreover,  if  we  put 

AkMk  =  uk,        AkMk+i=wk', 
the  abscissa-formula  (§  78)  for  the  kth  surface  may  be  writ- 
ten: 

nk+i _ nk  ,  nk+i— nk 
uk      uk  rk 

If  also  we  employ  the  symbol 

4  =  AkAk+i 
to  denote  the  distance  of  the  vertex  of  the  (k-\-l)th  surface 
from  that  of  the  A;th  surface  or  the  so-called  axial  thickness 
of  the  (7c+l)th  medium,  then,  evidently: 

uk+i  =  uk'—dk; 
which  enables  us  to  pass  from  one  surface  to  the  next. 

If  in  these  so-called  recurrent  formulae  we  give  k  in  suc- 
cession the  values  k  =  l,  2,  .  .  .  ,  (ra— 1),  and  if  also  in  the 
first  formula  we  put  finally  k  =  m,  we  shall  obtain  (2m— 1) 
equations;  and  if  the  constants  of  the  system  are  all  known, 
that  is,  if  the  values  of  all  the  magnitudes  denoted  by  n,  r 
and  d  are  given,  together  with  the  initial  value  ui,  which 
denotes  the  abscissa  of  the  axial  object-point,  these  (2m—  1) 
equations  will  enable  us  to  determine  the  value  of  each  of 
the  u's  in  succession.  The  position  of  the  image  point  Mm+i 
conjugate  to  the  axial  object-point  Mi  will  have  been  ascer- 
tained when  we  have  found  the  value  of  the  abscissa  um'. 

The  secondary  focal  point  of  the  system  is  the  point  F' 
where  a  paraxial  ray  which  is  parallel  to  the  axis  in  the  first 
medium  crosses  the  axis  in  the  last  medium;  and  if  we  put 
Ui  =  oo  ,  then  um'  =  AmF'  will  be  the  abscissa  of  the  second- 


§  118]  Lateral  Magnification  333 

ary  focal  point  with  respect  to  the  vertex  of  the  last  surface. 
Similarly,  the  primary  focal  point  is  the  point  F  where  a  par- 
axial ray  must  cross  the  axis  in  the  first  medium  if  it  is  to 
emerge  in  the  last  medium  in  a  direction  parallel  to  the  axis. 
In  this  case,  therefore,  we  must  put  umf=  oo  and  solve  for 
ui  =  AiF  in  order  to  obtain  the  abscissa  of  the  focal  point  F 
with  respect  to  the  vertex  of  the  first  surface  of  the  system. 

The  focal  planes  are  the  planes  at  right  angles  to  the  axis 
at  the  focal  points  F,  F'. 

Moreover,  if  we  put  2/k  =  MkQk,  then  according  to  the 
formula  for  the  lateral  magnification  in  a  spherical  refracting 
surface  (§  82),  we  can  write  for  the  kih  surface: 
2/k+i_  ftk    V. 
Vk       nk+\  Uk  ' 
and  if  we  give  k  all  integral  values  from  k  =  l  to  k  =  m,  we 
shall  obtain  m  equations,  one  for  each  surface,  wherein  the 
denominator  of  the  ratio  on  the  left-hand  side  of  each  of 
these  proportions  will  be  the  same  as  the  numerator  of  the 
corresponding  ratio  in  the  preceding  one  of  the  series.    Hence, 
if  we  multiply  together  all  of  these  equations,  and  if,  finally, 
we  put 

y=yi,      y'=ym+i,    '  n  =  nh      n'  =  nm+h 
we  shall  obtain : 

y'     nui.u2'.  .  .  Um' 


n'u\.ui. 


which  may  be  written  also : 


k= 

TT 

y      n' J-J-Wk  ' 
k=i 


y'  _n  1  TV 


where  the  symbol  IT  placed  in  front  of  an  expression  in  this 
way  means  merely  that  the  continued  product  of  all  terms 
of  that  type  is  to  be  taken.  Thus  having  found  the  values 
of  all  the  u's,  both  primed  and  unprimed,  we  can  calculate 
by  this  formula  the  lateral  magnification  produced  by  the 


334  Mirrors,  Prisms  and  Lenses  [§  119 

entire  centered  system  of  spherical  refracting  surfaces  for 
any  given  position  of  the  object-point. 

Moreover,  for  the  kth  surface  the  so-called  Smith- 
Helmholtz  formula  (§  86)  will  have  the  form: 

nk.yk.  6k  =  nk+i.yk+i.  dk+h 
where  0k  =  ZAkMkBk;  and  if  here  also  we  give  k  all  values 
in  succession  from  k  =  1  to  k  =  m,  we  shall  obtain : 

ni.yi.di  =  n2.y2-62=   •  •  •   =  nm+i.ym+i.  0m+i; 
and  finally: 

n'.y'.  6'  =  n.y.  6, 

where  n,  nr  and  y,  yf  have  the  same  meanings  as  above,  and 

6=  0i,        6'=  0m+i. 

119.  The  so-called  Cardinal  Points  of  an  Optical  System. 
The  methods  which  have  just  been  explained,  although 
perfectly  simple  in  principle,  involve  a  more  or  less  tedious 
process  of  tracing  the  path  of  a  paraxial  ray  from  one  surface 
to  the  next  throughout  the  entire  system.  We  have  now 
to  explain  the  celebrated  theory  of  Gauss  (1777-1855)  which 
was  developed  (1841)  in  order  to  avoid  as  much  of  this  labor 
as  possible,  by  keeping  steadily  in  view  the  fundamental  re- 
lations between  the  object-space  and  the  image-space.  It  is 
easy  to  show  that  the  imagery  produced  by  a  symmetrical  op- 
tical instrument  in  the  vicinity  of  the  axis  is  completely  de- 
termined so  soon  as  we  know  the  positions  of  the  focal  points 
and  one  pair  of  conjugate  points  on  the  axis,  together  with 
the  ratio  of  the  indices  of  refraction  of  the  first  and  last  media 
of  the  system.  However,  for  this  purpose  certain  pairs  of 
conjugate  axial  points  are  distinguished  above  others  on 
account  of  their  simple  geometrical  relations;  and  of  these 
the  most  important  are  the  principal  points  and  the  nodal 
points.  These  two  pairs  of  conjugate  points,  together  with 
the  focal  points,  are  sometimes  called  the  cardinal  points  of 
the  optical  system.  We  shall  explain  now  how  these  points 
are  defined. 

(1)  The  Focal  Planes  and  the  Focal  Points. — In  every 
centered  system  of  spherical  refracting  surfaces  there  are 


§  119]      Principal  Planes  and  Principal  Points         335 

two  (and  only  two)  transversal  planes  at  right  angles  to  the 
axis  which  are  characterized  by  the  following  properties: 

A  bundle  of  paraxial  object-rays  which  all  meet  in  a  point 
in  one  of  these  planes  {called  the  primary  focal  plane)  will 
emerge  from  the  system  as  a  cylindrical  bundle  of  parallel 
image-rays;  and,  similarly,  a  cylindrical  bundle  of  parallel 
object-rays  will  emerge  from  the  system  as  a  bundle  of  image- 
rays  which  all  meet  in  a  point  in  the  other  one  of  these  planes 
{called  the  secondary  focal  plane) .  The  points  in  which  these 
focal  planes  are  pierced  by  the  axis  are  the  primary  and  sec- 
ondary focal  points  F  and  F',  respectively. 

(2)  The  Principal  Planes  and  the  Principal  Points. — Again, 
in  every  symmetrical  optical  system  there  is  one  (and  only 
one)  pair  of  conjugate  transversal  planes  characterized  by 
the  property,  that  in  these  planes  object  and  image  are  con- 
gruent; and,  therefore,  any  straight  line  drawn  parallel  to  the 
axis  will  intersect  these  planes  in  a  pair  of  conjugate  points. 
These  are  the  so-called  principal  planes,  one  belonging  to 
the  object-space  {the  primary  principal  plane)  and  the  other 
belonging  to  the  image-space  {the  secondary  principal  plane). 
The  points  H,  H'  where  the  optical  axis  crosses  the  prin- 
cipal planes  are  the  principal  points  of  the  system.  Atten- 
tion was  first  directed  to  these  points  by  Moebius  in  1829, 
but  it  was  Gauss  who  recognized  their  significance  for  the 
development  of  simple  and  convenient  general  formulae  in 
the  theory  of  optical  imagery. 

In  the  principal  planes  the  lateral  magnification  is  unity, 
that  is,  y'  —  y.  (And  hence  the  principal  planes  and  principal 
points  are  called  also,  especially  by  English  writers,  the  unit 
planes  and  the  unit  points.)  Consider,  for  example,  the  case  of 
a  single  spherical  refracting  surface,  for  which  we  found  (§  85) 
y'_f   _f'+W 

V    f+u       r     ' 
If  we  put  y'=y,  we  find  u'  =  u  =  0]  which  means  that  the 
principal  points  of  a  spherical  refracting  surface  coincide  with 
each  other  at  the  vertex  of  the  surface  (§  81).    We  saw  likewise 


336 


Mirrors,  Prisms  and  Lenses 


[§H9 


that  these  points  coincided  with  each  other  at  the  optical 
center  of  an  infinitely  thin  lens  (§  94). 

A  useful  rule  is  as  follows: 

To  any  ray  in  one  region  (object-space  or  image-space) 
which  goes  through  the  focal  point  belonging  to  that  region, 


w 

w" 

s 

X 

/p 

N,^ 

H 

/y' 

?S 

V 

a 

w 

f 

W                   ^ 

F'                                H' 

H                                   ^-.F 

X 

\\              **  — . 

V' 

\ 

V 

ss 

Fig.  160,  a  and  b—  Focal  points  (F,F')  and  principal  points  (H,  H')  of 
(a)   convergent  and  (6)  divergent  optical  system. 

there  will  correspond  a  ray  in  the  other  region  which  is  par- 
allel to  the  axis,  and  the  rectilinear  portions  of  the  path  of 


119] 


Nodal  Planes  and  Nodal  Points 


337 


the  ray  in  these  two  regions  will  intersect  in  a  point  lying  in 
the  principal  plane  of  that  region  to  which  the  focal  point  in 
question  belongs;  as  is  illustrated  in  the  accompanying  dia- 
grams at  W  and  at  V'  (Fig.  160,  a  and  b). 

(3)  The  Nodal  Planes  and  the  Nodal  Points. — Finally,  in 
every  centered  system  of  spherical  refracting  surfaces  there 
is  also  a  pair  of  conjugate  transversal  planes  characterized 
by  the  property,  that  the  angle  between  any  pair  of  object- 


X 

U 

XT 

X 

A 

S 

X 

N 

H 

N* 

H' 

Fig.    161. — Principal   points   (H,   H')    and   nodal   points 
(N,  N'). 

rays  which  intersect  in  a  point  lying  in  the  so-called  primary 
nodal  plane  will  be  exactly  equal  to  the  angle  between  the  cor- 
responding pair  of  image-rays  which  meet  in  the  conjugate 
point  of  the  secondary  nodal  plane.  The  nodal  points  N,  N' 
where  the  axis  meets  these  planes  were  remarked  first  by 
Moser  in  1844,  but  they  were  brought  into  prominence 
through  the  work  of  Listing  (1845)  with  whose  name  there- 
fore they  are  generally  associated.  The  distinguishing  fea- 
ture of  this  pair  of  conjugate  axial  points  is  that  object-ray 
and  image-ray  cross  the  axis  at  the  nodal  points  at  exactly  the 
same  slope.  For  example,  if  the  straight  line  NU  (Fig.  161) 
represents  the  path  of  an  object-ray  which  crosses  the  axis 
at  the  primary  nodal  point  and  meets  the  primary  principal 
plane  in  the  point  marked  U,  the  path  of  the  corresponding 
image-ray  will  be  represented  by  a  straight  line  N'U'  which 
is  drawn  parallel  to  NU  and  which  meets  the  secondary  prin- 


'338  Mirrors,  Prisms  and  Lenses  [§  119 

cipal  plane  in  the  point  marked  U',  so  that  if  ZHNU=  0, 
ZH'N'U'=  0',then0'  =  0. 

Obviously,  the  quadrilateral  NUU'N'  is  a  parallelogram, 
and  hence  H'N'  =  HN;  that  is,  the  step  from  one  of  the  prin- 
cipal points  to  the  corresponding  nodal  point  is  identical  with 
the  step  from  the  other  principal  point  to  its  corresponding  nodal 
point.  The  nodal  points,  therefore,  lie  always  on  the  same 
side  of  the  corresponding  principal  points  and  at  equal  dis- 
tances from  them.  If  the  primary  nodal  point  and  principal 
point  coincide,  the  same  will  be  true  of  the  secondary  nodal 
point  and  principal  point.  Moreover,  since  NN/  =  UU,= 
HH',  the  interval  between  the  nodal  planes  is  precisely  the 
same  as  the  interval  between  the  principal  planes. 

If  in  the  Smith-Helmholtz  formula  (§  118)  we  put  0'  = 
0,  we  find  for  the  lateral  magnification  in  the  nodal  planes 
of  a  centered  system  of  spherical  refracting  surfaces 

yl=  — 

y  n" 
where  n  and  n'  denote  the  indices  of  refraction  of  the  first 
and  last  media,  respectively.  Applying  this  result  to  the 
case  of  a  single  spherical  refracting  surface,  we  obtain  for 
the  nodal  points  N,  N'  the  conditions  ur  =  u  =  r,  that  is, 
AN' =  AN  =  AC.  Consequently,  the  nodal  points  of  a  spher- 
ical refracting  surface  coincide  with  each  other  at  the  center 
C  of  the  surface;  as  might  have  been  inferred  at  once  from 
the  fact  that  a  central  ray  is  not  deviated  by  refraction  at  a 
spherical  surface. 

(4)  Various  writers  on  optics  have  distinguished  other 
pairs  of  conjugate  axial  points  besides  the  principal  points 
and  nodal  points,  but  none  of  these  can  be  said  to  have 
achieved  a  permanent  place  in  the  literature  of  the  subject. 
We  may  mention  the  so-called  negative  principal  points,  in- 
troduced by  Toepler  in  1871,  which  are  characterized  by 
the  fact  that  for  this  pair  of  points  the  lateral  magnification  is 
equal  to  —I;  that  is,  y'=—y,  so  that  the  image  is  inverted 
and  of  same  size  as  object.    Professor  S.  P.  Thompson,  hav- 


§120] 


Construction  of  Image 


339 


ing  this  property  in  view,  has  re-named  them  much  more 
happily  the  symmetric  points  of  the  optical  system. 

120.  Construction  of  the  Image-Point  Q'  conjugate  to  an 
Extra-axial  Object-Point  Q. — If  the  principal  planes  and 
focal  planes  have  been  determined,  it  will  not  be  necessary 
to  trace  the  path  of  a  ray  in  the  interior  of  the  system.    Sup- 


Fig.  162. — Construction  of  image-point  Q'  conjugate  to  object-point  Q 
in  an  optical  system. 

pose,  for  example,  that  Q  (Fig.  162)  designates  the  position 
of  an  object-point  not  on  the  axis;  the  position  of  the  point 
Q'  conjugate  to  Q  may  be  constructed  as  follows: 

Through  Q  draw  a  straight'  line  QV  parallel  to  the  axis 
meeting  the  secondary  principal  plane  in  the  point  marked 
V  and  also  another  straight  line  QF  meeting  the  primary 
principal  plane  in  the  point  marked  W.  The  required  point 
Q'  will  be  found  at  the  point  of  intersection  of  the  straight 
line  V'F'  with  the  straight  line  WQ'  drawn  parallel  to  the 
axis.  The  feet  of  the  perpendiculars  let  fall  from  Q,  Q'  on 
to  the  axis  will  locate  also  a  pair  of  conjugate  axial  points 
M,  M'.  The  construction  is  seen  to  be  entirely  similar  to 
that  given  in  §§71,  81  and  92.  The  case  represented  in  the 
figure  is  that  of  a  convergent  optical  system,  in  which  parallel 
object  rays  are  converged  to  a  real  focus  at  a  point  in  the 
secondary  focal  plane.    The  student  should  draw  for  him- 


340 


Mirrors,  Prisms  and  Lenses 


121 


self  the  corresponding  diagram  for  the  case  of  a  divergent 
optical  system. 

121.  Construction  of  the  Nodal  Points  N,  N\ — Having  de- 
termined the  position  of  the  point  Q'  conjugate  to  Q,  we  can 
easily  locate  the  positions  of  the  nodal  points  N,  N'.  For 
example,  on  the  straight  line  WQ'  (Fig.  162)  take  a  point  Z 
such  that  ZQ'  =  HH',  and  draw  the  straight  line  QZ  meeting 
the  primary  principal  plane  in  the  point  U.  Draw  UU'  par- 
allel to  the  axis  meeting  the  secondary  principal  plane  in 
the  point  U'.    Evidently,  the  straight  lines  QU  and  QTJ'  will 


Fig.   163. 


-Construction  of  nodal  points  (N,  N'),  and  proof  of 
relation  I'F'  =  FR. 


be  parallel,  and  the  points  where  they  cross  the  axis  will  be 
the  nodal  points  N,  N'  (§  119). 

A  simpler  way  of  constructing  the  nodal  points  N,  N'  is 
as  follows : 

Through  the  primary  focal  point  F  draw  a  straight  line 
FW  meeting  the  primary  principal  plane  in  the  point  marked 
W,  and  through  W  draw  a  straight  line  parallel  to  the  axis 
meeting  the  secondary  focal  plane  in  a  point  marked  I'  in 
Fig.  163.  This  point  I'  is  the  image-point  of  the  infinitely 
distant  point  I  of  the  straight  line  FW.  The  straight  line 
drawn  through  I'  parallel  to  FW  will  meet  the  axis  in  the 
secondary  nodal  point  N' ;  and  the  position  of  the  other  nodal 
point  N  can  be  found  immediately. 

The  diagram  shows  also  that 

FH  =  N'F'; 


§121] 


Construction  of  Image 


341 


whence  it  follows  (§  119)  that 

F'H'  =  NF. 
Accordingly,  the  step  from  one  nodal  point  to  the  correspond- 
ing focal  point  is  identical  with  the  step  from  the  other  focal 
point  to  its  corresponding  principal  point.  In  fact,  the  three 
segments  of  the  axis  FF',  HN'  and  H'N  all  have  a  common 
half-way  point. 

Incidentally,  another  useful  relation  may  be  seen  at  a 
glance  in  Fig.  163.  Let  R  designate  the  point  where  the  ray 
IH  which  passes  through  the  primary  principal  point  crosses 


a 

V 

V' 

X 

F 

fco^^ 

H 

H'                     \, 

F' 

X 

"-—~4sj^ 

Y'     \ 

W 

w 

a' 

Fig.  164. — Construction  of  image-point  Q'  conjugate  to  object-point  Q  in 
an  optical  system. 

the  primary  focal  plane;  the  corresponding  image-ray  will 
pass  through  the  secondary  principal  point  IT  and  cross  the 
secondary  focal  plane  at  I';  and,  obviously,  since  FRHW 
and  HWI'F'  are  both  parallelograms, 

I'F'  =  FR; 
Consequently,  a  pair  of  conjugate  rays  passing  through  the 
principal  points  H,  H'  will  cross  the  focal  planes  at  equal  dis- 
tances from  the  axis,  but  on  opposite  sides  thereof. 

This  result  may  be  utilized  in  the  construction  of  the 
point  Q'  (Fig.  164)  conjugate  to  the  object-point  Q.  Let 
X  designate  the  point  where  the  straight  line  QH  crosses 
the  primary  focal  plane ;  and  take  a  point  Y'  in  the  secondary 
focal  plane  such  that  F' Y'  =  XF.    Then  the  required  point  Q' 


342  Mirrors,  Prisms  and  Lenses  [§  122 

will  be  at  the  point  of  intersection  of  the  straight  line  H'Y' 
with  either  of  the  straight  lines  W'Q'  or  V'F'  shown  in  the 
figure. 

122.  The  Focal  Lengths  f,  f . — Let  us  employ  the  symbols 
co,  co'  to  denote  the  slopes  of  a  pair  of  conjugate  rays  which 
pass  through  the  principal  points  H,  H';  thus,  in  Fig.  164 
ZFHX=  co,  ZF/H/Y/=  co';  and  since  in  the  case  of  paraxial 
rays  we  may  write  co  andco'  in  place  of  tanco  and  tana/ 
(see  §  63),  we  have: 

FX  =  _  F^=     '  , 

FH         W'       F'H'         °)' 

Accordingly,  dividing  one  of  these  equations  by  the  other, 

and  taking  account  of  the  fact  that  F'Y'  =  XF  (§  121),  we 

obtain : 

FH       _&/ 
F'H'  co  ' 

Since  the  lateral  magnification  in  the  principal  planes  is 
equal  to  +1,  that  is,  since  y' '  —  y  (§  119),  the  Smith-Helm- 
holtz  formula  (§  118)  for  the  pair  of  conjugate  points 
H,H'  takes  the  form: 

n'.co'=n.co, 
where  n  and  n'  denote  the  indices  of  refraction  of  the  first 
and  last  media  of  the  optical  system. 

If,  therefore,  the  focal  lengths  of  the  optical  system  are  de- 
fined as  the  abscissa?  of  the  principal  points  with  respect  to  their 
corresponding  focal  points,  that  is,  if  we  put /=FH,  /'  =  F'H', 
where  /and/'  denote  the  primary  and  secondary  focal  lengths, 
respectively,  then  combining  the  relations  found  above  so 
as  to  eliminate  the  angles  co  and  co',  we  find: 

/'  n' ' 

which  may  be  put  in  words  as  follows:  The  focal  lengths  of 
a  centered  system  of  spherical  refracting  surfaces  are  propro- 
tional  to  the  indices  of  refraction  of  the  first  and  last  media, 
and  are  opposite  in  sign;  except  in  the  single  case  when  the 
optical  system  includes  an  odd  number  of  reflecting  surfacesf 


122] 


Focal  Lengths  of  Optical  System 


343 


in  which  case  the  focal  lengths  will  have  the  same  sign  (that  is, 
in  this  exceptional  case,  ///'  =  +w/n'). 
It  appears,  therefore,  that  the  formula, 
n'.f+n.f'=0, 
which  was  found  (§§  79  and  96)  to  hold  for  a  single  spherical 
refracting  surface  and  for  an  infinitely  thin  lens,  expresses, 
in  fact,  a  perfectly  general  relation  which  is  true  of  any 
centered  system  of  spherical  refracting  surfaces.    Consider, 
for  example,  the  optical  system  of  the  human  eye  in  which 
the  first  medium  is  air  (n  =  l)  and  the  last  medium  is  the 


To  I  at  cc 


To  E  at  oo- 


To  J'ata? 


Fig.  165. — Focal  lengths  (/,/')  of  an  optical  system. 


so-called  vitreous  humor  whose  index  of  refraction  is  n' = 
1.336.  In  Gullstrand's  schematic  eye  (see  §  130)  the 
primary  focal  length  is  found  to  be  /=  + 17.055  mm., 
whence,  according  to  the  above  formula,  the  secondary 
focal  length  is  /'=  -22.785  mm. 

In  particular,  when  the  media  of  object-space  and  image- 
space  are  identical  (nr  =  n) ,  the  focal  lengths  are  equal  in  mag- 
nitude, but  opposite  in  sign  ( /'  =  — /) .  This  is  the  case  with 
most  optical  systems,  since  they  are  usually  surrounded  by 
air.  According  to  the  definitions  of  the  focal  lengths  given 
above,  it  follows  from  §  121  that 

FH  =  N'F'  =/,      F'lT  =  NF  =/'; 

and  hence  we  see  that  the  nodal  points  (N,  N')  of  an  optical 

system  surrounded  by  the  same  medium  on  both  sides  coincide 

with  the  principal  points  (H,  H') ;  for  when  nf  =  n,  then 

FH=/=  -/'  =  FN,       F'H'=/'  =  -/=F'N'. 


344  Mirrors,  Prisms  and  Lenses  [§  123 

The  focal  lengths  of  a  centered  system  of  spherical  re- 
fracting surfaces  may  be  defined  also  exactly  as  in  §§  83 
and  95.  If  in  Fig.  165  we  put  ZHFW=  0,ZH'F'V'  =  0', 
we  can  write: 

HW  H'V 


tan  0  '    *  tan  0'  ' 

and  since  HW  =  FT,  H'V'  =  FJ,  tan0  =  0,  tan0'  =  0',  we 
have: 

Accordingly,  we  may  also  define  the  focal  lengths  as  follows : 
The  focal  length  of  the  object-space  (f)  is  equal  to  the  ratio  of 
the  linear  magnitude  of  an  image  formed  in  the  focal  plane 
of  the  image-space  to  the  apparent  (or  angular)  magnitude  of 
the  correspondingly  infinitely  distant  object;  and,  similarly,  the 
focal  length  of  the  image-space  (/')  is  equal  to  the  ratio  of  the 
linear  magnitude  of  an  object  lying  in  the  focal  plane  of  the 
object-space  to  the  apparent  (or  angular)  magnitude  of  its  in- 
finitely distant  image. 

The  focal  lengths  may  be  said,  therefore,  to  measure  the 
magnifying  power  of  the  optical  instrument,  for  if  the  appara- 
tus is  adapted  to  an  emmetropic  eye  (§  153),  the  image  will 
be  formed  at  infinity,  and  the  magnifying  power  will  be  deter- 
mined by  the  ratio  of  the  apparent  size  of  the  image  to  the 
actual  size  of  the  object  (see  Chapter  XIII). 

123.  The  Image-Equations  in  the  case  of  a  Symmetrical 
Optical  System. — The  image-equations  are  a  system  of  re- 
lations which  enable  us  to  find  the  position  of  an  image- 
point  Q'  (Fig.  162)  conjugate  to  a  given  object-point  Q. 
The  position  of  the  point  Q  will  be  given  by  its  two  co- 
ordinates referred  to  a  system  of  rectangular  axes  in  the 
object-space  in  the  meridian  plane  in  which  the  point  Q  lies. 
Naturally,  the  optical  axis  will  be  selected  as  the  axis  of 
abscissae  and  either  the  primary  focal  point  F  or  the  primary 
principal  point  H  as  the  origin.  Thus,  if  we  put 
FM  =  z;         HM=u,       MQ=2/, 


§  123]  The  Image-Equations  345 

the  object-point  Q  will  be  the  point  (x,  y)  or  the  point  (u,  y), 
according  as  we  take  the  origin  at  F  or  H,  respectively. 
Similarly,  in  the  image-space,  if  we  put 

F'M'  =  :r/,       H'M'  =  <       M'Q'  =  y', 
the  coordinates  of  Q'  will  be  denoted  by  (xf,  y')  or  (uf,  y') 
according  as  the  origin  of  this  system  of  axes  is  at  F'  or  H', 
respectively. 

a.  The  image-equations  referred  to  the  focal  points  F,  F'. — 
The  following  proportions  are  obtained  from  the  two  pairs 
of  similar  triangles  FHW,  FMQ  and  F'H'V,  F'M'Q': 

HW=FH        M'Q^F'M'. 
MQ    FM'       H'V'     F'H'' 
and  since 

HW  =  M'Q'  =  2/'f     H,V'  =  MQ  =  2/,     FH=/,      F'H'=/', 
we  find  immediately : 

y    x  f" 

whence  the  coordinates  x',  y'  can  be  found  in  terms  of  the 
given  coordinates  x,  y  and  the  focal  lengths  /,  /'. 

These  formulae,  which  were  obtained  formerly  for  cer- 
tain simple  special  cases  (§§  69,  85  and  93)  are  seen,  there- 
fore, to  be  entirely  general  and  applicable  always  to  any 
symmetrical  optical  system.  The  so-called  Newtonian 
form  of  the  abscissa-relation,  viz., 

x.x'=ff, 
shows  that  the  product  of  the  focal-point  abscissae  is  constant. 

b.  The  image-equations  referred  to  the  principal  points 
H,H'. — Again,  the  following  proportions  are  derived  from 
the  two  pairs  of  similar  triangles  FHW,  QVW  and  F'H'V, 
Q'W'V: 

WV  =  VQ  =  HM       VW  =  W^'  =  ITM'. 

HW    FH     FH  '     H'V    F'H'     FTT  ' 
and  since  WV  =  WH+HV  =  Q'M'+MQ  =-(?/-?/)  and 
V'W'  =  V'H'+H'W'  =  QM+M'Q'  =  (?/-?/),  we  find: 
y'—y_  _  u       y'-y _uf 

y'         f       y      /'* 


346  Mirrors,  Prisms  and  Lenses  [§  123 

These  relations  give  the  following  expressions  for  the  lateral 
magnification : 

y'_  f   _f'+W_     f  u^ 

y    f+u      f         f'u  ' 

Clearing  fractions,  we  obtain : 

f.uf+f.u+u.u'  =  0, 
and   dividing  through   by   u.u' ',   we  have  the   well-known 
abscissa-relation : 

£+4+1=0; 

u     u 

which  may  also  be  obtained  directly  by  substituting  x  =/+w, 
x'=f'-\-u'  in  the  equation  x.x'  =}.}'. 

By  means  of  these  formulae,  the  coordinates  v! ,  yf  may 
be  found  in  terms  of  the  given  coordinates  u,  y  and  the 
focal  lengths/,/'. 

Since  n'.f+n.f  =  0  (§  122),  we  have  also  another  expres- 
sion for  the  lateral  magnification,  viz., 

y'  _n.uf  m 
y     n'.u ' 
winch  has  likewise  been  obtained  already  in  the  special  case 
of  a  single  spherical  refracting  surface  (§  82). 

A  simple  and  convenient  method  of  locating  the  positions 
of  pairs  of  conjugate  axial  points  is  suggested  by  the  ab- 
scissa-relation 

£+4+1=0; 
u     u 

which  may  be  put  in  the  following  form : 
HF  ,  H'F' 
u         u 
Suppose,  therefore,  that  the  axial  line  segment  H'Fr  is  shoved 
along  the  optical  axis  until  the  secondary  principal  point  H' 
is  brought  into  coincidence  with  the  primary  principal  point 
H,  and  that  then  the  optical  axis  in  the  image-space  (xr)  is 
turned  about  H  until  it  makes  a  finite  angle  with  the  op- 
tical axis  in  the  object-space  (x),  as  represented,  for  example, 
in  Fig.  166.    Through  the  focal  points  F  and  F'  draw  the 


§123] 


The  Image-Equations 


347 


straight  lines  FS  and  F'S  parallel  to  H'F'  and  HF,  respec- 
tively, and  let  S  designate  their  point  of  intersection.  Then 
any  straight  line  drawn  through  S  will  intersect  x  and  x'  in 
a  pair  of  conjugate  axial  points  M,  M';  for  if  we  put  w  =  HM 
and   w'  =  H'M'   in   the  equation  above,  the  equation   will 


Fig.  166. — Construction   of  point   M'  conjugate  to 
axial  object-point  M  in  an  optical  system. 

evidently  be  satisfied.  The  vertex  S  of  the  parallelogram 
HF'SF  is  the  center  of  perspective  of  the  two  point-ranges 
x  and  x'. 

c.  The  image-equations  referred  to  any  pair  of  conjugate 
axial  points  0,  O'. 

If  the  origins  of  the  two  systems  of  rectangular  axes  are 
a  pair  of  conjugate  axial  points  0,  0'  whose  distances  from 
the  focal  points  F,  F'  are  denoted  by  a,  a',  respectively,  so 
that  FO =a;  F'O'  =  a';  and  if  we  put 

OM  =  z,       0'M'  =  z', 
then 

x  =  a-\-z,       x'  =  a'-\-z'; 
and  if  these  values  of  x  and  x'  are  substituted  in  the  equa- 
tions 


348  Mirrors,  Prisms  and  Lenses  [§  123 

we  obtain : 

y'_    f    _a'+z' 

Since  a.a'  =/./',  the  relation  between  z  and  zr  may  be  put  in 
the  form: 

^'+1=0, 

Z      Z 

where  the  constants  are  now  a  and  a'  instead  of  / and/'. 

Suppose,  for  example,  that  the  pair  of  conjugate  axial 
points  0,  O'  is  identical  with  the  pair  of  nodal  points  N,  N'; 
then 

a  =  FO  =  FN  =  -/',       a'  =  F'O'  =  F'N'  =  -/; 
so    that    the   image-equations   referred   to    the  nodal    points 
will  have  the  following  forms : 

1+1-1  =  0      t  =  J-=z^l 

z     zf  y     z-f      /'   ' 

where  z  =  NM,      2'=N'M'. 

d.  The  image-equations  in  terms  of  the  refracting  power 
and  the  reduced  vergences  (see  §§  105  and  106). 

The  refracting  power  of  the  optical  system  is  defined 
(§  105)  by  the  relations: 


/         T 

where  n,  n'  denote  the  indices  of  refraction  of  the  first  and 
last  media.  Similarly,  the  reduced  vergences  (§  106)  with 
respect  to  the  principal  points  are : 

u=n-     w-%. 

U  U 

If,  therefore,  in  the  image-equations  referred  to  the  prin- 
cipal points  we  eliminate  /,  /'  and  u,  u'  by  means  of  these 
two  pairs  of  formulae,  we  obtain  the  image-equations  in  the 
following  exceedingly  useful  and  convenient  form: 

v'     U' 

U'  =  U-\-F        —  =  — . 

If  the  linear  magnitudes  are  measured  in  terms  of  the  meter 


§  124]  Magnification-Ratios  349 

as  unit  of  length,  the  magnitudes  denoted  here  by  U,  JJ' 
and  F  will  all  be  expressed  in  dioptries  (§  107). 

124.  The  Magnification-Ratios  and  their  Mutual  Re- 
lations.— (a)  The  lateral  magnification  y.  This  has  al- 
ready been  defined  as  the  ratio  of  conjugate  line-segments 
lying  in  planes  at  right  angles  to  the  optical  axis.  The  fol- 
lowing expressions  were  obtained  for  this  ratio  in  §  123: 
_y' _f_x' _  f  J'+v! _J.v! _n.u' _XJ  . 
U    y     x    /'    J+u       f  f'.u    n'.u     U''' 

whence  we  see  that  the  lateral  magnification  is  a  function 
of  the  abscissa  of  the  object-point,  and  that  in  any  optical 
system  it  may  have  any  value  from  —  oo  to  +  oo  depending 
on  the  position  of  the  object. 

(b)  The  axial  magnification  or  depth-ratio  x.  If  x,  x'  de- 
note the  abscissae  with  respect  to  the  focal  points  of  a  pair 
of  conjugate  axial  points,  and  if  x-\-c,  xf-\-c'  denote  the  ab- 
scissae of  another  pair  of  such  points  immediately  adjacent 
to  the  former,  then,  since 

x.x'=f.f  =  (x+c)  (x'+cf), 

and  since  moreover  the  product  c.c'  is  a  small  magnitude  of 

the  second  order  as  compared  with  either  of  the  small  factors 

c  or  c',  and  is  therefore  negligible,  we  find : 

c.x'-\-c.x  =  0. 

The  ratio  c'  :  c  of  small  conjugate  segments  of  the  axis 
is  called  the  axial  or  depth-magnification.  If  this  ratio  is 
denoted  by  the  symbol  x,  then,  according  to  the  equation 
above : 

C  X  x2  ' 

so  that,  whereas  the  lateral  magnification  is  inversely  pro- 
portional to  the  abscissa  x,  the  depth-magnification  is  inversely 
proportional  to  the  square  of  x.  In  fact,  the  relation  between 
the  axial  magnification  and  the  lateral  magnification  may 
be  expressed  as  follows : 

U2        f     n  ' 


350 


Mirrors,  Prisms  and  Lenses 


124 


The  axial  magnification  or  " depth-elongation' '  of  a  small 
object  is  proportional  to  the  square  of  its  lateral  magnification. 
If,  therefore,  we  take  a  series  of  ordinates,  1,  2,  3,  4,  etc. 
(Fig.  167),  all  of  equal  height  and  at  equal  intervals  apart 


Fig.    167. — Relation    between    axial    or   depth-magnification    and   lateral 

magnification. 

(like  a  row  of  telegraph  poles),  their  images  will  be  of  un- 
equal heights  and  at  unequal  distances  apart;  but  the  in- 
tervals between  the  successive  images  will  increase  or  di- 
minish far  more  rapidly  than  the  corresponding  changes  in 
their  heights.  Accordingly,  the  image  of  a  solid  object  can- 
not, in  general,  be  similar  to  the  object,  but  will  be  distorted, 
since  the  dimension  parallel  to  the  axis  of  the  optical  system 
is  altered  very  much  more  than  the  dimensions  at  right  angles 
to  the  axis.    This  uneven  distribution  of  the  images  of  ob- 


J 

^^^ 

r 

-"""^l 

"\^ 

Te5"^ 

"M                      ^ 

F             H 

H'                  F' 

M'" 

Fig.  168. — Angular  magnification  or  convergence-ratio. 

jects  at  different  distances  explains  "the  curious  effect  no- 
ticeable in  modern  binocular  field-glasses  of  high  power, 
but  seen  also  in  opera-glasses  and  telescopes,  in  which  the 
successive  planes  of  landscapes  seem  exaggerated,  and  flat- 
tened almost  like  the  flat  scenery  of  the  theater.  Thin  trees 
and  hedges,  for  example,  seem  to  occupy  definite  planes;  and 


Ch.  X]  Problems  351 

the  more  distant  objects  appear  to  be  compressed  up  toward 
those  in  front  of  them  "    (Professor  S.  P.  Thompson). 

(c)  The  angular  magnification  or  so-called  convergence- 
ratio  z.  If  the  slopes  of  conjugate  rays  are  denoted  by  0,  0', 
that  is,  if  we  put  0  =  ZFMJ,  0'  =  ZF'MT  (Fig.  168), 
where  M,  M'  designate  the  points  where  the  ray  crosses  the 
axis  in  the  object-space  and  image-space,  respectively,  and 
J  and  V  designate  the  points  where  it  crosses  the  primary 
and  secondary  focal  planes,  then  evidently: 

tan0=lS'     tan(?'=5FF- 

But  the  focal  lengths  are  denned  by  the  equations  (§  122): 
FT  FJ 

'      tan0'      J      tan0" 
and  therefore : 

tan  0  =  —z- ,        tan  0'  =  -p- . 

Eliminating  the  intercepts  FJ  and  FT,  we  obtain: 
=  tan0_'=  _x_  =  _/ 
*~tan0~     /'        x" 
where  the  ratio   z=tan0'  :  tan  0   (or    0'  :  0)  is  called  the 
angular  magnification  or  the  convergence-ratio.    It  is  directly 
proportional  to  the  abscissa  x  of  the  object-point  M. 

The  three  magnification-ratios  jc,  y  and  z  are  connected 
by  the  following  relation: 

JL=1. 

x.z 


PROBLEMS 

1.  Taking  the  index  of  refraction  of  water  =  |,  show 
that  the  sun's  rays  passing  through  a  globe  of  water,  6  inches 
in  diameter,  will  be  converged  to  a  focus  6  inches  from  the 
center  of  the  sphere. 

2.  A  small  object  is  placed  at  a  distance  u  from  the  nearer 
side  of  a  solid  refracting  sphere  of  radius  r  and  of  refractive 


352  Mirrors,  Prisms  and  Lenses  [Ch.  X 

index  n.    Show  that  the  distance  of  the  image  from  the  other 
side  of  the  sphere  is 

,_  2r(u  —  r)  —  n.u.r 
U  ~2(n-l)u-(n-2)r' 
and  find  the  lateral  magnification. 

3.  A  luminous  point  is  situated  at  the  first  focal  point  of 
an  infinitely  thin  symmetric  double  convex  lens  made  of 
glass  (of  index  1.5)  and  surrounded  by  air.  The  radius  of 
each  surface  is  15  cm.  Show  that  the  image  formed  by  rays 
which  have  been  twice  reflected  in  the  interior  of  the  lens 
before  emerging  again  into  the  air  will  be  on  the  other  side 
of  the  lens  at  a  distance  of  2.5  cm.  from  it. 

4.  An  optical  system  is  composed  of  two  equal  double 
convex  lenses.  The  index  of  refraction  of  the  glass  is  n  = 
1.6202,  and  the  radii,  thicknesses,  etc.,  are  as  follows: 

ri=-r4= 47.92243;         r3=  -r2  =  9.39617; 
^  =  ^3  =  0.2;       d2  =  2.4287. 
If  an  incident  paraxial  ray  crosses  the  axis  at  a  distance 
u\——  7.31101  from  the  vertex  of  the  first  surface,  show 
that  the  emergent  ray  will  cross  the  axis  at  a  distance  u\  — 
33.65725  from  the  vertex  of  the  last  surface. 

5.  A.  Gleichen  in  his  Lehrbuch  der  geometrischen  Optik 
gives  the  following  data  of  P.  Goerz's  "double  anastigmat" 
photographic  objective,  composed  of  three  cemented  lenses, 
the  first  being  a  positive  meniscus  of  crown  glass,  the  second 
a  double  concave  flint  glass  lens,  and  the  third  a  double  con- 
vex crown  glass  lens : 

Indices  of  refraction: 

m  =  nb  =  l;    n2  =  1.5117;    w3  =  1.5478;    n4  =  1.6125 
Radii: 
n  =  -  0. 128965 ;      r2  =  -  0.049597 ;      r3  =  +0. 196423 ; 
r4= -0.1266629 
Thicknesses: 
dl=  +0.01277;  d2=  +0.00664;  d,= +0.02114. 
Show  that  the  second  focal  point  of  this  system  is  at  a  dis- 
tance of  +1.111095  from  the  vertex  of  the  last  surface.    (See 


Ch.  X]  Problems  353 

scheme  for  calculation  of  paraxial  ray  through  a  centered 
system  of  spherical  refracting  surfaces,  §  181). 

6.  Define  the  nodal  points  N,  N'  and  show  that  FN=  -/', 
F'N'  =  — /,  where  F,  F'  designate  the  positions  of  the  focal 
points  and  /,  /'  denote  the  focal  lengths  of  the  optical  system. 
Under  what  circumstances  are  the  nodal  points  identical 
with  the  principal  points? 

7.  Derive  the  image-equations  referred  to  the  principal 
points. 

8.  Given  the  positions  on  the  optical  axis  of  the  principal 
points  and  of  the  focal  points;  construct  the  nodal  points. 
Also,  construct  the  point  Q'  conjugate  to  a  given  object- 
point  Q.  Draw  diagrams  for  convergent  and  divergent 
systems. 

9.  Prove  that 

n'./+w./'  =  0, 
where  /  and  /'  denote  the  focal  lengths  of  the  optical  system, 
and  n  and  n'  denote  the  indices  of  refraction  of  the  first  and 
last  media. 

10.  A  small  cube  is  placed  on  the  axis  of  a  symmetrical 
optical  instrument  with  one  pair  of  its  faces  perpendicular 
to  the  axis.  Find  the  two  places  where  the  image  of  the  cube 
will  also  be  a  cube.  (Assume  that  the  instrument  is  sur- 
rounded by  the  same  medium  on  both  sides.) 

Ans.  At  the  points  for  which  the  lateral  magnification  is 
+  lor  -1. 

11.  An  object  is  placed  3  inches  in  front  of  the  primary 
focal  plane  of  a  convergent  optical  system.  Show  that  the 
image  will  be  one-and-a-half  times  as  large  as  it  was  at  first 
if  a  plate  of  glass  (n  =  1.5)  of  thickness  3  inches  is  interposed 
in  front  of  the  object. 

12.  Show  that  the  axial  magnification  at  the  nodal  points 
has  the  same  value  as  the  lateral  magnification  in  the  nodal 
planes. 

13.  A  symmetrical  optical  instrument  is  surrounded  by 
the  same  medium  on  both  sides.    If  the  images  of  two  small 


354  Mirrors,  Prisms  and  Lenses  [Ch.  X 

objects  A  and  B  on  the  axis  are  formed  at  A'  and  B',  show 
that  the  ratio  of  A'B'  to  AB  is  equal  to  the  product  of  the 
lateral  magnifications  for  the  pairs  of  conjugate  points  A,  A' 
and  B,  B'. 

14.  Show  that  in  a  symmetrical  optical  instrument  there 
are  two  pairs  of  conjugate  points  on  the  axis  for  which  an 
infinitely  small  axial  displacement  of  the  object  will  cor- 
respond to  an  equal  displacement  of  the  image ;  and  that  the 
focal  points  are  midway  between  these  points. 

15.  Show  that  in  a  symmetrical  optical  instrument  sur- 
rounded by  the  same  medium  on  both  sides  there  are  two 
points  on  the  axis  where  object  and  image  will  be  in  the  same 
plane;  and  that  if  a  denotes  the  distance  between  the  prin- 
cipal planes,  the  distance  between  these  two  points  will  be 

Va(a+4f). 

16.  In  a  centered  system  of  m  spherical  refracting  surfaces 
the  vertex  of  the  &th  surface  is  designated  by  Ak.  A  par- 
axial ray  crosses  the  axis  before  refraction  at  the  first  surface 
at  a  point  Mi  which  coincides  with  the  primary  focal  point  F 
of  the  optical  system.  Before  and  after  refraction  at  the 
fcth  surface  this  ray  crosses  the  axis  at  Mk  and  Mk+i,  re- 
spectively.   If  we  put  wk  =  AkMk,  wk'  =  AkMk+i,  show  that 

u2.us.  .  .  um      „A 

U\.U<i.    .    .   Wm-1 

where  /  denotes  the  primary  focal  length  of  the  optical 
system. 

17.  If  the  symbols  wk,  wk,  employed  in  the  same  sense  as 
in  the  preceding  problem,  refer  to  a  paraxial  ray  which  is 
incident  on  the  first  surface  of  the  system  in  a  direction 
parallel  to  the  optical  axis,  show  that 

71    u\.U2.    .    .    U^  ,_       UVU2.    ■    »    ^m 

n'  U2.U3.    .    .   Um  '  U2.U3.    .    .   um ' 

where  /,  /'  denote  the  focal  lengths  of  the  system  and  n,  n' 
denote  the  indices  of  refraction  of  the  first  and  last  media. 

18.  Employing  the  formulae  of  No.  17,  determine  the  focal 
lengths  of  a  hemispherical  lens  of  glass  of  refractive  index 


Ch.  X]  Problems  355 

1.5;  and  find  the  positions  of  the  principal  planes  and  the 
focal  planes. 

Ans.  If  r  denotes  the  radius  of  the  curved  surface,  and  if 
distances  are  measured  from  the  vertex  of  this  surface,  the 
distances  of  the  focal  points  are  —  2r  and  +7r/3,  and  the 
distances  of  the  principal  points  are  0  and  +r/3.  The  focal 
length  is  twice  the  length  of  the  radius. 

19.  If  a  paraxial  ray,  proceeding  originally  in  a  direction 
parallel  to  the  axis  of  a  centered  system  of  spherical  refract- 
ing surfaces  (as  in  No.  17),  crosses  the  axis  in  the  medium  of 
index  nk  at  a  point  Mk  whose  distance  from  the  vertex  of 
the  kih  surface  is  wk  =  AkMk  (Uk  =  nk/uk),  show  that 

Fhk  =  Fhk-i  (Uk+Fk)  (  — — — — —  — -), 

Vc7k-i+/'k-i       nk  I 

where  Fk  denotes  the  refracting  power  of  the  A;th  surface, 

Fi,k  denotes  the  refracting  power  of  the  system  of  surfaces 

bounded  by  the  1st  and  kth.  inclusive  {F\,\  =  Fi  and  Fi,0  =  0), 

and  dk_i  =  Ak_i  Ak  denotes  the  axial  thickness  between  the 

surfaces  bounding  the  medium  of  index  nk. 


CHAPTER  XI 

COMPOUND     SYSTEMS.         THICK    LENSES    AND     COMBINATIONS 
OF    LENSES    AND    MIRRORS 


125.  Formulae  for  Combination  of  Two  Optical  Systems 
in  terms  of  the  Focal  Lengths. — Suppose  that  the  optical 
system  consists  of  two  parts  I  and  II,  each  composed  of 
a  centered  system  of  spherical  refracting  surfaces  with  their 
optical  axes  in  the  same  straight  line.  On  a  straight  line 
parallel  to  this  common  optical  axis  take  two  points  P,  P' 
(Fig.  169),  which  we  shall  assume  to  be  a  pair  of  conjugate 
points  with  respect  to  the  compound  system  (I +11);  and 


X' 


P                              Vi' 

I 

V'                                 Kz 

JL 

Kl                     P 

\^\      H-, 

HI   \               / 

yf 

x  I 

I     f\       f,\^ 

F{\  /F?     H2 

H^F'/F'      H' 

\        w, 

<H    7 

L'2 

Fig.  169. — Combination  of  two  optical  systems.  Letters  with  subscripts 
refer  to  component  systems;  letters  without  subscripts  refer  to  com- 
pound or  resultant  system. 

since  these  points  are  on  the  same  side  of  the  optical  axis 
and  at  equal  distances  from  it,  evidently,  they  must  lie  in 
the  principal  planes  of  the  compound  system  (§  119).  Ac- 
cordingly, the  feet  of  the  perpendiculars  drawn  from  P,  P' 
to  the  optical  axis  will  be  the  pair  of  principal  points  H,  H' 
of  the  compound  system. 

On  the  optical  axis  select  a  point  Fi  for  the  position  of 

356 


§  125]       Combination  of  Two  Optical  Systems        357 

the  primary  focal  point  of  system  I ;  and  select  also  the  posi- 
tions of  the  principal  points  Hi,  Hi'  and  H2,  H2'  of  systems 
I  and  II,  respectively.  Through  Fi  draw  the  straight  line 
PWi  meeting  the  primary  principal  plane  of  system  I  in  the 
point  Wi;  take  Hi'Wi'  =  HiWi,  and  draw  the  straight  line 
Wi'G2  parallel  to  the  axis  meeting  the  primary  principal 
plane  of  system  II  in  the  point  G2;  take  H2'G2'  =  H2G2,  and 
draw  the  straight  line  G2/P/,  which  must  necessarily  cross 
the  optical  axis  at  the  secondary  focal  point  F2'  of  system  II. 

Let  the  straight  line  drawn  through  P  parallel  to  the  op- 
tical axis  meet  the  primary  and  secondary  principal  planes 
of  system  I  in  the  points  designated  by  Vi  and  Vi',  respec- 
tively; and  select  a  point  on  the  optical  axis  for  the  position 
of  the  secondary  focal  point  F/  of  system  I.  Through  Fi' 
draw  the  straight  line  V/Fi'  meeting  the  primary  principal 
plane  of  system  II  in  L2;  take  H2'L2'  =  H2L2,  and  draw  the 
straight  line  L2'P',  which  will  cross  the  optical  axis  in  the 
secondary  focal  point  F'  of  the  compound  system. 

Let  the  straight  line  drawn  through  P'  parallel  to  the  op- 
tical axis  meet  the  primary  and  secondary  principal  planes 
of  system  II  in  the  points  K2  and  K2',  respectively ;  and  let 
O  designate  the  point  of  intersection  of  the  pair  of  straight 
lines  W/G2  and  V/L2.  The  point  where  the  straight  line  K20 
crosses  the  optical  axis  will  be  the  position  of  the  primary 
focal  point  F2  of  system  II.  Let  the  straight  line  K2F2  meet 
the  secondary  principal  plane  of  system  I  in  the  point  T/, 
and  take  HiTi  =  Hi'T/;  then  the  straight  line  PTi  will  cross 
the  optical  axis  at  the  primary  focal  point  F  of  the  com- 
pound system. 

The  diagram  constructed  according  to  the  above  direc- 
tions represents  a  perfectly  general  case.  The  focal  lengths 
of  the  component  systems  are:  /i  =  FiHi,  /i/  =  Fi/H/  and 
/2  =  F2H2,  /2/=F2/H2/;  and  the  focal  lengths  of  the  compound 
system  are:  /=FH,  /'=F'H'.  The  step  from  the  secondary 
focal  point  of  the  first  system  to  the  primary  focal  point  of 
the  system  will  be  denoted  by  the  symbolA;  thus,  A  =  Fi'F2. 


358  Mirrors,  Prisms  and  Lenses  [§  125 

Now  if  we  know  the  positions  on  the  optical  axis  of  the 
focal  points  Fi,  Fi'and  F2,  F2'  of  the  two  component  systems, 
together  with  the  values  of  the  focal  lengths  fh  //  and  /2,  /2', 
it  is  easy  to  calculate  the  positions  of  the  focal  points  F,  F' 
and  the  values  of  the  focal  lengths  /,  /'  of  the  compound 
system;  as  will  now  be  shown. 

The  position  of  the  primary  focal  point  F  of  the  compound 
system  may  be  found  from  the  fact  that  F  and  F2  are  a  pair 
of  conjugate  axial  points  with  respect  to  system  I,  and  hence 
(§123,  a); 

FiF.  FiTi-A/i'. 
And,  similarly,  the  position  of  the  secondary  focal  point  F' 
may  be  found  from  the  fact  that  Fi'  and  F'  are  a  pair  of  con- 
jugate points  with  respect  to  system  II,  so  that 

F/"EV    77*    T7I    /         i*      J*   / 
2  r  .r  2r  i  =J2.J2  . 

Accordingly,  the  positions  of  the  focal  points  F,  F'  with  re- 
spect to  the  known  points  Fi,  F2',  respectively,  are  given  by 
the  following  f ormulse : 


FxF^f1', 

F2'F'= 

hU 

A 

A 

In  order  to  finti  the  focal  lengths 

/,  /',  we  may 

proceed 

as  follows : 

In  the  similar  triangles  FHP,  FHil 

\  we  have: 

FH 

HP 

FHi 

"HiTi ' 

and  since 

HP=H2K2, 

HiTi- 

-HxT/, 

the  proportion  above  may  be  written: 

FH      H2K2 

FHi    Hi'Ti'' 
Now  from  the  similar  triangles  F2H2K2,  F2H1,T/  we  have 


H2K2     F2H2 
H?T7=F2Hi' ; 


§  125]      Combination  of  Two  Optical  Systems        359 

and  hence: 

FH=g|.FH, 

Now  FH1=FF1+F1H1=-^+/1=-£(/1'-A); 

and  F2H1'=F2F1,-fFi,H1'=/1,-A. 
Accordingly,  putting  FH=/,  F2H2=/2,  we  obtain: 

f /i  >h 

}~    ~a~' 

whereby  the  primary  focal  length  of  the  compound  system 
may  be  calculated. 

Similarly,  from  the  figure  we  obtain  the  relations: 
FTT  _  IFF     H/Vi'^F/H/  . 
F,H2,~H2,L2,_  H2L2  ~  Fi'H2  ' 
and  since  F'H'  =/',       F^H/  =//, 

,      FHa/-FF,'+F,/H,'-^+/»/-^(/«+A)l 

A  A 

F/H2  =  Fi  F2-r-F2H2=/2-f-A, 
we  obtain  an  analogous  expression  for  the  secondary  focal 
length  of  the  compound  system,  as  follows : 

J        A     ' 

By  varying  the  interval  A,  which  is  the  common  denom- 
inator of  all  these  expressions,  it  is  obvious  that  it  is  possible 
with  two  given  component  systems  to  obtain  combinations 
of  widely  different  optical  effects.  In  particular,  when 
Fi'  coincides  with  F2,  so  that  the  interval  A  vanishes,  the 
focal  points  F,  F'  will  be  situated  both  at  infinity,  so  that 
the  focal  lengths  /,  /'  will  be  infinite  also.  This  is  the  case, 
for  example,  with  the  optical  instrument  known  as  the  tele- 
scope; and,  accordingly,  any  optical  system  which  trans- 
forms a  cylindrical  bundle  of  parallel  rays  into  another 
cylindrical  bundle  of  parallel  rays  is  called  a  telescopic  (or 
afocal)  system.  The  simplest  illustration  of  such  a  system 
is  afforded  by  a  single  plane  refracting  surface  or  by  a  plane 
mirror. 


360  Mirrors,  Prisms  and  Lenses  [§  126 

126.  Formulae  for  Combination  of  Two  Optical  Systems 
in  terms  of  the  Refracting  Powers. — Although  the  formulae 
derived  in  the  preceding  section  are  very  simple  and  con- 
venient, Gullstrand's  system  of  formulae  in  terms  of  the 
refracting  powers  possesses  certain  advantages  and  is  even 
more  useful.  The  latter  formulae  may  be  derived  immedi- 
ately from  the  former,  as  will  now  be  shown. 

In  Gullstrand's  system  the  interval  between  the  two 

component  optical  sj^stems  is  expressed,  not  by  A,  but  by 

the  reduced  distance  (§  104)  c  of  the  primary  principal  point 

H2  of  system  II  from  the  secondary  principal  point  H/  of 

system  I.     Thus,  if  nh  n2  and  n2,  n^  denote  the  indices  of 

refraction  of  the  first  and  last  media  of  systems  I  and  II, 

respectively,  then 

H/H2 
c  = . 

n2 

The  connection  between  the  two  magnitudes  c  and  A  is 
easily  obtained;  for  since 

F1T2=F1,H/+H1,H2+H2F2, 
we  find  immediately: 

A=/i'+n2.c-/2. 
Now  let  us  introduce  the  following  symbols: 

/i       /i  h      h  J        J 

where  F\,  F2  denote,  therefore,  the  refracting  powers  of  the 
component  systems  and  F  denotes  the  refracting  power  of 
the  compound  system  (§§  105  and  123,  d).     Hence,  since 


/l==~FiJ     /2=F2 


we  may  write: 


Now  if  this  value  of  A  is  substituted  in  either  of  the  formulae 
J~     "A"J      J  "    A    ' 


§  126]      Combination  of  Two  Optical  Systems        361 

and  if  the  focal  lengths  are  expressed  in  terms  of  the  refract- 
ing powers,  we  find : 

F=Fl+F2-c.Fl.F2; 
which  is  Gullstrand's  formula  for  the  refracting  power 
of  the  compound  system  in  terms  of  the  refracting  powers 
of  the  two  component  systems  and  of  the  interval  c  between 
them. 

Likewise,  if  in  the  formulae 

-p  -p  _/i*/i        -p  /-p/  _  _h-h 
A    '        2  A 

F 

we  eliminate  /i,  //  and  f2,  //  and  put  A= — n2ri   „  ,  we  ob- 

t  \.t2 

tain  for  the  reduced  steps  FiF  and  F2/F/  the  following  ex- 
pressions: 

FiF  =  J^_        FVF=       F± 
m  ~F.Fi  7i3    "     F.F2 

The  positions  of  the  focal  points  F,  F'  of  the  compound  sys- 
tem with  respect  to  Hi,  H2',  respectively  are  obtained  as 
follows : 

HXF  =  H1F1+F1F  =  FiF  - m/Fi, 
H2'F'  =  H2,F2,+F2,F/ =F2'F,+n3/>2; 
and  if  herein  the  values  of  FiF  and  F2'F'  are  substituted, 
and  if  also  we  note  that 

F-F^F^l-c.Fi),        F-F2=Fi(l-c.F2), 
we  obtain  finally: 

HiF        l-c.F2        H2'F'    1-c.ffi 
m  F      '  m    ~      F       ' 

Moreover,  since 

H1H  =  H1F+FH  =  HiF+m/F, 
H2'H'  =  H2'F'+F'H'  =  WF'+rh/F, 
the  Gullstrand  system  of  formulae  for  the  combination  of 
two  optical  systems  may  be  written  as  follows : 
HiH    F2  H2  H         F\ 

~W  =  ~F'C}       ~^~=~FX' 
F=Fl+F2-c.Fl.F2. 
Accordingly,  if  the  positions  of  the  principal  points  Hi,  H/ 


362  Mirrors,  Prisms  and  Lenses  [§  127 

and  H2,  H2'  of  the  two  component  systems,  the  refracting 
powers  F\y  F2  and  the  indices  of  refraction  m,  n2  and  n3  are 
known,  we  can  calculate  the  reduced  interval  c  and  find  the 
refracting  power  F  of  the  compound  system  and  the  posi- 
tions of  the  principal  points  H,  H'.  We  shall  see  numerous 
applications  of  these  formulae  in  the  succeeding  sections  of 
this  chapter. 

127.  Thick  Lenses  Bounded  by  Spherical  Surfaces. — 
When  a  centered  system  of  spherical  refracting  surfaces  con- 
sists of  two  surfaces,  it  constitutes  a  spherical  lens  involving 
three  media,  viz.,  the  medium  of  the  incident  rays  (ni),  the 
medium  comprised  between  the  two  spherical  surfaces, 
sometimes  called  the  lens-medium  (n2),  and  the  medium  of 
the  emergent  rays  (n3),  which  is  generally  but  not  necessarily 
the  same  as  that  of  the  incident  rays.  Usually,  a  lens  is  de- 
scribed by  assigning  the  values  of  the  three  indices  of  re- 
fraction and  the  positions  of  the  centers  Ci,  C2  and  the  ver- 
tices Ai,  A2  on  the  optical  axis;  the  usual  data  being  the 
radii  ri=AiCi,  r2  =  A2C2  and  the  thickness  d=A\A2.  The 
lens  may  be  regarded,  therefore,  as  a  combination  of  two 
spherical  refracting  surfaces  whose  refracting  powers  Fi,  F2 
are  given  by  the  formulae  (§  105) 

v     fh-ni        „     n3-n2 
r  \  = ,      r2  = . 

n  r2 

Since  the  principal  points  of  a  spherical  refracting  surface 
coincide  with  each  other  at  the  vertex  of  the  surface  (§§  81 

and  119),  the  interval  c= — - — -=    *   * ,  and  therefore 

n2  n2 

d 
c= — . 

n2 

Accordingly,  if,  by  way  of  abbreviation,  we  introduce  the 
special  symbol 

N  =  n2\  (n2-ny)r2-(n2-n3)ri}+(n2-n3)(n2-ni)d 

to  denote  a  constant  of  the  lens,  we  obtain,  by  substituting 


§  127]  Thick  Lens  Formulae  363 

the  values  of  Fh  F2  and  c  in  the  formula  F=Fi+F2  —  c.Fi.F2, 
the  following  expression  for  the  refracting  power  F  of  a  lens: 

n2.r1.r2 

where  the  value  of  F  will  be  given  in  dioptries  in  case  the 

distances  n,  r2  and  d  are  all  measured  in  meters  (§  107). 

The  positions  of  the  principal  points  (H,  H')  of  a  lens  are 

determined  in  the  same  way  by  the  formulae : 

AiH        n2-n3       ,          A2H'        n2-ni      , 
= ^7—  n.d,         = ^r—  r2.d; 

and  the  positions  of  the  focal  points  (F,  F')  may  likewise  be 
calculated  from  the  following  expressions: 


—  =  -^{712.7-24 


A2F'    r2 


= —  \n2.r\  —  (712  —  n{)  d 


713         N  % 

When,  as  is  usually  the  case,  the  lens  is  surrounded  by  the 
same  medium  on  both  sides,  we  may  put 

m  =  n3  =  n,        n2=n'; 
and  then  the  above  formulae  become : 

N  =  (n'  -  n)  { n'  (r2  -  r  1)  +  (n'  -  n)  d } ; 

N 


F  = 


n  .n.r2 


AiH        n'  —  n       ,  A2H'        nf  —  n      1 

= Tf-  n.d,       =  -  -1rf-  r2.d; 

n             N  n             N 

AiF        n\   ,        ft      NJ]  A2F'    r2\   ,        f  ,      NJ 

- ^  n'.r2+(nf -n)d\,  — — =~\ n'.n- {n'—n) d 


n  N{  j         n       N  [ 

The  nodal  points  (N,  Nr)  of  a  lens  surrounded  by  the  same 
medium  on  both  sides  coincide  with  the  principal  points 
(§  122). 

The  positions  of  the  focal  points  and  principal  points 
may  be  exhibited  in  the  case  of  a  thick  convergent  lens  in 
the  following  manner,  as  described  in  Grimsehl's  Handbuch 
der  Physik: 

Two  thin  piano-lenses,  each  4  cm.  in  diameter,  are  ce- 
mented with  Canada  balsam  to  the  opposite  faces  of  a  glass 


\ 

\ 

F 

_A^ 

H 

T^^—       1 

A2    ' 

(b)  / 

i==^z^_ 

F  ^ 

_Ai| 

H 

H' 

A2^-— 

— ^F 

(c)/ 

Fig.  170,  a,  6,  c,  and  d. — Double  convex  lens:  (a)  Location  of  second  focal 
point  (F')  and  principal  point  (H') ;  (6)  Location  of  first  focal  point  (F) 
and  principal  point  (H) ;  (c)  Location  of  focal  points  (F,  F')  and  principal 
points  (H,  H').  (d)  Meniscus  convex  lens:  location  of  principal  points 
and  focal  points,  showing  their  unsymmetrical  positions  with  respect  to 
the  surfaces  of  the  lens. 


§128]  "Vertex  Refraction"  of  Lens  365 

cube  of  edge  4  cm.  and  made  of  the  same  glass,  so  as  to  form 
a  thick  symmetric  double  convex  lens,  as  represented  in 
Fig.  170,  a,  b  and  c.  A  diaphragm  with  three  parallel  horizon- 
tal slits  is  placed  in  the  path  of  a  cylindrical  beam  of  parallel 
rays  so  as  to  separate  it  into  three  smaller  beams,  and  the 
lens  is  adjusted  so  that  the  middle  beam  proceeds  along 
the  axis  of  the  lens.  The  paths  of  the  rays  in  air  can  be 
rendered  visible  by  tobacco-smoke  and  may  be  photo- 
graphed. In  this  way  figures  will  be  obtained  similar  to 
those  shown  in  the  diagrams.  The  position  of  the  second- 
ary focal  point  F'  is  shown  by  the  point  of  convergence  of 
the  rays  on  emergence  (Fig.  170,  a).  A  point  in  the  second 
principal  plane  of  the  lens  may  be  located  by  rinding  the 
point  of  intersection  of  an  incident  ray  parallel  to  the  axis 
with  the  corresponding  emergent  ray  (§  119),  as  indicated 
by  the  dotted  lines  in  the  figure;  and  the  second  principal 
point  H'  will  be  at  the  foot  of  the  perpendicular  dropped 
from  this  point  on  to  the  axis.  If  the  rays  are  sent  through 
the  lens  from  the  opposite  side  (that  is,  from  right  to  left  in 
the  drawing,  Fig.  170,  6),  they  will  intersect  on  emergence 
in  the  primary  focal  point  F;  and  the  position  of  the  primary 
principal  point  H  may  be  found  in  exactly  the  same  way 
as  above.  The  two  diagrams  Figs.  170,  a  and  b,  are  com- 
bined in  one  in  Fig.  170,  c.  In  Fig.  170,  d,  the  lens  is  con- 
cave towards  the  incident  light  and  convex  when  viewed 
from  the  other  side;  and  this  figure  shows  very  clearly  how 
the  focal  points  F,  F'  and  the  principal  points  H,  H'  may  be 
both  unsymmetrically  placed  with  respect  to  the  lens,  al- 
though here  also  we  have,  as  before,  FH  =  H'F'. 

128.  So-called  "  Vertex  Refraction  "  of  a  Thick  Lens  — 
The  step  from  the  second  vertex  (A2)  of  a  lens  to  the  second 
focal  point  (F'),  which  may  be  denoted  by  v,  is  sometimes 
called  the  "back  focus"  of  the  lens;  that  is,  v=A2~F'.  If 
the  lens  is  surrounded  by  the  same  medium  (n)  on  both 
sides,  then  v/n  =  (l  —  c.Fi)/F,  where  F  denotes  the  refract- 
ing power  of  the  lens,  F\  denotes  the  refracting  power  of 


366  Mirrors,  Prisms  and  Lenses  [§  129 

the  first  surface,  and  c=d\n'  denotes  the  reduced  thickness. 
The  reciprocal  of  this  magnitude  v/n  is  called  the  vertex  re- 
fraction of  the  lens  (  —  =  VJ  and  its  relation  to  the  re- 
fracting power  is  given  by  the  formula: 

F  F 


V  = 


1-c.Fi         n'—nd 


ri  n 
If  F  is  given  in  dioptries,  the  values  of  d  and  n  must  be  ex- 
pressed in  meters;  and  then  the  expression  above  will  give 
the  value  of  V  in  dioptries.  The  importance  of  this  function 
V  in  the  theory  of  modern  spectacle  lenses  has  been  pointed 
out  by  Von  Rohr;  it  is  measured  from  the  second  face  of  the 
lens  because  that  is  the  side  next  the  eye.  When  a  lens 
(with  spherical  surfaces)  is  reversed  by  turning  it  through 
180°  around  any  line  perpendicular  to  its  axis,  the  refracting 
power  F  remains  the  same,  whereas  the  vertex  refraction  V 
will  be  different  unless  the  lens  is  a  symmetric  lens  or  in- 
finitely thin,  in  which  latter  case  d  =  0  and  V—F.  Thus, 
whereas  the  refracting  power  of  a  lens  is  the  same  whether 
the  light  traverses  it  from  one  side  or  the  other,  the  vertex 
refraction  depends  essentially  on  which  side  of  the  lens  is 
presented  to  the  incident  rays. 

129.  Combination  of  Two  Lenses. — Let  us  take  the  sim- 
plest case,  and  suppose  that  the  system  is  composed  of  two 
infinitely  thin  co-axial  lenses,  each  surrounded  by  air.  Let 
Ai  and  A2  designate  the  points  where  the  optical  axis  meets 
the  two  lenses,  and  let  the  interval  between  them  be  denoted 
by  c;  that  is,  put  c  =  AiA2.  Since  the  principal  points  of 
an  infinitely  thin  lens  coincide  with  each  other  at  the 
point  A  where  the  axis  crosses  the  lens,  and  since  the  inter- 
vening medium  is  assumed  to  be  air  of  index  unity,  this 
distance  c  has  here  the  same  meaning  as  the  reduced  in- 
terval c  =  Hi'H2/n2  in  the  general  formulae  of  §  126.  Ac- 
cordingly, we  may  write  immediately  the  following  system 
of  formulae  for  a  combination  of  two  thin  lenses  of  refracting 


129] 


Combination  of  Two  Lenses 


367 


powers  Fi,  F2,  surrounded  on  both  sides  by  air  and  sepa- 
rated by  the  distance  c: 

F=Fl+F2-c.F1.F2; 

AiH=-^r>       A2H'=— -^-; 


AiF=- 


1-c.Fo 


A.F-1"^1 


F       '        ""  F 

These  formulae  may  also  be  expressed  in  terms  of  the  focal 
lengths /i  and/2,  as  follows: 

f_   Mi 

J  /1+/2-C' 

AM-U  A&—U  AlF=-^p^,  A2F-^p). 

J2  /l  J2  /l 

The  positions  of  the  focal  points  F,  F'  and  the  princi- 
K       J  L 


Fig.  171,  a. — Combination  of  two  thin  lenses.  Graphical  method 
of  determining  the  positions  of  the  first  focal  point  (F)  and 
principal  point  (H) :  Case  when  both  lenses  are  convex. 

pal  points  H,  H'  of  a  combination  of  two  infinitely  thin 
lenses  surrounded  by  air  may  be  constructed  geometrically 
as  follows : 


368 


Mirrors,  Prisms  and  Lenses 


[§129 


Draw  a  straight  line  to  represent  the  common  axis  of  the 
pair  of  thin  lenses,  and  mark  the  points  Ai  and  A2  (Fig.  171, 
a,  b,  and  c)  where  the  axis  crosses  the  lenses,  and  also  the 
positions  of  the  primary  focal  points  Fi  and  F2.  Through 
F2  draw  a  straight  line  perpendicular  to  the  axis,  and  take 
on  it  a  point  K  such  that  F2K  =  F2A2=/2;  this  point  K  lying 


Fig.  171,  b. — Combination  of  two  thin  lenses.  Graphical  method 
of  determining  the  positions  of  the  first  focal  point  (F)  and 
principal  point  (H) :  Case  when  first  lens  is  concave  and  second 
lens  convex. 

above  or  below  the  axis  according  as  the  second  lens  is  con- 
vex or  concave,  respectively.  Through  K  draw  a  straight 
line  parallel  to  the  axis  and  through  Ai  a  straight  line  per- 
pendicular to  the  axis;  and  let  L  designate  the  point  where 
these  two  lines  intersect.  Moreover,  let  P  designate  the 
point  of  intersection  of  the  pair  of  straight  lines  LFi  and  KAi. 
The  foot  of  the  perpendicular  let  fall  from  P  on  to  the  axis 
will  be  the  primary  focal  point  F  of  the  compound  system; 
and  the  ordinate  FP  will  be  equal  to  the  primary  focal  length 
f  of  the  compound  system;  and  hence  if  the  quadrant  of  a 


129] 


Combination  of  Two  Lenses 


369 


circle  is  described  around  F  as  center  with  radius  FP,  it  will 
cut  the  axis  at  the  primary  principal  point  H,  which  lies  to 
the  right  or  left  of  F  according  as  the  point  P  falls  above  or 
below  the  axis. 

According  to  this  construction,  the  points  P  and  K  are 
a  pair  of  conjugate  extra-axial  points  with  respect  to  the 


Fig.  171,  c. — Combination  of  two  thin  lenses.  Graphical  method  of  de- 
termining the  positions  of  the  first  focal  point  (F)  and  principal  point 
(H) :  Case  when  first  lens  is  convex  and  second  lens  concave. 

first  lens;  so  that  the  construction  really  consists  in  locating 
the  object-point  P  which  is  imaged  by  the  first  lens  in  the 
point  K.  This  will  help  the  student  to  remember  the  con- 
struction. 

In  order  to  show  that  the  construction  is  correct,  let  J 
designate  the  point  of  intersection  of  the  pair  of  straight 
lines  FP  and  LK.    Then  since  JP  and  FP  are  corresponding 
altitudes  of  the  similar  triangles  PLK  and  PFiAi,  we  have: 
JP  =L  K  ^AiF2  =  AiA2+A2F2^c-/2 
FP"FiArFiAi"       FiAi  /i 

Now  JP=JF+FP=KF2+FP  =  FP-/2,  and  therefore: 
FP-/2    c-/2 
FP         /i     ' 
and  if  this  equation  is  solved  for  FP,  we  find : 


FP 


/1+/2-C 


■/, 


370  Mirrors,  Prisms  and  Lenses  [§  130 

in  agreement  with  the  formula  found  above.    Moreover ,  in 
the  similar  triangles  AiFP  and  AiF2K, 

AiF=AxF2 . 

FP  ~F2K  ; 
and  since  AiF2  =  c-/2,    F2K=/2,    FP=/,    we  find: 

which  is  likewise  in  agreement  with  the  formula  found  above. 

Similarly,  mark  the  positions  of  the  secondary  focal  points 
F/  and  F2',  and  through  F/  draw  a  straight  line  perpendic- 
ular to  the  optical  axis,  and  take  on  it  a  point  O  such  that 
Fi'O  =  F/Ai  =//.  Through  O  draw  the  straight  line  OR  par- 
allel to  the  axis,  and  through  A2  a  straight  line  perpendicular 
to  the  axis;  and  let  R  designate  the  point  where  these  two 
lines  intersect.  Then  if  Q  designates  the  point  of  intersec- 
tion of  the  straight  lines  F2'R  and  A20,  that  is,  if  Q  is  the 
image  of  O  in  the  second  lens,  the  secondary  focal  point  F'  of 
the  combination  will  be  at  the  foot  of  the  perpendicular 
drawn  from  Q  to  the  optical  axis,  and  the  secondary  prin- 
cipal point  H'  will  lie  on  the  axis  at  a  distance  F'H^F'Q. 
This  construction  may  be  proved  in  a  manner  entirely  an- 
alogous to  the  proof  given  above. 

130.  Optical  Constants  of  Gullstrand's  Schematic  Eye.— 
As  a  further  illustration  of  the  use  of  the  formulae  for  the 

CORNEA  , ^^ , 


AQUEOUS  /       /  CORE  \        \VITREOUS 


■"3  \  txA  tic, 


Fig.  172. — Schematic  eye. 


combination  of  two  optical  systems,  let  us  apply  them  to 
the  calculations  of  the  refracting  power  (F)  of  the  human 
eye,  together  with  the  positions  of  the  principal  points  (H,H') 


130] 


Constants  of  Schematic  Eye 


371 


and  the  focal  points  (F,  F').  For  this  purpose  we  shall  use 
the  data  of  Gullstrand's  schematic  eye  (in  its  passive 
state,  accommodation  entirely  relaxed)  which  are  given 
in  the  third  edition  of  Helmholtz's  Handbuch  der  physiolo- 
gischen  Optik,  Bd.  I  (Hamburg  u.  Leipzig,  1909),  pages  300 
and  301,  as  follows  (see  Fig.  172) : 

Indices  of  refraction: 

Cornea n2  =  1.376 

Aqueous  and  vitreous  humors  n3  =  n7  =  1.336 

Lens n4  =  n6  =  1 .386 

Lens-core n5  =  1.406 

Position  of  ssur faces: 

mm. 


Posterior  surface  of  cornea : 

AiA2  =  0.5 

Anterior  surface  of  lens : 

AiA3  =  3.6 

n 

u 

"  lens-core: 

AiA4 =4.146 

Posterior 

a 

"  lens-core: 

AiA5  =  6.565 

a 

u 

"  lens: 

AiA6  =  7.2 

Radii  of  surfaces: 

Anterior  surface  of  cornea : 

ri=+  7.7 

Posterior 

a 

a           a 

r2  =  +  6.8 

Anterior 

a 

lens: 

r3=+10.0 

it 

a 

lens-core: 

r4=+  7.911 

Posterior 

a 

u       a          a 

r5=-  5.76 

a 

it 

lens: 

r6=-  6.0 

mm. 


Consider,  first,  the  cornea-system  composed  of  the  an- 
terior and  posterior  surfaces  of  the  cornea.  The  refracting 
power  of  the  anterior  surface  is : 

n2-ni 


Fi  = 


n 


+48.831  dptr.; 


and  that  of  the  posterior  surface  is : 


F2  = 


n%  —  n<i 
r2 


=  -5.882  dptr. 


The  reduced  interval  between  the  two  surfaces  is: 

AxA2    0.0005 


Cl  = 


m 


1.376 


372  Mirrors,  Prisms  and  Lenses  [§  130 

Hence,  if  Fn  denotes  the  refracting  power  of  the  cornea- 
system,  where 

Fi2=Fi+F2  —  C1.F1.F2, 
we  find: 

F12  =  +43.053  dptr. 
The  positions  of  the  principal  points  of  the  cornea-system 
are  given  by  the  formulae: 

AiHi2_Ci.Fa         A2Hi2' g.Fi  # 

m     ~  Fu  '  n3  Fn  ' 

whence  we  find: 

A1H12  =  -  0.0496  mm.,        AiHi2'  =  -  0.0506  mm. 
The  lens-system  is  composed  of  four  refracting  surfaces. 
The  first  two  surfaces  form  the  so-called  anterior  cortex  and 
the  last  two  surfaces  the  posterior  cortex.     The  refracting 
power  of  the  anterior  surface  of  the  lens  is : 

ft .*=*.. +5  dptr.; 

and  that  of  the  anterior  surface  of  the  lens-core  is : 

FJ*Zl** +2.528  dptr. 

r4 

The  reduced  interval  between  these  two  surfaces  is 
A3A4  ^0.000546 
C3~   ru   ~    1.386     ' 
Hence,  if  F34  denotes  the  refracting  power  of  the  combina- 
tion, that  is,  if 

Fs^=F3+F4  —  C3.F3.F4, 
we  find:  F34  =4-7.523  dptr. 

If  the  principal  points  of  the  anterior  cortex  are  designated 
by  H34,  H34',  then 

A3H34    C3.F4        A4H34/=     C3.F3  . 
m        F34  '  nh  F34   ' 

whence  we  obtain: 

AiH34  =  +3.777  mm.,        A1H34'  =  +3.778  mm. 
so  that  the  principal  points  of  the  anterior  cortex  are  coin- 
cident with  each  other, 


§  130]  Constants  of  Schematic  Eye  373 

Proceeding  in  the  same  way  with  the  posterior  cortex,  we 
have: 

F5=^^-5= +3.472  dptr.,  F6=^^-6= +8.333  dptr., 
rb  r& 

_A5A6_  0.000635  . 

C5~   n6         1.386     ' 

and  hence  if 

F66  =  F5+F6~C&.F5.F6) 

we  find :      Fb6  =  + 1 1 .792  dptr. 
Moreover,  since 

rib        FbQ  '  rn  Fb& 

we  have  finally  for  the  positions  of  the  principal  points  of 
the  posterior  cortex: 

AiH56  =  +7.0202  mm.,      AiH56'  =  +7.0198  mm. ; 
so  that  H56  and  H56'  may  also  be  regarded  as  coincident. 

If  the  refracting  power  of  the  lens-system  as  a  whole  is 
denoted  by  L,  then 

L  =7^34+^56^5.^34-^56, 

where 

_H84H56/__  0.0032422  . 
S~      rib      ""1.406 
and  if  P,  P'  designate  the  principal  points  of  the  lens-system, 
then 

B^jJF^  Hse'P'^        5.^34 

7i3         L  ni  L 

Accordingly,  we  find: 

L=  +19.110  dptr.; 
AiP=  +5.6780  mm.,       AiP'  =  +5.8070  mm. 
Lastly,  combining  the  cornea-system  and  the  lens-system, 
we  obtain  for  the  refracting  power  of  the  entire  optical  sys- 
tem of  the  eye : 

F=Fl2+L-c.F12.L, 
where 

==Hi2/P  ^0.0057285 
C~    m  1.336     ' 


374  Mirrors,  Prisms  and  Lenses  [§  131 

Also, 

Hi2H  =  cX       P/H/  =  _c.Fi2 
m       F  m  F    ' 

where  H,  H'  designate  the  positions  of  the  principal  points  of 
the  eye.    Thus,  we  find : 

F  =+58.64  dptr.; 
AiH=  +1.348  mm.,      AiH'  =  +1.602  mm. 
If  the  focal  lengths  of  the  eye  are  denoted  by  /  and  /',  then, 
since  f= rii/ F  and  /'  =  -  n7/F,  we  obtain : 

/=  +17.055  mm.,         /'  =  -22.785  mm. 
The  focal  points  F,  F'  are  located  as  follows : 

AiF  =  -  15.707  mm.,  AiF'  =  +24.387  mm. 
In  Gullstrand's  schematic  eye  the  length  of  the  eyeball 
is  taken  as  24  mm.,  and  therefore  the  second  focal  point  F' 
is  not  on  the  retina  but  0.387  mm.  beyond  it;  so  that  the 
schematic  eye  is  not  emmetropic  but  hypermetropic  (see 
§  153)  to  the  extent  of  1  dptr. 

131.  Combination  of  Three  Optical  Systems. — It  is  fre- 
quently the  case,  especially  in  problems  connected  with 
physiological  optics,  that  we  desire  to  find  the  resultant  of 
three  co-axial  optical  systems  of  known  refracting  powers 
Fi9  Fi  and  Fs  separated  by  given  intervals  ch  c2,  where 

Hi  H2  Ho  H3 

Ci  = ,         c2  = , 

the  principal  points  of  the  component  systems  being  desig- 
nated by  Hi,  Hi';  H2,  H2';  and  H3,  H3'.  The  indices  of  re- 
fraction of  the  first  and  last  media  of  system  I  are  denoted 
by  nit  n2;  °f  system  II  by  ti2,  ^3,'  and  of  system  III  by  n3,  n4. 

Here  let  us  employ  the  symbol  D  to  denote  the  refracting 
power  of  the  compound  system  (I +11),  and  the  letters  G,  G' 
to  designate  the  positions  of  the  principal  points  of  this  par- 
tial combination.  Evidently,  according  to  the  formulae 
derived  in  §  126,  we  may  write: 

D=F1+F2-c1.F1.F2  ; 
HiG=c±F2         H2/G^     c1.F1 

Til  D      '   •  713  D 


§  131]      Combination  of  Three  Optical  Systems         375 

Now  let  F  denote  the  refracting  power  of  the  combination 
of  systems  I,  II  and  III,  and  let  H,  H'  designate  the  posi- 
tions of  the  principal  points  of  this  compound  system.  Then 
if  the  reduced  interval  between  (I +11)  and  III  is  denoted 
by  k,  that  is,  if 

G'H3 


k  = 


nz 


then  also 


Since 


we  find: 


F  =  D+Fz-k.D.F3, 

GH    k.F3  HS'K'        k.D 

m       F    '  m         "  F 

Gr  H3      Cj  H2        H2  H3 

nz  nz  nz 

k=c1.F1+c2.D 


D 

If  now  these  equations  are  combined  so  as  to  eliminate  D 
and  k,  the  following  system  of  formulae  for  the  combination  of 
three  optical  systems  will  be  obtained  finally : 

F=Fi(l  -  c2.F3)  +F2(1  - c1.F1)  (1  -  c2F,)  +F8(1  -  C1.F1) ; 
HiH         ci  C2.F3-C1.F1. 

m        1-ci.Fi     F(l-ci.Fi)' 
H3/H/=_      c2         c2.F3-c1.F1 

7i4  I-C2.F3      F(1-C2.F3)    ' 

In  the  special  case  when  the  compound  system  is  symmet- 
rical with  respect  to  system  II,  that  is,  when  n3  =  n2  and  n±  =  n\ 
and  c2  =  ci  =  c  and  F3=Fi,  the  formulae  above  will  be  simpli- 
fied as  follows: 

F=(l-c.F0  (2Fi+F2-c.Fi.F2), 
HiH  =  HTV=      c 
ni  n\        1  -  c.Fi ' 

Thus,  if  an  optical  system  is  symmetrical  with  respect  to 
a  middle  component  part  of  the  system,  the  principal 
points  (H,  H')  will  be  symmetrically  placed,  and  their  posi- 
tions will  be  independent  of  the  refracting  power  F2  of  the 


376  Mirrors,  Prisms  and  Lenses  [§  132 

middle  system.  These  latter  formulae  should  be  compared 
with  the  formulae  for  a  " thick  mirror"  to  be  developed  in 
the  following  section. 

132.  "  Thick  Mirror." — The  general  formulae  which  have 
been  derived  in  this  chapter  are  applicable  also  when 
the  centered  system  of  spherical  surfaces  includes  one  or 
more  reflecting  surfaces,  provided  that  reflection  is  treated 
as  a  special  case  of  refraction,  according  to  the  method  ex- 
plained in  §  75.  Thus,  for  example,  if  the  rays  are  reflected 
at  the  kth  surface  of  the  system,  we  must  put  nk+1=  -nk; 
and,  consequently,  if  the  reflecting  power  of  this  surface  is 
denoted  by  Fk,  we  shall  have  Fk  =  nk/fk=nk/fk,  in  accord- 
ance with  the  characteristic  requirement  that  the  focal 
lengths  of  a  spherical  mirror  are  identical,  that  is,  /=/' 
(see  §77). 

A  special  case  of  much  interest  and  practical  importance 
occurs  when  the  last  surface  of  the  system  acts  as  a  mirror, 
the  rays  of  light  arriving  there  being  reflected  back  through 
the  system  as  so  to  emerge  finally  at  the  first  surface  into 
the  medium  of  index  n\  where  they  originated.  For  ex- 
ample, this  happens  always  in  the  case  of  an  ordinary  glass 
mirror  which  is  silvered  at  the  back.  The  rays  return  into 
the  air  in  front  of  a  mirror  of  this  kind  after  having  twice 
traversed  the  thickness  of  the  glass,  and  the  failure  to  take 
account  of  the  refractions  from  air  to  glass  and  from  glass 
to  air  is  sometimes  responsible  for  serious  errors  in  the 
measurement  of  the  focal  length  of  a  glass  mirror  silvered 
at  the  back.  The  image  produced  by  rays  which  have  been 
partially  reflected  from  the  second  surface  of  an  ordinary 
lens  is  often  very  disturbing,  although  the  intensity  of  the 
reflected  fight  is  usually  comparatively  feeble  unless  the 
second  surface  of  the  lens  has  been  silvered. 

The  name  "thick  mirror"  has  been  applied  by  Dr.  Searle* 
to  any  combination  of  centered  spherical  refracting  surfaces 

*  G.  F.  C.  Searle:  The  determination  of  the  focal  length  of  a  thick 
mirror.    Proc.  Cambr.  Phil.  Soc,  xviii,  Part  iii,  1915,  115-126. 


132] 


"  Thick  Mirror 


377 


wherein  the  rays  are  supposed  to  be  reflected  at  the  last  sur- 
face and  to  return  through  the  system  #in  the  opposite  sense. 
It  may  easily  be  shown  that  a  " thick  mirror"  as  thus  de- 
fined acts  exactly  like  a  single  spherical  reflecting  surface 
(or  "thin  mirror,"  as  we  may  calhit,  having  in  mind  a  cer- 
tain analogy  which  exists  here  between  lenses  and  mirrors), 
whose  vertex  and  center  have  perfectly  definite  and  calcu- 
lable positions  depending  on  the  constants  of  the  "thick 
mirror."  This  is  proved  by  Dr.  Searle  in  a  simple  manner 
as  follows : 

In  Fig.   173  the  system  is  represented  as  consisting  of 
three  spherical  surfaces,  the  first  two  forming  a  thick  lens 


LENS 


MIRROR 


Fig.  173. — Diagram  of  "thick  mirror"  system. 

and  the  last  surface  being  a  spherical  mirror  with  its  vertex 
at  a  point  A  on  the  axis  of  the  lens.  Draw  the  straight  line 
QV  parallel  to  the  axis  of  the  system  to  represent  the  path  of 
an  incident  ray;  which  after  traversing  the  lens  and  being 
reflected  at  the  mirror  will  again  emerge  from  the  lens  and 
cross  the  axis  at  the  secondary  focal  point  (F')  of  the  system. 
The  point  V  designates  the  point  of  intersection  of  the  in- 
cident ray  QV  and  the  corresponding  emergent  ray  VF', 
and  hence  this  point  must  lie  in  the  secondary  principal 
plane  of  the  system  (§  119).  Consequently,  the  foot  of  the 
perpendicular  let  fall  from  V  on  to  the  axis  will  be  the  sec- 
ondary principal  point  H'.    But  by  the  principle  of  the  re- 


378  Mirrors,  Prisms  and  Lenses  [§  132 

versibility  of  the  light-path  (§  29),  if  the  straight  line  F'V  is 
regarded  as  an  incident  ray,  then  VQ  will  be  the  path  of  the 
corresponding  emergent  ray,  and  since  in  this  case  the  emer- 
gent ray  is  parallel  to  the  axis,  the  corresponding  incident 
ray  F'V  must  cross  the  axis  at  the  primary  focal  point  F, 
so  that  the  two  focal  points  F  and  F'  will  be  coincident. 
Moreover,  the  point  V  must  lie  in  the  primary  principal 
plane,  and  hence  the  two  principal  planes  are  coincident. 
But  these  are  the  characteristics  of  a  spherical  mirror,  and 
it  is  evident  that  the  " thick  mirror"  is  equivalent  to  a  "thin 
mirror"  with  its  vertex  at  H  (or  IF)  and  its  center  at  a  point 
K  such  that  HK=2HF. 

The  four  images  of  Puekinje  are  the  catoptric  images 
formed  in  the  eye  by  reflection  at  the  anterior  and  posterior 
surfaces  of  the  cornea  and  the  crystalline  lens;  which  are  of 
fundamental  importance  in  determining  the  curvatures  and 
positions  of  the  refracting  surfaces  in  the  optical  system  of 
the  eye.  The  first  image  is  produced  by  direct  reflection  at 
the  anterior  surface  of  the  cornea,  but  the  optical  systems 
which  give  rise  to  the  three  other  images  are  more  or  less 
complicated.  However,  according  to  the  above  explanation, 
each  of  these  systems  may  be  reduced  to  a  single  reflecting 
surface  of  appropriate  radius  with  its  center  at  a  certain 
definite  place  to  be  ascertained  by  the  conditions  of  the 
problem.  One  of  these  cases  will  be  investigated  presently, 
as  soon  as  the  formulae  for  a  thick  mirror  have  been  devel- 
oped. 

The  radius  and  positions  of  the  vertex  and  center  of  the 
equivalent  "thin  mirror"  may  easily  be  calculated  by  means 
of  the  general  formulae  which  were  obtained  in  the  preced- 
ing section  for  a  combination  of  three  optical  systems.  Here 
the  first  system  (I)  of  refracting  power  F\  may  be  regarded 
as  composed  of  the  entire  lens-system  lying  in  front  of  the 
reflecting  surface;  while  the  mirror  itself  of  reflecting  power 
F2  may  be  regarded  as  the  second  system  (II).  In  this  case 
the  third  system  (III)  will  be  the  lens-system  reversed,  and 


§132]  "  Thick  Mirror "  379 

its  refracting  power  will  be  the  same  as  that  of  system  I, 
that  is,  Fz  =  F2;  but  the  principal  points  H3  and  H3'  of  system 
III  will  coincide  with  the  principal  points  HY  and  Hi,  re- 
spectively, of  system  I.  Above  all  we  must  impose  here 
the  conditions  that 

7i3=  —ri2=—nf,  ft4  =  — fti=— ft, 
where  n  denotes  the  index  of  refraction  of  the  medium  of 
the  object-space  and  n'  denotes  the  index  of  refraction  of- 
the  medium  in  contact  with  the  reflecting  surface.  These 
conditions  take  account  of  the  fundamental  fact  that  the 
sense  of  propagation  of  the  light  is  reversed  by  the  mirror. 
The  principal  points  H2,  H2'  of  the  mirror  coincide  with  each 
other  at  its  vertex  which  will  be  designated  here  by  the 
letter  A'.  If  therefore  Ci,  Oi  denote  the  reduced  intervals 
between  the  first  system  and  the  mirror  and  between  the 
mirror  and  the  third  system,  we  have : 

H/A'          A'H3    Hi'A' 
Ci  =  — — ,   c2  = = — — , 

n  m         n 

and  hence 

ci  =  c2  =  c,  say. 

Moreover,  if  the  radius  of  the  reflecting  surface  is  denoted 

2n' 
by  r',  then  F  =  ——.     Introducing  these  relations  in  the 

general  formulae  for  the  combination  of  three  optical  sys- 
tems (§  131),  we  obtain  the  following  expressions  for  finding 
the  reflecting  power  (Fu)  and  the  positions  of  the  principal 
points  H13,  Hi3'  of  a  "thick  mirror": 

Fi3  =  (l-c.F1)  (2F1+F2-c.F1.F2) 

=  a-c.F1){2F1-'^-(l-c.F1)}; 

H,H13    H^,/         c 


ft  ft         1  —  c.Fi 

Accordingly,  we  see  not  only  that  the  principal  points  of 
a  " thick  mirror"  are  coincident  with  each  other,  but  that 
the  position  of  the  vertex  Hi3  of  the  equivalent  "thin  mirror  " 
is  entirely  independent  of  the  power  F2  or  the  curvature  of 


H13K 


380  Mirrors,  Prisms,  and  Lenses  [§  132 

the  actual  mirror.  The  position  of  Hi3  does  depend  on  the 
position  of  the  vertex  A'  of  the  actual  mirror;  but  for  any 
mirror  placed  at  A'  the  vertex  of  the  equivalent "  thin  mir- 
ror" will  be  at  the  same  point  Hi3.  It  may  be  noted  that 
the  formula  for  the  reflecting  power  of  a  " thick  mirror"  is 
identical  in  form  with  the  expression  for  the  refracting  power 
of  a  compound  system  which  is  symmetric  with  respect  to 
a  middle  member  (see  end  of  §  131). 

If  the  center  of  the  equivalent  "thin  mirror"  is  designated 
by  K,  then  its  radius  will  be 

2n 

and  hence 

HiK_         c.F2-2 
n       2F1+F2-C.F1.F2' 

If  the  surface  of  the  mirror  (II)  is  plane,  then  F2=0,  and 
in  this  case  the  formulae  for  the  equivalent  "thin  mirror" 
become : 

F1,=2F1(l-c.F1),    Mi!=Sl5il=      c  H1K=_1 

n  n         1-c.Fi        n  Fi 

The  distinguishing  characteristic  of  the  imagery  in  a 
spherical  mirror  is  that  a  pair  of  conjugate  axial  points  M,  M' 
is  harmonically  separated  by  the  vertex  H  and  the  center  K 
of  the  mirror,  that  is,  (KHMM')  =  -1(§  68).  An  interest- 
ing special  case  occurs  when  one  of  the  points  K  or  H  is  at 
infinity;  for  in  that  case  the  reflecting  power  of  the  mirror 
vanishes  (F=0).  When  the  center  of  the  mirror  is  at  an 
infinite  distance  from  it,  the  mirror  lies  midway  between 
object  and  image  (MH  =  HM')  and  the  lateral  magnifica- 
tion is  equal  to  +1  (y'  =  y);  which  is  the  case  of  an  ordinary 
plane  mirror.  But,  on  the  other  hand,  if  the  mirror  itself 
is  at  an  infinite  distance,  while  the  center  K  remains  in  the 
region  of  finite  space,  it  is  the  center  of  the  mirror  in  this 
case  that  is  always  midway  between  object  and  image,  that 
is,  MK  =  KM',  and  now  the  lateral  magnification  will  be 
equal  to  —  1,  that  is,  the  image  will  be  of  the  same  size  as 


§  132]  " Thick  Mirror"  381 

the  object  but  inverted  {yf=—y).  Both  of  these  special 
cases  may  be  realized  by  a  "thick  mirror";  for  the  condi- 
tion that  the  reflecting  power  of  the  equivalent  "thin  mirror" 
shall  vanish  (Fu=0)  requires  that  either 

2^1+^2-0.^1.^2=0, 

or 

1-0.^1=0. 

In  the  former  case  the  center  of  the  mirror  (K)  is  at  infinity, 
and  in  the  latter  case  the  vertex  of  the  mirror  (Hi3)  is  at 
infinity.  If  therefore  the  distance  between  the  anterior  lens- 
system  and  the  final  reflecting  surface  of  a  "thick  mirror" 
is  c  =  l/Fh  the  system  will  produce  an  inverted  image  of  the 
same  size  as  the  object,  no  matter  where  the  object  is  placed. 
As  an  illustration  of  the  use  of  the  formulae  for  a  "thick 
mirror,"  consider  the  optical  system  in  the  eye  which  pro- 
duces the  third  of  the  so-called  Purkinje  images,  to  which 
allusion  was  made  earlier  in  this  section.  The  third  image  is 
formed  by  rays  which  coming  from  an  external  source  enter 
the  eye,  and  after  having  traversed  the  cornea  system  and 
the  aqueous  humor  are  reflected  at  the  anterior  surface  of  the 
crystalline  lens;  whence  returning  through  the  same  media 
in  the  reverse  order  they  issue  again  into  the  air.  In  order 
to  find  the  "thin  mirror"  which  is  equivalent  to  this  system, 
we  shall  employ  the  constants  of  Gullstrand's  schematic 
eye  as  given  in  §  130.  The  vertex  of  the  anterior  surface  of 
the  cornea  will  be  designated  by  Ax  and  the  principal  points 
of  the  cornea-system  by  Hi  and  H/.  We  found  that  AiHi  = 
-0.0496  mm.  and  AiH/= -0.0506  mm.;  also,  Fi  =  +43.05 
dptr.,  where  F\  denotes  the  refracting  power  of  the  cornea- 
system.  The  reflecting  power  of  the  anterior  surface  of  the 
lens  is  given  by  the  formula: 

r3 
where  n3  =  1.336  and  r3= +0.010  m.;  accordingly,  we  find: 
F2  =  ~  267.2  dptr. 


382  Mirrors,  Prisms  and  Lenses  [§  132 

The  reduced  distance  between  the  cornea-system  and  the 

first  surface  of  the  lens  is : 

H/A3 
c  = , 

nz 

where   A3   designates    the   vertex  of    this   surface;  AiA3  = 
0.00036  m.    Thus,  we  obtain : 

c= 0.0027325. 
Substituting  these  numerical  values  in  the  system  of  for- 
mulae for  a  "thick  mirror, "  we  find  for  the  reflecting  power  of 
the  equivalent  "thin  mirror"  in  this  case: 
F13=~  132.062  dptr.; 
and  for  the  positions  of  its  vertex  Hi3  and  its  center  K: 
HiH13=  +3.0968  mm.,     HiK= +18.2412  mm. 
Accordingly,   the  system  that  produces  the  third  of  the 
Purkinje  images  in  Gullstrand's  passive  schematic  eye 
is  equivalent  to  a  convex  mirror  of  radius  15.14  mm.  with 
its  vertex  at  a  distance  of  3.047  mm.  from  the  vertex  of  the 
anterior  surface  of  the  cornea. 

Formulas  for  calculating  the  reflecting  power  Fu  of  a 
"thick  mirror"  may  also  be  obtained  in  terms  of  different 
data  from  those  employed  in  the  expressions  which  have 
been  deduced  above.  Suppose,  for  example,  that  we  are 
given  the  refracting  power  (F)  of  a  centered  system  of 
spherical  refracting  surfaces,  the  positions  of  the  principal 
points  of  the  system  (H,  H')>  and  the  indices  of  refraction 
of  the  first  and  last  media  (n,  n');  together  with  the  radius 
(r')  and  the  position  of  the  vertex  (A')  of  the  last  surface; 
and  that  it  is  required  to  determine  in  terms  of  these  data  the 
characteristics  of  the  imagery  produced  by  light  which  pro- 
ceeding from  the  object-space  through  the  system  is  partially 
reflected  at  the  last  surface  and  again  partially  refracted  at 
the  first  surface  into  the  original  medium.  In  order  to  solve 
this  problem  in  the  simplest  way,  it  is  convenient  to  employ 
a  mathematical  artifice  which  will  be  found  to  be  serviceable 
in  other  optical  problems.  The  refracting  power  of  an  in- 
finitely thin  concentric  lens  is  equal  to  zero,  and  it  is  easy  to 


§132]  " Thick  Mirror"  383 

show  that  such  a  lens  may  be  inserted  anywhere  in  an  opti- 
cal system  without  affecting  at  all  the  resultant  imagery 
(see  §  90).  Let  us  suppose,  therefore,  that  the  given  optical 
system  is  terminated  by  an  infinitely  thin  layer  of  material  of 
index  n',  bounded  by  two  concentric  spherical  surfaces,  the 
first  of  which  coincides  with  the  last  surface  of  the  given 
system.  Under  these  circumstances  the  resultant  system  may 
be  considered  as  compounded  of  three  component  systems, 
namely,    (1)  the  given  system  of  refracting  power  Fi  =  F, 

2n' 
(2)  a  mirror  of  reflecting  power  F2  = ,  and  (3)  the  given 

system  reversed  (F3  =  F).    Hence,  if 

H'A' 

the  following  formulae  will  be  obtained  in  the  same  way  as 
above : 


F^il-c.F) 

HH13    HHi,/ 


2F-  £5.(1- c.F) 


n  n         l-c.F   ' 

which  are  similar  in  form  to  the  previous  expressions,  but 
c  here  has  a  different  meaning  and  F  denotes  the  refracting 
power  of  the  entire  lens-system  and  not  merely  of  that  part 
of  the  system  which  is  in  front  of  the  reflecting  surface. 

A  problem  of  considerable  interest,  especially  in  connec- 
tion with  the  optical  system  of  the  human  eye,  is  the  inves- 
tigation of  the  procedure  of  the  light  which  after  being  par- 
tially reflected  at  the  last  surface  of  the  system  (as  in  the 
case  above)  is  also  partially  reflected  at  the  first  surface,  so 
that  it  emerges  finally  into  the  last  medium  of  index  n' .  The 
imagery  in  this  case  may  be  determined  by  adding  a  second 
infinitely  thin  concentric  lens,  which  is  assumed  to  be  made 
of  material  of  index  n  and  whose  second  surface  coincides 
with  that  of  the  first  surface  of  the  system.  Accordingly, 
now  we  shall  have  five  systems  in  all,  namely,  the  first  three 
systems  whose  reflecting  power  Fu  was  obtained  above, 


384  Mirrors,  Prisms  and  Lenses  [Ch.  XI 

a  fourth  system  consisting  of  the  first  surface  of  the  lens- 

2n 
system  acting  as  a  mirror,  whose  reflecting  power  is  F4  =  —  , 

r 

where  r  denotes  the  radius  of  this  surface,  and  a  fifth  system 
of  refracting  power  F&  =  F.  The  entire  system,  whose  refract- 
ing power  may  be  denoted  by  Fu,  and  whose  principal  points 
may  be  designated  by  Hi5,  Hi5',  may,  therefore,  be  considered 
as  compounded  of  3  systems  of  powers  Fu,  2n/r  and  F,  sep- 
arated by  the  intervals  Ci  and  c2,  where  (if  A  designates  the 
vertex  of  the  first  surface  of  the  lens-system) 

AH13  AH 

ci= »       c2  = . 

n  n 

Accordingly,  by  substituting  Fu  in  place  of  Fh  2n/r  in  place 

of  F2,  and  F  in  place  of  F3  in  the  formulae  of  §  131  for  the 

combination    of    three    optical    systems,    we    obtain    here: 

Fn=Fu  (1-C2.F)+^  (I-C1.F13)  (l-c.F)+F  (1-d.Fu); 

H13H15  d  c2.F-c1.Fu    . 


n  1-ci.Fn    Fu(l-ci.Fi3)  ' 

ITHil=___C2_     c^.F-d.Fu 
n'  1-C2.F+Fn(l-C2.F)' 

Being  given  the  magnitudes  denoted  by  n,  n',  r,  r',  and  F 
and  the  positions  of  the  points  designated  by  A,  A'  and  H,  H', 
and  having  found  by  means  of  the  previous  formulae  the 
magnitude  denoted  by  Fu  and  the  position  of  the  point 
designated  by  Hi3  (or  Hi/),  we  can  introduce  these  data  and 
results  in  the  expressions  above  and  thus  determine  the  re- 
fracting power  Fit  and  the  distances  AH15,  A'His'  of  the  prin- 
cipal points  H15,  Hi5'  from  the  vertices  A,  A'  of  the  first  and 
last  surfaces,  respectively. 

PROBLEMS 

1.  Find  the  refracting  power  and  the  positions  of  the  focal 
points  and  principal  points  of  each  of  the  following  glass 
lenses  surrounded  by  air  (n  =  l,  n'  =  1.5);  and  make  an  ac- 


Ch.  XI]  Problems  385 

curate  sketch  of  each  lens,  marking  the  positions  of  the 
points  mentioned. 

(a)  Double  convex  lens  of  radii  10  cm.  and  15  cm.  and  of 
thickness  3  cm. 

(&)  Double  concave  lens  with  same  data  as  above. 

(c)  Meniscus  lens  for  which  ri=+5cm.,  r2=+10cm., 
and  d=+3  cm. 

(d)  Meniscus  lens  for  which  ri=+6  cm.,  r2=+3  cm.,  and 
d=+2.52  cm. 

(e)  A  plano-convex  lens  with  its  curved  surface,  of  radius 
5  cm.,  turned  towards  the  incident  light;  d  =  +0.5  cm. 

(/)  Symmetric  convex  lens,  the  radius  of  each  surface 
being  5  cm.;  d  =  +0.5  cm. 

(g)  Symmetric  concave  lens  with  same  data  as  above. 

(h)  A  meniscus  lens  with  radii  ri=+5cm.,  r2=+8cm., 
and  thickness  d=+0.5  cm. 

(i)  A  meniscus  lens  with  radii  ri=+8cm.,  r2  =  +5cm., 
and  thickness  d=  +0.5  cm. 

(j)  A  meniscus  lens  with  radii  ri  =  +8cm.,  r2=+7cm., 
and  thickness  d=  +3  cm. 

(k)  A  plano-convex  lens  with  its  curved  surface,  of  radius 
5  cm.,  turned  towards  the  incident  light;  d=+5  cm. 

Answers : 


F 

in  clptr. 

AiF 
in  cm 

A2F' 
in  cm. 

AiH 
in  cm. 

A2H' 
in  cm. 

(a)  +  8.000 

-11.667 

+  11.250 

+0.833 

-1.250 

(b)   -  8.667 

+  12.307 

-12.692 

+0.769 

-1.154 

(c)   +  6.000 

- 18.333 

+  13.333 

-1.667 

-3.333 

(d)  -  6.000 

+21.333 

- 14.333 

+4.667 

+2.333 

(e)  +10.000 

-  10.000 

+  9.667 

0.000 

-0.333 

(/)   +19.666 

-  4.915 

+  4.915 

+0.170 

-0.170 

(g)  -20.335 

+  5.082 

-  5.082 

+0.164 

-0.164 

(h)  +  3.959 

-25.789 

+24.421 

-0.526 

-0.842 

(i)   -   3.542 

+  29.176 

-27.647 

+0.941 

+0.588 

(j)        0.000 

00 

00 

00 

00 

(k)  +10.000 

-10.000 

+  6.667 

0.000 

-3.333 

2.  In  a  symmetric  lens  (n  =  —r<i=r)  surrounded  by  the 


386  Mirrors,  Prisms  and  Lenses  [Ch.  XI 

same  medium  (index  n)  on  both  sides,  show  that  we  have 

the  following  sj^stem  of  formulae: 

N=(:n,-n)  \  (n'-n)  d-2n'.r  } ; 

„         N      AiF        A2F'         r\(,      w       , 

F=——r-  ;  =  — =  —  —\  (n'—n)  d—n' 

n'r*       n  n  N  { 

AiH__A2H/_  _n'-n    , 

n  n  N 

3.  If  the  first  face  of  a  lens  is  plane,  and  if  the  radius  of 
the  curved  face  is  denoted  by  r,  show  that 

r  n       n 

AiF_    r       d   m     A2Fr=        r 

n      n'—n    n' '         n  n'—n ' 

And  if  the  second  face  of  the  lens  is  plane, 

„    n'—n       .  TT    .      A2H'        d 

F= ;   AiH=0;    — — =—  -.;  etc. 

r  n  n 

If  either  face  of  a  lens  is  plane,  the  refracting  power  of  the 
lens  is  equal  to  that  of  the  curved  surface  and  is  entirely  in- 
dependent of  the  thickness  of  the  lens;  and,  moreover,  one 
of  the  principal  points  coincides  with  the  vertex  of  the  curved 
face. 

4.  If  the  radii  n  and  r2  of  the  two  surfaces  of  a  lens  are 
both  positive,  and  if  r2  is  greater  than  n,  show  that  the  lens 
is  convergent,  provided  the  lens-medium  is  more  highly  re- 
fracting than  the  surrounding  medium. 

5.  A  "lens  of  zero-curvature"  is  a  crescent-shaped  menis- 
cus for  which  r2  =  ri=r.  Show  that  such  a  lens  is  always 
convergent  unless  it  is  infinitely  thin;  and  that  this  is  the 
case  whether  the  lens-medium  is  more  or  less  highly  refract- 
ing than  the  surrounding  medium. 

6.  Show  that  a  meniscus  lens  for  which 

ri>r2>0  and  n'>n 

is  divergent  provided  its  thickness  is  less  than 

n'(ri—r2) 
n' — n 


Ch.  XI]  Problems  387 

7.  Show  that  in  any  meniscus  lens  surrounded  by  air  at 
least  one  of  the  principal  points  must  lie  outside  the  lens. 

8.  A  so-called  concentric  lens  is  one  for  which  the  centers 
of  curvature  of  the  two  faces  are  coincident  (d  =  r\  —  r2).  It 
may  be  double  convex  or  meniscus.  Show  that  the  refract- 
ing power  of  a  concentric  lens  surrounded  by  the  same  me- 
dium on  both  sides  is 

w_n(n'-n)(l  _1_\ 
n'       \n     rj' 
and  that  the  principal  points  coincide  at  the  common  center 
of  the  two  surfaces. 

9.  Find  the  refracting  power  and  the  positions  of  the  focal 
points  and  principal  points  of  each  of  the  following  concentric 
glass  lenses  (ft' =  1.5)  surrounded  by  air  (ft  =  l);  and  draw 
accurate  sketch  of  each  lens  showing  the  positions  of  the 
points  named : 

(a)  Double  convex  lens  with  radii  n.  =  + 10  cm.,  r2  =  —  2  cm. 

(b)  Meniscus  lens  with  radii  n  =  +5  cm.,  r2  =  +2  cm. 

Ans.  (a)  F=+20  dptr.,  AiF=+5cm.,  A2F=+3  cm., 
AiH  =  +10  cm.,  A2H'  =  -2  cm.;  (b)  F=-10  dptr.,  A]F  = 
+15  cm.,  A2F'=-8  cm.,  AiH  =  +5  cm.,  A2H'=+2  cm. 

10.  Find  the  focal  length  and  the  positions  of  the  prin- 
cipal points  of  a  concentric  glass  lens  surrounded  by  air 
(ft  =  l,  n' =  1.5),  with  radii  ri=+8  cm.,  r2  =  +5  cm. 

Ans.  /=— 40  cm.,  AiH  =  +8  cm.,  A2H'=+5  cm. 

11.  What  is  the  refracting  power  of  a  concentric  glass 
meniscus  lens  surrounded  by  air  (n  =  l,  ft' =  1.5),  the  radii 
being  5  cm.  and  3  cm.?  Ans.  F=  —  4.44  dptr. 

12.  The  radius  of  the  second  surface  of  a  concentric  glass 
lens  surrounded  by  air  (ft  =  l,  ft' =  1.5)  is  +3  cm.,  and  its  re- 
fracting power  is  — 2  dptr.    Determine  its  thickness. 

Ans.  6.59  mm.  If  it  were  not  too  heavy,  this  would 
be  a  fairly  good  form  of  spectacle  glass  for  a  near-sighted 
person. 

13.  If  the  two  principal  points  of  a  lens  surrounded  by 
the  same  medium  on  both  sides  coincide  with  each  other  at 


388  Mirrors,  Prisms  and  Lenses  [Ch.  XI 

a  point  midway  between  the  two  vertices,  what  is  the  form 
of  the  lens?  Ans.  A  solid  sphere. 

14.  The  refracting  power  of  a  symmetric  glass  lens  sur- 
rounded by  air  (n  =  l,  w'  =  1.5)  is  +10  dptr.,  and  its  thick- 
ness is  0.5  cm.    Determine  the  radius  of  the  first  surface. 

Ans.   +9.916  cm. 

15.  A  solid  sphere  is  a  symmetric  concentric  lens.  If  the 
radius  is  denoted  by  r  (r  =  AiC),  show  that  we  have  the  fol- 
lowing system  of  formulae  for  a  solid  sphere  surrounded  by 
the  same  medium  (ri)  on  both  sides : 

2n(n'-n)  .  H'A  -r  •    A,F'    FA,  S2n~n'> 

16.  If  the  plane  surface  of  a  glass  hemisphere,  of  index  n' 

and  surrounded  on  both  sides  by  a  medium  of  index  n,  is 

turned  towards  the  incident  light,  and  if  r  denotes  the  radius 

of  the  curved  surface,  show  that 

n'—n       .  TT        n.r       .  TT,    _       A  „         n2r 
F=- ,    AiH= T,     A2H'  =  0,     AiF  = 


n 
A2F' 


n'in'  -n) ' 


n.r 
n'-n 


17.  An  object  is  placed  in  front  of  the  plane  surface  of 
a  glass  hemisphere,  of  index  1.5  and  radius  3  inches,  at  a 
distance  of  10  inches  from  this  surface.  Find  the  position, 
nature  and  size  of  the  image. 

Ans.  A  real,  inverted  image,  of  same  size  as  object,  will 
be  formed  at  a  distance  of  25  inches  from  the  object. 

18.  What  is  the  refracting  power  of  a  glass  sphere  (n'  = 
1.5),  16|  cm.  in  diameter,  (a)  surrounded  by  air  (w=l), 
and  (b)  surrounded  by  water  (n=|)? 

Ans.  (a)  +8  dptr.;  (b)  +3|  dptr. 

19.  The  radius  of  each  surface  of  a  symmetric  convex 
glass  lens  (n'  =  1.5)  is  10  cm.,  and  the  thickness  of  the  lens 
is  5  mm.  What  is  its  refracting  power  (a)  when  the  thick- 
ness is  neglected,  and  (6)  when  the  thickness  is  taken  into 
account?  Ans.  (a)  +10  dptr.;  (b)  +9||  dptr. 

20.  The  radii  of  a  convex  meniscus  glass  lens  (n'  =  1.5) 


Ch.  XI]  Problems  389 

surrounded  by  air  (n  =  l)  are  2.5  cm.  and  5  cm.  (a)  If  the 

lens  is  infinitely  thin,  what  is  its  refracting  power?  (6)  If 

the  thickness  of  the  lens  is  1  cm.,  what  is  its  refracting  power? 

Ans.  (a)F=+10dptr.;  (6)  F=+ll£  dptr. 

21.  Determine  the  focal  length  (/)  of  a  glass  lens  of  in- 
dex 1.5  surrounded  by  air  for  which  n=+10,  r2=+9, 
(1)  when  thickness  d  =  0,  and  (2)  when  thickness  d= +1. 

Ans.   (1)  /=-180;  (2)  f=-270. 

22.  A  plane  object  is  placed  at  right  angles  to  the  axis  of 
a  plano-convex  lens  at  a  distance  of  8.77  cm.  in  front  of  its 
curved  surface.  The  lens  is  made  of  glass  of  index  1.52, 
and  the  thickness  of  the  lens  is  0.5  cm.  The  radius  of  the 
curved  surface  is  4.56  cm.  Show  that  the  image  will  be  at 
infinity,  and  that,  in  order  to  see  distinctly  the  image  of  a 
point  in  the  object  which  is  2  cm.  from  the  axis,  an  eye  be- 
hind the  lens  must  look  in  a  direction  inclined  to  the  axis 
of  the  lens  at  an  angle  of  nearly  12°  51'. 

23.  The  refracting  power  of  a  meniscus  spectacle  glass  is 
+6  dptr.,  and  r2  =2rh  d  =  6  mm.  The  index  of  refraction 
is  1.5.  Find  the  radii  n  and  r2  and  the  vertex  refrac- 
tion V. 

Ans.  ri  =  +4.36  cm.,  r2  =  +8.72  cm.,  7= +6.29  dptr. 

24.  The  thickness  of  a  spectacle  glass  is  4.75  mm.,  and 
the  index  of  refraction  is  1.5.  The  refracting  power  of  the 
first  surface  is  +15.4  dptr.,  and  that  of  the  second  surface 
is  —9.1  dptr.  Find  the  refracting  power  of  the  lens  and  its 
vertex  refraction.         Ans.  F=  +6.74  dptr.;  V  =+7.09  dptr. 

25.  A  paraxial  ray  is  incident  on  the  cornea  of  Gull- 
strand's  schematic  eye  (§  130)  in  a  direction  parallel  to  the 
axis.  Trace  the  path  of  this  ray  through  the  eye  and  de- 
termine the  position  of  the  secondary  focal  point  F'  (see 
calculation-scheme,  §  181) ;  and  calculate  the  focal  lengths 
/,  f  according  to  the  formulae  derived  in  problem  No.  17 
at  the  end  of  Chapter  X. 

Ans.  Distance  of  F'  from  the  vertex  of  the  cornea  is 
24.387  mm.;  /=  +17.055  mm.,  f=  -22.785  mm. 


390  Mirrors,  Prisms  and  Lenses  [Ch.  XI 

26.  The  reduced  thickness  of  a  symmetric  spectacle  glass 
is  denoted  by  c.  If  V  denotes  its  vertex  refraction,  show 
that 

r     V(Vl+c2.V2-c.V) 

? ' 

27.  A  hollow  globe  of  glass  is  filled  with  water.  The  di- 
ameter of  the  water  sphere  is  8.5  inches  and  the  thickness  of 
the  glass  shell  is  0.25  inch.  Show -that  a  narrow  beam  of 
parallel  rays  directed  towards  the  center  of  the  globe  will  be 
converged  to  a  point  4.68  inches  from  the  outside  surface, 
the  indices  of  refraction  of  glass  and  water  being  #  and 
-|,  respectively. 

28.  What  is  the  focal  length  of  a  combination  of  two  thin 
convex  lenses,  each  of  focal  length  /,  placed  at  a  distance 
apart  equal  to  2//3?  Ans.  3//4. 

29.  An  optical  system  is  composed  of  two  thin  convex 
lenses  of  refracting  powers  +10  dptr.  and  +6§  dptr., 
the  stronger  lens  being  towards  the  incident  light.  Find 
the  refracting  power  of  the  combination  and  the  positions 
of  the  principal  points  and  focal  points  when  the  dis- 
tance between  the  lenses  is:  (a)  5  cm.;  (b)  25  cm.;  and 
(c)  40  cm. 

Ans.  (a)  Convergent  system:  F  =  1S~  dptr.;  AiH  = 
+2.5  cm. ;  A2H'  =  -  3.75  cm. ;  AiF  =  -  5  cm. ;  A2F'  =  +3.75  cm. ; 
(b)  Telescopic  system:  F  =  0;  focal  and  principal  points 
all  at  infinity;  (c)  Divergent  system:  F=  — 10  dptr.;  AiH  = 
-26|  cm.;  A2H,=  +40  cm.;  AiF=-16f  cm.;  A2F'  = 
+30  cm. 

30.  An  optical  system  is  composed  of  two  thin  lenses, 
namely,  a  front  concave  lens  of  power  — 10  dptr.  and  a  rear 
convex  lens  of  power  +6|  dptr.  Find  the  refracting  power 
of  the  combination  and  the  positions  of  the  focal  points  and 
principal  points,  when  the  interval  between  the  lenses  is: 
(a)  2.5  cm.;  (b)  5  cm.;  (c)  6.25  cm.;  (d)  20  cm. 

Ans.  (a)  Divergent  system :  F  =  — 1|  dptr. ;  AiH  =  — 10  cm. ; 
A2H'=-15  cm.;  AiF  =+50  cm.;  A2F'=-75  cm.;  (6)  Tele- 


Ch.  XI]  Problems  391 

scopic  system:  F=0,  focal  and  principal  points  all  at 
infinity;  (c)  ^=+|  dptr.;  AiH  =  +50  cm.;  A2H'=+75  cm.; 
AiF=— 70  cm.;  A2F'  =  +195  cm.;  (d)  Convergent  system: 
^=+10  dptr.;  AiH  =  +13^  cm.;  A2H'  =  +20  cm.;  AiF  = 
-f3^cm.;A2F'  =  +30cm. 

31.  Two  thin  convex  lenses  of  focal  lengths  /i  and  /2  are 
separated  by  an  interval  equal  to  2/2.  If  /i  =  3/2,  what  is 
the  focal  length  of  the  combination? 

Ans.  Convergent  system  of  focal  length  3/2/2. 

32.  Two  lenses,  one  convex  and  the  other  concave,  are 
separated  by  an  interval  2a.  The  convex  lens  is  the  front 
lens,  and  its  focal  length  is  a,  while  that  of  the  concave  lens 
is  —a.  Find  the  focal  length  of  the  combination  and  the 
positions  of  the  principal  points  and  focal  points. 

Ans.  /=a/2;  AiH  =  A2H'  =  -a;  AiF  =  3A2F'  =  -3a/2. 

33.  Where  are  the  principal  planes  of  a  system  of  two  thin 
convex  lenses  of  focal  lengths  2  inches  and  6  inches,  separated 
by  an  interval  of  4  inches? 

Ans.  The  principal  planes  coincide  with  the  focal  planes 
of  the  stronger  lens. 

34.  The  objective  of  a  compound  microscope  may  be  re- 
garded as  a  thin  convex  lens  of  focal  length  0.5.  inch.  The 
ocular  may  also  be  regarded  as  a  thin  convex  lens  of  focal 
length  1  inch.  The  distance  between  the  two  lenses  is  6 
inches.  Where  must  an  object  be  placed  in  order  that  its 
image  may  be  seen  distinctly  by  a  person  whose  distance  of 
distinct  vision  is  8  inches? 

Ans.  If  inch  in  front  of  the  objective. 

35.  The  focal  lengths  of  the  objective  and  ocular  of  a  com- 
pound microscope  are  0.5  inch  and  1  inch,  respectively.  If 
the  distance  of  distinct  vision  is  12  inches,  find  the  distance 
between  the  objective  and  ocular  when  the  object  viewed  is 
0.75  inch  from  the  objective.  Ans.  2.42  inches. 

36.  A  thin  convex  lens,  of  focal  length  5  inches,  is  placed 
midway  between  two  thin  concave  lenses  each  of  focal  length 
10  inches.    The  distance  between  the  first  lens  and  the  second 


392  Mirrors,  Prisms  and  Lenses  [Ch.  XI 

is  5  inches.    Find  the  focal  length  of  the  system  and  the  po- 
sitions of  the  principal  points. 

Ans.  /=+6|  inches;  the  principal  points  are  3^-  inches 
from  the  outside  lenses. 

37.  In  the  preceding  problem,  suppose  that  the  two 
outside  lenses  are  concave,  everything  else  remaining  the 
same. 

Ans.  /=+6|  inches.  The  principal  points  are  on  op- 
posite sides  of  the  middle  lens  and  If-  inches  from  it. 

38.  A  thin  convex  lens,  of  focal  length  10  inches,  is  placed 
in  front  of  a  concave  mirror  of  focal  length  5  inches,  the  dis- 
tance between  them  being  5  inches.  The  light  traverses  the 
lens,  is  reflected  at  the  mirror,  and  again  passes  through  the 
lens.  Find  the  focal  length  of  this  so-called  "  thick  mirror  " 
and  the  positions  of  the  principal  points. 

Ans.  /=+6§  inches;  the  principal  points  coincide  with 
each  other  at  a  point  5  inches  behind  the  vertex  of  the  mirror. 

39.  In  the  preceding  problem,  suppose  that  the  lens  is 
concave,  everything  else  remaining  the  same. 

Ans.  /=  +6|  inches;  the  principal  points  coincide  with 
each  other  at  a  point  between  the  lens  and  the  mirror  and 
3|  inches  from  the  former. 

40.  In  front  of  each  of  the  systems  described  in  Nos.  35, 
36,  37  and  38,  an  object,  one  inch  high,  is  placed  at  a  distance 
of  5  inches  from  the  first  member  of  the  system.  Find  the 
position,  size  and  nature  of  the  image  in  each  case. 

Ans.  In  No.  35:  A  real,  inverted  image,  2  inches  beyond 
the  third  lens  and  0.8  in.  high.  In  No.  36:  A  real,  inverted 
image,  30  inches  beyond  the  third  lens  and  4  inches  high. 
In  No.  37:  A  real,  inverted  image,  2  inches  in  front  of  the 
lens  and  0.8  in.  high.  In  No.  38:  A  real,  inverted  image, 
30  inches  in  front  of  the  lens  and  4  inches  high. 

41.  The  center  of  a  concave  mirror,  of  radius  r,  coincides 
with  the  optical  center  of  a  thin  lens,  of  focal  length  /,  and 
the  axes  of  lens  and  mirror  are  in  the  same  straight  line.  The 
light  traverses  the  lens,  is  reflected  at  the  mirror,  and  again 


Ch.  XI]  Problems  393 

traverses  the  lens.  Show  that  the  system  is  equivalent  to  a 
thin  mirror  of  radius  r.f/(r+f),  with  its  center  at  the  same 
place  as  the  center  of  the  given  mirror. 

42.  A  centered  system  of  lenses  (I)  is  placed  in  front  of 
a  spherical  mirror  (II),  and  the  whole  constitutes  a  "thick 
mirror/'  as  explained  in  §  132.  Show  that  the  vertex  A  and 
the  center  C  of  the  actual  mirror  are  the  images  of  the  vertex 
H  and  the  center  K,  respectively,  of  the  equivalent  "thin 
mirror/'  which  are  produced  by  the  lens-system  I  in  the 
medium  of  index  n2  between  systems  I  and  II. 

43.  A  "thick  mirror"  consists  of  a  thin  lens  of  focal  length 
/i  and  a  spherical  mirror  of  focal  length  /2  placed  co-axially 
so  that  the  focal  point  of  the  mirror  coincides  with  the  opti- 
cal center  Ai  of  the  thin  lens.  Show  that  the  focal  length  of 
the  equivalent  "thin  mirror"  is 

t       /l-/2      . 

Si- ft' 

and  that  the  positions  of  the  vertex  H  and  the  center  K  are 
given  by  the  following  expressions : 

AlH  =  -A4,       A1K=-/^4. 

/l-/2'  /1+/2 

Does  it  make  any  difference  whether  the  lens  is  convex  or 
concave? 

44.  At  each  of  the  focal  points  of  a  thin  convex  lens  of 
focal  length  /2  is  placed  a  thin  lens  of  focal  length  /1.  Find 
the  focal  length  of  the  combination  of  the  three  lenses  and 
the  positions  of  the  principal  points.  Does  it  make  any  dif- 
ference whether  the  two  equal  outside  lenses  are  convex  or 
concave? 

/l/2       .        A     XT       XT/  A  /l-/2 


Ans./=-^  ;    AiH  =  H'A3=- 


Jl—fl  /1-/2 

45.  A  thin  convex  lens  of  focal  length  10  cm.  is  placed  in 
front  of  a  plane  mirror  at  a  distance  of  8  cm.  from  it.  Find 
the  radius  of  the  equivalent  "thin  mirror"  and  the  position 
of  its  vertex  H. 


394  Mirrors,  Prisms  and  Lenses  [Ch.  XI 

Ans.  The  equivalent  "thin  mirror"  is  a  concave  mirror  of 
radius  50  cm.  with  its  vertex  32  cm.  behind  the  plane  mirror. 

46.  The  axes  of  three  thin  convex  lenses  are  all  in  the 
same  straight  line,  the  interval  between  the  first  and  second 
lenses  being  one  inch  and  the  interval  between  the  second 
and  third  lenses  being  half  an  inch.  The  focal  lengths  of  the 
first,  second  and  third  lenses  are  f,  -^  and  f  inch,  re- 
spectively. A  plane  object  is  placed  at  right  angles  to  the 
axis  of  the  lens-system;  show  that  an  inverted  image  of  the 
same  size  as  the  object  will  be  formed  in  the  plane  of  the 
object. 

47.  A  plano-concave  flint  glass  lens  of  index  1.618  is  ce- 
mented to  a  double  convex  crown  glass  lens  of  index  1.523. 
The  radii  and  thicknesses  are  as  follows :  r\  =  oo ,  r<i  = 
+50.419  mm.,  r3  =  — 74.320  mm.;  c?i  =  +2.15  mm.,  d2  = 
+4.65  mm.  Find  the  focal  length  of  the  combination  and 
the  positions  of  the  principal  points. 

Ans.  /= +192.552  mm.;  distances  of  principal  points  from 
the  plane  surface,  +5.466  and  +7.908  mm. 

48.  A  plano-concave  flint  glass  lens  of  index  1.618  is  ce- 
mented to  a  double  convex  crown  glass  lens  of  index  1.523. 
The  radii  and  thicknesses  are  as  follows:  r\=  +22.00  mm., 
r2=-19.65  mm.,  r3=  oo  ;  di=+2.60  mm.,  d2  =  +2.00  mm. 
Find  the  focal  length  of  the  combination  and  the  positions  of 
the  principal  points. 

Ans.  /= +52.26  mm.;  distances  of  principal  points  from 
plane  surface,  —5.03  and  —3.36  mm. 

49.  The  radii  and  thickness  of  a  symmetric  double  convex 
lens  are  10  cm.  and  1  cm.,  respectively.  The  lens  is  made  of 
glass  of  index  1.5  and  surrounded  by  air  of  index  unity. 
A  portion  of  the  light  which  enters  the  lens  will  be  reflected 
at  the  second  surface  and  partially  refracted  at  the  first 
surface  from  glass  back  into  the  air.  Find  the  radius,  re- 
flecting power  and  position  of  the  vertex  of  the  equivalent 
"thin  mirror." 

Ans.  Concave  mirror  of  radius  —53.050  mm.,  reflecting 


Ch.  XI]  Problems  395 

power  +37.7  dptr.,  with  its  vertex  +6.90  mm.  from  the  ver- 
tex of  the  first  face  of  the  lens. 

50.  In  the  case  of  the  lens  in  the  preceding  problem,  as- 
sume that  the  light  is  reflected  internally  twice  in  succession 
and  issues  finally  at  the  second  face  into  the  air.  Find  the 
refracting  power  and  the  positions  of  the  principal  points 
for  the  imagery  produced  by  these  rays. 

Ans.  Refracting  power,  +53.08  dptr.;  distances  of  prin- 
cipal points  from  vertex  of  first  surface  of  the  lens,  +10.94 
and  —  0.94  mm. 

51.  In  Gullstrand's  schematic  eye  in  its  state  of  maxi- 
mum accommodation  the  crystalline  lens  consists  of  an  outer 
symmetric  double  convex  lens  of  index  ^4  =  n6  =  1.386  (see 
§  130),  enclosing  an  inner  symmetric  double  convex  "core" 
lens  of  index  n5  =  1.406;  the  inner  portion  being  symmetri- 
cally placed  with  respect  to  the  surrounding  outer  part. 
The  radii  of  the  surfaces  are  as  follows : 

Outer  portion:  r3=A3C3  = +5.3333  mm.=  -r6  =  C6A6; 

Inner  portion:  r4  =  A4C4=  +2.6550  mm.=  -r5  =  C5A5. 
Moreover, 

A3A4  =  A3C5  =  A5A6  =  C4A6  =  0.6725  mm.; 
A4A5  =  C5C4  =  A4C4  =  C5A5  =  2.6550  mm. 
The  entire  lens  is  surrounded  by  a  medium  of  index  n3  = 
717  =  1.336.  Show  (1)  that  the  refracting  power  of  the  inner 
portion  or  "core"  lens  is  F4b  =  +14.959  dptr.,  and  that  its 
principal  points  are  1.9905  mm.  from  the  anterior  and  pos- 
terior surfaces.  Moreover,  employing  the  formulae  of  §  131, 
show  (2)  that  the  refracting  power  of  the  entire  lens  in  case 
of  maximum  accommodation  is  F3Q  =  +33.056  dptr.  and  that 
A3H36  =  H36'A6=  +1.9449  mm. 

52.  Using  the  data  of  the  preceding  problem,  find  the 
refracting  power  (F)  and  the  positions  of  the  principal  points 
(H,  H')  of  Gullstrand's  schematic  eye  in  its  state  of  maxi- 
mum accommodation:  being  given,  according  to  the  results 
of  §  130,  that  the  refracting  power  of  the  cornea  system  is 
Fn =43.053  dptr.  and  that  AiHi2= -0.0496  mm.,  AiH'i2  = 


396  Mirrors,  Prisms  and  Lenses  [Ch.  XI 

—  0.0506  mm.,  aifld  also  that  for  maximum  accommodation 
AiA3=+3.2  mm. 

Ans.  ^=+70.575  dptr.;  AiH  =  + 1.772  mm.;  AiH^ 
+2.086  mm. 

53.  Two  thin  lenses  of  focal  lengths /i  and/2  are  placed  on 
the  same  axis  with  the  second  focal  point  (F/)  of  the  first 
lens  coincident  with  the  first  focal  point  (F2)  of  the  second 
lens,  so  as  to  form  an  afocal  or  telescopic  system.  Show 
that  the  lateral  magnification  is  constant  and  equal  to 
— /2//1,  and  that  the  angular  magnification  is  likewise  con- 
stant and  equal  to  the  reciprocal  of  the  lateral  magnification. 

54.  If  (as  in  Huygens's  ocular)  two  thin  lenses  are  placed 
on  the  same  axis  with  their  second  focal  points  in  coincidence, 
show  that  the  second  focal  point  of  the  combination  is  mid- 
way between  this  common  focal  point  and  the  second  lens, 
and  that  the  deviation  produced  by  the  second  lens  is  twice 
that  produced  by  the  first  (assuming  that  the  angles  are 
small) . 


CHAPTER  XII 

APERTURE  AND  FIELD  OF  OPTICAL  SYSTEM 

133.  Limitation  of  Ray-Bundles  by  Diaphragms  or  Stops. 
— The  geometrical  theory  of  optical  imagery  which  has  been 
developed  in  Chapter  X  was  based  on  the  assumption  of 
punctual  correspondence  between  object-space  and  image- 
space,  whereby  each  point  of  the  object  is  reproduced  by 
one  point,  and  by  one  point  only,  in  the  image;  and  on  this 
hypothesis  simple  relations  in  the  form  of  the  so-called 
image-equations  (§  123)  were  obtained  for  determining  the 
position  and  size  of  the  image  in  terms  of  the  focal  lengths 
of  the  optical  system.  When  we  attempted  to  realize  the 
imagery  expressed  by  these  equations,  we  were  obliged  to 
confine  ourselves  to  the  so-called  paraxial  rays  comprised 
within  the  narrow  cylindrical  region  immediately  surround- 
ing the  axis  of  symmetry  or  optical  axis  of  the  centered  sys- 
tem of  spherical  refracting  or  reflecting  surfaces.  Based  on 
the  same  assumptions,  certain  rules  were  given  for  con- 
structing the  image-point  Q'  corresponding  to  a  given  object- 
point  Q.  For  example,  a  pair  of  straight  lines  was  drawn 
through  Q  (Fig.  174),  one  parallel  to  the  optical  axis  and 
meeting  the  second  principal  plane  of  the  system  in  a  point 
V,  and  the  other  going  through  the  primary  focal  point  F 
and  meeting  the  first  principal  plane  in  a  point  W.  The 
required  point  Q'  was  shown  to  lie  at  the  point  of  intersec- 
tion of  the  straight  line  V'Q',  drawn  through  the  second 
focal  point  F',  with  the  straight  line  WQ'  drawn  parallel  to 
the  axis.  The  position  of  the  point  Q'  having  been  located, 
the  problem  was  considered  as  solved,  and  we  were  not  par- 
ticularly concerned  with  inquiring  whether  the  straight 
lines  used  in  the  construction  represented  the  paths  of  ac- 

397 


398 


Mirrors,  Prisms  and  Lenses 


133 


tual  rays  that  formed  the  image  at  Q\  As  a  matter  of  fact, 
the  pair  of  geometrical  lines  which  is  employed  here  will 
generally  not  belong  to  the  bundle  of  optical  rays  by  which 
the  imagery  is  actually  produced;  and  a  glance  at  the  dia- 
gram will  show  how  the  diameter  of  the  lens  and  the  size  of 


Fig.  174. — Effective  rays  as  distinguished  from  rays  used  in  making  geo- 
metrical constructions. 

the  object  control  the  selection  of  the  rays  that  are  really 
effective  in  producing  the  image. 

In  Chapter  1  attention  was  called  to  the  fact  that  every 
optical  instrument  is  provided  with  some  means  of  cutting 
out  such  portions  of  a  bundle  of  rays  as  for  one  reason  or 
another  are  not  desirable;  which  is  usually  accomplished, 
as  has  been  explained,  by  interposing  in  the  paths  of  the 
rays  at  some  convenient  place  a  plane  opaque  screen  at 
right  angles  to  the  axis  containing  a  circular  aperture  with 
its  center  on  the  axis.  There  may,  indeed,  be  several  such 
diaphragms  or  stops  disposed  at  various  places  along  the 
axis  of  the  instrument.  A  perforated  screen  of  this  kind  is 
called  a  front  stop,  a  rear  stop  or  an  interior  stop,  according 
as  it  lies  in  front  of,  behind  or  within  the  system,  respect- 
ively. The  rims  and  fastenings  of  the  lenses  act  in  the  same 
way  as  the  diaphragms  to  limit  the  ray-bundles.  The  stops 
have  various  duties  to  perform,  their  chief  functions  being 


§  134]  Aperture-Stop  399 

to  cut  off  the  view  of  indistinct  parts  of  the  image  (limita- 
tion of  the  field  of  view) ,  to  cut  out  such  rays  as  would  tend 
to  mar  the  perfection  of  that  part  of  the  image  which  is  to 
be  inspected  (limitation  of  the  aperture  of  the  system),  and, 
finally,  to  nullify  injurious  reflections  from  the  sides  of  the 
tube  or  other  parts  of  the  instrument. 

134.  The  Aperture-Stop  and  the  Pupils  of  the  System. — 
To  an  eye  looking  into  the  instrument  from  the  side  of  the 
object,  a  front  stop  (which  may  be  the  rim  of  the  first  lens 
of  the  system)  will  be  the  only  one  that  will  be  visible  di- 
rectly. Any  other  stop  or  lens-rim  will  be  seen  only  by  means 
of  the  real  or  virtual  image  of  it  that  is  cast  by  that  part  of 
the  optical  system  which  is  between  it  and  the  eye.  Simi- 
larly, if  the  eye  is  directed  towards  the  instrument  from  the 
image-side,  an  interior  stop  or  a  front  stop  may  be  seen  by 
means  of  the  image  of  it  that  is  produced  by  the  part  of  the 
system  that  lies  between  it  and  the  eye.  Now  these  impal- 
pable stop-images,  whether  visible  or  not,  are  just  as  effect- 
ive in  cutting  out  the  rays  as  if  they  were  actual  material 
stops;  because,  obviously,  any  ray  that  goes  through  an 
actual  stop  must  necessarily  pass  either  really  or  virtually 
through  the  corresponding  point  of  the  stop-image;  whereas 
a  ray  that  is  obstructed  by  a  stop  will  not  go  through  the 
opening  in  the  stop-image. 

That  one  of  the  stops  which  by  virtue  of  its  size  and  po- 
sition with  respect  to  the  radiating  object  is  most  effective 
in  cutting  out  the  rays  is  distinguished  as  the  aperture-stop 
of  the  system  (§  11),  and  in  order  to  determine  which  of  the 
several  stops  performs  this  office,  it  is  necessary,  first  of  all, 
to  assign  the  position  of  the  axial  object-point  M,  without 
which  the  aperture  of  the  system  can  have  no  meaning.  Ac- 
cordingly, we  must  suppose  that  the  instrument  is  focused 
on  some  selected  point  M  on  the  axis,  which  is  reproduced 
by  an  image  at  the  conjugate  point  M'.  The  transversal 
planes  at  right  angles  to  the  axis  at  M  and  M'  will  be  a  pair 
of  conjugate  planes,  for  it  is  assumed  here  that  the  imagery 


400  Mirrors,  Prisms  and  Lenses  [§  134 

is  ideal  and  of  the  same  character  as  that  produced  by  par- 
axial rays.  Now  this  pair  of  conjugate  planes  plays  a  very 
important  role  in  the  theory  of  an  optical  instrument,  so 
that  hereafter  we  shall  refer  to  the  object-plane  as  the  focus- 
plane  (or  the  plane  which  is  in  focus  on  the  screen)  and  to 
the  conjugate  plane  in  the  image-space  as  the  screen-plane. 

Now  if  the  eye  is  supposed  to  be  placed  on  the  axis  at  the 
point  M  and  directed  towards  the  instrument,  the  stop  or 
stop-image  whose  aperture  subtends  the  smallest  angle  at  M 
is  called  the  entrance-pupil  of  the  system.  All  the  effective 
rays  (§11)  in  the  object-space  must  be  directed  towards 
points  which  He  within  the  circumference  of  the  circular 
opening  of  the  entrance-pupil.  In  general,  the  entrance- 
pupil  is  the  image  of  the  aperture-stop  as  seen  by  looking  into 
the  instrument  in  the  direction  of  the  light  coming  from  the 
object;  but  if  the  aperture-stop  is  a  front  stop,  it  will  also 
be  the  entrance-pupil. 

On  the  other  hand,  when  the  eye  is  placed  on  the  axis  at 
the  point  M'  so  as  to  look  into  the  instrument  through  the 
other  end,  the  stop  or  stop-image  which  subtends  the  smallest 
angle  at  M'  is  called  the  exit-pupil,  and  all  the  effective  rays 
when  they  emerge  from  the  instrument  must  go,  really  or 
virtually,  through  the  opening  of  the  exit-pupil.  In  this 
statement  it  is  tacitly  assumed  that  M'  is  a  real  image  of  M; 
otherwise,  it  would  not  be  possible  for  the  eye  placed  at  M' 
to  look  into  the  instrument  through  the  end  from  which  the 
rays  emerge.  But  in  any  case  the  exit-pupil  is  the  stop  or 
stop-image  which  subtends  the  smallest  angle  at  M'.  Gen- 
erally, the  exit-pupil  will  be  the  image  of  the  aperture-stop 
as  seen  by  looking  into  the  instrument  from  the  image-side; 
but  if  the  aperture-stop  is  a  rear  stop,  it  will  be  itself  the 
exit-pupil. 

Since  the  effective  rays  enter  the  system  through  the 
entrance-pupil  in  the  object-space  and  leave  it  through  the 
exit-pupil  in  the  image-space,  it  is  evident  that  the  exit- 
pupil  is  the  image  of  the  entrance-pupil,  so  that  the  pupil- 


§  135]  Entrance-Pupil  of  Eye  401 

centers,  designated  03-  0  and  0',  are  a  pair  of  conjugate  axial 
points  with  respect  to  the  entire  system. 

The  apertures  of  the  ray-bundles  in  the  object-space  are 
determined  by  the  entrance-pupil  of  the  system;  and  the 
exit-pupil  has  a  similar  office  in  the  image-space.  Each  of 
the  pupils  is  the  common  base  of  the  cones  of  effective  rays 
in  the  region  to  which  it  belongs. 

135.  Illustrations. — The  name  "pupil"  applied  to  these 
apertures  by  Abbe  was  suggested  by  an  analogy  with  the 
optical  system  of  the  eye.  The  pupil  of  the  eye  is  the  con- 
tractile aperture  of  the  colored  iris,  the  image  of  which 
produced  by  the  cornea  and  the  aqueous  humor  is  the  en- 
trance-pupil of  the  eye  corresponding  to  what  is  popularly 
called  the  " black  of  the  eye,"  because  it  looks  black  on  the 
dark  background  of  the  posterior  chamber  of  the  eye.  Since 
the  center  O  of  the  entrance-pupil  is  the  image  of  the  center 
K  of  the  iris-opening  formed  by  rays  that  are  refracted  from 
the  aqueous  humor  through  the  cornea  into  the  air,  then, 
by  the  principle  of  the  reversibility  of  the  light-path,  we 
may  also  regard  K  as  the  image  of  O  formed  by  rays  which 
are  refracted  from  air  (n  =  l)  through  the  cornea  into  the 
aqueous  humor  (n'  =  1.336).  The  apparent  place  of  the  eye- 
pupil  varies  slightly  in  different  individuals  and  in  the  same 
individual  at  different  ages.  If  we  assume  that  the  point  O 
is  3.03  mm.  from  the  vertex  (A)  of  the  cornea,  that  is,  if  we 
put  u  =  0.00303  m.,  then  U  =  n/u  =  330  dptr.  And  if  we  take 
the  refracting  power  of  the  cornea  asF  =  42  dptr.  (§  130), 
then,  since  U'=U+F,  we  find  U'  =  372  dptr.  and  consequently 
u'  =  AK  =  n'/U'  =  0.0036  m.;  so  that  with  these  data  the 
plane  of  the  iris  is  found  to  be  at  a  distance  of  3.6  mm.  from 
the  vertex  of  the  cornea.  Thus  we  see  that  the  entrance- 
pupil  of  the  eye  is  very  nearly  0.6  mm.  in  front  of  the  iris. 

As  a  simple  illustration  of  these  principles,  consider  an 
optical  system  which  consists  of  an  infinitely  thin  convex 
lens,  with  a  stop  placed  a  little  in  front  of  it.  In  the  dia- 
gram (Fig.  175)  the  straight  line  DG  perpendicular  to  the 


402 


Mirrors,  Prisms  and  Lenses 


[§135 


axis  of  the  lens  represents  the  diameter  of  the  lens  which 
lies  in  the  meridian  plane  of  the  paper.  The  diameter  of 
the  stop-opening  is  shown  by  the  straight  line  BC  parallel 
to  DG.    The  centers  of  the  lens  and  stop  are  designated  by 


Screen 
Plovxc 


Exit  Pupil 

Fig.  175. — Optical  system  composed  of  thin  convex  lens  with  front 

stop. 


A  and  K,  respectively.  The  position  of  the  focus-plane  is 
determined  by  the  axial  object-point  M,  which  in  the  figure 
is  represented  as  lying  in  front  of  the  lens  beyond  the  pri- 
mary focal  plane.  The  solid  angle  subtended  at  M  by  the 
opening  in  the  stop  is  supposed  to  be  smaller  than  that 
subtended  by  the  rim  of  the  lens;  that  is,  as  here  shown, 
Z  AMC<Z  AMG;  and,  consequently,  the  front  stop  acts  here 
both  as  aperture-stop  and  entrance-pupil,  so  that  the  center  K 
of  the  aperture-stop  is  likewise  the  center  O  of  the  entrance- 
pupil.  Looking  through  the  lens  from  the  other  side,  one 
will  see  at  0'  a  virtual,  erect  image  B'C  of  the  aperture-stop 
BC,  and  hence  this  image  is  the  exit-pupil  of  the  system. 
The  angle  BMC  is  the  aperture-angle  of  the  cone  of  rays 
that  come  from  the  axial  object-point  M  in  the  focus-plane; 
after  passing  through  the  system,  these  rays  meet  at  M'  in 
the  screen-plane,  the  aperture-angle  of  the  bundle  of  rays 


135] 


Pupils  of  Optical  System 


403 


in  the  image-space  being  ZB'M'C.  The  effective  rays 
coming  from  a  point  Q  in  the  focus-plane  are  comprised 
within  ZBQC  in  the  object-space  and  ZB'Q'C  in  the 
image-space.  If  the  object-point  does  not  he  in  the 
focus-plane,  and  yet  not  too  far  from  it,  the  opening  BOC  will 
act  as  entrance-pupil  for  this  point  also.  Thus,  for  example, 
in  order  to  construct  the  point  R'  conjugate  to  an  object- 
point  R  which  does  not  he  exactly  in  the  focus-plane,  we 
have  merely  to  draw  the  straight  lines  RB,  RO,  RC  until 
they  meet  the  lens,  and  connect  these  latter  points  with  B', 
0',  C,  respectively,  by  straight  lines  which  will  intersect  in 
the  image-point  R'. 

Again,   consider  a  system  composed  of  two  equal  thin 
convex  lenses  whose  centers  are  at  Ai  and  A2  (Fig.  176), 


Fig.  176, 


-Optical  system  composed  of  two  equal  thin  convex  lenses  with 
interior  stop  placed  midway  between  the  two  lenses. 


with  a  stop  UV  placed  midway  between  them;  if  the  center 
of  the  stop  is  designated  by  K,  then  AiK  =  KA2.  The  image 
of  the  stop  as  seen  through  the  front  lens  is  BOC,  and  its 
image  as  seen  by  looking  through  the  other  lens  in  the  op- 
posite direction  is  B'O'C;  these  images  being  equal  in  size 
and  symmetrically  situated  with  respect  to  the  stop  itself. 
The  image  of  the  rim  of  each  lens  cast  by  the  other  lens 
should  also  be  constructed,  but  for  the  sake  of  simplicity 
these  images  are  not  drawn  in  the  figure,  because  the  di- 


404  Mirrors,  Prisms  and  Lenses  [§  136 

ameters  of  the  lenses  are  taken  sufficiently  large  as  com- 
pared with  the  diameter  of  the  stop  interposed  between 
them  at  K  to  insure  that  the  latter  acts  as  aperture-stop 
with  respect  to  the  axial  object-point  M  on  which  the  in- 
strument is  supposed  to  be  focused.  Consequently,  since 
the  stop-image  BC  subtends  at  M  an  angle  less  than  that 
subtended  by  the  rim  of  the  front  lens  or  by  the  image  of 
the  rim  of  the  second  lens,  it  will  be  the  entrance-pupil  of 
the  system;  and,  similarly,  B'C  which  is  the  image  of  BC 
formed  by  the  system  as  a  whole  will  be  the  exit-pupil. 
Thus,  in  order  to  construct  the  image-point  M'  conjugate 
to  the  axial  object-point  M,  we  have  merely  to  draw  the 
straight  line  MC  and  to  determine  the  point  where  this  line 
meets  the  first  lens;  and  from  the  latter  point  draw  a  straight 
line  through  the  point  V  in  the  edge  of  the  stop  to  meet  the 
second  lens;  and,  finally,  draw  the  straight  line  which  joins 
this  latter  point  with  the  point  C  in  the  edge  of  the  exit- 
pupil;  this  fine  will  cross  the  axis  at  the  required  point  M' 
in  the  screen-plane.  Similarly,  drawing  from  the  object- 
point  Q  the  three  rays  QB,  QO  and  QC,  we  can  continue 
the  paths  of  these  rays  from  the  first  lens  to  the  second 
through  the  points  U,  K  and  V,  respectively,  in  the  stop- 
opening;  and  since  the  rays  must  issue  from  the  second  lens 
so  as  to  go  through  B',  0'  and  C,  respectively,  in  the  exit- 
pupil,  their  common  point  of  intersection  in  the  image- 
space  will  be  the  point  Q'  conjugate  to  Q.  In  the  diagram 
the  point  Q  is  taken  in  the  focus-plane;  but  the  same  con- 
struction will  apply  also  to  determine  the  position  of  an 
image-point  R'  conjugate  to  an  object-point  R  which  does 
not  lie  in  the  focus-plane. 

136.  Aperture- Angle.  Case  of  Two  or  More  Entrance- 
Pupils.— The  angle  OMC=t?  (Figs.  175  and  176)  subtended 
at  the  axial  object-point  M  by  the  radius  OC  of  the  entrance- 
pupil  is  called  the  aperture-angle  of  the  optical  system.  If 
we  put  OC  =  p  (where  p  is  to  be  reckoned  positive  or  neg- 
ative according  as  the  point  C  lies  above  or  below  the  axis) 


§  136]  System  with  Two  Entrance-Pupils  405 

and  OM=2,  then  tsn\r)=—p/z.  In  like  manner,  if  r)'  = 
Z  O'M'C  denotes  the  angle  subtended  at  the  point  M'  con- 
jugate to  M  by  the  corresponding  radius  of  the  exit-pupil 
(0'C'  =  p'),  and  if  also  O'MW,  then  tan  77'=  -p'/z'. 

The  pupils  of  an  optical  system  depend  essentially,  as  has 
been  stated,  on  the  position  of  the  axial  object-point  M  on 


Fig.  177. — Case  of  two  entrance-pupils. 

which  the  instrument  is  focused.  In  the  diagram  (Fig.  177) 
I  and  II  represent  a  pair  of  stops  or  stop-images  as  seen 
by  an  eye  looking  into  the  front  end  of  the  instrument.  Join 
one  end  of  the  diameter  of  one  of  these  openings  by  straight 
lines  with  both  ends  of  the  diameter  of  the  other  opening; 
and  let  the  points  where  the  straight  lines  cross  the  axis  be 
designated  by  X  and  Y.  The  two  apertures  subtend  equal 
angles  at  these  points,  and  hence  if  the  object-point  M  co- 
incides with  either  X  or  Y,  the  entrance-pupil  of  the  system 
may  be  either  I  or  II ;  in  fact,  for  these  two  special  positions 
of  M  there  will  be  two  entrance-pupils,  and,  of  course,  also 
two  exit-pupils.  If  the  object-point  M  lies  between  X  and 
Y,  then  in  the  case  represented  in  the  figure  the  opening  II 
will  subtend  a  smaller  angle  at  M  than  the  opening  I  so 


406  Mirrors,  Prisms  and  Lenses  [§  137 

that  the  former  will  act  as  the  entrance-pupil.  But  for  any- 
other  position  of  the  axial  object-point  M  besides  those 
above  mentioned  the  opening  I  will  be  the  entrance-pupil. 

137.  Field  of  View. — The  limitation  of  the  apertures  of 
the  bundles  of  effective  rays  is  not  the  only  office  of  the 
stops  and  lens  fastenings.  One  of  their  most  important 
functions  is  to  define  the  extent  of  the  object  that  is  to  be 
reproduced  in  the  instrument  as  has  been  pointed  out  in 
several  simple  illustrations  in  the  earlier  pages  of  this  book 
(see  §§  9,  16,  73  and  98).  In  the  adjoining  diagram 
(Fig.  178),  where  the  entrance-pupil  of  the  system  is  repre- 
sented by  the  opening  BC,  the  other  stops  or  stop-images  in  the 
object-space  act  like  circular  windows  or  port-holes  through 
which  the  rays  that  are  directed  from  the  various  parts  of 
the  object  towards  points  in  the  open  space  of  the  entrance- 
pupil  will  have  to  pass  if  they  are  to  succeed  in  getting 
through  the  instrument  without  being  intercepted  on  the 
way.  Evidently,  that  one  of  these  openings  which  subtends 
the  smallest  angle  at  the  center  O  of  the  entrance-pupil  will 
limit  the  extent  of  the  field  of  view  in  the  object-space.  This 
opening  which  is  represented  in  the  figure  by  GH  is  called 
the  entrance-port;  and  the  material  stop  or  lens-rim  which 
is  responsible  for  it  is  called  the  field-stop  (§9). 

Let  the  straight  line  CH  drawn  through  the  upper  extrem- 
ities of  the  diameters  BC  and  GH  of  the  entrance-pupil  and 
entrance-port  meet  the  optical  axis  in  the  point  designated 
by  L  and  the  focus-plane  in  the  point  designated  by  U.  If 
this  straight  line  is  revolved  around  the  axis  of  the  instru- 
ment, the  point  U  will  describe  a  circle  in  the  focus-plane 
around  the  axial  object-point  M  as  center;  and  it  is  obvious 
that  any  point  in  this  plane  within  the  circumference  of  this 
circle,  or,  indeed,  any  object-point  contained  inside  the 
conical  surface  generated  by  the  revolution  of  the  straight 
line  passing  through  C  and  H,  may  send  rays  to  all  parts 
of  the  entrance-pupil  which  will  not  be  intercepted  any- 
where in  the  instrument.    Thus,  the  entire  aperture  of  the 


137] 


Field  of  View 


407 


entrance-pupil  will  be  the  common  base  of  the  cones  of  ef- 
fective rays  emanating  from  sources  which  lie  within  this 
region  of  the  object-space. 


Entr-auce 
Pupil 


Fig.  178. 


-Field  of  view  of  optical  system  on  side  of  object,  determined 
by  the  entrance-pupil  and  the  entrance-port. 


Again,  the  straight  line  OH  drawn  through  the  center  of 
the  entrance-pupil  and  the  upper  edge  of  the  entrance-port 
will  determine  a  second  limiting  point  V  in  the  focus-plane 
which  is  farther  from  the  optical  axis  than  the  first  point  U; 


408  Mirrors,  Prisms  and  Lenses  [§  137 

and  in  case  of  object-points  lying  in  the  focus-plane  between 
U  and  V  the  sections  of  the  bundles  of  effective  rays  made 
by  the  plane  of  the  entrance-pupil  will  have  areas  that  are 
comprised  between  the  entire  area  of  the  opening  of  the 
entrance-pupil  and  half  that  area;  and  this  will  be  true  like- 
wise with  respect  to  all  those  points  in  the  object-space  that 
are  contained  between  the  two  conical  surfaces  generated 
by  the  revolution  of  the  straight  lines  CH  and  OH  around 
the  axis  of  symmetry.  Such  points  will  not  lie  outside  the 
field  of  view,  but  although  they  can  utilize  more  than  half 
the  opening  of  the  entrance-pupil,  they  are  not  in  a  position 
to  take  advantage  of  the  entire  opening. 

Finally,  the  straight  line  BH  drawn  through  the  lower 
edge  of  the  entrance-pupil  and  the  upper  edge  of  the  entrance- 
port,  which  crosses  the  optical  axis  at  the  point  marked  J, 
will  determine  an  extreme  point  W  in  the  focus-plane  which 
is  more  remote  from  the  axis  than  the  point  V;  and  it  is  evi- 
dent from  the  figure  that  object-points  in  the  focus-plane 
which  lie  in  the  annular  space  between  the  two  circles  de- 
scribed around  M  as  center  with  radii  MV  and  MW  are 
even  more  unfavorably  situated  for  sending  rays  into  the 
entrance-pupil,  because  they  cannot  utilize  as  much  as  half 
of  the  pupil-opening.  In  fact,  the  effective  rays  which  come 
from  the  farthest  point  W  pass  through  the  circumference 
of  the  pupil,  and  any  point  lying  beyond  W  will  be  wholly 
invisible,  that  is,  entirely  outside  the  field  of  view  of  the 
instrument. 

Thus,  we  see  that  the  focus-plane  is  divided  into  zones  by 
three  concentric  circles  of  radii  MU,  MV  and  MW.  Object- 
points  lying  in  the  interior  central  zone  send  their  light 
through  the  entrance-pupil  without  let  or  hindrance  on  the 
part  of  the  field-stop;  so  that  this  is  the  brightest  part  of 
the  field.  But  in  the  two  outer  zones  there  is  a  gradual  fad- 
ing away  of  light  until  we  reach  finally  the  border  of  complete 
darkness.  The  three  regions  of  the  field  of  view  in  the  object- 
space  are  usually  defined  by  the  angles  271,  27,  and  272 


§  138]  Lens  and  Eye  409 

whose  vertices  are  on  the  optical  axis  at  the  points  L,  O  and 
J,  respectively;  so  that  yi  =  ZSLH,  7  =  ZS0H  and  72  = 
ZSJH.  If  the  radii  of  the  entrance-pupil  and  entrance- 
port  are  denoted  by  p=OC  and  6=SH,  and  if  the  distance 
of  the  entrance-pupil  from  the  entrance-port  is  denoted  by 
c= SO,  then 

b  —  p  b  b+p 

tan7i=— — -,    tan7=--,  tan72=— — -. 
c  c  c 

The  field  of  view  in  the  image-space  is  determined  in  like 
manner.  The  image  of  the  entrance-port  GH  with  its  center 
at  S,  which  is  produced  by  the  entire  optical  system,  is  the 
exit-port  G'H'  with  its  center  at  S';  and  by  priming  all  the 
letters  in  the  expressions  above  a  similar  system  of  equations 
will  be  obtained  for  defining  the  three  regions,  27/,  27' 
and  272',  of  the  field  of  view  in  the  image-space.  Generally, 
the  edge  of  the  field  is  considered  as  determined  by  the  cen- 
ter of  the  pupil,  that  is,  by  the  angle  27  in  the  object-space 
and  the  angle  27'  in  the  image-space. 

138.  Field  of  View  of  System  Consisting  of  a  Thin  Lens 
and  the  Eye. — A  simple  but  very  instructive  illustration  of 
the  principles  explained  in  the  foregoing  section  is  afforded 
by  an  ordinary  convex  lens  used  as  a  magnifying  glass.  In 
order  to  obtain  a  virtual,  magnified  image  with  a  lens  of 
this  kind,  the  distance  of  the  glass  from  the  object  must  not 
exceed  the  focal  length  of  the  lens,  and  then  when  the  image 
is  viewed  through  the  glass,  the  iris  of  the  observer's  eye 
will  act  as  the  aperture-stop  of  the  system,  no  matter  where 
the  eye  is  placed,  provided  the  diameter  of  the  pupil  of  the 
eye  is  less  than  that  of  the  lens,  as  is  practically  nearly  al- 
ways the  case.  Moreover,  since  the  pupil  of  the  eye  is  the 
common  base  of  the  bundles  of  rays  which  come  to  it  from 
the  various  parts  of  the  image,  it  is  the  exit-pupil  of  the 
system,  and  its  image  in  the  glass  is,  therefore,  the  entrance- 
pupil.  If  the  eye  is  placed  on  the  axis  of  the  lens  between 
the  lens  and  its  second  focal  point  (Fig.  179),  the  entrance- 
pupil  will  be  a  virtual  image  of  the  pupil  of  the  eye  and  will 


410 


Mirrors,  Prisms  and  Lenses 


138 


lie  on  the  same  side  of  the  lens  as  the  eye;  if  the  eye  is  placed 
at  the  second  focal  point  of  the  glass,  the  entrance-pupil  will 
be  at  infinity  (see  §  144) ;  and,  finally,  if,  as  represented  in 


Fig.  179. — Field  of  view  of  thin  convex  lens  when  the  eye 
is  between  the  lens  and  its  second  focal  plane. 

Fig.  180,  the  eye  is  placed  at  a  point  O'  beyond  the  second 
focal  point  of  the  convex  lens  GH,  the  center  of  the  entrance- 
pupil  will  be  at  a  point  O  on  the  same  side  of  the  lens  as  the 
object  MQ.  The  distance  between  the  eye  and  the  second 
focal  point  of  a  convex  lens  used  as  a  magnifying  glass  is 
never  very  great,  and,  consequently,  the  distance  of  the  cen- 
ter O  of  the  entrance-pupil  from  the  first  focal  point  is  rela- 
tively always  quite  large.  The  rim  of  the  glass  acts  as  the 
field-stop,  and  it  is  at  the  same  time  both  the  entrance-port 
and  the  exit-port  of  the  system;  and  hence  the  field  of  view 
exposed  to  the  eye  in  the  image-space  is  entirely  analogous 
to  the  field  which  would  be  seen  by  an  eye  looking  through 
a  circular  window  of  the  same  form,  dimensions  and  position 
as  the  lens.    Since  the  exit-port  is  represented  here  as  being 


138] 


Lens  and  Eye 


411 


at  a  considerable  distance  from  the  exit-pupil,  the  field  of 
view  will  appear  vignetted,  that  is,  the  border  will  not  be 
sharply  outlined,  but  the  field  will  fade  out  imperceptibly 


M' 


-".;>. 

H 

\?^ 

""]].■■'.■ 

":::!* 

N. 

^\^~^^ 

Cr 
D' 

M 

A                         \ 

B' 

1 

Fig.  180. — Field  of  view  of  thin  convex  lens  when  the  eye 
beyond  the  second  focal  plane. 


placed 


towards  the  edges.  If  the  diameter  of  the  lens  is  denoted 
by  2b,  and  if  the  distance  of  the  eye  from  the  lens  is  denoted 
by  c  =  AO',  then  tany=— 5/c,  where  y'  =  ZAO'H.  The 
extent  of  the  field  as  measured  by  the  angle  2  7'  is  indepen- 
dent of  the  size  of  the  pupil  of  the  eye.  If  the  focus-plane 
coincides  with  the  first  focal  plane  of  the  magnifying  glass, 
the  diameter  of  the  visible  portion  of  the  object  will  be  2y  = 
-2/.tan7/. 

In  a  compound  microscope  or  in  an  astronomical  telescope 
the  object-glass  produces  a  real  inverted  image  of  the  object, 
and  this  image  is  magnified  by  the  ocular,  which  is  essen- 
tially a  convergent  optical  system  on  the  order  of  a  convex 
lens  used  as  a  magnifying  glass.  In  the  interior  of  the  in- 
strument between  the  object-glass  and  the  ocular,  at  the 
place  where  the  real  image  is  cast  by  the  object-glass, 
there  is  usually  inserted  a  material  stop,  which  cuts  off  the 


412 


Mirrors,  Prisms  and  Lenses 


[§138 


" ragged  edge"  of  the  field  of  view,  so  that  only  the  central 
portion  which  sends  complete  bundles  of  rays  through  the 
instrument  is  visible  to  the  eye. 

In  the  Dutch  telescope  the  ocular  is  a  divergent  optical 
system  which  may  be  represented  in  a  diagram  by  a  con- 


Fig.  181. — Ocular  system  of  Galileo's  telescope  represented  in  the  dia- 
gram by  a  thin  concave  lens.  Diagram  shows  how  the  rays,  after 
having  passed  through  the  object-glass,  enter  the  .pupil  of  the  observ- 
er's eye  B'C.  Inverted  image  of  distant  object  in  the  object-glass  of  the 
telescope  is  formed  at  MQ;  M'Q'  is  the  image  of  MQ  in  the  ocular. 
G'H'  is  the  image  of  the  rim  of  the  object-glass  in  the  ocular.  B'C  is 
the  image  of  BC  in  the  ocular. 

cave  lens  (Fig.  181)  which  is  placed  between  the  object- 
glass  and  the  real  image  of  the  object  in  the  object-glass; 
so  that  so  far  as  the  ocular  is  concerned,  this  image  is  a  vir- 
tual object,  shown  in  the  figure  by  the  line-segment  MQ. 
The  eye  in  this  case  is  usually  adjusted  very  close  to  the 
concave  lens.  The  pupil  of  the  eye  is  represented  in  the 
figure  by  the  opening  B'C'  with  its  center  on  the  axis  at  0'; 
its  image  in  the  lens  is  BC.  Here  also,  just  as  in  the  case  of 
a  convergent  ocular,  the  pupil  of  the  eye  will  act  as  the  exit- 
pupil  unless  the  diameter  of  the  lens  is  so  small  that  the 
lens-rim  itself  performs  this  office.  The  image  of  MQ  is 
M'Q',  which  latter  will  be  erect  if  MQ  is  inverted,  and  since 
MQ  is  always  inverted  in  the  simple  telescope,  the  final 
image  in  the  Dutch  telescope  is  erect.  In  the  case  of  the 
Dutch  telescope  the  rim  of  the  ocular  lens  does  not  limit 


§  139]  Chief  Rays  413 

the  field  of  view,  but  this  is  limited  by  the  rim  of  the  object- 
glass,  which  is  the  entrance-port  of  the  telescope.  Hence, 
the  image  of  the  object-glass  in  the  ocular  is  the  exit-port. 
This  image  (called  the  " eye-ring,"  §  159)  is  represented  in 
the  diagram  by  the  opening  G'H'  with  its  center  on  the 
axis  at  S'.  The  object-point  Q,  as  shown  in  the  figure,  is 
just  at  the  edge  of  the  field,  because  the  image-ray  coming 
from  Q'  which  is  directed  towards  the  center  O'  of  the  exit- 
pupil  is  made  to  pass  through  the  edge  of  the  exit-port  (7'  = 
ZS'OTT). 

139.  The  Chief  Rays. — Every  bundle  of  effective  rays 
emanating  from  a  point  of  the  object  contains  one  ray  which 
'in  a  certain  sense  is  the  central  or  representative  ray  of  the 
configuration  and  which  may  therefore  be  distinguished  as 
the  chief  ray  (see  §  11).  The  ray  which  is  entitled  to  this 
preeminence  is  evidently  that  one  which  in  traversing  the 
medium  in  which  the  aperture-stop  lies  passes  through  the 
center  K  of  this  stop.  If  the  optical  system  is  free  from  the 
so-called  aberrations,  both  spherical  and  chromatic  (as  is 
assumed  in  the  present  discussion),  the  chief  ray  of  the 
bundle  may  also  be  defined  as  that  ray  which  in  the  object- 
space  passes  through  the  center  0  of  the  entrance-pupil; 
but  the  first  definition  is  preferable  because  it  is  applicable 
to  actual  as  well  as  to  ideal  optical  systems. 

The  totality  of  the  chief  rays  coming  from  all  parts  of  the 
object  constitute,  therefore,  a  homocentric  bundle  of  rays 
in  the  medium  where  the  aperture-stop  lies,  and  these  rays 
proceed  exactly  as  though  they  had  originated  from  a  lu- 
minous point  at  K. 

If  the  aperture-stop  is  very  narrow,  comparable,  say,  with 
the  dimensions  of  a  pin-hole,  the  apertures  of  the  bundles  of 
effective  rays  will  be  correspondingly  small;  and  in  the  limit 
when  the  opening  in  the  stop  may  be  regarded  as  reduced  to 
a  mere  point  at  its  center  K,  the  ray-bundles  will  have  col- 
lapsed into  mere  skeletons,  so  to  speak,  each  one  represented 
by  its  chief  ray.     It  is  because  the  chief  rays  are  the  last 


414 


Mirrors,  Prisms  and  Lenses 


[§140 


survivors  of  the  ray-bundles  that  it  is  particularly  impor- 
tant in  nearly  all  optical  problems  to  investigate  the  pro- 
cedure of  these  more  or  less  characteristic  rays. 

140.  The  so-called  "  Blur-Circles  »  (or  Circles  of  Dif- 
fusion) in  the  Screen-Plane. — Now  if  the  cardinal  points  of 
the  optical  system  are  assigned,  the  image-relief  correspond- 
ing to  a  three-dimensional  object  may  be  constructed  point 
by  point,  according  to  the  methods  which  have  been  ex- 
plained.   But,  as  a  matter  of  fact,  the  image  produced  by 


Fig.  182. — Diagram  showing  how  object-relief  and  image-relief  are  pro- 
jected in  focus-plane  and  screen-plane  from  entrance-pupil  and  exit- 
pupil,  respectively;  and  the  "blur  circles"  in  these  planes. 

an  optical  instrument,  instead  of  being  left,  as  it  were, 
floating  in  space,  is  almost  invariably  received  on  a  surface 
or  screen  of  some  kind,  as,  for  example,  the  ground-glass 
plate  of  a  photographic  camera.  In  case  the  image  is  vir- 
tual, as  in  a  microscope  or  telescope,  it  is  intended  to  be 
viewed  by  the  eye  looking  into  the  instrument,  so  that  here 
also  in  the  last  analysis  the  image  is  projected  on  the  sur- 
face of  the  retina  of  the  observing  eye.  This  receiving  sur- 
face is  called  technically  the  "screen,"  which  affords  also 
an  explanation  of  the  name  screen-plane  (§  134)  as  applied 
to  the  plane  conjugate  to  the  focus-plane. 

In  the  diagram  (Fig.  182)  the  screen-plane  is  placed  at 


§140]  "Blur-Circles"  415 

right  angles  to  the  axis  at  the  point  marked  M'  which  is 
conjugate  to  the  axial  object-point  M,  so  that  this  point  is 
seen  sharply  focused  on  the  screen.  Evidently,  however, 
the  optical  system  cannot  be  in  focus  for  all  the  different 
points  of  the  object-relief  at  the  same  time,  because  the 
screen-plane  is  conjugate  to  only  one  transversal  plane  of 
the  object-space,  namely,  the  focus-plane  perpendicular  to 
the  axis  at  M.  Thus,  for  example,  the  reproduction  of  a 
solid  object  such  as  an  extended  view  of  a  landscape  on  the 
ground-glass  plate  of  a  camera  is  not  an  image  at  all  in  the 
strict  optical  sense  of  the  term,  inasmuch  as  it  is  not  con- 
jugate to  the  entire  object  with  respect  to  the  photographic 
objective.  Only  such  points  of  the  object  as  lie  in  the  focus- 
plane  will  be  reproduced  by  sharp  clear-cut  image-points 
in  the  screen-plane  (as,  for  example,  the  point  marked  1  in 
the  figure);  whereas  object-points  situated  to  one  side  or 
the  other  of  the  focus-plane  will  be  depicted  more  or  less  in- 
distinctly on  the  screen-plane  by  small  luminous  areas  which 
are  sections  cut  out  by  this  plane  from  the  cones  of  image- 
rays  emanating  originally  from  points  of  the  object  such  as 
those  marked  2,  3  in  the  diagram.  These  little  patches  of 
light  on  the  screen,  which  are  usually  elliptical  in  form,  and 
whose  dimensions  depend  on  obvious  geometrical  factors, 
such  as  the  diameter  and  position  of  the  exit-pupil,  etc., 
are  the  so-called  circles  of  diffusion  or  "blur-circles ,"  in 
consequence  of  which  details  of  the  image  as  projected  on 
the  screen  are  necessarily  impaired  to  a  greater  or  less 
degree. 

It  is  a  simple  matter  to  reconstruct  the  object-figure 
which  is  optically  conjugate  to  this  configuration  of  image- 
points  and  " blur-circles"  in  the  screen-plane,  which  will 
obviously  be  a  similar  configuration  of  object-points  and 
"blur-circles"  all  lying  in  the  focus-plane.  Moreover,  since 
the  exit-pupil  is  conjugate  to  the  entrance-pupil,  the  cones 
of  rays  in  the  object-space  corresponding  to  those  in  the 
image-space  may  be  easily  constructed  by  taking  the  points 


416 


Mirrors,  Prisms  and  Lenses 


141 


of  the  object-relief  as  vertices  and  the  entrance-pupil  as  the 
common  base  of  these  cones.  The  tout  ensemble  of  the  sec- 
tions of  all  these  bundles  of  object-rays  made  by  the  focus- 
plane  will  evidently  be  the  figure  in  the  object-space  that 
corresponds  to  the  representation  on  the  screen,  and  ac- 
cording to  the  theory  of  optical  imagery  these  two  plane 
configurations  will  be  similar.  This  " vicarious"  object  in 
the  focus-plane  is  sometimes  called  the  projected  copy  of  the 
object-relief,  because  it  is  obtained  by  projecting  the  points 
of  the  object  from  the  entrance-pupil  on  the  focus-plane. 

141.  The  Pupil-Centers  as  Centers  of  Perspective  of 
Object-Space  and  Image-Space. — It  hardly  needs  to  be 
pointed  out  that  the  "  blur-circles "  which  arise  from  this 


Fig.  183. — Projection  of  object-relief  and  image-relief  in  focus-plane  and 
screen-plane  from  the  centers  of  entrance-pupil  and  exit-pupil,  respectively. 

mode  of  reproducing  a  solid  object  on  a  plane  (or  curved) 
surface  are  due  to  no  faults  of  the  optical  system  itself,  but 
are  necessary  consequences  of  the  mode  of  representation, 
having  their  origin,  in  fact,  in  the  object-space  by  virtue  of 
the  process  employed.  The  only  possible  way  of  diminish- 
ing or  eliminating  the  indistinctness  or  lack  of  detail  in  the 
reproduction  of  parts  of  the  object  that  do  not  lie  in  the 
focus-plane  consists  in  reducing  the  diameter  of  the  aperture- 
stop,  or  in  " stopping  down"  the  instrument,  as  it  is  called. 


§  142]  Distance  of  Photograph  417 

If  the  stop-opening  is  contracted  more  and  more  until  finally 
it  is  no  larger  than  a  fine  pin-hole,  the  pupils  likewise  will 
tend  to  become  mere  points  at  their  centers  O,  O'  (Fig.  183), 
and  the  "  blur-circles "  both  in  the  focus-plane  and  in  the 
screen-plane  will  diminish  in  area  pari  passu  and  ultimately 
collapse  also  into  the  points  where  the  chief  rays  cross  this 
pair  of  conjugate  planes.  The  points  marked  I,  II,  III, 
etc.,  where  the  chief  rays  belonging  to  the  object-points  1, 
2,  3,  etc.,  cross  the  focus-plane,  and  which  are  the  centers  of 
the  so-called  " blur-circles"  in  this  plane,  are  obtained, 
therefore,  by  projecting  all  the  points  of  the  object  from  the 
center  of  the  entrance-pupil  on  to  the  focus-plane.  This 
mode  of  representing  a  three-dimensional  object  is,  however, 
in  no  wise  peculiar  to  the  optical  system  itself,  but  is  the 
old  familiar  process  of  perspective  reproduction  by  central 
projection  on  a  plane.  Thus,  the  pupil-centers  O,  0'  are  to 
be  regarded  as  the  centers  of  perspective  of  the  object-space 
and  image-space,  respectively. 

142.  Proper  Distance  of  Viewing  a  Photograph. — These 
principles  explain  why  it  is  necessary  to  view  a  photograph 
at  a  certain  distance  from  the  eye  in  order  to  obtain  a  cor- 
rect impression  of  the  object  which  is  depicted.  Suppose, 
for  example,  that  O,  0'  (Fig.  184)  designate  the  centers  of 
the  pupils  of  a  photographic  lens,  and  that  an  object  NR  is 
reproduced  in  the  screen-plane  by  the  perspective  copy 
M'Q'  whose  size  is  one  kth.  of  that  of  the  projection  MQ  of 
the  object  in  the  focus-plane.  Now  if  the  picture  is  to  pro- 
duce the  same  impression  as  was  produced  by  the  original 
itself  on  an  observer  with  his  eye  placed  at  O,  the  photo- 
graph must  be  held  in  front  of  the  eye  at  a  place  P  such  that 
the  visual  angle  KOP  which  it  subtends  at  the  center  of 
rotation  of  the  eye  shall  be  equal  to  the  angle  QOM ;  that  is, 
the  distance  PO  in  the  figure  must  be  equal  to  one  kth.  of 
the  distance  of  the  center  of  the  entrance-pupil  from  the 
focus-plane,  or  PO  =  MO//c.  If  (as  is  usually  the  case  with 
a  landscape  lens)  the  focus-plane  is  at  infinity,  then  PO  will 


418 


Mirrors,  Prisms  and  Lenses 


[§142 


be  equal  to  the  focal  length  (/)  of  the  objective.  Generally 
speaking,  we  may  say,  therefore,  that  the  correct  distance 
for  viewing  a  photograph  of  a  distant  object  is  equal  to  the 
focal  length  of  the  objective,  this  distance  being  measured 


Screen 
Plane 


Fig.  184. — Correct  distance  of  viewing  photograph. 


from  the  picture  to  the  center  of  rotation  of  the  observer's 
eye.  Accordingly,  if  the  focal  length  is  less  than  the  dis- 
tance between  the  near  point  of  the  eye  and  the  center  of 
rotation,  which  in  the  case  of  a  normal  emmetropic  eye  of 
an  adolescent  is  about  10  or  12  cm.,  it  will  be  impossible  to 
see  the  picture  distinctly  with  the  naked  eye  and  at  the  same 
time  under  the  correct  visual  angle.  Moreover,  even  if  the 
focal  length  of  the  photographic  lens  were  not  less  than  this 
least  distance  of  distinct  vision,  the  effort  of  accommodation 
which  the  eye  has  to  make  in  order  to  focus  the  image  sharply 
on  the  retina  under  the  correct  visual  angle  will  superinduce 
an  illusion  which  will  be  different  from  the  impression  of 
reality  which  it  is  the  purpose  of  the  picture  to  convey.  In 
the  case  of  a  photograph  made  by  an  objective  of  very  short 
focal  length  it  is  possible  indeed  to  make  an  enlarged  copy 
which  may  be  viewed  at  the  correct  distance,  but  this  is 
always  more  or  less  troublesome  and  expensive.  Dr.  Von 
Rohr  has  invented  an  instrument  called  a  verant  which  is 
ingeniously  designed  to  oyercome  as  far  as  possible  the  dif- 
ficulties above  mentioned;  so  that  viewed  through  this  ap- 


143] 


Perspective  Elongation  of  Image 


419 


paratus  the  photograph  is  seen  more  or  less  exactly  as  the 
object  appeared. 

143.  Perspective   Elongation  of  Image. — If  the   screen- 
plane  is  not  focused  exactly  on  the  image-point  R'  (Fig.  185), 


Fig.  185. — Perspective  elongation  of  image. 

this  point  will  be  shown  on  the  screen  by  a  "blur-circle'7 
whose  center  will  be  at  the  point  Q'  which  is  the  projection 
of  R'  from  the  center  O'  of  the  exit-pupil.  Let  e  =  L'M' 
denote  the  distance  of  the  screen-plane  M'Q'  from  the  image- 
plane  L'R',  where  L',  M'  designate  the  feet  of  the  perpen- 
diculars dropped  from  R',  Q',  respectively,  on  the  axis. 
From  the  diagram  we  obtain  the  proportion : 

M'Q'^0'M'_       Q'M' 

TTW'MZ  ~0'M'+M'L'; 
which  may  be  written: 

y'ly'^z'Kz'-e), 
where  y' =  M'Q',  y"  =  L'R'  and  z'  =  0'M'.    Moreover,  since 
e  may  be  regarded  as  small  in  comparison  with  zf,  we  obtain  : 

y'  —  y" =—y" i  approximately. 

The  difference  (y' —y")  is  the  measure  of  the  perspective 
elongation  due  to  imperfect  focusing. 

If  the  exit-pupil  is  at  infinity,  then  R'Q'  will  be  parallel 
to  the  axis  and  yf  =  y" ;  and  under  these  circumstances,  the 
perspective  reproduction  in  the  screen-plane  will  be  of  the 
same  size  as  the  image,  no  matter  how  much  it  is  out  of  focus. 


420 


Mirrors,  Prisms  and  Lenses 


;§  144 


144.  Telecentric  Systems. — A  common  laboratory  use  of 
an  optical  instrument  is  to  ascertain  the  size  of  an  inacces- 
sible or  intangible  object  from  the  measured  dimensions  of 
its  image  as  determined  by  means  of  a  scale  on  which  the 


Fig.  186. — Telecentric  optical  system:  Case  of  a  thin  convex  lens  with  front 
stop  in  first  focal  plane.  Object  represented  by  LR;  blurred  image 
M'Q'  appears  of  the  same  size  as  sharp  image  L'R'. 

image  is  projected;  but,  in  general,  unless  the  scale  is  exactly 
in  the  same  plane  as  the  image,  there  will  be  a  parallax  error 
in  the  measurement  of  the  image  due  to  its  perspective 
elongation.  However,  if  the  chief  rays  in  the  image-space 
are  parallel  to  the  axis,  which  may  be  effected  by  placing  the 
aperture-stop  so  that  the  entrance-pupil  lies  in  the  primary 
focal  plane  of  the  instrument,  as  illustrated  in  Fig.  186,  the 
perspective  elongation  vanishes  (y'—y"  =  0,  as  explained 
in  §  143);  and,  consequently,  the  image  y"  =  L'R'  will  ap- 
pear of  the  same  size  as  its  projection  ?/  =  M'Q',  no  matter 
whether  it  lies  in  the  same  plane  as  the  scale  or  not. 

Similarly,  if  the  aperture-stop  is  placed  so  that  the  en- 
trance-pupil is  at  infinity  and  the  exit-pupil  lies  therefore  in 
the  secondary  focal  plane,  the  chief  rays  in  the  object-space 
will  then  all  be  parallel  to  the  optical  axis. 

Systems  of  this  description  in  which  one  or  other  of  the 
two  projection  centers  0,  0'  is  at  infinity  are  said  to  be 


§  144]  Keratometer  421 

telecentric.     This  is  the  principle  of  nearly  all  systems  for 
micrometer  measurements  of  optical  images. 

A  simple  illustration  of  a  device  of  this  kind  that  is  tele- 
centric  on  the  side  next  the  object  is  afforded  by  the  oph- 
thalmic instrument  called  a  keratometer,  which,  as  the  name 
implies,  is  intended  primarily  to  measure  the  diameter  of  the 
cornea  or  the  apparent  diameter  of  the  eye-pupil.  It  is  used 
also  to  measure  the  distance  of  a  correction-glass  (§  154) 
from  an  ametropic  eye  (§  153),  which  is  an  important  factor 
in  the  prescription  of  spectacles.  The  instrument  consists 
essentially  of  a  long  narrow  tube,  near  the  middle  of  which 
is  mounted  a  convex  lens  of  low  power  adjusted  so  that  its 
second  focal  point  F'  coincides  exactly  with  the  center  of  a 
small  aperture  in  a  metal  disk  placed  at  the  end  of  the  tube 
where  the  observer  puts  his  eye.  At  the  opposite  end  of  the 
tube  a  scale  graduated  in  half-millimeters  is  mounted  so  that 


eye  of 
Patient; 


r^i; 


Spectacle  Glass 

Scale 

Fig.  187. — Diagram  of  instrument  called  keratometer,  as  used  to  measure 
the  distance  of  spectacle  glass  from  the  cornea  of  the  eye. 

its  upper  edge  coincides  with  a  horizontal  diameter  of  the 
tube  at  this  place.  The  upper  part  of  this  end  of  the  tube 
is  cut  away  in  order  to  admit  sufficient  light  to  illuminate  the 
scale. 

When  the  keratometer  is  used  to  measure  the  distance 
between  the  vertex  of  the  cornea  and  the  vertex  of  the  cor- 
rection-glass, it  is  placed  with  its  axis  at  right-angles  to  the 
line  of  sight  of  the  patient,  as  represented  in  the  diagram 
(Fig.  187),  the  scale  being  brought  as  near  as  possible  to 


422 


Mirrors,  Prisms  and  Lenses 


l§  144 


the  patient  between  his  eye  and  the  spectacle-glass.  The 
distance  AB  to  be  measured  is  projected  on  the  scale  by 
rays  that  are  parallel  to  the  axis  of  the  lens,  so  that  when 
the  observer  looks  through  the  instrument  he  can  read  off 
this  distance  on  the  image  of  the  scale. 

Practically  the  same  principle  is  employed  also  in  Badal's 
optometer  for  measuring  the  visual  acuity  of  the  eye.     It 


Fig.  188. — Badal's  optometer,  with  second  focal  point  (F)'  of 
convex  lens  at  first  nodal  point  of  patient's  eye;  forming  in 
conjunction  with  the  eye  a  telecentric  system. 

consists  of  a  single  convex  lens  mounted  at  one  end  of  a  long 
graduated  bar  which  is  provided  with  a  movable  carrier 
holding  a  test-chart  of  some  kind.  If  the  lens,  which  usually 
has  a  refracting  power  of  about  10  dioptries,  is  adjusted 
about  9  cm.  in  front  of  the  cornea  so  that  its  second  focal 
point  F'  coincides  with  the  nodal  point  of  the  eye  (Fig.  188), 
a  ray  meeting  the  lens  in  a  direction  parallel  to  the  axis  will 
emerge  from  it  so  as  to  go  through  the  nodal  point  of  the 


Fig.  189. — Badal's  optometer,  with  second  focal  point  (F')  of 
convex  lens  at  first  focal  point  of  patient's  eye;  forming  in 
conjunction  with  the  eye  a  telescopic  system. 

eye  and  thence  to  the  retina  without  change  of  direction. 
Accordingly,  just  as  though  a  narrow  aperture  were  placed 


Ch.  XII]  Problems  423 

at  the  nodal  point  of  the  eye,  the  size  of  the  retinal  image 
will  not  be  altered  whether  the  object  or  chart  on  the  bar  be 
far  or  near;  whereas  the  distinctness  with  which  the  details 
of  the  object  are  seen,  which  affords  the  measure  of  the  visual 
acuity,  will  depend  on  the  distance  of  the  object. 

Another  method  of  using  this  optometer  is  to  place  the 
lens  about  2  cm.  farther  from  the  eye,  as  shown  in  Fig.  189, 
so  that  now  its  second  focal  point  lies  in  the  anterior  focal 
plane  of  the  eye.  Under  these  circumstances  an  incident 
ray  proceeding  parallel  to  the  axis  will  emerge  from  the  lens 
and  cross  the  axis  at  the  anterior  focal  point  of  the  eye,  so 
that  after  traversing  the  eye-media  it  will  again  be  parallel 
to  the  axis.  Consequently,  here  also  the  image  formed  on 
the  retina  will  be  of  the  same  size  no  matter  where  the  object 
is  placed  on  the  bar  in  front  of  the  lens,  just  as  if  there  were 
a  narrow  stop  at  the  anterior  focal  point  of  the  eye.  In  this 
latter  adjustment  the  lens  and  the  eye  together  constitute 
an  optical  system  which  is  telecentric  on  both  sides,  that  is, 
a  telescopic  system  (§  125). 

PROBLEMS 

1.  A  cylindrical  tube,  2  cm.  in  diameter  and  10  cm.  long, 
is  closed  at  one  end  by  a  thin  convex  lens  of  focal  length  4  cm. 
If  this  end  of  the  tube  is  pointed  towards  a  distant  object, 
what  will  be  the  position  and  diameter  of  the  entrance- 
pupil?  Ans.  6|  cm.  in  front  of  the  lens;  diameter,  1^  cm. 

2.  In  the  preceding  problem,  where  would  the  object  have 
to  be  in  order  that  the  lens  itself  might  act  as  entrance- 
pupil? 

Ans.  In  front  of  the  lens,  not  more  than  20  or  less  than 
4  cm.  away. 

3.  If  in  No.  1  the  other  end  of  the  tube  is  closed  by  a  thin 
eye-lens  whose  focal  length  is  such  that  when  the  combina- 
tion is  pointed  at  an  object  24  cm.  from  the  object-glass,  the 
bundles  of  rays  issuing  from  the  eye-lens  are  cylindrical,  find 


424  Mirrors,  Prisms  and  Lenses  [Ch.  XII 

the  positions  of  the  pupils  of  the  system  and  the  focal  length 
of  the  eye-lens. 

Ans.  Entrance-pupil  6|  cm.  in  front  of  object-glass; 
exit-pupil  coincides  with  eye-glass;  focal  length  of  eye-glass, 
5.2  cm. 

4.  In  the  preceding  problem  what  will  be  the  answers  on 
the  supposition  that  the  object  is  12  cm.  from  the  object- 
glass? 

Ans.  Entrance-pupil  coincides  with  object-glass;  exit- 
pupil  is  6|  cm.  beyond  eye-glass;  focal  length  of  eye- 
glass, 4  cm. 

5.  A  real  inverted  image  of  an  extended  object  is  formed 
by  the  object-glass  of  a  simple  astronomical  telescope  in  the 
primary  focal  plane  of  the  eye-glass.  The  focal  lengths  of 
the  object-glass  and  eye-glass  are  2  feet  and  1.5  inches,  re- 
spectively, and  their  diameters  are  6  inches  and  1  inch, 
respectively.  If  the  distance  of  the  object  from  the  object- 
glass  is  240  feet,  find  the  position  and  diameter  of  the  en- 
trance-port and  the  diameter  of  the  portion  of  the  object 
that  is  completely  visible  through  the  telescope. 

Ans.  Entrance-port  is  30.21  feet  from  object-glass,  and 
its  diameter  is  1.175  feet;  diameter  of  visible  portion  of  ob- 
ject, 5.865  feet. 

6.  A  thin  convex  lens  of  focal  length  10  cm.  and  diameter 
4  cm.  is  used  as  a  magnifying  glass.  If  an  eye  adapted  for 
parallel  rays  is  placed  at  a  distance  of  5  cm.  from  the  lens, 
what  will  be  the  diameter  of  the  portion  of  the  object  that 
can  be  seen  distinctly?  Ans.  8  cm. 

7.  The  diameter  of  a  thin  convex  lens  is  1  inch,  and  its 
focal  length  is  10  inches.  The  lens  is  placed  midway  between 
the  eye  and  a  plane  object  which  is  10  inches  from  the  eye. 
How  much  of  the  object  is  visible  through  the  lens? 

Ans.  1|  inch. 


CHAPTER  XIII 

OPTICAL  SYSTEM  OF  THE  EYE.      MAGNIFYING  POWER 
OF  OPTICAL  INSTRUMENTS 

145.  The  Human  Eye. — The  organ  of  vision  is  composed 
of  the  eye-ball,  wherein  the  visual  impulses  are  produced  by 
the  impact  of  light;  the  optic  nerve  which  transmits  these 
excitations  to  the  brain;  and  the  visual  center  in  the  brain 
where  the  sensation  of  vision  comes  to  consciousness. 

The  eye-ball  (Fig.  190)  lying  in  a  bony  socket  on  a  cushion 
of  fat  and  connective  tissue,  in  which  it  is  free  to  turn  in  all 
directions  with  little  or  no  friction,  consists  of  an  almost 
spherical  dark  chamber,  filled  with  transparent  optical  media 
which  form  the  optical  system  of  the  eye  (Fig.  191).  The 
outer  protecting  envelope  of  the  eye-ball  is  the  tough,  white 
membrane  called,  from  its  hardness,  the  sclerotic  coat  or  sclera, 
popularly  known  as  the  " white  of  the  eye."  This  opaque 
membrane  is  continued  in  front  by  a  round  opening  or  win- 
dow called,  on  account  of  its  horny  texture,  the  cornea.  The 
cornea  is  beautifully  transparent,  and  its  mirror-like  surface 
forms  a  slight  protuberance  shaped  something  like  a  watch- 
glass  or  a  prolate  spheroid.  In  the  interior  of  the  eye  the 
sclerotic  coat  is  overlaid  with  the  dark-colored  choroid  which 
contains  the  blood-vessels  that  nourish  the  eye  and  also  a 
layer  of  brown  pigment  acting  to  protect  the  dark  chamber 
of  the  eye  from  diffused  light.  Behind  the  cornea  lies  the 
anterior  chamber  filled  with  transparent  fluid  called  the 
aqueous  humor.  This  anterior  chamber  is  limited  behind  the 
iris,  which,  rich  in  blood-vessels,  imparts  to  the  eye  its  char- 
acteristic color.  This  is  an  opaque  screen  or  curtain  which 
contains  a  central  hole,  the  pupil,  which  is  circular  in  the 
human  eye.    The  aperture  of  a  bundle  of  rays  entering  the 

425 


426 


Mirrors,  Prisms  and  Lenses 


L§  145 


eye  from  a  luminous  point,  in  proportion  to  the  dimensions 
of  the  eye,  is  enormous  as  compared,  for  example,  with  the 
same  magnitude  in  a  telescope;  and  the  office  of  the  pupil  is 


to  stop  down  this  aperture  to  suitable  proportions.  The 
pupil  contracts  or  dilates  involuntarily  and  regulates  the 
quantity  of  light  that  is  admitted  to  the  eye.  In  the  struc- 
ture of  the  iris  there  are  two  sets  of  fibers,  the  circular  and 


§145] 


Human  Eye 


427 


the  radiating;  when  the  circular  fibers  contract,  the  pupil 
contracts,  and  when  the  radiating  fibers  contract,  the  pupil 


dilates.  In  the  front  part  of  the  eye  the  choroid  lining  is 
bordered  at  the  edge  of  the  cornea  by  a  kind  of  folded  drapery 
the  so-called  ciliary  body,  which  is  hidden  from  without  be- 


428  Mirrors,  Prisms  and  Lenses  [§  145 

hind  the  iris  and  which  contains  the  delicate  system  of 
muscles  which  control  the  mechanism  of  accommodation. 
The  crystalline  lens  composed  of  a  perfectly  transparent 
substance  is  indirectly  attached  to  the  ciliary  body  by  a 
band  which  surrounds  the  edge  of  the  lens  like  a  ring  and 
which  is  disposed  in  radial  folds  somewhat  after  the  manner 
of  a  neck-frill.  This  band  is  the  suspensory  ligament  or  zonule 
of  Zinn.  The  lens  itself  is  double  convex,  the  posterior  sur- 
face being  more  strongly  curved  than  the  anterior  surface. 
The  substance  of  the  lens  consists  of  layers  of  different  in- 
dices of  refraction  increasing  towards  the  center  or  core  of  the 
lens.  The  entire  space  behind  the  lens  is  filled  with  a  trans- 
parent jelly-like  substance  called  the  vitreous  humor,  which 
has  the  same  index  of  refraction  as  the  aqueous  humor, 
namely,  1.336. 

The  light-sensitive  retina  lying  on  the  inside  of  the  choroid 
is  exceedingly  delicate  and  transparent.  In  spite  of  its 
slight  thickness  which  nowhere  exceeds  0.4  mm.,  the  struc- 
ture of  the  retina  is  very  complicated,  and  no  less  than  ten 
layers  have  been  distinguished  (Fig.  192).  The  layer  next 
the  vitreous  humor  is  composed  of  nerve-fibres  spreading 
out  radially  from  the  optic  nerve.  This  layer  is  connected 
with  the  following  layer  containing  the  large  ganglion  or 
nerve-cells,  and  this  in  turn  is  connected  by  an  apparatus 
of  fibers  and  cells  with  the  peculiar  light-sensitive  elements 
of  the  retina,  the  so-called  visual  cells  which  form  the  "bacil- 
lary  layer."  These  visual  cells  consist  of  characteristic 
elongated  bodies  which  are  distinguished  as  rods  and  cones. 
The  rods  are  slender  cylinders,  while  the  cones  or  bulbs  are 
somewhat  thicker  and  flask-shaped.  They  are  all  disposed 
perpendicularly  to  the  surface  of  the  retina,  closely  packed 
together,  so  as  to  form  a  mosaic  layer  at  the  back  of  the 
retina. 

Near  the  center  of  the  retina  at  the  back  of  the  eye,  a  little 
to  the  temporal  side,  is  located  the  yellow  spot  or  macula  lutea, 
where  the  visual  cells  are  composed  mostly  of  cones.    This 


145] 


Human  Eye 


429 


is  the  most  sensitive  part  of  the  retina,  especially  the  minute 
pit  or  depression  at  the  center  of  this  area,  called  the  fovea 
centralis,  which  consists  entirely  of  cones  densely  crowded 
together. 

As  compared  with  an  artificial  optical  instrument,  the 

2.  &€nu  <****  £>***** 


^flfe  J  &****•*   Store  J*y 


7-  J'wWw^  J^«  **»-(« 


A  WVu*.  3l&>*   *<^8-r 
Fig.  192. — Structure  of  the  retina  of  the  human  eye. 

field  of  view  of  the  immobile  eye  is  very  extensive,  amounting 
to  about  150°  laterally  and  120°  vertically.  The  diameter 
of  the  fovea  centralis  corresponds  in  the  field  of  vision  of  the 
eye  to  an  angular  space  which  may  be  covered  by  the  nail 
of  the  fore  finger  extended  at  arm's  length.  In  this  part  of 
the  field  vision  is  so  acute  that  details  of  an  object  can  be 


430  Mirrors,  Prisms  and  Lenses  J§  145 

distinguished  as  separate  provided  their  angular  distance  is 
not  less  than  one  minute  of  arc  (cf.  §10).  If  the  apparent  size 
of  an  object  is  so  small  that  its  image  formed  on  the  retina  at 
the  fovea  centralis  covers  only  a  single  visual  cell,  the  object 
ceases  to  have  any  apparent  size  at  all  and  cannot  be  dis- 
tinguished from  a  point.  The  size  of  the  retinal  image  corre- 
sponding to  an  object  whose  apparent  size  is  one  minute  of 
arc  is  found  by  calculation  from  the  known  optical  constants 
of  the  eye  to  be  0.00487  mm.  Anatomical  measurements 
give  a  similar  value  for  the  diameter  of  a  visual  cell. 

The  inverted  image  cast  on  the  retina  of  the  eye  has  been 
compared  to  a  sketch  which  is  roughly  outlined  in  the  outer 
parts,  but  which  is  more  and  more  finely  executed  in  towards 
the  center,  until  at  the  fovea  centralis  itself  the  details  are 
exquisitely  finished.  Thus,  only  a  comparatively  small 
portion  of  an  external  object  can  be  seen  distinctly  by  the 
eye  at  any  one  moment.  If  all  the  parts  of  the  field  of  view 
were  portrayed  with  equal  vividness  at  the  same  time  and 
came  to  consciousness  at  once,  the  spectator  would  be  com- 
pletely bewildered  and  unable  to  concentrate  his  attention 
on  a  particular  spot  or  phase  of  the  object. 

The  ends  of  the  rods  next  the  choroid  contain  a  coloring 
matter  which  is  sensitive  to  light,  the  so-called  visual  purple, 
which  is  bleached  white  by  exposure  to  bright  light,  but 
which  is  renewed  in  darkness  by  the  layer  of  cells  lying  be- 
tween the  choroid  and  the  retina.  The  light-disturbance 
arriving  at  the  retina  penetrates  it  as  far  as  the  bacillary 
layer  of  rods  and  cones,  and  the  stimulus  is  transmitted  back 
through  the  interposed  apparatus  to  the  layer  of  nerve- 
fibers  and  thence  conducted  to  the  optic  nerve  in  communi- 
cation with  the  brain. 

Not  far  from  the  center  of  the  retina,  a  little  to  the  nasal 
side,  the  optic  nerve  pierces  the  eye-ball  through  the  sclera 
and  choroid.  Here  the  retina  is  interrupted,  so  that  any  light 
which  falls  on  the  optic  nerve  itself  cannot  be  perceived. 
This  is  the  place  of  the  so-called  blind-spot  (punctum  ccecum) 


§  146]  Optical  Constants  of  Eye  431 

of  the  eye.  Corresponding  to  the  area  of  the  blind  spot, 
there  is  a  gap  in  the  field  of  vision  of  the  eye  amounting  to 
about  6°  horizontally  and  8°  vertically.  The  dimensions  of 
the  blind  spot  are  great  enough  to  contain  the  retinal  im- 
ages of  eleven  full  moons  placed  side  by  side.  The  optic 
nerve  leaves  the  eyeball  through  a  bony  canal  and  passes 
thence  to  the  visual  center  of  the  brain. 

The  mobility  of  the  eye  is  produced  by  six  muscles,  the 
four  recti  and  the  two  oblique  muscles  (Fig.  190).  The  recti 
originate  in  the  posterior  part  of  the  socket  and  are  attached 
by  their  tendons  to  the  sclera  so  as  to  move  the  eye  up  or 
down  and  to  the  right  or  left.  The  procedure  of  the  oblique 
muscles  is  more  complicated.  The  superior  oblique,  which 
also  arises  in  the  posterior  part  of  the  socket,  passes  in  the 
front  of  the  eye  through  a  loop  or  kind  of  pulley  lying  on 
the  upper  nasal  side  of  the  socket  and  then  turns  downwards 
to  attach  itself  to  the  sclera.  The  inferior  oblique  muscle  has 
its  origin  on  the  front  lower  nasal  side  of  the  eye-socket, 
and  passes  to  the  posterior  surface  of  the  eye-ball,  being  at- 
tached to  the  sclera  on  the  temporal  side.  The  superior  ob- 
lique turns  the  eye  downwards  and  outwards,  and  the  inferior 
oblique  turns  it  upwards  and  outwards. 

The  motor  muscles  of  the  two  eyes  act  together  so  that 
both  eyes  turn  always  in  the  same  sense,  to  the  right  or  to 
the  left,  up  or  down.  It  is  impossible  to  turn  one  eye  up 
and  the  other  down  at  same  time,  so  as  to  look  up  to  the 
sky  with  one  eye  and  down  at  the  earth  with  the  other. 

146.  Optical  Constants  of  the  Eye. — The  optical  axis  of  the 
eye  may  be  defined  as  the  normal  to  the  anterior  surface  of 
the  cornea  which  goes  through  the  center  of  the  pupil.  This 
line  passes  approximately  through  the  centers  of  curvature 
of  the  refracting  surfaces.  The  schematic  eye  (see  §  130)  is 
a  centered  system  of  spherical  refracting  surfaces  symmetric 
with  respect  to  the  optical  axis.  The  point  where  the  optical 
axis  meets  the  anterior  surface  of  the  cornea  is  called  the 
cornea  vertex  or  anterior  pole  of  the  eye  and  is  designated 


432  Mirrors,  Prisms  and  Lenses  [§  146 

by  A ;  and  the  point  where  the  optical  axis  meets  the  retina 
is  called  the  posterior  pole  of  the  eye  and  is  designated  by  B. 
In  Gullstrand's  schematic  eye  the  distance  from  A  to  B 
is  equal  to  24  mm.,  therefore  somewhat  less  than  an  inch. 

The  motor  muscles  of  the  eye  (§  145) ,  acting  in  pairs,  turn 
the  eye-ball  around  axes  of  rotation  which  all  pass  through 
a  fixed  point  or  pivot  called  the  center  of  rotation  of  the  eye 
and  designated  by  Z.  This  point  may  be  considered  as 
lying  also  on  the  optical  axis  in  the  medium  of  the  vitreous 
humor  about  13  or  14  mm.  from  the  vertex  of  the  cornea  or 
about  10.5  mm  behind  the  pupil.  All  the  excursions  of  the 
eye  are  performed  around  this  point. 

The  object-point  which  is  sharply  imaged  on  the  retina  at 
the  fovea  centralis  (§  145)  is  called  the  point  of  fixation,  and  the 
straight  line  which  joins  the  point  of  fixation  with  the  centre 
of  rotation  is  called  the  line  of  fixation.  This  line  indicates 
the  direction  in  which  the  eye  is  looking.  The  field  of  fixa- 
tion is  measured  by  the  greatest  angular  distance  through 
which  the  line  of  fixation  can  be  turned;  which  amounts  to 
about  a  right  angle  both  vertically  and  horizontally. 

In  Gullstrand's  schematic  eye,  as  was  shown  in  §  130, 
the  primary  focal  point  F  lies  in  front  of  the  eye  at  a  dis- 
tance of  15.707  mm.  from  the  anterior  vertex  of  the  cornea, 
while  the  secondary  focal  point  F'  lies  on  the  other  side  of 
the  cornea  at  a  distance  of  24.387  mm.  The  principal  points 
(H,  H')  lie  in  the  aqueous  humor  slightly  beyond  the  cor- 
nea system  at  distances  AH  =  +1.348  mm.,  AH' =  +1.602 
mm.  Thus  the  focal  lengths  are:  /= +17.055  mm.  /'  = 
—  22.785  mm.;  the  ratio  between  them  being  equal  to  1.336, 
which  is  therefore  the  value  of  the  index  of  refraction  (n') 
of  the  vitreous  humor.  Accordingly,  the  refracting  power 
of  Gullstrand's  schematic  eye  is  F= 58.64  dptr.  The 
nodal  points  (N,  N')  lie  close  to  the  posterior  vertex  of  the 
crystalline  lens,  on  opposite  sides  of  it,  at  the  following  dis- 
tances from  the  vertex  of  the  cornea:  AN  =  +7.078  mm., 
AN'=  +7.332  mm.    The  straight  line  which  joins  the  point 


147] 


Accommodation  of  the  Eye 


433 


of  fixation  with  the  anterior  nodal  point  of  the  eye  is  called 
the  visual  axis.  It  is  parallel  to  the  straight  line  which  joins 
the  posterior  nodal  point  with  the  fovea  centralis.  Since  the 
nodal  points  are  so  close  together,  for  many  problems  con- 
nected with  the  eye  they  may  be  regarded  as  coincident;  so 
that  then  the  visual  axis  may  be  defined  as  the  line  drawn 
from  the  point  of  fixation  to  the  fovea  centralis.  The  visual 
axis  meets  the  cornea  a  little  to  the  nasal  side  of  the  anterior 
vertex  and  slightly  above  it,  forming  with  the  optical  axis  an 
angle  between  3°  and  5°. 

The  above  values  are  all  given  for  the  passive,  unaccommo- 
dated eye.  By  the  act  of  accommodation  the  positions  of 
the  focal  points,  principal  points  and  nodal  points  are  all  dis- 
placed, and  accordingly  the  focal  lengths  and  the  refracting 
power  of  the  eye  can  be  varied  within  certain  limits  depend- 
ing on  the  power  of  accommodation,  as  will  be  explained  in 
the  following  section. 

147.  Accommodation  of  the  Eye. — When  the  eye  is  at  rest, 
as  when  one  gazes  pensively  into  space,  it  is  adapted  for  far 


Fig.  193. — Accommodation  of  the  human  eye; 
indicating  how  the  crystalline  lens  is  changed 
from  far  vision  to  near  vision. 


vision,  so  that  in  order  to  see  distinctly  objects  which  are  close 
at  hand,  an  effort  has  to  be  made  which  will  be  greater  in 


434  Mirrors,  Prisms  and  Lenses  [§  148 

proportion  as  the  object  fixed  is  nearer  to  the  eye.  This  proc- 
ess whereby  the  normal  eye  is  enabled  to  focus  on  the  retina 
in  succession  sharp  images  of  objects  at  different  distances 
is  called  accommodation,  and  it  is  this  marvelous  adapt- 
ability of  the  human  eye,  together  with  its  mobility,  which 
perhaps  more  than  any  other  quality  entitles  it  to  superior- 
ity over  the  most  perfectly  constructed  artificial  optical  in- 
struments. The  power  of  accommodation  is  achieved  by 
changes  in  the  form  of  the  crystalline  lens,  consisting  chiefly 
in  a  change  in  the  convexity  of  the  anterior  surface,  produced 
through  the  mechanism  of  the  ciliary  muscle.  According 
to  the  generally  accepted  theory,  so  long  as  the  eye  is  passive, 
the  elastic  substance  of  the  lens  is  held  flattened  in  front  by 
the  suspensory  ligament;  but  in  the  act  of  accommodation 
the  ciliary  muscle  contracts,  and  this  is  accompanied  by  a 
relaxation  of  the  ligament  of  the  lens,  which  is  thereby 
permitted  to  bulge  forward  by  virtue  of  its  own  elasticity 
(Fig.  193). 

148.  Far  Point  and  Near  Point  of  the  Eye. — The  far  point 
of  the  eye  {punctum  remotum)  is  that  point  (R)  on  the  axis 
which  is  sharply  focused  at  the  posterior  pole  of  the  eye 
when  the  crystalline  lens  has  its  least  refracting  power;  it 
is  the  point  which  is  seen  distinctly  when  the  accommodation 
is  entirely  relaxed.  On  the  other  hand,  the  near  point  (or 
punctum  proximum)  is  that  point  (P)  on  the  axis  which  is 
seen  distinctly  when  the  crystalline  lens  has  its  greatest  re- 
fracting power,  that  is,  when  the  accommodation  is  exerted 
to  the  utmost.  The  region  of  distinct  vision  within  which  an 
object  must  lie  in  order  that  its  image  can  be  sharply  fo- 
cused on  the  retina  of  the  naked  eye  is  comprised  between 
two  concentric  spherical  surfaces,  the  far  point  sphere  and 
the  near  point  sphere,  described  around  the  center  of  ro- 
tation of  the  eye  (Z)  with  radii  equal  to  ZR  and  ZP,  re- 
spectively. If  the  far  point  lies  at  infinity,  as  is  the  case  in  the 
normal  eye,  the  far  point  sphere  is  identical  with  the  infinitely 
distant  plane  of  space  {cf.  §  83),  as  represented  in  Fig.  194; 


149] 


Presbyopia 


435 


whereas  the  near  point  sphere  will  be  real  and  at  a  finite 
distance  in  front  of  the  eye.  In  such  a  case  the  eye  can  be 
directed  towards  any  point  in  the  field  of  fixation  (§  146) 
lying  on  or  beyond  the  near  point  sphere  and  accommodate 


Neat-  Point 
Sphere 


Fig.  194. — Region  of  accommodation  of  emmetropic  eye. 

itself  to  see  this  point  distinctly.  In  a  near-sighted  eye  both 
far  point  and  near  point  are  real  points  lying  at  finite  dis- 
tances in  front  of  the  eye;  but  the  far  point  of  a  far-sighted 
eye  is  a  "  virtual"  point  lying  at  a  finite  distance  behind  the 
eye,  and  hence  an  unaided  far-sighted  eye  cannot  see  dis- 
tinctly a  real  object  without  exerting  its  accommodation  to 
a  greater  or  less  degree. 

149.  Decrease  of  the  Power  of  Accommodation  with 
Increasing  Age. — The  faculty  of  accommodation  is  greatest 
in  youth  and  diminishes  rapidly  with  advancing  years. 
The  near  point  of  the  eye  gradually  recedes  farther  and  far- 
ther away,  which  is  commonly  supposed  to  be  due  to  a  pro- 
gressive diminution  of  the  elasticity  of  the  crystalline  lens. 
Thus,  at  the  ages  of  10,  20  and  40  years  the  punctum  proxi- 
mum  of  a  normal  eye,  according  to  Donders,  is  in  front  of 
the  eye  at  distances  from  the  primary  principal  point  equal 
to  7.1,  10  and  22.2  cm.,  respectively.  When  the  near  point 
has  retreated  to  a  distance  of  22  cm.,  so  that  it  is  no  longer 
possible  to  read  or  write  or  do  "near  work"  conveniently 
without  the  aid  of  spectacles,  the  condition  of  presbyopia 
or  old-age  vision  has  begun  to  set  in.    Meantime,  while  the 


436  Mirrors,  Prisms  and  Lenses  [§  150 

power  of  accommodation  of  the  eye  thus  continually  dimin- 
ishes as  the  near  point  recedes  farther  and  farther  away, 
the  position  of  the  far  point  remains  practically  fixed  until 
well  after  middle  life;  but  between  the  ages  of  55  and  60 
years  it  too  begins  to  separate  farther  from  the  eye,  and 
thereafter  both  the  near  point  and  the  far  point  travel  out- 
wards along  the  axis  of  the  eye,  the  former,  however,  con- 
stantly gaining  on  the  latter;  until  at  last  in  extreme  old  age 
the  near  point  actually  overtakes  the  far  point,  and  from 
that  time  until  death  they  remain  together,  the  power  of 
accommodation  having  been  entirely  lost.  Both  points  are 
displaced  along  the  axis  always  in  the  same  direction,  that  is, 
opposite  to  that  of  the  incident  light.  For  example,  the  far 
point  of  a  normal  eye  is  infinitely  distant  up  to  about  55 
years  of  age,  whereas  ten  years  later,  according  to  Donders, 
this  point  will  be  about  133  cm.  behind  the  eye,  having 
moved  out  through  infinity,  so  to  speak,  and  approached 
the  eye  from  behind.  At  the  sanie  age,  namely,  65  years, 
the  near  point  will  also  be  behind  the  eye  at  a  distance  of 
400  cm.  At  75  years  of  age  the  two  points  will  be  together 
at  a  distance  of  57.1  cm.  behind  the  eye.  Various  theories 
have  been  advanced  to  account  for  the  senile  recession  of 
the  far  point  of  the  eye.  It  is  probably  due  to  a  combina- 
tion of  causes,  partly  to  a  change  in  the  form  of  the  lens  pro- 
duced by  the  increased  resistance  of  the  enveloping  coat  of 
the  eye-ball  and  the  decreased  pressure  of  the  surrounding 
tissue,  and  partly  also  to  senile  changes  in  the  lens-substance 
itself  whereby  the  " total  index"  of  the  lens  is  lowered  in 
value. 

150.  Change  of  Refracting  Power  in  Accommodation. — 
It  was  remarked  above  (§  146)  that  the  positions  of  the  car- 
dinal points  of  the  optical  system  of  the  eye  are  all  altered 
in  the  act  of  accommodation.  Thus,  for  example,  in  Gull- 
strand's  schematic  eye,  which  is  calculated  for  an  adoles- 
cent youth,  the  near  point  is  at  a  distance  AP  =  — 10.23  cm. 
from  the  vertex  of  the  cornea;  and  for  this  state  of  maxi- 


§  151]  Amplitude  of  Accommodation  437 

mum  accommodation  the  positions  of  the  focal  points  and 
principal  points  are  found  to  be  as  follows : 

AF  =  - 12.397  mm.,  AF '  =  +21.016  mm., 
AH  =  +    1.772  mm.,  AH'  =  +  2.086  mm.; 
and,  accordingly,  the  focal  lengths  and  the  refracting  power 
are: 

i=+14.169mm.,  /'= -18.930  mm.,  F=  +70.57  dptr. 
It  will  be  observed  that,  whereas  the  focal  points  have  un- 
dergone considerable  displacements  from  their  positions  in 
the  passive  eye,  the  corresponding  displacements  of  the 
principal  points  are  less  than  half  a  millimeter;  and  since  in 
most  physiological  measurements  half  a  millimeter  is  within 
the  limit  of  error,  we  can  usually  afford  to  neglect  altogether 
the  accommodative  displacement  of  the  principal  points  of 
the  eye,  that  is,  we  may  regard  the  positions  of  the  princi- 
pal points  H,  H'  as  practically  fixed  and  independent  of 
the  state  of  accommodation.  This  is  one  reason,  among 
others,  why  the  principal  points  of  the  eye  have  super- 
seded the  other  cardinal  points  as  points  of  reference.  Their 
proximity  to  the  cornea  is  another  advantage,  inasmuch  as 
measurements  referred  to  them  are  easily  related  to  an  ex- 
ternal, visible  and  tangible  point  of  the  eye.  In  the  so- 
called  "reduced  eye,"  which  consists  of  a  single  spherical 
refracting  surface  separating  the  outside  air  from  the  vitre- 
ous humor  and  so  placed  that  its  vertex  lies  at  the  primary 
principal  point  of  the  schematic  eye,  the  two  principal  points 
are,  in  fact,  coincident  with  each  other  on  the  surface  of  this 
simplified  cornea. 

151.  Amplitude  of  Accommodation. — The  far  point  dis- 
tance(a)  and  the  near  point  distance  (b)  are  the  distances  of 
the  far  point  and  near  point,  respectively,  measured  from 
the  primary  principal  point  of  the  eye;  thus,  a  =  HR,  b  = 
HP;  it  being  tacitly  assumed  here  that  the  position  of  the 
point  H  remains  sensibly  stationary  during  accommodation, 
as  was  explained  above.  Each  of  these  distances  is  to  be 
reckoned  negative  or  positive  according  as  the  point  in  ques- 


438  Mirrors,  Prisms  and  Lenses  [§  151 

tion  lies  in  front  of  the  eye  or  behind  it,  respectively.  The 
reciprocals  of  these  magnitudes,  namely,  A  =  l/a,  B  =  l/b, 
are  termed  the  static  refraction  (A),  or  the  refraction  of  the 
eye  when  the  accommodation  is  completely  relaxed,  and  the 
dynamic  refraction  (B),  or  the  refraction  of  the  eye  when 
the  accommodation  is  exerted  to  the  highest  degree.  If  the 
distances  a  and  b  are  given  in  meters,  the  reciprocal  magni- 
tudes will  be  expressed  in  dioptries,  as  is  generally  the  case. 

The  range  of  accommodation  is  denned  to  be  the  distance 
of  the  near  point  from  the  far  point,  that  is,  RP  =  b  — a; 
whereas  the  amplitude  of  accommodation  is  the  value  obtained 
by  subtracting  algebraically  the  magnitude  of  the  dynamic 
refraction  from  that  of  the  static  refraction,  thus: 
Amplitude  of  Accommodation  =  A— B. 
Imagine  a  thin  convex  lens  placed  in  the  primary  principal 
plane  of  the  eye  with  its  axis  in  the  same  line  as  the  optical 
axis  of  the  eye,  and  of  such  strength  that  it  produces  at  the 
far  point  of  the  eye  an  image  of  the  near  point;  according  to 
the  above  definition,  the  amplitude  of  accommodation  of 
the  eye  is  equal  to  the  refracting  power  of  this  lens.  For  ex- 
ample, in  the  normal  eye  at  30  years  of  age,  a=  oo,  b  = 
— 14.3  cm.,  so  that  the  amplitude  of  accommodation  in  this 
case  amounts  to  7  dptr.;  whereas  at  60  years  of  age  a  = 
+200  cm.,  b  =—200  cm.,  and  hence  the  amplitude  of  ac- 
commodation will  have  been  reduced  to  1  dptr. 

The  distance  from  the  secondary  principal  point  (H')  to 
the  posterior  pole  (B)  where  the  optical  axis  meets  the  retina 
may  be  regarded  as  a  measure  of  the  length  of  the  eye-axis, 
especially  since  the  position  of  H'  is  sensibly  independent 
of  the  state  of  accommodation,  as  has  been  explained,  (§  150). 
If  this  distance  is  denoted  by  a',  that  is,  if  we  put  a' '= 
H'B,  and  if  also  we  put  A! '  =  n'/a' ',  where  n'  denotes  the 
index  of  refraction  of  the  vitreous  humor,  then  we  may  write : 

A'=A+F, 
where  F  denotes  here  the  refracting  power  of  the  passive, 
unaccommodated  eye.      Similarly,  if  the  symbol  Fa  is  em- 


§153]  Emmetropia  and  Ametropia  439 

ployed  to  denote  the  refracting  power  of  the  eye  in  its  state 
of  maximum  accommodation,  we  shall  have: 

A'  =  B+F&. 
Consequently,  we  may  also  say  that  the  power  of  accommo- 
dation (A  —  B)  is  equal  to  the  difference  (F^  —  F)  between 
the  greatest  and  least  values  of  the  refracting  power  of  the 
eye. 

152.  Various  Expressions  for  the  Refraction  of  the  Eye. 
— The  refraction  of  the  eye  in  a  given  state  of  accommoda- 
tion is  measured  by  the  reciprocal  of  the  distance  from  the 
eye  of  the  axial  object-point  M  for  which  the  eye  is  accom- 
modated. Thus,  if  w=HM,  x  =  FM  denote  the  distances  of 
M  from  the  primary  principal  point  and  the  primary  focal 
point,  respectively,  the  magnitudes  U  =  l/u  and  X  =  l/x, 
usually  expressed  in  dioptries,  are  the  measures  of  the  prin- 
cipal point  refraction  and  the  focal  point  refraction.  The 
relation  between  U  and  X  may  be  given  in  terms  of  the  re- 
fracting power  of  the  eye  (F)  when  it  is  accommodated  for 
the  object-point  M,  as  follows: 

TJ_  F.X  v_  F.U 

U~F-X'  F+U' 

If  an  arbitrary  point  O  on  the  axis  of  the  eye  is  selected 
as  the  point  of  reference,  and  if  we  put  OM  =  2,  the  refrac- 
tion of  the  eye,  referred  to  the  point  O,  will  be  measured  by 
Z  =  1/z.  If  the  distances  of  the  points  H  and  F  from  0  are 
denoted  by  b  and  g,  that  is,  if  6  =  OH,  g  =  OF,  then  since 
z  =  u+b  =  x+g,  we  can  obtain  also  the  following  useful  re- 
lations between  U,  X  and  Z  in  terms  of  b  and  g: 
Z  X 


x= 


l-b.Z  l-(b-g)X' 

Z  u 

1-g.Z  l+(b-g)U' 

U  X 


1+b.U    1+g.X' 
153.  Emmetropia  and  Ametropia. — When  the  static  re- 
fraction of  the  eye  is  equal  to  zero  (A=0),  that  is,  when 


440  Mirrors,  Prisms  and  Lenses  [§  153 

the  far  point  (R)  is  infinitely  distant,  the  eye  is  said  to  be 
emmetropic.  If  in  the  equation  A'=A+F,  we  put  ^4  =  0, 
we  obtain  A'  =  F,  which  therefore  may  be  said  to  be  the 
condition  of  emmetropia.  Here  F  denotes  the  refracting 
power  of  the  eye  when  accommodation  is  entirely  relaxed. 
In  emmetropia,  therefore,  the  second  focal  point  (F')  of 
the  passive  eye  lies  on  the  retina  at  the  posterior  pole  (B) ; 


To  Rat  co 


Fig.  195. — Diagram  of  emmetropic  eye. 

so  that  in  a  passive  emmetropic  eye  incident  parallel  rays 
are  converged  to  a  focus  on  the  retina,  as  represented  in 
Fig.  195,  and  the  length  of  the  eye-axis  is  a'  =  —  /'.  The 
normal  position  of  the  far  point  is  to  be  regarded  as  at  in- 
finity; and  in  this  sense  an  emmetropic  eye  is  a  normal  eye, 
although,  strictly  speaking,  an  emmetropic  eye  may  at  the 
same  time  be  abnormal  in  various  ways. 

On  the  other  hand,  if  the  static  refraction  of  the  eye  is 
different  from  zero  (A^O),  that  is,  when  the  far  point  (R) 
is  not  infinitely  distant,  the  eye  is  said  to  be  ametropic 
Thus,  the  condition  of  ametropia  may  be  said  to  be  charac- 
terized by  the  fact  that  the  refracting  power  (F)  of  the 
unaccommodated  eye  is  not  equal  to  A'}  which  is  equiva- 
lent to  saying  that  the  length  of  the  eye-axis  (a')  is  numer- 
ically different  from  the  value  of  the  second  focal  length 
(/').  In  other  words,  the  second  focal  point  (F')  of  an 
ametropic  eye  in  a  state  of  repose  does  not  fall  on  the  retina. 

Two  general  divisions  of  ametropia  are  distinguished  de- 
pending, on  whether  the  far  point  (R)  lies  on  one  side  or  the 


153] 


Myopia  and  Hypermetropia 


441 


other  of  the  primary  principal  point  (H).  Thus,  if  A<Q, 
that  is,  if  the  far  point  lies  at  a  finite  distance  in  front  of  the 
eye,  the  ametropia  in  this  case  is  called  myopia  (Fig.  196). 
In  a  myopic  eye  in  a  state  of  repose  the  second  focal  point 


Fig.  196. — Ametropic  eye:  myopia. 

(F')  lies  in  front  of  the  retina  (in  the  vitreous  humor),  so 
that  parallel  incident  rays  will  be  brought  to  a  focus  be- 
fore reaching  the  retina.  On  the  other  hand,  if  A>0,  the 
far  point  will  lie  at  a  finite  distance  beyond  (or  behind)  the 


Fig.  197. — Ametropic  eye:  hypermetropia. 

eye,  and  this  form  of  ametropia  is  known  as  hypermetropia 
(Fig.  197).  In  a  hypermetropic  eye  in  a  state  of  repose  the 
second  focal  point  (F')  falls  beyond  the  retina,  so  that  in- 
cident parallel  rays  arrive  at  the  retina  before  coming  to  a 
focus.  A  myopic  eye  cannot  focus  for  a  distant  object  with- 
out the  aid  of  a  glass,  and  it  lacks  therefore  an  important 
part  of  the  capacity  of  an  emmetropic  eye.  On  the  other 
hand,  a  hypermetropic  eye  must  make  an  effort  of  accom- 
modation each  time  in  order  to  focus  on  the  retina  the  image 


442  Mirrors,  Prisms  and  Lenses  [§  153 

of  a  real  object;  which  frequently  produces  various  troubles, 
sometimes  very  annoying.  Accordingly,  both  conditions 
included  under  the  general  name  of  ametropia  are  disad- 
vantageous for  practical  vision. 

Theoretically,  ametropia  may  be  considered  as  due  to 
some  abnormality  in  the  values  of  one  or  of  both  of  the  mag- 
nitudes denoted  by  A'  and  F'  on  which  the  value  of  the 
static  refraction  (^4)  depends;  so  that  the  following  cases 
are  possible: 

(1)  The  length  of  the  eye-ball  (a')  may  be  too  great 
(axial  myopia,  af>  —/')  or  too  small  (axial  hypermetropia, 
a'  <  — /'),  whereas  the  refracting  power  (F)  is  normal.  This, 
by  far  the  most  common,  type  is  known  as  axial  ametropia. 

(2)  On  the  other  hand,  while  the  length  of  the  eye-ball 
may  be  normal,  the  magnitude  of  the  refracting  power  (F) 
may  be  abnormally  great  or  small.  In  general,  this  form  of 
ametropia,  which  is  comparatively  rare,  is  due  to  abnormal 
curvatures  of  the  refracting  surfaces  {curvature  ametropia). 
Or  the  indices  of  refraction  of  the  eye-media  may  have  ab- 
normal values  (indicial  ametropia).  Here  also  may  be  men- 
tioned the  condition  known  as  aphakia  produced  by  the 
extraction  of  the  crystalline  lens  in  the  operation  for  cataract. 

(3)  Finally,  it  may  happen  that  the  refracting  power  and 
the  length  of  the  eye-ball  are  both  abnormal.  In  fact,  these 
two  anomalies  might  exist  together  in  exactly  the  degree 
necessary  to  counteract  each  other,  so  that,  in  spite  of  its 
abnormalities,  the  eye  in  such  a  case  would  be  emmetropic. 

In  the  case  of  axial  ametropia,  the  relation  between  the 
static  refraction  (A)  and  the  length  (I)  of  the  eye-ball  is 
given  by  the  following  formula: 

<=AB=AH+^ 

and  if  the  values  for  Gullstrand's  schematic  eye  (§  146) 
are  substituted  in  this  formula,  it  may  be  written  as  follows: 

1  =  1.602+— — — — -  millimeters. 
^4 +58.64 


§154] 


Correction  Eye-Glasses 


443 


According  to  this  formula,  the  length  of  the  eye  varies  from 
about  21.07  mm.  in  extreme  axial  hypermetropia  (A  = 
+  10  dptr.)  to  about  36.18  mm.  in  case  of  the  highest  degree 
of  axial  myopia  (A  =—20  dptr.).  The  length  of  an  axially 
emmetropic  eye  (.4=0)  is  24,38  mm.     The  length  of  Gull- 


tnttt 

35 

30 
25 

^~ 

20 

Mi 

Hyper 

mctropia 

UJ 

-90 


-lb 


-10 


-5 


+  5 


+10 


Fig.  198. 


-Curve  showing  connection  between  the  length  of 
the  eye-axis  and  the  static  refraction. 


strand's  schematic  eye  is  24.01  mm.,  and  hence  this  eye 
has  1  dptr.  of  hypermetropia  (A=+l  dptr.).  The  accom- 
panying diagram  (Fig.  198)  exhibits  graphically  the  relation 
between  the  magnitudes  denoted  by  I  and  A.  The  heights 
of  the  ordinates  indicate  the  axial  length  of  the  eye-ball  in 
millimeters  for  values  of  the  static  refraction  of  the  eye  com- 
prised between  —20  and  +10  dioptries. 

154.  Correction  Eye-Glasses. — When  a  spherical  spectacle 
lens  is  placed  in  front  of  the  passive,  unaccommodated  eye, 
with  the  axis  of  the  lens  in  the  same  straight  line  as  the  opti- 
cal axis  of  the  eye,  there  will  be  a  certain  axial  point  M  whose 
image  in  the  lens  will  fall  at  the  far  point  (R)  of  the  eye; 
and  hence  the  eye  looking  through  the  lens  will  see  distinctly 
the  image  of  an  object  placed  at  M.    If  the  positions  of  the 


444  Mirrors,  Prisms  and  Lenses  [§154 

principal  points  of  the  lens  are  designated  by  Hi  and  Hi', 
and  if  we  put 

tH  =  1/Ui  =  HiM,     Mi'  =  1/Ui'  =  Hi'R, 
then 

Ui'-U+Fh 

where  F\  denotes  the  refracting  power  of  the  lens.  Let  the 
distance  of  the  primary  principal  point  (H)  of  the  eye  from 
the  secondary  principal  point  (Hi7)  of  the  lens  be  denoted 
by  c,  that  is,  c  =  Hi'H;  then  since  a  =  ui'—c,  where  a  de- 
notes the  far  point  distance  of  the  eye,  the  following  ex- 
pression for  the  static  refraction  (A  =  l/a)  may  be  derived 
immediately: 

A         Ui+Ft 
l-c(Ui+Fi)  ' 

In  case  the  axial  object-point  M  is  infinitely  far  away,  the 
lens  is  called  a  correction-glass,  because  it  enables  the  pas- 
sive ametropic  eye  to  see  distinctly  a  very  distant  object 
on  the  axis  of  the  lens,  so  that  to  this  extent  the  lens  inter- 
posed in  front  of  the  eye  endows  it  with  the  characteristic 
faculty  of  an  unaccommodated,  naked,  emmetropic  eye. 
The  condition  that  M  shall  be  infinitely  distant  is  Ui  =  0; 
and  hence  the  relation  between  the  static  refraction  of  the 
eye  and  the  refracting  power  of  a  correction-glass  is  given  as 
follows : 

a.*     Fl=,  A 


1-c.Fi  1+c.A 

If  the  distance  c  between  the  correction-glass  and  the  eye  is 
neglected  entirely,  then  Fi  =  A,  that  is,  the  power  of  the 
correction-glass  is  approximately  equal  to  the  static  refrac- 
tion of  the  eye.  The  distance  c,  which  must  be  expressed 
in  meters  in  case  the  magnitudes  denoted  by  F\  and  A  are 
given  in  dioptries,  is  always  a  comparatively  small  magni- 
tude, which  in  actual  spectacle  glasses  is  comprised  between 
0.008  and  0.016  m.;  so  that  if,  without  neglecting  c  entirely, 


§  154]        Vertex  Refraction  of  Spectacle  Lens  445 

we  neglect  only  the  second  and  higher  powers  thereof,  the 
formulae  above  may  be  written  in  the  following  convenient 
approximate  forms: 

A  =  F1(l+c.F1),      F^Ail-cA); 
which  for  nearly  all  practical  purposes  will  be  found  to  be 
sufficiently  accurate. 


f^ 


Fig.  199. — Correction  of  myopia  with  concave  spectacle-glass. 

The  condition  that  a  spectacle-lens  shall  be  a  correction- 
glass  may  be  expressed  simply  by  saying  that  the  second  focal 
point  (Fir)  of  the  glass  must  coincide  with  the  far  point  (R)  of 
the  eye.    Thus,  in  case  of  a  myopic  eye  the  correction-glass 


Fig.  200. — Correction  of  hypermetropia  with  convex  spectacle-glass. 

will  be  concave  (Fig.  199)  and  in  case  of  a  hypermetropic 
eye  it  will  be  convex  (Fig.  200). 

Instead  of  describing  the  power  of  a  spectacle  glass  by 
means  of  its  refracting  power,  it  is  really  more  convenient 
and  logical  to  express  it  in  terms  of  its  vertex  refraction  (V), 
as  defined  in  §  128.  If  the  vertex  of  the  lens  which  lies  next 
the  eye  is  designated  by  L,  and  if  the  distance  of  the  eye  from 
the  glass  is  denoted  by  k,  that  is,  if  we  put  fc  =  LH,  then, 
since  the  points  designated  by  Fi'  and  R  must  be  coincident, 


446  Mirrors,  Prisms  and  Lenses  [§  155 

v  =  a+k,  where  a  denotes  the  "back  focus"  of  the  lens,  that 
is,  v=l/7  =  LF/  =  LR;  and  hence: 

v    ,     v-  A 


l-k.V  1+k.A' 

or  approximately: 

A  =  V(l+k.V),  V=A(l-kA). 
It  may  be  seen  from  the  above  formulae  how  the  power  of  a 
correction-glass  depends  essentially  on  the  location  of  the 
glass  in  front  of  the  eye.  The  distance  k,  being  referred  to  a 
tangible,  external  point  of  the  glass,  is  more  easily  measured 
than  the  interval  denoted  by  c. 

155.  Visual  Angle. — The  apparent  size  of  an  object,  as 
was  explained  in  §  10,  is  measured  by  the  visual  angle  co 
which  it  subtends  at  the  eye;  thus,  if  the  vertex  of  this  angle 
is  designated  by  0  and  if  ?/  =  MQ  denotes  a  diameter  of  the 
object  at  right  angles  to  the  line  of  vision,  the  apparent  size 
of  the  object  in  the  direction  of  this  dimension  is  co  =  Z  MOQ. 
Accordingly,  if  the  distance  of  the  object  from  the  eye  is  de- 
noted by  z,  that  is,  if  2  =  0M,  then  tana? =y/z.  As  the  im- 
mobile eye  looking  in  a  fixed  direction  can  see  distinctly 
only  that  comparatively  small  portion  of  the  object  whose 
image  falls  on  the  sensitive  part  of  the  retina  in  the  immedi- 
ate vicinity  of  the  fovea  centralis  (§  145),  the  rays  concerned 
in  the  production  of  the  retinal  image  in  this  so-called  case 
of  " indirect  vision"  may  be  regarded  as  paraxial  rays.  Ac- 
cordingly, the  value  of  the  angle  co  in  radians  may  be  sub- 
stituted here  for  the  tanco,  so  that  we  may  write: 

cc  =  y/z  =  y.Z, 
where  Z  =  l/z.     On  the  assumption  that  y  is  reckoned  as 
positive,  a  negative  value  of  the  angle  co  indicates  that  the 
object  is  real  and  therefore  in  front  of  the  point  O  where  the 
eye  is  supposed  to  be. 

The  exact  meaning  to  be  attached  to  the  visual  angle  co 
will  depend,  of  course,  on  the  precise  location  with  respect 
to  the  eye  of  the  vertex  of  this  angle.  To  be  sure,  so  long  as 
the  object  is  quite  remote  from  the  eye,  as  is  often  the  case, 


§  155]  Visual  Angle  447 

it  will  not  generally  be  necessary  to  define  particularly  the 
position  of  the  vertex  O  of  the  visual  angle.  For  example, 
to  take  a  somewhat  extreme  instance,  the  apparent  size  of 
the  moon  will  not  be  sensibly  altered  by  removing  the  ver- 
tex of  the  visual  angle  as  much  as  a  mile  or  more  away  from 
the  eye.  And,  in  general,  provided  the  object  is  not  less 
than,  say,  10  meters  away,  it  will  be  sufficient  to  know  that 
the  vertex  of  the  visual  angle  is  in  the  eye  without  specifying 
its  position  more  exactly.  On  the  other  hand,  especially 
when  the  eye  has  to  exert  its  power  of  accommodation  in 
order  to  focus  the  object,  it  is  sometimes  a  matter  of  much 
importance  to  define  the  visual  angle  with  the  utmost  pre- 
cision. In  such  a  case  several  meanings  of  this  term  are  to 
be  specially  distinguished.  For  example,  when  the  vertex 
of  the  visual  angle  is  at  the  primary  principal  point  of  the 
eye,  it  is  called  the  principal  point  angle  (coh  =  ^MHQ),  so 
that  we  may  write : 

ccH  =  ij!u  =  y.U, 
where  w=l/[/=HM  denotes  the  distance  of  the  object  from 
the  primary  principal  point.     Similarly,  the  so-called  focal 
point  angle   (coF  =  ZMFQ)   is  the  angle  subtended  by  the 
object  at  the  primary  focal  point  of  the  eye;  and  hence: 

uF  =  y/x  =  y.X, 
where  x=l/I  =  PM  denotes  the  distance  of  the  object  from 
the  primary  focal  point  of  the  eye. 

According  to  the  definitions  of  these  angles  and  the  rela- 
tions between  the  magnitudes  denoted  by  X,  U  and  Z,  as 
given  in  §  152,  we  may  write  therefore: 
co  :  cor-  :  coF  =  Z  :  U  :  X 

=  1  :(l+b.U)  :(l+g.X) 
=  (1-6.Z)  :1  :(1-X/F) 
=  (l-g.Z):(l+U/F):l; 
where  F  denotes  here  the  refracting  power  of  the  eye  when 
it  is  accommodated  for  the  point  M. 

The  apparent  size  of  an  object  may  be  measured  also  at 
other  points  of  the  eye,  for  example,  at  the  center  of  the 


448  Mirrors,  Prisms  and  Lenses  [§  156 

entrance-pupil,  at  the  anterior  nodal  point,  at  the  center  of 
rotation,  etc.  The  center  of  rotation  or  eye-pivot  is  the 
point  of  reference  in  the  estimate  of  the  apparent  size  of  an 
object  in  the  case  of  ordinary  so-called  "direct  vision"  with 
the  mobile  eye,  when  the  gaze  is  directed  in  quick  succession 
to  the  different  parts  of  an  extended  object.  Especially,  in 
viewing  an  image  through  an  optical  instrument,  it  is  nearly 
always  desirable,  if  practicable,  to  adjust  the  eye  in  such  a 
position  that  the  center  of  rotation  coincides  with  the  center 
of  the  exit-pupil  of  the  instrument,  so  as  to  command  as 
large  an  extent  of  the  field  of  view  of  the  image-space  as 
possible.  Anyone  who  has  ever  tried  to  look  through  a  key- 
hole in  a  door  will  realize  how  the  field  of  view  would  have 
been  widened  if  the  eye  could  have  been  placed  in  the  hole 
itself. 

156.  Size  of  Retinal  Image. — If  the  eye  is  accommodated 
to  see  an  object  y  situated  at  a  distance  u  (  =  1/U)  from  its 
primary  principal  point,  the  size  of  the  image  (yf)  formed  on 
the  retina  is  given  by  the  relation: 
y.U=y'.A', 

where  A'  =  n'/a'  denotes  the  reciprocal  of  the  reduced  length 
of  the  eye-axis  measured  from  the  secondary  principal  point 
of  the  eye.  Since  2/.J7=coH  (§155),  the  above  equation 
may  be  put  in  the  following  form : 

coh     n'  ' 

Since  the  positions  of  the  principal  points  remain  sensibly 
stationary  in  the  act  of  accommodation  (§  150),  the  reduced 
length  of  the  eye-axis  (a'jnf)  may  be  considered  as  constant 
in  the  same  individual.  And  hence  the  peculiar  significance 
of  the  principal  point  angle  consists  in  the  fact  that,  ac- 
cording to  this  formula,  this  angle  (coH )  may  be  taken  as 
a  measure  of  the  size  of  the  retinal  image  {y')  which  is  in- 
dependent of  the  state  of  accommodation  of  the  eye.  Thus, 
for  a  given  individual,  all  objects  which  have  the  same  ap- 


157] 


Apparent  Size  of  Image 


449 


parent  size  as  measured  at  the  principal  point  of  the  eye  will 
produce  retinal  images  of  equal  size. 

On  the  other  hand,  since  y'.F=y.X=aiF  (§  155),  it  ap- 
pears that,  for  a  given  value  of  the  refracting  power  (F), 
the  size  of  the  image  on  the  retina  of  the  eye  is  proportional 
to  the  focal  point  angle.  And  since  the  variations  of  the  re- 
fracting power  are,  generally  speaking,  independent  of  axial 
ametropia  (§  153),  the  focal  point  angle  will  be  particularly 
useful  in  comparing  the  apparent  size  of  an  object  as  seen 
by  different  individuals  under  the  same  external  conditions. 

157.  Apparent  Size  of  an  Object  seen  Through  an  Op- 
tical Instrument. — Let  the  principal  points  of  the  optical 
instrument  be  designated  by  H,  IT  (Fig.  201);  and  for  the 


Fig.  201. — Apparent  size  of  object  seen  through  an  optical 
instrument. 

sake  of  simplicity,  let  us  assume  that  the  instrument  is  sur- 
rounded by  air  so  that  the  straight  lines  HQ,  H'Q'  joining 
the  principal  points  with  corresponding  points  of  object  and 
image  will  be  parallel;  and  let  ?/  =  MQ,  ?/'  =  M'Q'  denote  the 
linear  magnitudes  of  object  and  image,  respectively.  Let 
the  distance  of  the  image  from  the  eye  be  denoted  by  z= 
O'M',  where  O'  designates  the  position  of  the  eye  on  the 
axis.    Then  the  apparent  size  of  the  image  will  be 

co=2/'.Z, 
where  co  =  ZM'0'Q'  (expressed  here  in  radians)  andZ  =  l/z. 
The  angle  co  may  be  increased  by  reducing  the  distance  be- 


450  Mirrors,  Prisms  and  Lenses  [§  157 

tween  the  image  and  the  eye,  that  is,  by  increasing  Z;  but 
this  distance  cannot  be  diminished  below  the  near  point 
distance  of  the  eye,  because  then  distinct  vision  would  not 
be  possible  for  the  naked  eye. 

If  the  distances  of  object  and  image  from  the  principal 
points  are  denoted  by  u  and  u',  that  is,  if  w=HM,  w'  =  H'M', 
then 

y'.U'=y.U, 
where  U=l/u,  U'  =  l/u';  and  hence 

In  general  (except  when  the  rays  undergo  an  odd  number 
of  reflections) ,  the  sign  of  Z  as  here  defined  will  be  negative, 
and  therefore  the  sign  of  co  will  depend  on  the  sign  of  the 
ratio  U  :  U'.  Accordingly,  if  object  and  image  lie  on  the 
same  side  of  their  corresponding  principal  points,  the  sign 
of  co  will  be  negative,  that  is,  the  image  will  be  erect. 

Let  the  distance  of  the  eye  from  the  instrument  be  de- 
noted by  c  =  H'0/;  then  since  u'  =  c+z,  we  may  write: 

U'  =     Z      . 

1+c.Z 

Accordingly,  if  the  refracting  power  of  the  instrument  is 
denoted  by  F,  so  that  U=Uf—F,we  may  write  also: 

F-Z(l-c.F) 
1+c.Z 
Introducing  these  expressions,  we  obtain  therefore  the  fol- 
lowing formula  for  the  apparent  size  of  the  image: 
a>=-y\F-Za-c.F)\  . 
Thus,  we  see  that  the  apparent  size  of  the  image  may  be 
varied  in  one  of  two  ways,  either  by  changing  the  position 
of  the  eye  (that  is,  by  varying  c)  or  else  by  displacing  the 
object  so  that  Z  is  varied.     There  are  two  cases  of  special 
practical  importance,  namely:  (1)  When  the  eye  is  adjusted 
so  that  l  —  c.F  =  0,  and    (2)  When  the  object  is  focused  so 
that  Z  =  0.     In  both  cases  the  second  term  inside  the  large 
brackets   vanishes,    and  hence   oo  =  —y.F.     The  condition 


§  157]  Apparent  Size  of  Image  451 

c  =  —  l/F  means  that  the  eye  is  placed  at  the  second  focal 
point  (F')  of  the  instrument  (which  might  easily  be  practi- 
cable if  the  optical  system  were  convergent) ;  so  that  under 
such  circumstances  the  apparent  size  of  the  image  would  be 
the  same  for  all  positions  of  the  object,  because  evidently 
the  highest  point  (Q')  of  the  image  will  always  lie  on  the 
straight  line  which  crosses  the  axis  at  the  second  focal  point 
at  the  constant  angle  6=  —y.F.  On  the  other  hand,  the 
condition  Z  =  0  means  simply  that  the  object  lies  in  the 
first  focal  plane  of  the  instrument.  Now  this  is  the  natural 
adjustment  for  a  normal,  unaccommodated,  emmetropic 
eye,  because  then  the  rays  flow  into  the  eye  in  cylindrical 
bundles.  This  is  the  reason  why  the  image  produced  by 
the  object-glass  of  a  telescope  or  microscope  is  usually  fo- 
cused in  the  primary  focal  plane  of  the  eye-piece  or  ocular. 
Accordingly,  when  Z  =  0,  the  apparent  size  of  the  image 
will  be  independent  of  the  position  of  the  eye. 

An  experienced  observer  who  wishes  to  obtain  the  best 
results  with  an  optical  instrument  will  ordinarily  adjust  it 
to  his  eye  in  such  a  way  that  the  image  can  be  seen  distinctly 
without  his  having  to  make  an  effort  of  accommodation. 
This  will  be  the  case  if  the  image  is  formed  at  the  far  point 
(R)  of  the  eye  (§  148).  If,  therefore,  the  static  refraction 
of  the  eye  is  denoted  by  A  (§151),  then  (assuming  that  the 
point  O'  in  Fig.  201  is  coincident  with  the  anterior  principal 
point  of  the  eye)  we  may  put  Z  =  A;  and  hence  the  apparent 
size  of  an  object  as  seen  in  an  optical  instrument  by  an  eye 
with  relaxed  accommodation  is  given  by  the  expression: 

coK=-y\F-A(l-c.F)\  . 
Thus,  it  is  evident  how  the  apparent  size  of  the  image  de- 
pends not  only  on  the  refracting  power  of  the  instrument, 
but  essentially  also  on  the  adjustment  and  idiosyncrasies 
of  the  eye  of  the  individual  who  looks  through  it. 

It  may  be  remarked  that  these  formulae  have  been  derived 
on  the  tacit  assumption  that  the  eye  is  at  rest,  and  conse- 
quently only  a  small  portion  of  the  external  field  is  sharply 


452  Mirrors,  Prisms  and  Lenses  [§  158 

in  focus  at  the  sensitive  part  of  the  retina.  Otherwise,  we 
should  have  had  to  write  tanco  instead  of  co;  nor  should  we 
have  been  justified  in  assuming  that  the  effective  rays  were 
paraxial.  If  the  eye  turns  in  its  socket  to  inspect  the  image, 
the  apparent  size  of  the  image  will  depend  essentially  on 
the  angular  movement  of  the  eye,  and  in  this  case  the  visual 
angle  must  be  measured  at  the  center  of  rotation  of  the  eye. 
These  are  considerations  that  are  too  often  overlooked  in 
discussions  of  this  kind. 

158.  Magnifying  Power  of  an  Optical  Instrument  Used 
in  Conjunction  with  the  Eye. — An  object  may  be  so  remote 
that  its  details  are  indistinguishable,  or,  on  the  other  hand, 
it  may  be  so  close  to  the  eye  that  not  even  by  the  greatest 
effort  of  accommodation  can  a  sharp  image  of  it  be  focused 
on  the  retina.  Under  such  circumstances  one  has  recourse 
to  the  aid  of  a  suitable  optical  instrument  whereby  the  ob- 
ject is  magnified  to  such  an  extent  that  the  parts  of  it  which 
were  obscure  or  entirely  invisible  to  the  naked  eye  will  be 
revealed  to  view.  The  magnifying  power  is  usually  expressed 
by  an  abstract  number  M,  which  in  the  case  of  an  optical 
instrument  on  the  order  of  a  microscope  is  defined  to  be  the 
ratio  of  the  apparent  size  of  the  image  as  seen  in  the  instrument 
to  the  apparent  size  of  the  object  as  it  would  appear  at  the  so- 
called  "distance  of  distinct  vision."  This  latter  term  is  a 
somewhat  unfortunate  form  of  expression  for  several  rea- 
sons, not  only  because  the  distance  at  which  an  object  is 
ordinarily  placed  in  order  to  be  seen  distinctly  is  different 
for  different  persons,  but  because  the  same  person,  accord- 
ing to  the  extent  of  his  power  of  accommodation,  usually 
possesses  the  ability  of  seeing  distinctly  objects  at  widely 
different  distances.  The  expression  appears  to  have  arisen 
from  a  confusion  of  ideas,  and  its  origin  may  probably  be 
traced  to  the  fact  that  even  nowadays  many  people  have 
difficulty  in  conceiving  how  the  eye  can  be  "  focused  for 
infinity,"  although,  indeed,  as  has  been  explained,  that  is 
to  be  regarded  as  the  natural  state  of  the  normal  eye  in  re- 


§  158]    Magnifying  Power  of  Optical  Instrument      453 

pose.  However,  the  phrase  has  become  too  deeply  rooted 
in  optical  literature  ever  to  be  eradicated,  and  no  harm  will 
be  done  by  continuing  to  use  it,  provided  it  is  not  taken 
literally,  but  is  considered  merely  as  the  designation  of  a 
more  or  less  arbitrary  conventional  projection-distance. 
Accordingly,  if  the  so-called  " distance  of  distinct  vision" 
is  denoted  by  I,  the  apparent  size  of  the  object  (y)  as  seen 
at  this  distance  from  the  eye  will  be  —y/l,  and  hence  if  the 
apparent  size  of  the  image  in  the  instrument  is  denoted  by 
co,  the  magnifying  power,  as  above  defined,  will  be: 

y 

The  actual  value  of  this  conventional  distance  I  is  usually 
taken  as  10  inches  or  25  centimeters,  which  is  large  enough 
for  the  convenient  accommodation  of  most  human  beings 
who  are  not  already  past  the  prime  of  life  and  yet  not  so 
large  that  the  size  of  the  image  on  the  retina  differs  much 
from  its  greatest  dimensions.  If  distances  are  all  measured 
in  meters,  the  conventional  value  of  the  magnifying  power 
will  be  given,  therefore,  by  the  formula: 

M=     " 

The  explanation  of  the  minus  sign  in  front  of  the  fraction 
is  to  be  found  in  the  mode  of  reckoning  the  visual  angle  co, 
which,  as  we  have  pointed  out  (§  157),  is  negative  in  case 
the  image  of  the  object  y  is  erect,  as,  for  example,  with  an 
ordinary  convex  lens  used  as  a  magnifying  glass.  Thus, 
according  to  the  above  formula,  a  positive  value  of  the  mag- 
nifying power  means  magnification  without  inversion.  Or- 
dinarily, what  is  meant  by  the  magnifying  power  of  an  op- 
tical instrument  is  the  value  of  this  abstract  number  M; 
which  gives  the  ratio  of  the  sizes  of  the  retinal  images  when 
an  emmetropic  eye  views  one  and  the  same  object,  first,  in  the 
instrument  without  effort  of  accommodation,  and  then  with- 
out the  instrument  with  an  accommodation  of  four  dioptries. 
If  the  expression  for  the  visual  angle  co  which  was  ob- 


454  Mirrors,  Prisms  and  Lenses  [§  158 

tained  in  §  157  is  introduced  here,  we  shall  derive,  therefore, 
the  following  formula  for  the  magnifying  power  (M)  in  terms 
of  the  refracting  power  (F)  of  the  instrument,  the  distance 
(c)  of  the  eye  from  the  instrument,  and  the  distance  (2= 
1/Z)  of  the  image  (yf)  from  the  eye: 

M  =  l\F-Z(l-c.F)\. 
This  expression  is  really  a  measure  of  the  individual  mag- 
nifying power,  since  it  involves  not  merely  the  instrument 
itself  but  the  characteristic  peculiarities  of  the  eye  of  the 
observer.  In  order  to  obtain  a  measure  of  the  absolute  mag- 
nifying power  of  the  instrument,  the  second  term  inside  the 
large  brackets  must  be  made  to  vanish.  Thus,  if  the  object 
is  placed  in  the  primary  focal  plane,  so  that  the  image  is 
infinitely  distant,  then  Z  =  0,  and  now  M  =  l.F  denotes  the 
absolute  magnifying  power.  If  1  =  0.25  meter,  then  F  =  4M; 
and  usually,  therefore,  when  we  say  that  the  magnifying 
power  of  a  lens  or  microscope  is  M,  this  means  simply  that 
its  refracting  power  is  equal  to  4M  dioptries. 

If  the  image  in  the  instrument  is  formed  at  the  "  distance 
of  distinct  vision"  (I),  then  Z=—l/l.  and 

M  =  l +  (l-c)F. 
The  distance  (c)  between  the  instrument  and  the  eye  is  usu- 
ally small  in  comparison  with  I,  so  that  it  is  often  entirely 
neglected.  Assuming  that  (l—c)  is  positive,  we  may  say 
that  in  a  convergent  optical  system  (F>0),  the  object  will 
appear  magnified  (M>1);  whereas  in  a  divergent  optical 
system  (F<0),  the  object  appears  to  be  diminished  in  size 
(M<1). 

In  order  to  avoid  the  use  of  an  arbitrary  projection- 
distance,  (Z),  Abbe  proposed  to  define  the  magnifying  power 
as  the  ratio  of  the  apparent  size  ( 00)  of  the  image  in  the  instru- 
ment to  the  actual  size  (y)  of  the  object  (compare  with  Abbe's 
definition  of  focal  length,  §  122) ;  so  that  if  this  ratio  is  de- 
noted by  P,  then 

^  CO 

p=--. 

y 


§  159]  Magnifying  Power  of  Telescope  455 

This  measure  of  the  magnifying  power  is  not  an  abstract 
number  like  M,  but  a  quantity  of  the  same  physical  dimen- 
sions as  the  refracting  power  of  the  instrument.  The  two 
definitions  are  connected  by  the  simple  relation 

M  =  Z.P; 
so  that  if  we  put  1  =  0.25  m.,  the  value  of  P  will  be  obtained 
by  multiplying  M  by  the  number  four  (P  =  4M).  Thus,  for 
example,  in  the  case  of  a  convex  lens  of  refracting  power  F 
used  as  a  magnifying  glass,  if  the  object  is  placed  in  the  first 
focal  plane,  we  have  P  =  F. 

159.  Magnifying  Power  of  a  Telescope. — In  the  case  of 
a  telescope,  which  is  an  instrument  for  magnifying  the  ap- 
parent size  of  a  distant  object,  neither  of  the  definitions  of 
magnifying  power  given  in  the  foregoing  section  is  appli- 
cable. An  infinitely  distant  object  (like  the  moon,  for  ex- 
ample) can  be  seen  distinctly  by  an  emmetropic  eye  without 
any  effort  of  accommodation,  but  its  apparent  size  may  be 
so  minute  that  the  distinguishing  features  cannot  be  made 
out  by  the  naked  eye.  This  same  eye  looking  at  the  object 
through  a  telescope  will  see  an  infinitely  distant  image  of 
it,  but  presented  to  the  eye  under  a  larger  visual  angle,  so 
that  it  appears  magnified.  Essentially,  a  telescope  may  be 
regarded  as  a  combination  of  two  optical  systems,  one  of 
which — the  part  pointed  towards  the  object — is  a  con- 
vergent system,  generally  of  relatively  long  focus  and  large 
aperture  (so  as  to  intercept  a  large  quantity  of  light),  called 
the  object-glass;  while  the  other,  composed  of  the  lenses 
next  the  eye,  and  called  therefore  the  ocular  or  eye-piece, 
may  be  a  convergent  or  divergent  system  depending  on  the 
type  of  telescope.  The  object-glass  which  is  at  one  end  of 
a  large  tube  forms  a  real  inverted  image  of  the  object  in  its 
second  focal  plane  or  not  far  from  it;  and  this  image  is  in- 
spected through  the  ocular,  which  is  usually  fixed  in  a  smaller 
tube  inserted  in  the  larger  one  so  that  the  focus  can  be  ad- 
justed to  suit  different  eyes  and  different  circumstances. 
A  simple  schematic  telescope  may  be  regarded  as  composed 


456 


Mirrors,  Prisms  and  Lenses 


[§  159 


of  two  thin  lenses,  one  of  which,  of  focal  length  /i  (refracting 
power  Fi)  acts  as  the  object-glass  while  the  other,  of  focal 
length  f2  (refracting  power  F2)  performs  the  part  of  the  oc- 
ular. When  the  telescope  is  adjusted  for  an  emmetropic, 
unaccommodated  eye,  the  second  focal  point  (F/)  of  the  ob- 
ject-glass coincides  with  the  first  focal  point  (F2)  of  the  ocular; 
and  hence  the  focal  length  of  the  entire  system  is  infinite 
(/=  oo  or  F  =  0),  that  is,  the  system  is  afocal  or  telescopic 
(§125).  In  this  case  the  telescope  is  said  to  be  in  normal 
adjustment. 

The  first  telescope  appears  to  have  been  invented  by  one 
of  two  Dutch  spectacle-makers  named  Zacharias  Jansen 
and  Franz  Lippershey  (circa  1608).  Galileo  (1564- 
1642),  having  heard  of  this  Dutch  toy,  was  led  to  experiment 


To  J  at  oc 


To  J'atco 


Fig.  202. — Diagram  of  simple  Dutch  or  Galilean  telescope. 

with  a  combination  of  two  lenses  and  he  soon  succeeded 
(1609)  in  making  a  telescope  with  which  he  made  a  number 
of  renowned  astronomical  discoveries.  The  so-called  Dutch 
or  Galilean  telescope,  represented  schematically  in  Fig.  202, 
consists  of  a  large  convex  object-glass  (Ai)  combined  with 
a  small  concave  eye-piece  (A2),  which  intercepts  the  con- 
verging rays  before  they  come  to  a  focus  and  adapts  them  to 
suit  the  eye  of  the  observer.  The  other  type  of  telescope 
(Fig.  203)  is  composed  of  two  convex  lenses.  It  is  called  the 
astronomical  telescope  or  Kepler  telescope,  because  the  idea 


159] 


Magnifying  Power  of  Telescope 


457 


occurred  first  to  John  Kepler  (1611);  but  the  first  instru- 
ment of  this  kind  was  made  by  the  celebrated  Jesuit  father, 
Christian  Scheiner  (1615),  who  also  conceived  the  idea 
of  using  a  third  lens  to  erect  the  image  as  is  done  in  the  so- 
called  terrestrial  telescope. 

If  the  telescope  is  in  normal  adjustment,  then  from  each 
point  J  of  the  infinitely  distant  object  there  will  issue  a  bundle 


Fig.  203. — Diagram  of  simple  astronomical  telescope. 

of  parallel  rays  whose  inclination  to  the  axis  of  the  telescope 
may  be  denoted  by  6.  Falling  on  the  object-glass,  these 
rays  are  converged  to  a  focus  at  a  point  P  lying  in  the  com- 
mon focal  plane  of  object-glass  and  eye-piece;  and  conse- 
quently they  will  emerge  from  the  eye-piece  and  enter  the 
eye  as  a  bundle  of  parallel  rays  proceeding  from  the  infi- 
nitely distant  image-point  J'  in  a  direction  which  makes  an 
angle  6'  with  the  axis.  The  slope-angles  6  and  6'  have 
a  constant  relation  to  each  other,  as  may  easily  be  shown; 
for  from  the  right  triangles  F/AiP  and  F2A2P  (Figs.  202 
and  203),  where  AiF/  =  AiF2=/i,  ZFi'AiP  =  0,  and  F2A2  = 
Fi'A2=/2,  ZF2A2P=  6',  we  obtain  immediately: 

tan  0'        /i 

—  =  —  -  =  constant. 

tan  Q         j 2 


458  Mirrors,  Prisms  and  Lenses  [§  159 

Now  the  angles  denoted  here  by  6  and  6f  are  the  measures 
of  the  apparent  sizes  of  corresponding  portions  of  the  in- 
finitely distant  object  and  image,  and  the  ratio  of  these 
angles  (or  of  their  tangents)  is  defined  to  be  the  magnifying 
power  of  the  telescope;  so  that  if  this  ratio  is  denoted  by  M, 
we  shall  have: 

h 
Accordingly,  the  magnifying  power  of  a  telescope  focused 

on  an  infinitely  distant  object  and  adjusted  for  distinct 
vision  for  an  unaccommodated,  emmetropic  (or  corrected 
ametropic)  eye  is  measured  by  the  ratio  of  the  focal  lengths 
of  the  objective  and  ocular.  In  the  astronomical  telescope 
/i  and  /2  are  both  positive,  and  consequently  the  ratio  M  is 
negative,  which  means  that  the  image  is  inverted ;  whereas  in 
the  Dutch  telescope  /i  is  positive  and  f2  is  negative,  and 
hence  *M  is  positive,  that  is,  the  final  image  is  erect. 

Another  convenient  expression  for  the  magnifying  power 
of  a  telescope,  as  defined  above,  may  easily  be  obtained. 
All  the  effective  rays  which  fall  on  the  object-glass  will  after 
transmission  through  the  instrument  pass  through  a  certain 
circular  aperture  called  the  eye-ring  (or  Ramsden  circle), 
which  is  the  image  of  the  object-glass  in  the  ocular.  If  the 
object-glass  is  brightly  illuminated  (for  example,  if  the  tele- 
scope is  pointed  towards  the  bright  sky),  this  image  appears 
as  a  luminous  disk  floating  in  the  air  not  far  from  the  ocular 
and  can  easily  be  perceived  by  placing  the  eye  at  a  suitable 
distance.  In  the  astronomical  telescope  the  eye-ring  is  a 
real  image  which  can  be  received  on  a  screen,  and  in  this 
instrument  it  usually  acts  as  the  exit-pupil  (§  134).  In  the 
case  of  the  Dutch  telescope  the  eye-ring  is  a  virtual  image 
on  the  other  side  of  the  ocular  from  the  eye;  and  generally 
its  effect  is  to  limit  the  field  of  view  in  the  image-space,  that 
is,  its  office  is  that  of  the  exit-port  of  the  system  (§§  137, 
138).  Now  if  the  telescope  is  in  normal  adjustment,  then 
the  distance  of  the  ocular  from  the  object-glass  is  equal  to 


§  159]  Magnifying  Power  of  Telescope  459 

the  algebraic  sum  (/1+/2)  of  the  focal  lengths  of  the  two 
components ;  and  it  may  easily  be  shown  that 
M  _  /1  _  diameter  of  object-glass 
J2  diameter  of  eye-ring 
The  advantage  of  this  latter  form  of  expression  is  to  be 
found  in  the  fact  that  even  if  the  telescope  is  not  in  normal 
adjustment,  it  may  still  be  considered  in  a  certain  sense  as 
a  measure  of  the  magnifying  power  of  the  instrument.  Sup- 
pose, for  example,  that  the  optical  system  is  not  telescopic, 
so  that  the  interval  between  the  second  focal  point  (Fi')  of 
the  object-glass  and  the  first  focal  point  (F2)  of  the  ocular 
is  not  negligible,  as  frequently  happens  in  focusing  the  eye- 
piece to  suit  the  eye  of  the  individual,  especially  if  the  object 
itself  is  not  infinitely  distant.  Consider  a  ray  which  is  di- 
rected originally  from  the  extremity  of  the  object  towards 
a  point  O  on  the  axis  of  the  telescope  and  which  emerges 
so  as  to  enter  the  eye  at  the  conjugate  point  O'.  If  the  angles 
which  the  ray  makes  with  the  axis  at  O  and  O'  are  denoted 
by  6  and  6' ',  respectively,  then  the  ratio  tan#'  :  tan#  will 
be  a  measure  of  the  magnifying  power  of  the  telescope  for 
this  adjustment  and  position  of  the  eye.  But  according  to 
the  Smith-Helmholtz  formula  (§§  86  and  118),  since  the 
telescope  is  surrounded  by  the  same  medium  on  both  sides, 
we  shall  have  here : 

tan#'  :  tand  =  y  :  y', 
where  y  and  y'  denote  the  linear  magnitudes  of  an  object  and 
its  image  in  conjugate  transversal  planes  at  O  and  O'  (the 
planes  of  the  pupils).  Now  if  the  point  O'  is  at  the  center 
of  the  eye-ring,  the  point  O  will  lie  at  the  center  of  the  object- 
glass,  and  the  ratio  y  :  y'  will  be  equal  to  the  ratio  of  the 
diameters  of  object-glass  and  eye-ring.  Hence,  provided 
the  eye  is  placed  at  the  eye-ring,  the  magnifying  power  of  the 
telescope  will  be 

lvr_  diameter  of  object-glass 
diameter  of  eye-ring 
In  an  astronomical  telescope  the  best  adjustment  for  com- 


460  Mirrors,  Prisms  and  Lenses  [§159 

manding  a  wide  extent  of  the  field  of  view  is  to  place  the  eye 
with  its  center  of  rotation  at  the  center  of  the  eye-ring,  but 
in  a  Dutch  telescope  this  is  not  practicable,  because  the  eye- 
ring  is  not  accessible. 

In  order  to  obtain  a  general  formula  for  the  magnifying 
power  of  a  telescope,  let  us  fix  our  attention  on  the  inverted 
image  of  the  object  which  is  formed  by  the  object-glass. 
If  u=l/U  denotes  the  distance  of  the  object  from  the  object- 
glass  and  if  q  denotes  the  linear  size  of  the  image,  the  appar- 
ent size  of  the  object  as  seen  from  the  center  of  the  object- 
glass  will  be 

tan0 =q(U+Fi), 

where  F\  denotes  the  refracting  power  of  the  object-glass. 
On  the  other  hand,  according  to  the  formula  deduced  in 
§  157,  the  apparent  size  of  the  image  seen  in  the  telescope 
will  be  , 

ttmd'=-q{F2-Z(l-c.F2)}  , 

where  F2  denotes  the  refracting  power  of  the  ocular,  z—\\Z 
denotes  the  distance  of  the  image  in  the  ocular  from  the  eye, 
and  c  denotes  the  distance  of  the  eye  from  the  ocular  itself 
(or  from  its  second  principal  point).  Accordingly,  we  obtain 
the  following  expression  for  the  magnifying  power  of  the 
telescope: 

tanfl^      F2-Z(\-c.F2) 

tanfl  U+Fi 

which  is  applicable  to  all  cases.  If  the  object  is  infinitely 
distant,  then  £7  =  0;  and  if  the  telescope  is  in  normal  adjust- 
ment, then  the  image  is  also  infinitely  distant,  that  is,  Z  =  0, 
andM=-F2/Fi. 


Ch.  XIII]  Problems  461 

PROBLEMS 

1.  If  the  refracting  power  of  a  correction  spectacle-glass 
is  +10  dptr.,  and  if  the  distance  of  the  anterior  principal 
point  of  the  eye  from  the  second  principal  point  of  the  glass 
is  12  mm.,  find  the  static  refraction  of  the  eye. 

Ans.  +11.36  dptr. 

2.  Take  the  refracting  power  of  the  eye  equal  to  58.64  dptr., 
the  distances  of  the  principal  points  from  the  vertex  of  the 
cornea  as  1.348  and  1.602  mm.,  and  the  index  of  refraction 
of  the  vitreous  humor  equal  to  1.336.  If  the  refracting  power 
of  a  correction  spectacle-glass,  whose  second  principal  point 
is  14  mm.  from  the  anterior  principal  point  of  the  eye,  is 
+  5.37  dptr.,  show  that  the  total  length  of  the  eye-ball  is 
26.5  mm. 

3.  In  Gtjllstrand's  schematic  eye,  with  accommodation 
relaxed,  the  distance  from  the  vertex  of  the  cornea  to  the 
point  where  the  optical  axis  meets  the  retina  is  24  mm.  The 
other  data  are  the  same  as  those  given  in  No.  2  above.  Find 
the  position  of  the  far  point  and  determine  the  static  refrac- 
tion. 

Ans.  The  far  point  is  99.34  cm.  from  the  vertex  of  the 
cornea,  and  the  static  refraction  is  + 1 .008  dptr. 

4.  In  Gullstrand's  schematic  eye  in  its  state  of  maxi- 
mum accommodation  the  distances  of  the  principal  points 
from  the  vertex  of  the  cornea  are  1.7719  and  2.0857  mm.,  and 
the  refracting  power  is  70.5747  dptr.  The  length  of  the  eye- 
ball is  24  mm.,  as  stated  in  No.  3.  Find  the  position  of  the 
near  point  and  determine  the  dynamic  refraction  of  the 
eye. 

Ans.  The  near  point  is  10.23  cm.  from  the  vertex  of  the 
cornea;  the  dynamic  refraction  is  —9.609  dptr.  Accordingly, 
with  the  aid  of  the  result  obtained  in  No.  3,  we  obtain  for 
the  amplitude  of  accommodation  10.62  dptr. 

5.  Taking  the  refracting  power  of  the  eye  as  equal  to 
59  dptr.,  show  that  the  size  of  the  retinal  image  of  an  object 


462  Mirrors,  Prisms  and  Lenses        [Ch.  XIII 

1  meter  high  at  a  distance  of  10  meters  from  the  eye  will  be 
1.7  mm. 

6.  The  apparent  size  of  a  distant  air-ship  is  one  minute  of 
arc.  Taking  the  refracting  power  of  the  eye  as  equal  to 
58.64  dptr.,  show  that  the  size  of  the  image  on  the  retina 
will  be  0.00495  mm. 

7.  What  is  the  magnifying  power  of  a  convex  lens  of  focal 
length  5  cm.?  Ans.  5. 

8.  A  myope  of  10  dptr.  uses  a  convex  lens  of  focal  length 
5  cm.  as  a  magnifying  glass.  Find  the  individual  magnify- 
ing power,  neglecting  the  distance  of  the  eye  from  the  glass. 

Ans.  7|. 

9.  In  the  preceding  example,  what  will  be  the  individual 
magnifying  power  of  the  same  glass  in  the  case  of  an  hyper- 
metrope  of  10  dptr.?  Ans.  2£. 

10.  A  certain  person  cannot  see  distinctly  objects  which 
are  nearer  his  eye  than  20  cm.  or  farther  than  60  cm.  Within 
what  limits  of  distance  from  his  eye  must  a  concave  mirror 
of  focal  length  15  cm.  be  placed  in  order  that  he  may  be  able 
to  focus  sharply  the  image  of  his  eye  as  seen  in  the  mirror? 

Ans.  In  order  to  see  a  real  image  of  his  eye,  the  distance 
of  the  mirror  must  be  between  43.23  cm.  and  78.54  cm.;  in 
order  to  see  a  virtual  image,  the  distance  of  the  mirror  must 
be  between  6.97  cm.  and  11.46  cm. 

11.  The  magnifying  power  of  a  telescope  12  inches  long 
is  equal  to  8:  determine  the  focal  lengths  of  object-glass 
and  eye-glass  (1)  when  it  is  an  astronomical  telescope  and 
(2)  when  it  is  a  Galileo's  telescope. 

Ans.  (1)  /i  =  +10|,  /2=+l|  inches;  (2)/i=+13|, 
/2  =  —  ly  inches. 

12.  The  focal  lengths  of  the  object-glass  and  eye-glass  of 
an  astronomical  telescope  are  /1  and  /2)  and  their  diameters 
are  2hi  and  2h2)  respectively.  Show  that  the  radius  of  the 
stop  which  will  cut  off  the  " ragged  edge"  (§  138)  is  equal  to 

M2—M1 


Ch.  XIII]  Problems  463 

13.  A  telescope  is  pointed  at  an  infinitely  distant  object, 
and  the  eye-piece  is  focused  so  that  the  image  is  formed  at 
the  distance  I  of  distinct  vision  of  the  eye.  If  the  distance  of 
the  eye  from  the  eye-piece  is  neglected,  show  that  the  mag- 
nifying power  is  M=  — /i(7+/2)/Z./2,  where  /i,  f2  denote  the 
focal  lengths  of  the  object-glass  and  eye-glass. 

14.  A  Ramsden  ocular  consists  of  two  thin  convex  lenses 
each  of  focal  length  a  separated  by  an  interval  equal  to  2a/3. 
Show  that  the  magnifying  power  of  an  astronomical  tele- 
scope furnished  with  a  Ramsden  ocular  is  4/i/3a,  where  /i 
denotes  the  focal  length  of  the  object-glass. 

15.  The  object-glass  of  an  astronomical  telescope  has  a 
focal  length  of  50  inches,  and  the  focal  length  of  each  lens 
of  the  Ramsden  ocular  is  2  inches.  The  distance  between 
the  two  lenses  in  the  ocular  is  ^  inch.  Show  that  the  dis- 
tance between  the  object-glass  and  the  first  lens  of  the  oc- 
ular is  50.5  inches,  and  that  the  magnifying  power  is  equal 

to  ir- 

16.  If  a  Galileo's  telescope  is  in  normal  adjustment, 
show  that  the  angular  diameter  of  the  field  of  the  image  as 
measured  at  the  vertex  of  the  concave  eye-glass  is  2tanY'  = 
—  2/ii/(/i+/2),  where  hi  denotes  the  radius  of  the  object-glass 
and  /i,  /2  denote  the  focal  lengths  of  object-glass  and  eye- 
glass. 

17.  The  focal  length  of  the  object-glass  and  eye-glass  of 
an  astronomical  telescope  are  36  and  9  inches,  respectively. 
If  the  object  is  infinitely  distant  and  if  the  eye  is  placed  in 
the  eye-ring  at  a  distance  of  9  inches  from  the  image,  show 
that  the  magnifying  power  is  equal  to  3. 

18.  'The  magnifying  power  of  a  simple  astronomical  tele- 
scope in  normal  adjustment  is  M,  and  the  focal  length  of  the 
object-glass  is  /i.  Show  that  if  the  eye-glass  is  pushed  in  a 
distance  x  and  the  eye  placed  in  the  eye-ring,  the  magnifying 
power  will  be  diminished  by  x.M/fi. 

19.  An  astronomical  telescope  is  pointed  towards  the  sun, 
and  a  real  image  of  the  sun  is  obtained  on  a  screen  placed 


464  Mirrors,  Prisms  and  Lenses         [Ch.  XIII 

beyond  the  eye-lens  at  a  distance  d  from  it.  If  the  diameter 
of  this  image  is  denoted  by  26,  and  if  the  apparent  diameter 
of  the  sun  is  denoted  by  2  6,  show  that  the  magnifying  power 
of  the  telescope  is  M  =  6. cot  6/d. 

20.  The  eye  is  placed  at  a  distance  c  from  the  eye-glass  of 
a  Galileo's  telescope  in  normal  adjustment.  The  length 
of  the  telescope  as  measured  from  the  object-glass  to  the 
eye-glass  is  denoted  by  d,  the  radius  of  the  object-glass  is 
denoted  by  hi,  and  the  radius  of  the  pupil  of  the  eye  is  de- 
noted by  g  (it  being  assumed  that  g  is  less  than  the  radius  of 
the  eye-glass).  Show  that  the  semi-angular  diameters  of 
the  three  portions  of  the  field  of  view  on  the  image-side  are 
given  by  the  following  expressions : 

hi— gM  ,  h  ,        hx+gM 

where  M  denotes  the  magnifying  power  of  the  telescope. 


CHAPTER  XIV 


DISPERSION  AND  ACHROMATISM 


160.  Dispersion  by  a  Prism. — When  a  beam  of  sunlight  is 
admitted  into  a  dark  chamber  through  a  small  circular  hole  A 
(Fig.  204)  in  the  window  shutter,  a  round  spot  of  white  light 
will  be  formed  on  a  vertical  wall  or  screen  opposite  the  win- 
dow, which  will  be,  indeed,  an  image  of  the  sun  of  the  same 
kind  as  would  be  produced  by  a  pinhole  camera  (§  3) ;  its 


Fig.  204. — Prism  dispersion:  Newton's  experiment. 


angular  diameter,  therefore,  being  equal  to  that  of  the  sun, 
namely,  about  half  a  degree.  In  the  track  of  such  a  beam 
Newton  inserted  a  prism  with  its  refracting  edge  horizontal 
and  at  right  angles  to  the  direction  of  the  incident  light; 
whereupon  the  white  spot  on  the  screen  vanished  and  in  its 
stead  at  a  certain  vertical  distance  above  or  below  the  place 
that  was  first  illuminated  there  was  displayed  an  elongated 

465 


466  Mirrors,  Prisms  and  Lenses  [§  160 

vertical  band  or  spectrum,  exhibiting  the  colors  of  the  rain- 
bow in  an  endless  variety  of  tints  shading  into  each  other  by 
imperceptible  gradations.  This  spectrum  was  rounded  at 
the  ends  and  its  vertical  dimension,  depending  on  how  the 
prism  was  tilted,  was  about  4  or  5  times  as  great  as  its  hori- 
zontal dimension,  the  latter  being  equal  to  the  diameter  of 
the  spot  of  white  light  that  was  formed  on  the  screen  before 
the  interposition  of  the  prism.  For  convenience  of  descrip- 
tion, Newton  distinguished  seven  principal  or  " primary" 
colors  arranged  in  the  following  order  from  one  end  of  the 
spectrum  to  the  other,  namely,  red,  orange,  yellow,  green, 
blue,  indigo,*  and  violet;  of  which  the  violet  portion  of  the 
spectrum  is  the  longest  and  the  orange  the  shortest.  The 
red  end  of  the  spectrum  was  the  part  of  the  image  on  the 
screen  that  was  least  displaced  by  the  interposition  of  the 
prism. 

This  phenomenon  was  explained  by  Newton  on  the  as- 
sumption that  ordinary  sunlight  is  composite  and  consists 
in  reality  of  an  innumerable  variety  of  colors  all  blended 
together;  and  that  the  index  of  refraction  (n)  of  the  prism, 
instead  of  having  a  definite  value,  has  in  fact  a  different 
value  for  light  of  each  color,  being  greatest  for  violet  and 
least  for  red  light  and  varying  between  these  limits  for  light 
of  other  colors. 

The  resolution  of  white  light  into  its  constituent  colors 
by  refraction  is  called  dispersion.  If  a  puff  of  tobacco-smoke 
is  blown  across  the  beam  of  light  where  it  issues  from  the 
prism,  only  the  outer  parts  of  the  beam  will  show  any  very 
pronounced  color,  because  the  central  parts  at  this  place  will 

*  There  has  been  much  discussion  as  to  what  Newton  understood 
by  the  color  which  he  named  " indigo"  and  which  lies  somewhere  be- 
tween the  blue  and  the  violet.  Indigo,  as  we  understand  it,  is  more 
nearly  an  inky  blue  rather  than  a  violet  blue,  more  like  green  than  like 
violet;  and  hence  it  has  been  suggested  that  Newton's  color  vision 
may  have  been  slightly  abnormal.  In  this  connection  see  article  en- 
titled "Newton  and  the  Colours  of  the  Spectrum"  by  Dr.  R.  A.  Hous- 
toun,  Science  Progress,  Oct.  1917. 


160] 


Monochromatic  Light 


467 


not  have  been  sufficiently  dispersed  to  exhibit  their  individ- 
ual effects.  At  some  little  distance  away  from  the  prism  the 
entire  section  of  the  beam  will  be  brilliantly  colored. 

Having  pierced  a  small  hole  through  the  screen  at  that 
part  of  it  where  the  spectrum  was  formed  (Fig.  205) ,  Newton 
was  able  by  rotating  the  prism  around  an  axis  parallel  to 


Fig,  205. — Newton's  experiment  with  two  prisms;  showing  that  light  of 
a  definite  color  traverses  the  second  prism  without  further  dispersion. 


its  edge  to  transmit  rays  of  each  color  in  succession  through 
the  opening  to  a  second  prism  placed  with  its  edge  parallel 
to  that  of  the  first  prism;  and,  agreeably  to  his  expectations, 
he  found  that  while  these  rays  were  again  deviated  in  tra- 
versing the  second  prism,  there  was  no  further  dispersion  of 
the  light.  This  experiment  demonstrated  that  the  single 
colors  of  the  spectrum  were  irreducible  or  elementary  and 
not  a  mixture  of  still  simpler  colors,  and  that  the  light  which 
had  been  separated  in  this  fashion  from  the  beam  of  sun- 
light was  monochromatic  light. 

If  all  the  various  components  of  the  incident  light  which 
has  been  resolved  by  the  prism  are  re-united  again,  the  effect 
will  be  the  same  as  that  of  the  light  before  its  dispersion. 


468 


Mirrors,  Prisms  and  Lenses 


160 


The  simplest  way  to  achieve  this  result  is  to  cause  the  rays 
to  traverse  a  second  prism  precisely  equal  to  the  first,  but 
inverted  so  that  the  dihedral  angle  between  the  planes  of 
the  adjacent  faces  of  the  two  prisms  is  equal  to  180°,  the 
edges  of  the  prisms  being  parallel.  Indeed,  if  the  two  prisms 
were  placed  in  contact  in  this  way,  they  would  form  a  slab 
of  the  same  material  throughout  with  a  pair  of  plane  parallel 
faces,  for  which  the  resultant  dispersion  is  zero;  because  the 
colored  rays  would  all  emerge  in  a  direction  parallel  to  that 
of  the  incident  ray  which  was  the  common  path  of  all  these 


Fig.  206. — Light  is  not  dispersed  in  traversing  a  plate  with 
plane  parallel  faces  surrounded  by  same  medium  on  both 
sides. 


rays  before  they  were  separated  by  refraction  at  the  first 
face  of  the  plate  (Fig.  206). 

Another  and  essentially  different  way  of  re-uniting  the 
colored  rays  is  to  converge  them  to  a  single  point  by  means 
of  a  so-called  achromatic  lens,  as  represented  diagrammati- 
cally  in  the  accompanying  drawing  (Fig.  207);  so  that  the 
effect  at  the  focus  C  where  the  colored  rays  meet  is  the  same 
as  that  of  light  from  the  source.  Beyond  C  the  rays  sepa- 
rate again,  so  that  if  they  are  received  on  a  screen  the  same 
succession  of  colors  will  be  exhibited  as  before,  only  in  the 
reverse  order.  If  some  of  the  rays  are  intercepted  before 
arriving  at  C,  the  color  at  C  will  be  the  resultant  effect  of 
the  residual  rays.  The  point  B  where  the  rays  are  separated 
on  entering  the  prism  and  the  point  C  where  they  are  re- 
united by  the  lens  are  a  pair  of  conjugate  points  with  re- 
spect to  the  prism-lens  system. 


§160]  Spectrum  469 

The  solar  spectrum  which  Newton  obtained  in  his  cele- 
brated prism-experiments,  described  in  1672,  had  one  serious 
defect,  due  to  the  fact  that  the  colors  in  it  were  not  in  reality 
pure  but  consisted  of  a  blending  of  two  or  more  simple  colors. 
When  the  light  passes  through  a  round  hole  before  falling 
on  the  prism,  the  spectrum  on  the  screen  will  be  composed 
of  a  series  of  colored  disks,  each  one  overlapping  the  one  next 
to  it.  The  colors,  therefore,  are  partly  superposed  on  each 
other,  and  the  eye  is  so  constituted  with  respect  to  color 
vision  that  it  cannot  distinguish  the  separate  effects  and 


Fig.  207. — Achromatic  lens  used  to  re-unite  the  colored  light  after  it  has 
been  dispersed  by  prism. 

analyze  them  but  obtains  only  a  general  resultant  impression 
of  the  whole. 

Wollaston's  experiments  in  1802  differed  essentially 
from  Newton's  only  in  the  form  and  dimensions  of  the  beam 
of  sunlight  that  was  dispersed  by  the  prism,  but  this  simple 
modification  represented  a  distinct  advance  in  the  mode  of 
investigation  of  the  spectrum.  Wollaston  admitted  the 
sunlight  through  a  narrow  slit  *  whose  length  was  parallel  to 

*  Dr.  Houstoun,  in  the  article  already  referred  to,  calls  attention 
to  the  fact  that  in  some  of  his  prism-experiments  Newton  also  em- 
ployed an  opening  in  the  form  of  a  narrow  slit,  and  was  aware  of  its 
advantages  with  respect  to  the  purity  of  the  spectrum;  for  Newton 
states  that  "instead  of  the  circular  hole,"  "it  is  better  to  substitute  an 
oblong  hole  shaped  like  a  long  Parallelogram  with  its  length  Parallel 
to  the  Prism.    For  if  this  hole  be  an  Inch  or  two  long,  and  but  a  tenth 


470 


Mirrors,  Prisms  and  Lenses 


160 


the  prism-edge;  and  in  order  to  diminish  still  more  the  di- 
vergence of  the  incident  beam,  a  screen  with  a  second  slit 
parallel  to  the  first  was  interposed  in  front  of  the  prism,  as 
represented  in  the  accompanying  diagram  (Fig.  208).  The 
spectrum  formed  in  this  way  is  far  purer  than  that  obtained 
with  a  round  opening  in  the  shutter.    But  a  difficulty  that 


Fig.  208. — Pure  spectrum  obtained  by  causing  sunlight  to  pass  through  two 
narrow  slits  before  traversing  prism. 

inheres  in  both  methods  arises  from  the  fact  that  the  image 
formed  by  a  prism  is  always  virtual,  and  therefore  a  homo- 
centric  bundle  of  monochromatic  divergent  rays  will  nec- 
essarily be  divergent  after  traversing  a  prism,  so  that  if 
they  are  received  on  a  screen  they  will  illuminate  a  certain 
area  on  it  which  is  the  cross-section  of  the  ray-bundle  and 
not  in  any  strict  sense  an  optical  image  of  the  original  source. 

or  twentieth  part  of  an  Inch  broad  or  narrower;  the  Light  of  the  Image, 
or  spectrum,  will  be  as  Simple  as  before  or  simpler,  and  the  Image  will 
become  much  broader,  and  therefore  more  fit  to  have  Experiments 
tried  in  its  Light  than  before."  The  fact  that  Newton  did  not  dis- 
cover the  Fraunhofer  lines  of  the  solar  spectrum  (§  161)  is  probably 
to  be  explained  on  the  supposition  that  his  prisms  were  of  an  inferior 
quality  of  glass  and  that  possibly  also  the  surfaces  were  not  as  highly 
polished  as  they  might  have  been. 


§  160]  Spectrum  471 

Consequently,  if  the  source  sends  out  light  of  different  colors, 
the  effect  on  the  screen  will  correspond  to  the  sections  of  all 
the  bundles  of  colored  rays,  and  since  these  sections  will 
overlap  each  other  to  a  greater  or  less  extent,  the  spectrum 
will  not  be  pure.  The  narrower  the  apertures  of  the  bundles 
of  rays  and  the  farther  the  screen  is  from  the  prism,  the  less 


s 

Fig.   209. — Pure  spectrum  obtained  by  slit,  prism  and  achromatic  lens. 

will  be  the  overlapping  of  the  adjacent  colors,  and  therefore 
the  purer  the  spectrum;  but  on  the  other  hand,  the  less  also 
will  be  the  illumination. 

A  much  more  satisfactory  method  consists  in  making  these 
divergent  bundles  of  rays  convergent  by  means  of  an  achro- 
matic convex  lens,  as  represented  in  Fig.  209;  whereby  the 
blue  rays  proceeding  apparently  from  a  virtual  focus  at  B 
are  brought  to  a  real  focus  on  the  screen  at  B',  and,  similarly, 
the  red  rays  are  united  at  R\  The  plane  of  the  diagram 
represents  a  principal  section  of  the  prism.  The  light  orig- 
inates in  a  luminous  line  or  narrow  illuminated  slit  at  S  par- 
allel to  the  prism-edge,  and  the  spectrum  R'B'  on  the  screen 
consists  of  a  series  of  colored  images  of  this  slit  and  is  ap- 
proximately pure,  except  in  so  far  as  the  slit  must  necessarily 
have  a  certain  width.  Moreover,  in  the  case  of  a  very  nar- 
row slit,  there  are  certain  so-called  diffraction-effects  (§  7) 
which  are  indeed  of  very  great  importance  in  any  thorough 
scientific  discussion  of  the  condition  of  the  purity  of  the 
spectrum. 


472  Mirrors,  Prisms  and  Lenses  [§  162 

161.  Dark  Lines  of  the  Solar  Spectrum. — Wollaston 
himself  observed  that  the  spectrum  of  sunlight  was  not  ab- 
solutely continuous,  but  that  there  were  certain  narrow  gaps 
or  dark  bands  in  it  parallel  to  the  slit.  Fraunhofer  (1787- 
1826),  with  his  rare  acumen  and  experimental  skill,  was  able 
to  obtain  spectra  of  far  higher  purity  than  any  of  his  prede- 
cessors, and  he  discovered,  independently,  that  the  solar  spec- 
trum was  crossed  by  a  very  great  number  of  dark  lines,  the 
so-called  Fraunhofer  lines,  from  which  he  argued  that  sun- 
light was  deficient  in  light  of  certain  colors.  Fraunhofer 
counted  more  than  600  of  these  lines,  but  there  are  now 
known  to  be  several  thousand.  One  great  advantage  of  this 
remarkable  discovery,  which  Fraunhofer  was  quick  to 
realize,  consists  in  the  fact  that  these  lines  are  especially 
suitable  and  convenient  for  enabling  us  to  specify  particular 
regions  or  colors  of  the  spectrum,  because  each  of  them  cor- 
responds to  a  certain  degree  of  refrangibility,  that  is,  to  a 
perfectly  definite  color  of  light.  An  explanation  of  the  origin 
of  the  dark  lines  of  the  solar  spectrum  may  be  found  in 
treatises  on  physics  and  physical  optics. 

The  dark  lines  are  distributed  very  irregularly  over  the  en- 
tire extent  of  the  solar  spectrum.  In  some  cases  they  are 
sharp  and  fine  and  isolated;  some  of  them  are  exceedingly 
close  together  so  as  to  be  hardly  distinguishable  apart;  others 
again  are  quite  broad  and  distinct.  In  order  to  describe 
their  positions  with  respect  to  each  other,  Fraunhofer  se- 
lected eight  prominent  lines  distributed  in  the  different 
regions  of  the  spectrum,  which  he  designated  by  the  capital 
letters  A  (dark  red),  B  (bright  red),  C  (orange),  D  (yellow), 
E  (green),  F  (dark  blue),  G  (indigo),  and  H  (violet).  This 
notation  is  still  in  use,  and  has  since  been  extended  beyond 
the  limits  of  the  visible  spectrum. 

162.  Relation  between  the  Color  of  the  Light  and  the 
Frequency  of  Vibration  of  the  Light- Waves.— According  to 
the  undulatory  theory  of  light,  a  luminous  body  sets  up 
disturbances  or  " vibrations''  in  the  ether  which  are  prop- 


§  162]  Light-Waves  and  Color  473 

agated  in  waves  in  all  directions  with  prodigious  velocities. 
The  velocity  of  light  in  the  free  ether  is  about  300  million 
meters  per  second.  When  a  train  of  light-waves  traverses 
a  rectilinear  row  of  ether-particles  all  lying  in  the  same  me- 
dium, the  distance  between  one  particle  and  the  nearest  one 
to  it  that  is  in  precisely  the  same  phase  of  vibration  is  called 
the  wave-length;  and  the  number  of  waves  which  pass  a 
given  point  in  one  second  or  the  frequency  of  the  undulation 
will  be  equal  to  the  velocity  of  propagation  of  the  wave 
divided  by  the  wave-length.  The  reciprocal  of  the  frequency 
will  be  the  time  taken  by  a  single  wave  in  passing  a  given 
point,  which  is  called  the  period  of  the  vibration.  If  the 
wave-length  is  denoted  by  X,  the  velocity  of  propagation 
by  v,  the  frequency  by  N,  and  the  period  by  T=l/N,  the 
relations  between  these  magnitudes  is  expressed  as  fol- 
lows: 

\  =  v/N  =  v.T. 
When  ether- waves  fall  on  the  retina  of  the  eye,  they  may 
excite  a  sensation  of  light  provided  their  frequencies  are 
neither  too  small  nor  too  great,  the  limits  of  visibility  being 
confined  to  waves  whose  frequencies  lie  between  about  392 
and  757  billions  of  vibrations  per  second.  Just  as  the  pitch 
of  a  musical  note  is  determined  by  its  frequency,  so  also  the 
sensation  which  we  call  color  appears  to  be  more  or  less  in- 
explicably associated  with  the  frequency  of  the  vibrations 
of  the  luminiferous  ether;  so  that  to  each  frequency  between 
the  limits  named  there  corresponds  a  perfectly  definite  kind 
of  light  or  color.  Absolutely  monochromatic  light  due  to 
ether-waves  of  one  single  frequency  of  vibration  is  difficult 
to  obtain.  In  general,  the  light  which  is  emitted  by  a  lumi- 
nous body  is  more  or  less  complex,  and  the  sensation  which 
it  produces  in  the  eye  is  due  to  a  variety  of  impulses.  The 
yellow  light  which  is  characteristic  of  the  flame  of  a  Bunsen 
burner  when  a  trace  of  common  salt  is  burned  in  it  is  a  sen- 
sation excited  by  the  impact  of  two  kinds  of  ether-waves 
corresponding  to  the  double  D-line  of  the  solar  spectrum 


474  Mirrors,  Prisms  and  Lenses  [§  162 

which  have  frequencies  of  about  509  and  511  billions  of  vi- 
brations per  second.  Red  light  corresponds  to  the  lowest 
and  violet  light  to  the  highest  frequency. 

It  is  known  that  the  velocity  of  light  of  a  given  color  de- 
pends on  the  medium  in  which  the  light  is  propagated;  and 
it  has  also  been  established  that  the  velocity  of  light  in  a  given 
medium  depends  on  the  color  of  the  light.  However,  appar- 
ently light  of  all  colors  is  transmitted  with  equal  velocities 
in  vacuo;  and  also  in  air,  on  account  of  its  slight  dispersion, 
there  is  practically  no  difference  in  the  velocity  of  propaga- 
tion of  light  of  different  colors.* 

One  reason  for  inferring  that  the  frequency  of  the  ether- 
vibrations  is  the  physical  explanation  of  the  phenomenon  of 

*  "When  white  light  enters  a  transparent  medium,  the  long  red  waves 
forge  ahead  of  the  green  ones,  which  in  their  turn  get  ahead  of  the  blue. 
If  we  imagine  an  instantaneous  flash  of  white  light  traversing  a  re- 
fracting medium,  we  must  conceive  it  as  drawn  out  into  a  sort  of  linear 
spectrum  in  the  medium,  that  is,  the  red  waves  lead  the  train,  the 
orange,  yellow,  green,  blue,  and  violet  following  in  succession.  The 
length  of  this  train  will  increase  with  the  length  of  the  medium  traversed. 
On  emerging  again  into  the  free  ether  the  train  will  move  on  without 
any  further  alteration  of  its  length. 

"We  can  form  some  idea  of  the  actual  magnitudes  involved  in  the 
following  way.  Suppose  we  have  a  block  of  perfectly  transparent  glass 
(of  ref.  index  1.52)  twelve  miles  in  thickness.  Red  light  will  traverse 
it  in  1/10000  of  a  second,  and  on  emerging  will  be  about  1.8  miles  in 
advance  of  the  blue  light  which  entered  at  the  same  time.  If  white 
light  were  to  traverse  this  mass  of  glass,  the  time  elapsing  between  the 
arrival  of  the  first  red  and  the  first  blue  light  at  the  eye  will  be  less  than 
1/6000  of  a  second.  Michelson's  determination  of  the  velocity  of  light 
in  carbon  bisulphide  showed  that  the  red  rays  gained  on  the  blue  in 
their  transit  through  the  tube  of  liquid.  The  absence  of  any  change  of 
color  in  the  variable  star  Algol  furnished  direct  evidence  that  the  blue 
and  red  rays  traverse  space  with  same  velocity.  In  this  case  the  dis- 
tance is  so  vast,  and  the  time  of  transit  so  long,  that  the  white  light 
coming  from  the  star  during  one  of  its  periodic  increases  in  brilliancy 
would  arrive  at  the  earth  with  its  red  component  so  far  in  advance  of 
the  blue  that  the  fact  could  easily  be  established  by  the  spectro- 
photometer or  even  by  the  eye."— R.  W.  Wood:  Physical  Optics, 
Second  Edition  (New  York,  1911),  page  101. 


§  162]  Wave-Lengths  of  Light  475 

color  is  found  in  the  fact  that  the  color  of  monochromatic 
light  remains  unaltered  when  the  light  passes  from  one  me- 
dium into  another;  and  since  the  vibrations  in  the  second 
medium  are  excited  and  forced  by  those  in  the  first  medium, 
it  is  natural  to  suppose  that  the  vibration-frequency  is  the 
same  in  both  media. 

Accordingly,  it  is  the  ratio 

that  remains  constant  in  the  transmission  of  monochromatic 
light  through  different  media.  And  hence  if  the  velocities  of 
light  in  two  media  are  denoted  by  v,  v' ,  and  if  the  wave-lengths 
in  these  two  media  are  denoted  by  X,  X',  then  v/\  =  v'/\' 
or  X/X'  =  ^/V/  that  is,  the  wave-length  of  light  of  a  given  color 
varies  from  medium  to  medium,  and  is  proportional  to  the  ve- 
locity of  propagation  of  light  of  that  color  in  the  medium  in 
question.  Thus,  the  wave-length  of  yellow  light  is  shorter 
in  glass  than  it  is  in  air,  because  light  travels  more  slowly  in 
glass  than  in  air. 

Generally,  therefore,  when  we  speak  of  the  wave-length  of 
a  given  kind  of  light,  wTe  mean  its  wave-length  measured  in 
vacuo.  The  lengths  of  waves  of  light  are  all  relatively  very 
short,  the  longest,  corresponding  to  the  extreme  red  end  of 
the  spectrum,  being  less  than  one  13-thousandth  of  a  centi- 
meter, and  the  shortest,  belonging  to  the  extreme  violet  end 
of  the  visible  spectrum,  being  less  than  one  25-thousandth 
of  a  centimeter.  These  magnitudes  are  usually  expressed  in 
terms  of  a  special  unit  called  a  " tenth-meter"  which  is  one 
10-billionth  part  of  a  meter  (10~10  meter)  or  in  terms  of  a 
"micromillimeter"  which  is  equal  to  the  millionth  part  of 
a  millimeter  and  for  which  the  symbol  fxfji  is  employed 
(l/x/x  =  lCT6  mm.).  Thus,  the  wave-lengths  of  light  cor- 
responding to  the  red  and  violet  ends  of  the  visible  spectrum 
are  about  767/x/x  and  397/^/x,  respectively.  The  Fraun- 
hofer  line  A  is  a  broad,  indistinct  line  at  the  beginning  of  the 
red  part  of  the  spectrum,  wave-length  759.4  fifi;  the  B-line 


476  Mirrors,  Prisms  and  Lenses  [§  163 

in  the  red  part  corresponds  to  light  of  wave-length  686.7/x/x; 
the  C-line  in  the  orange  corresponds  to  light  of  wave-length 
656.3/x/x;  the  D-line  in  the  yellow  is  a  double  line,  cor- 
responding to  light  of  wave-lengths  589.6/x/a  and  589.0/x/a; 
the  E-line  in  the  green  corresponds  to  light  of  wave- 
length 527. OfJLfx;  the  F-line  in  the  blue  corresponds  to  light 
of  wave-length  486.1^/a;  the  G-line  in  the  indigo  corre- 
sponds to  light  of  wave-length  430.8ju/x;  and  the  H-line, 
consisting  of  two  broad  lines  in  the  violet,  corresponds  to 
light  of  wave-lengths  396.8/x/x  and  393.3/a/x. 

163.  Index  of  Refraction  as  a  Function  of  the  Wave- 
Length. — Now  according  to  the  wave-theory  of  light,  the 
absolute  index  of  refraction  (n)  of  a  medium  for  light  of  a 
definite  color  is  equal  to  the  ratio  of  the  velocity  of  light 
in  vacuo  (V)  to  its  velocity  (y)  in  the  medium  in  question 
(§33);  that  is, 

V 

n  =  —  . 

v 

Strictly  speaking,  therefore,  the  index  of  refraction  of  a  me- 
dium, without  further  qualification,  is  a  perfectly  vague  ex- 
pression, because  each  medium  has  as  many  indices  of  re- 
fraction as  there  are  different  kinds  of  monochromatic  light. 
When  the  term  is  used  by  itself,  it  is  generally  understood 
to  mean  the  index  of  refraction  corresponding  to  the  D-line 
in  the  bright  yellow  part  of  the  solar  spectrum,  which  is 
characteristic  of  the  light  of  incandescent  sodium  vapor. 
Hence, 

velocity  of  yellow  light  in  vacuo 


nD  = 


velocity  of  yellow  light  in  the  medium  in  question 
wave-length  of  j^ellow  light  in  vacuo 


wave-length  of  yellow  light  in  the  given  medium 

In  the  following  table  the  values  of  the  indices  of  refraction 
of  several  transparent  liquids  are  given  for  light  correspond- 
ing to  the  Fraunhofer  lines  A,  B,  C,  D,  E,  F,  G,  and  H. 


164] 


Irrationality  of  Dispersion 


477 


A 

B 

C 

D 

E 

F 

G 

H 

Wave-length 
in  fifi 

759.4 

686.7 

1.360 
1.495 
1.616 
1.331 

656.3 

589.0 

527.0 

4£6.1 

430.8 

396.8 

Alcohol 
Benzene 
Sulphuric  Acid 
Water 

1.359 
1.493 
1.610 
1.329 

1.361 
1.497 
1.620 
1.332 

1.363 
1.503 
1.629 
1.334 

1.365 
1.507 
1.642 
1.336 

1.367 
1.514 
1.654 
1.338 

1.371 
1.524 
1.670 
1.341 

1.374 
1.536 
1.702 
1.344 

It  may  be  remarked  that,  in  general,  the  shorter  the  wave- 
length, the  greater  will  be  the  index  of  refraction  of  a  sub- 
stance. But  the  exact  relation  between  the  index  of  refrac- 
tion and  the  wave-length  of  the  light  has  to  be  determined 
empirically  for  each  substance.  There  is,  indeed,  a  certain 
group  of  substances  which  form  an  exception  to  the  general 
statement  made  above,  and  which  yield  refraction-spectra 
with  the  order  of  the  colors  partially  or  entirely  reversed. 
This  phenomenon  is  called  anomalous  dispersion. 

164.  Irrationality  of  Dispersion. — Other  things  being 
equal,  the  length  of  the  spectrum  or  the  interval  between 


laau  tola  h  m 


Fig.  210. — Irrationality  of  dispersion. 

a  given  pair  of  Fratjnhofer  lines  depends  essentially  on  the 
nature  of  the  refracting  medium,  so  that,  in  general,  as  shown 
by  the  table  in  the  preceding  section,  the  dispersion  of  two 
colors  will  be  found  to  be  different  for  different  substances. 


478 


Mirrors,  Prisms  and  Lenses 


[§164 


For  example,  the  dispersion  of  glass  is  greater  than  that  of 
water,  and  the  dispersion  of  so-called  flint  glass  is  higher 
that  of  so-called  crown  glass.  In  Fig.  210  are  exhibited  the 
relative  lengths  of  the  different  regions  of  the  solar  spectra 
cast  on  the  same  screen  under  precisely  the  same  cir- 
cumstances by  prisms  of  equal  refracting  angles  made  of 
water,  crown  glass  and  flint  glass.  The  length  of  the  spec- 
trum may  be  increased  by  shifting  the  screen  farther  from 


■asassa 


Fig.  211. — Irrationality  of  dispersion. 

the  prism,  and  Fig.  211  shows  the  relative  positions  of  the 
Fraunhofer  lines  B,  C,  D,  E,  F,  G  and  H,  when  the  lengths 
of  the' spectra  of  the  crown  glass  prism  and  the  water  prism 
have  been  elongated  in  this  manner  until  their  lengths  are 
both  equal  to  the  length  of  the  spectrum  of  the  flint  glass 
prism  for  the  interval  between  the  Fraunhofer  lines  B  and 
H.  The  other  lines  in  the  three  spectra  do  not  coincide  at  all. 
Moreover,  it  appears  that  the  dispersion  of  water  for  the 
colors  towards  the  red  end  of  the  spectrum  is  relatively  high, 
whereas  the  dispersion  of  the  flint  glass  is  relatively  high 
towards  the  blue  end.  In  the  spectrum  of  flint  glass  the  in- 
terval between  G  and  H,  and  in  the  spectrum  of  water  the  in- 
terval between  B  and  F,  is  greater  than  it  is  in  either  of  the 
other  spectra.    If  the  law  of  the  variation  of  the  index  of  re- 


§  165]  Dispersive  Power  479 

fraction  with  the  color  of  the  light  has  been  found  empirically 
for  one  substance,  this  will  not  afford  any  clue  to  the  corre- 
sponding law  in  the  case  of  another  substance.  Diamond,  for 
example,  is  very  highly  refracting  but  shows  comparatively 
little  dispersion,  whereas  flint  glass  which  has  a  much  lower 
index  of  refraction  gives  a  much  higher  dispersion;  on  the 
other  hand,  fluorite  has  a  low  index  of  refraction  and  at  the 
same  time  a  low  dispersion.  This  phenomenon  which  is 
characteristic  of  refraction-spectra  is  known  as  the  irration- 
ality of  dispersion. 

165.  Dispersive  Power  of  a  Medium. — In  the  case  of  a 
prism  of  small  refracting  angle  fi  the  deviation  is  given  by 
the  formula  e  =  (n— 1)/3,  as  was  explained  in  §60.  Let 
the  letters  P  and  Q  be  used  to  designate  two  colors,  and  let 
nF  and  nQ  denote  the  indices  of  refraction  of  the  prism- 
substance  for  these  colors.  If  the  angles  of  deviation  are 
denoted  by  eP  and  €q  then  eQ—  eF  =  (nQ— nF)/3,  and,  con- 
sequently, for  a  thin  prism  the  angular  magnitude  of  the 
interval  in  the  spectrurn  between  the  colors  P  and  Q  is  pro- 
portional to  the  difference  of  the  values  of  the  indices  of  re- 
fraction. This  difference  (wq  — nF)  is  called  the  partial  dis- 
persion of  the  substance  for  the  spectrum-interval  P,  Q. 
Thus,  in  the  brightest  part  of  the  spectrum  comprised  be- 
tween the  Fratjnhofer  lines  C  and  F,  the  partial  dispersion 
is  (nF—nc).  The  deviation  of  a  prism  of  small  refracting 
angle  /3  for  light  corresponding  to  the  D-line  which  lies 
between  C  and  F  is  €D  =  (nD  — 1)/3,  and  since  eF—  €c  = 
(nF  —  nc)  fi,  we  obtain : 

€f~  ec=nF—nc 
€d  nD-l 
This  ratio  of  the  angular  dispersion  of  two  colors  to  their 
mean  dispersion  is  called  the  dispersive  power  or  the  relative 
dispersion  of  the  substance  for  the  two  colors,  which  are  usu- 
ally red  (C)  and  blue  (F);  so  that  the  dispersive  power  of 
an  optical  medium  with  respect  to  the  visible  spectrum  may 
be*  defined  to  be  the  quotient  of  the  difference   (nF— nc) 


480  Mirrors,  Prisms  and  Lenses  [§  165 

between  the  indices  of  refraction  for  red  and  blue  light  by 
(nD  — 1),  where  nD  denotes  the  index  of  refraction  for  yellow 
light.  The  values  of  the  dispersive  powers  of  the  various 
kinds  of  optical  glass  that  are  of  chief  practical  importance 
in  the  construction  of  optical  instruments  vary  from  about 
~  to  about  ~;  although  there  are  compositions  of  glass 
with  values  of  the  dispersive  power  not  comprised  within 
these  limits.  Instead  of  assigning  the  value  of  the  dispersive 
power  of  a  substance,  it  is  more  convenient  to  adopt  Abbe's 
method  and  employ  the  reciprocal  of  this  function,  which  is 
denoted  by  the  Greek  letter  v,  and  which  is  known,  there- 
fore, as  the  i>-value  of  the  substance;  thus, 

n?-nc 
If  the  rvalue  of  one  substance  is  less  than  that  of  another, 
the  dispersive  power  of  the  former  will  be  correspondingly 
greater  than  that  of  the  latter. 

It  is  this  constant  v  that  is  the  essential  factor  to  be  con- 
sidered in  the  selection  of  different  kinds  of  glass  suitable  to 
be  used  in  making  a  so-called  achromatic  combination  of 
lenses  or  prisms.  Curiously  enough,  Newton  persisted  in 
maintaining  that  the  dispersion  of  a  substance  was  propor- 
tional to  the  refraction,  which  is  equivalent  to  saying  that 
the  dispersive  powers  of  all  optical  media  are  equal;  and, 
consequently,  he  despaired  of  constructing  an  achromatic 
combination  of  lenses  which  would  refract  the  rays  without 
at  the  same  time  dispersing  the  constituent  colors.  This 
condition,  however,  is  an  essential  requirement  in  the  object- 
glass  of  a  telescope,  and  it  was  just  because  Newton  and  his 
followers  believed  that  a  lens  of  this  kind  was  in  the  nature 
of  things  unattainable  that  they  expended  their  efforts  in 
the  direction  of  perfecting  the  reflecting  telescope  in  which 
the  convex  lens  was  replaced  by  a  concave  mirror.  On  the 
other  hand,  from  the  assumption  that  the  optical  system  of 
the  human  eye  is  free  from  color-faults  (which  is  by  no  means 
true),  it  was  argued,  notably  by  James  Gregory  in  England 


§  166]  Optical  Glass  481 

(about  1670)  and  long  afterwards  by  Euler  in  Germany 
(1747),  that  Newton's  conclusions  as  to  the  impossibility 
of  an  achromatic  combination  of  refracting  media  were  er- 
roneous. In  fact,  an  English  gentleman  named  Hall  suc- 
ceeded in  1733  in  constructing  telescopes  which  yielded 
images  free  from  serious  color  faults.  Klingenstierna  in 
Sweden  in  1754  demonstrated  the  feasibility  of  combining 
a  pair  of  prisms  of  different  kinds  of  glass  and  of  different  re- 
fracting angles  so  as  to  obtain,  in  one  case,  deviation  without 
dispersion  and,  in  another  case,  dispersion  without  deviation. 

But  in  its  practical  results  the  most  important  advance 
along  this  line  was  achieved  by  the  painstaking  and  original 
work  of  the  English  optician  John  Dollond.  Impressed 
by  the  force  of  Klingenstierna's  demonstration,  he  care- 
fully repeated  Newton's  crucial  experiment  in  which  a  glass 
prism  was  inclosed  in  a  water  prism  of  variable  refracting 
angle;  and  having  found  that  the  results  of  this  experiment 
were  exactly  contrary  to  those  stated  by  Newton,  he  was 
led  also  to  the  opposite  conclusion.  After  much  persever- 
ance Dollond  had  succeeded  by  1757  in  making  achromatic 
combinations  of  several  different  types,  which  produced  a 
more  or  less  colorless  image  of  a  point-source  on  the  axis  of 
the  system.  In  its  original  form  the  combination  consisted 
of  a  double  convex  " crown  glass"  lens  cemented  to  a  double 
concave  " flint  glass"  lens.  As  a  rule,  the  focus  of  the  blue 
rays  will  be  nearer  a  convex  lens  and  farther  from  a  concave 
lens  than  the  focus  of  the  red  rays;  and  hence  by  combining 
a  convex  crown  glass  lens  of  relatively  lower  refractive  index 
(shorter  focus)  and  less  dispersive  power  with  a  concave  flint 
glass  lens  of  higher  refractive  index  and  higher  dispersive 
power,  a  resultant  system  may  be  obtained  which  still  has 
a  certain  finite  focal  length  and  in  which  at  the  same  time 
the  opposed  color-dispersions  for  two  colors,  say,  red  and 
blue,  are  compensated. 

166.  Optical  Glass.— Newton's  error  in  supposing  that  for 
all  substances  the  dispersion  was  proportional  to  the  index 


482  Mirrors,  Prisms  and  Lenses  [§  166 

of  refraction  retarded  the  development  of  technical  optics 
for  a  long  time  to  come.  Although  Dollond's  achievement, 
mentioned  above,  was  one  of  far-reaching  importance  for  the 
practical  construction  of  optical  instruments,  the  great  diffi- 
culty in  the  way  of  utilizing  and  applying  the  principle  was 
to  be  found  in  the  fact  that  the  actual  varieties  of  optical 
glass  at  the  disposal  of  the  optician  were  exceedingly  limited 
in  number;  although  from  time  to  time  systematic  efforts 
were  made,  notably  by  Fraunhofer  (about  1812)  in  Ger- 
many and  by  Faraday  (1824),  Harcourt  (1834)  and  Stokes 
(about  1870)  in  England,  to  remedy  this  deficiency,  by  dis- 
covering and  manufacturing  new  compositions  of  glass  suit- 
able for  optical  purposes.  For  a  long  time  after  Fraun- 
hofer's  epoch  the  art  of  making  optical  glass  was  confined 
almost  exclusively  to  France  and  England.  It  was  a  for- 
tunate coincidence  that  just  about  the  time  when  E.  Abbe 
had  reached  the  conclusion  that  no  further  progress  in  op- 
tical construction  could  be  expected  unless  totally  new  va- 
rieties of  optical  glass  were  forthcoming,  0.  Schott  was 
already  beginning  to  experiment  with  new  chemical  combina- 
tions and  processes  of  manufacture  in  his  glass  works  at  Jena. 
Thanks  to  the  systematic  and  indefatigable  efforts  of  these 
two  collaborators,  who  were  also  encouraged  by  the  Prus- 
sian government,  the  obstacle  which  had  stood  so  long  in  the 
way  of  the  improvement  and  development  of  optical  instru- 
ments was  at  length  triumphantly  overcome  by  the  successful 
production  of  an  entire  new  series  of  varieties  of  optical  glass 
with  properties  in  some  instances  almost  beyond  the  highest 
expectations.  The  first  catalogue  of  the  Glastechnisches 
Laboratorium  at  Jena  was  issued  in  1885;  which  marked 
the  beginning  of  the  manufacture  of  the  renowned  Jena  glass, 
to  which  more  than  to  any  other  single  factor  the  remarkable 
development  of  modern  optical  instruments  is  due.  From 
that  time  to  the  present  the  great  province  of  applied  optics 
may  almost  be  said  to  have  become  a  German  territory. 
The    earlier    so-called    "  ordinary"    varieties   of   optical 


§  166]  Jena  Glass  483 

glass  were  silicates  in  which  the  basic  constituents  were 
lime  (crown  glass)  or  lead  (flint  glass)  combined  with  soda 
(Na2C03)  or  potash  (K2C03)  or  both.  The  newer  kinds  of 
optical  glass  have  been  produced  by  employing  a  much 
greater  variety  of  chemical  substances,  including,  in  addi- 
tion to  those  named  above,  hydrated  oxide  of  aluminum 
(A1203,H20),  barium  nitrate  (BaN206),  zinc  oxide  (ZnO), 
etc.,  and  boric  acid  (H3B03)  or  phosphoric  acid  which  to  a 
greater  or  less  extent  replace  the  silica  (Si02)  in  the  older 
types.  Some  of  the  new  compounds  have  been  found  to 
have  slight  durability,  and  for  this  and  other  reasons  cer- 
tain products  formerly  listed  in  the  Jena  glass  catalogue 
have  been  discontinued.  At  present,  besides  the  old  "  or- 
dinary" silicate  crown  and  flint,  the  chief  varieties  are  ba- 
rium and  zinc  silicate  crown,  boro-silicate  crown,  dense 
baryta  crown,  baryta  flint,  antimony  flint,  borate  glass  and 
phosphate  glass.  The  table  on  the  following  page  contains 
a  list  of  certain  varieties  of  Jena  glass  arranged  in  the  order 
of  their  ^-values.  In  the  Jena  glass  catalogue  the  values  of 
the  dispersion  are  given  also  for  the  spectrum-intervals 
^d— ^a'>  nF~  %>>  nG'—nF  (where  A'  and  G'  are  the  lines 
corresponding  to  the  wave-lengths  768  and  434yuju,  re- 
spectively) ,  together  with  the  values  of  the  so-called  relative 
partial  dispersions  obtained  by  dividing  each  of  these  num- 
bers by  the  value  of  (nF— nc). 

It  has  recently  been  proposed  to  describe  an  optical 
glass  by  means  of  two  numbers  of  3  digits  each,  separated 
by  an  oblique  line.  The  first  number  gives  the  first  three 
figures  after  the  decimal  point  in  the  value  of  nD,  .while  the 
second  number  is  equal  to  10  times  the  value  of  v. 
Thus,  for  example,  the  second  glass  in  the  table  would  be 
described  as  crown  glass  No.  559/669. 


484  Mirrors,  Prisms  and  Lenses  (§  166 

SELECTED  VARIETIES  OF  JENA  GLASS 


Index  of 

Mean 

n    —  1 

Description 

Refraction 

Dispersion 

"d  .   x 

nD 

nF-?iG 

nF-nc 

Light  phosphate  crown 

1.5159 

0.007  37 

70.0 

Medium    phosphate    crown 

1 . 5590 

0.008  35 

66.9 

Boro-silicate  crown 

1.5141 

0.008  02 

64.1 

Boro-silicate  crown 

1.5103 

0.008  05 

63.4 

Silicate  crown 

1.5191 

0.008  60 

60.4 

Silicate  crown 

1.5215 

0.008  75 

59.6 

Silicate  crown 

1.5127 

0.008  97 

57.2 

Densest  baryta  crown 

1.6112 

0.010  68 

57.2 

Barium  crown 

1 . 5726 

0.009  95 

57.5 

Dense'  baryta  crown 

1.6130 

0.010  87 

56.4 

Dense  baryta  crown 

1.6120 

0.010  98 

55.7 

Baryta  flint 

1.5664 

0.010  21 

55.5 

Borate  flint 

1.5503 

0.009  96 

55.2 

Baryta  flint 

1.5489 

0.010  25 

53.6 

Baryta  flint 

1.5848 

0.011  04 

53.0 

Antimony  flint 

1.5286 

0.010  25 

51.6 

Boro-silicate  flint 

1.5503 

0.011  14 

49.4 

Extra  light  flint 

1.5398 

0.011  42 

47.3 

Baryta  flint 

1 . 5825 

0.012  55 

46.4 

Ordinary  light  flint 

1 . 5660 

0.013  19 

42.9 

Silicate  flint 

1.5794 

0.014  09 

41.1 

Baryta  flint 

1.6235 

0.015  99 

39.1 

Heavy  borate  flint 

1.6797 

0.017  87 

38.0 

Silicate  flint 

1.6138 

0.016  64 

36.9 

Silicate  flint 

1.6489 

0.019  19 

33.8 

Dense  silicate  flint 

1.7174 

0.024  34 

29.5 

Densest  silicate  flint 

1.9626 

0.048  82 

19.7 

In  recent  years  in  France,  England  and  the  United  States 
much  attention  has  been  bestowed  on  the  study  of  the  com- 
position and  manufacture  of  optical  glass,  and  according  to 
the  1916-17  report  of  the  British  Committee  of  the  Privy 
Council  for  Scientific  and  Industrial  Research  (summarized 
in  Nature,  Vol.  100,  pp.  17-20),  Professor  Jackson  in  England 
"has  succeeded  in  defining  the  composition  of  the  bath 
mixtures  necessary  for  the  production  of  several  glasses 
hitherto  manufactured  exclusively  in  Jena,  including  the 
famous  fluor-crown  glass,"  and,  moreover,  "he  has  also 
discovered  three  completely  new  glasses  with  properties 
hitherto    unobtainable,"     However,   it   seems    improbable 


§  166]  -Manufacture  of  Optical  Glass  485 

that  any  essential  changes  in  the  optical  properties  of  glass 
are  to  be  obtained  by  the  use  of  materials  that  have  not  al- 
ready been  tried.  The  index  of  refraction  of  all  glasses  at 
present  available  are  comprised  between  1.45  and  1.96.  The 
mineral  fluorite  (calcium  fluoride),  which  is  used  in  the  best 
modern  microscope  objectives,  has  an  index  of  refraction  of 
1.4338  and  a  rvalue  of  95.4,  so  that  in  both  respects  it 
lies  beyond  the  limits  attainable  with  glass.  Other  crystal- 
line transparent  minerals,  notably  rock  crystal  or  quartz, 
have  already  been  employed  in  lens-systems,  and  any  es- 
sential improvement  in  the  range  of  optical  instruments 
in  the  future  is  more  likely  to  come  from  an  adaptation  of 
these  mineral  substances  than  from  the  production  of  new 
kinds  of  glass. 

The  difficulties  involved  in  the  manufacture  of  high-grade 
optical  glass  are  very  great,  and  the  utmost  care  has  to  be 
exercised  throughout  every  stage  of  the  process.  Not  only 
must  the  raw  materials  themselves  be  free  from  impurities  as 
far  as  possible,  but  the  physical  and  chemical  nature  of  the 
fireclays  used  in  the  pots  or  crucibles  also  requires  the  most 
painstaking  care  and  preparation.  The  empty  crucible  is 
dried  slowly  and  then  heated  gradually  for  several  days  until 
it  comes  to  a  bright  red  glow.  Fragments  of  glass  left  over 
from  a  previous  melting  and  of  the  same  chemical  composi- 
tion as  the  glass  which  is  in  process  of  making  are  introduced 
into  the  pot  and  melted.  The  raw  materials,  pulverized  and 
mixed  in  definite  proportions,  are  placed  in  the  pot  in  layers 
little  by  little  at  a  time,  and  the  pot,  which  is  covered  to 
protect  the  contents  from  the  furnace  gases  is  maintained 
at  a  sufficiently  high  temperature  (between  about  800  and 
1000°  C.)  until  the  contents  are  all  melted  together.  The 
molten  mass  is  usually  full  of  bubbles  of  all  sizes,  and  the 
temperature  must  be  raised  until  these  are  all  gotten  rid 
of  as  far  as  possible.  This  entire  process  takes  a  longer  or 
shorter  time  depending  on  circumstances,  say,  from  24  to 
36  hours  or  more.    After  skimming  off  the  impurities  on  the 


486  Mirrors,  Prisms  and  Lenses  [§  166 

surface,  the  mixture  is  allowed  to  cool  gradually,  and  at  the 
same  time  it  is  kept  constantly  stirred  in  order  to  make  the 
glass  as  homogeneous  as  possible.  This  part  of  the  process 
requires  constant  care.  When  the  glass  in  cooling  has  be- 
come quite  viscous,  so  that  it  is  no  longer  possible  to  con- 
tinue the  stirring,  it  is  allowed  to  cool  very  slowly  over  a 
period  of  days  or  even  weeks.  Usually  at  the  end  of  the 
cooling  process  the  solid  contents  of  the  pot  will  be  found 
to  be  broken  into  irregular  fragments  of  optical  glass  in  the 
first  stage  of  its  manufacture.  These  fragments  are  care- 
fully examined  to  see  whether  they  are  homogeneous  and 
above  all  free  from  striae;  but  the  broken  surfaces  are  so 
irregular  that  this  preliminary  examination  is  necessarily 
very  imperfect.  The  pieces  which  pass  muster  in  this  way 
are  selected  for  molding  and  annealing.  The  lumps  of  glass 
are  placed  in  suitable  molds  made  of  iron  or  fireclay  and 
heated  until  the  glass  becomes  soft  like  wax,  so  that  it  takes 
the  form  of  the  mold  usually  with  the  aid  of  external  press- 
ure. The  molded  pieces  are  then  annealed  by  being  cooled 
gradually  for  a  week  or  longer.  They  are  in  the  form  of 
disks  or  rectangular  blocks  of  approximately  the  right  size 
for  being  made  into  lenses  and  prisms.  At  this  stage  the 
glass  has  to  be  subjected  to  the  most  rigid  testing  to  see  if 
it  is  really  suitable  for  optical  purposes.  Two  opposite  faces 
on  the  narrow  sides  are  ground  flat  and  parallel  and  polished 
so  that  the  slab  can  be  inspected  in  the  direction  of  its  greatest 
diameter.  If  any  striae  or  other  imperfections  are  found,  the 
piece  will  have  to  be  rejected  and  melted  over  again.  Even 
in  case  there  are  no  directly  visible  defects,  there  may  be  in- 
ternal strains  which  will  be  revealed  by  examination  with 
polarized  light.  Slight  strains  are  not  always  serious,  but  even 
these  will  impair  the  image  in  a  large  prism  or  lens.  These 
strains  can  be  gotten  rid  of  by  heating  the  glass  to  a  tempera- 
ture between  350  and  480°  C,  depending  on  the  composition, 
and  then  cooling  very  slowly  and  uniformly  over  a  period  of 
about  six  weeks.    It  is  very  difficult  to  obtain  pieces  of  op- 


§  167]  Achromatism  487 

tical  glass  which  do  not  contain  minute  bubbles,  and  indeed 
they  are  often  to  be  found  in  the  best  kinds  of  glass. 

Of  course,  the  process  as  above  described  varies  in  details 
according  to  the  special  nature  of  the  glass,  but  enough  has 
been  said  to  enable  the  reader  to  form  some  idea  of  the  pa- 
tience and  skill  which  are  required  in  the  manufacture  of 
optical  glass.  A  yield  of  20  per  cent,  of  the  total  quantity  of 
glass  melted  is  considered  good.  The  glass  to  be  used  for 
photographic  lenses  has  to  fulfill  the  most  exact  requirements 
and  must  be  of  the  highest  quality. 

167.  Chromatic  Aberration  and  Achromatism. — Since  the 
index  of  refraction  varies  with  the  color  of  the  light,  and  since 
this  function  enters  in  one  form  or  another  in  all  optical  cal- 
culations, it  is  obvious,  for  example,  that  the  positions  of 
the  cardinal  points  of  a  lens-system  will,  in  general,  be  differ- 
ent for  light  of  different  colors;  and  that  there  will  be  a  whole 
series  of  colored  images  of  a  given  object  depending  on  the 
nature  of  the  light  which  it  radiates,  these  images  being  all 
more  or  less  separated  from  each  other  and  of  varying  sizes. 
This  phenomenon  is  called  chromatic  aberration,  and  unless 
it  is  at  least  partially  corrected,  the  definition  of  the  resultant 
image  is  very  seriously  impaired.  In  an  optical  system  which 
was  absolutely  free  from  chromatic  aberration  all  these 
colored  images  would  coalesce  into  a  single  composite  image 
which,  so  far  as  the  quality  of  the  light  was  concerned,  would 
be  a  faithful  reproduction  of  the  object.  But  nothing  at  all 
comparable  to  this  ideal  condition  of  achromatism  can  be 
achieved  in  the  case  of  any  actual  lens-system.  In  fact,  the 
term  achromatism  by  itself  and  without  any  further  explana- 
tion is  entirely  vague,  for  an  optical  system  may  be  achro- 
matic in  one  sense  without  being  at  all  so  in  other  senses.  For 
example,  the  images  corresponding  to  different  colors  may 
all  be  formed  in  the  same  plane  and  yet  be  of  different  sizes, 
or  vice  versa.  Fortunately,  however,  the  fact  that  it  is  im- 
possible to  achieve  at  best  more  than  a  partial  achromatism 
is  not  such  a  serious  matter  after  all.    The  kind  of  achromat- 


488  Mirrors,  Prisms  and  Lenses  [§  167 

ism  which  is  adapted  for  one  type  of  optical  instrument  may 
be  entirely  unsuited  to  another  type.  Thus,  it  is  absolutely 
essential  that  the  colored  images  formed  by  the  object-glass 
of  a  telescope  or  microscope  shall  be  produced  as  nearly  as 
possible  at  one  and  the  same  place  (achromatism  with  re- 
spect to  the  location  of  the  image),  whereas,  since  the  images 
in  this  case  do  not  extend  far  from  the  axis,  the  unequal 
color-magnifications  are  comparatively  unimportant.  On 
the  other  hand,  in  the  case  of  the  ocular  systems  of  the  same 
instruments,  the  main  consideration  will  be  a  partial  achro- 
matism with  respect  to  the  magnification  or  the  apparent 
sizes  of  the  colored  images.  The  object-glass  of  a  telescope 
must  be  achromatic  with  respect  to  the  position  of  its  focal 
point,  and  the  ocular  must  be  achromatic  with  respect  to  its 
focal  length. 

An  optical  system  which  produces  the  same  definite  effect 
for  light  of  two  different  wave-lengths,  no  matter  what  that 
special  effect  may  be,  is  to  that  extent  an  achromatic  system. 
A  combination  which  is  achromatic,  even  in  its  limited  sense, 
for  a  certain  prescribed  distance  of  the  object  will,  in  general, 
not  be  achromatic  when  the  object  is  placed  at  a  different 
distance.  No  lens  composed  of  two  kinds  of  glass  only  can 
be  achromatic  for  light  of  all  different  colors.  It  can  be  con- 
structed, for  example,  so  that  it  will  bring  the  red  and  violet 
rays  accurately  to  the  same  focus  at  a  prescribed  point  on 
the  axis;  but  then  the  yellow,  green  and  blue  rays  will,  in 
general,  all  have  different  foci,  some  of  which  will  be  nearer 
the  lens  than  the  point  of  reunion  of  the  red  and  violet  light 
while  others  will  lie  farther  away.  Accordingly,  when  achro- 
matism has  been  attained  in  the  case  of  two  chosen  colors, 
there  will  usually  remain  an  uncorrected  residual  dispersion 
or  so-called  secondary  spectrum,  which  under  certain  circum- 
stances may  impair  the  definition  of  the  image  to  such  a 
degree  as  to  be  very  injurious  and  annoying.  It  is  neces- 
sary to  abolish  the  secondary  spectrum  in  the  object-glass  of 
a  microscope.     This  may  be  done  by  using  more  than  two 


§  16S]  Optical  and  Actinic  Achromatism  489 

kinds  of  glass.  There  is  also  the  possibility  of  diminishing  the 
secondary  spectrum  try  employing  two  kinds  of  glass  whose 
relative  partial  dispersions  (§  166)  are  very  nearly  the  same 
for  all  the  spectrum-intervals;  and,  in  fact,  one  of  the  prin- 
cipal items  in  the  Abbe-Schott  programme  for  the  manu- 
facture of  optical  glass  was  the  production  of  various  pairs 
of  flint  and  crown  glass  suitable  for  such  combinations,  so 
that  the  dispersions  in  the  different  regions  of  the  spectrum 
should  be,  for  each  pair,  as  nearly  as  possible  proportional. 
This  purpose  was  satisfactorily  accomplished,  and  we  have 
now  achromatic  lenses  of  a  far  more  perfect  kind  than  could 
be  made  out  of  the  older  kinds  of  glass.  This  higher  degree 
of  achromatism  is  called  apochromatism.  An  apochromatic 
photographic  lens  is  absolutely  essential  in  the  three-color 
process  of  photography  in  which  the  three  images  taken 
through  light-filters  on  a  plate  of  medium  or  large  size  must 
be  superposed  as  exactly  as  possible.  In  most  ordinary  op- 
tical systems,  however,  the  secondary  spectrum  is  relatively 
unimportant,  and  achromatism  with  respect  to  two  prin- 
cipal colors  will  usually  be  found  to  be  sufficient. 

168.  "  Optical  Achromatism  "  and  "  Actinic  Achromat- 
ism."— The  character  and  extent  of  the  secondary  spectrum 
(§  167)  of  an  achromatic  combination  of  lenses  will  evidently 
depend  essentially  on  the  choice  of  the  two  principal  colors  for 
which  the  achromatism  is  to  be  achieved.  This  choice  will 
be  determined  by  the  purpose  for  which  the  instrument  is 
intended  and  the  mode  of  using  it.  Thus,  if  the  system  is 
to  be  an  optical  instrument  in  the  strict  literal  sense  of  the 
word,  that  is,  if  it  is  constructed  to  be  used  subjectively  in 
conjunction  with  the  eye,  we  shall  be  concerned  primarily 
with  the  physiological  action  of  the  rays  on  the  retina  of  the 
human  eye;  whereas  in  the  case  of  a  photographic  lens  which 
is  used  to  focus  an  image  on  a  prepared  sensitized  plate,  it 
is  important  to  have  achromatism  with  respect  to  the  so- 
called  actinic  rays  corresponding  to  the  violet  and  ultra- 
violet regions  of  the  spectrum,  because  these  are  the  rays 


490  Mirrors,  Prisms  and  Lenses  [§  168 

which  are  most  active  on  the  ordinary  bromo-silver  gelatine 
plate. 

The  retina  of  the  human  eye  is  most  sensitive  to  the  kind 
of  light  which  is  comprised  within,  the  interval  between  the 
lines  C  and  F,  with  a  distinct  maximum  of  visual  effect  cor- 
responding to  wave-lengths  lying  somewhere  between  the 
lines  D  and  E.  Accordingly,  in  an  optical  instrument  which 
is  to  be  applied  to  the  eye,  it  is  usually  desirable  to  unite  the 
red  and  blue  rays  as  nearly  as  possible  at  the  focus  of  the 
yellow  rays.  If,  for  example,  the  system  is  assumed  to  be 
a  convergent  combination  of  two  thin  lenses  in  contact  (as 
in  the  case  of  the  object-glass  of  a  telescope),  it  will  be  found 
that  the  focal  points  corresponding  to  the  colors  (say,  green 
and  yellow)  between  C  and  F  will  lie  nearer  the  lens  and  the 
focal  points  corresponding  to  the  other  colors  (dark  red, 
dark  blue  and  violet)  will  lie  farther  from  it  than  the  com- 
mon focal  point  of  the  two  principal  colors  C  and  F.  More- 
over, the  residual  color-error  or  secondary  spectrum  in  this 
case  will  be  least  for  some  color  very  nearly  corresponding  to 
the  D-line,  which  is  a  favorable  circumstance,  since,  as  above 
stated,  this  is  the  region  of  the  brightest  part  of  the  visible 
spectrum.    Achromatism  with  respect  to  the  colors  C  and  F 


omatism. 


I  v  =  — )  is  sometimes  called  optical  achr< 

\       riF-nc/ 

On  the  other  hand,  in  the  construction  of  a  photographic 
lens  a  kind  of  compromise  must  be  effected  between  the  con- 
vergence of  the  visual  rays  and  the  so-called  actinic  rays, 
because  the  image  has  to  be  focused  first  on  the  ground  glass 
plate  by  the  eye  and  afterwards  it  has  to  be  received  on  the 
sensitized  plate  or  film  which  is  inserted  for  exposure  in  the 
camera  in  the  place  of  the  translucent  focusing  screen.  Ac- 
cordingly, for  ordinary  photographic  practice  an  exact  co- 
incidence of  the  "optical"  and  " actinic"  images  is  de- 
manded. Here  it  is  found  that  the  best  results  are  obtained 
by  uniting  the  colors  corresponding  to  the  D-line  and  the 
violet  band  in  the  spectrum  of  hydrogen,  which,  since  it  is 


§  169]  Achromatic  Prism  491 

not  far  from  the  G-line,  may  be  designated  by  G'  (434^/0- 
This  is  sometimes  called  actinic  or  photographic  achromat- 
ism for  which  the  function  v  has  a  special  value,  namely : 

_=   flD-1 


riG'-nj) 

If  the  photographic  lens  is  a  combination  of  two  thin 
lenses  in  contact,  which  is  achromatic  for  the  colors  D  and 
G',  the  focus  of  the  rays  corresponding  to  the  blue-green 
region  of  the  spectrum  will  be  nearer  the  lens  than  the  com- 
mon focus  of  the  two  principal  colors  and  the  focus  of  the 
bright  red  rays  will  be  farther  from  the  lens.  In  an  achro- 
mat  of  this  kind  the  residual  dispersion  will  usually  be  quite 
large  for  both  the  " optical"  and  the  "actinic"  image,  but 
for  most  practical  purposes  the  definition  of  the  image  in 
either  case  is  good  enough.  In  astrophotography  the  focus 
of  the  camera  is  determined  once  for  all,  and  a  lens  for  stellar 
photography  is  usually  designed  to  have  an  entirely  actinic 
achromatism,  the  two  principal  colors  in  this  case  correspond- 
ing to  the  F-line  (486/f/x)  and  the  violet  line  in  the  mercury- 
spectrum  (405 fifi).  The  rays  belonging  in  these  two  colors 
are  made  to  unite  as  nearly  as  possible  at  the  fociis  of  the 
rays  corresponding  to  the  G'-line,  which  is  approximately 
the  place  of  maximum  actinic  action.  In  a  photographic 
achromat  of  this  kind  the  foci  of  the  green,  yellow  and  red 
rays  will  lie  beyond  the  actinic  focus  in  the  order  named. 

169.  Achromatic  Combination  of  Two  Thin  Prisms. — 
Two  prisms  of  different  substances  may  be  combined  so  as 
to  obtain  achromatism  in  the  sense  that  rays  of  light  cor- 
responding to  a  definite  pair  of  colors  will  issue  from  the 
system  in  parallel  directions,  as  represented  in  Fig.  212. 
When  an  object  is  viewed  through  the  combination,  the  red 
and  blue  rays,  for  example,  will  be  fused  or  superposed  and 
the  residual  color-effect  will  be  comparatively  slight.  By 
employing  a  greater  number  of  prisms  a  more  perfect  union 
of  colors  could  be  obtained,  but  usually  two  prisms  are  suf- 
ficient. 


492  Mirrors,  Prisms  and  Lenses  [§  169 

The  problem  is  simplified  by  assuming  that  the  refracting 
angles  of  the  prisms,  denoted  here  by  /3  and  7,  are  both 
small;  so  that  the  deviation  produced  by  each  prism  may  be 
considered  as  proportional  to  its  refracting  angle,  accord- 
ing to  the  approximate  formula  deduced  in  §  60.    Usually, 


Fig.  212. — Achromatic  combination  of  two  thin  prisms. 

the  two  prisms  are  cemented  together  with  their  edges  par- 
allel but  oppositely  turned,  as  shown  in  the  diagram,  so  that 
the  thicker  portions  of  one  prism  are  adjacent  to  the  thinner 
portions  of  the  other;  accordingly,  the  total  deviation  (e) 
will  be  equal  to  the  arithmetical  difference  of  the  deviations 
produced  by  each  prism  separately. 

Let  P,  Q  and  R  designate  three  elementary  colors,  the 
color  Q  being  supposed  to  lie  between  P  and  R  in  the  spec- 
trum; and  let  the  indices  of  refraction  for  these  three  colors 
be  denoted  by  nP',  Uq  and  nR'  for  the  first  prism  and  by 
n~p",  nQf  and  nR"  for  the  second  prism.  The  total  devia- 
tions for  the  three  colors  will  be: 
€P = (V  - 1)  j8  -  (V  -1)7,        eQ  =  (V  -1)0-  ( V  - 1)  y, 

€R=(nR'-l)i3-(nR"-l)7. 
Now  if  the  system  is  to  be  achromatic  with  respect  to  the 
colors  P  and  R,  the  condition  is  that  eP=  eR,  which,  there- 
fore, is  equivalent  to  the  following : 

/3_nR"-y . 

7      nR'-nF'    ' 
that  is,  the  refracting  angles  of  the  prisms  must  be  inversely 


€Q  =  (WR/-Wp/) 


P- 


§  170]  Direct  Vision  Prism  493 

proportional  to  the  partial  dispersions  of  the  two  media  for 
the  two  given  colors. 

Moreover,  the  deviation  of  the  rays  of  the  intermediate 
color  Q  will  be : 

'  V-l  _nQ"-l)  * 
ftR'-ttp'     WR//-Wp/' 

Actually  the  colors  P,  Q  and  R  are  usually  chosen  to  cor- 
respond to  the  Fraunhofer  lines  C,  D  and  F,  respectively, 
in  which  case  the  combination  will  be  achromatic  with  re- 
spect to  C  (red)  and  F  (blue) .  Thus,  the  fractions  inside  the 
large  brackets  are  the  ^-values  of  the  two  kinds  of  glass. 
Accordingly,  for  an  achromatic  combination  of  two  thin 
prisms  for  which  the  deviation  €D  has  a  finite  value,  whereas 
the  dispersion  ( ec  —  €p )  is  abolished,  we  have  the  following 
formulae : 

l=n*"7nc"  ,      £D=(nF'-nc')  {v>-v")P. 
7      riF-nc 

Consider,  for  example,  a  combination  of  two  kinds  of  Jena 

glass  as  follows: 

nD  nF— uq        v 

Light  Phosphate  Crown      1.5159      0.007  37     70.0 

Borate  Flint  1.5503      0.009  96     55.2 

Assuming  that  the  angle  of  the  crown  glass  prism  is  (3  =  20°, 

we  find:  7  =  14.8°,    €D  =  2.18°.      Generally  speaking,    those 

pairs  of  glasses  in  which  the  partial  dispersions  are  more 

nearly  equal  will  be  .found  to  be  best  adapted  for  achromatic 

combinations. 

170.  Direct  Vision  Combination  of  Two  Thin  Prisms. — 

In  the  case  of  an  ordinary  prism-spectroscope  the  rays  are 

deflected  in  passing  through  the  system,  so  that  in  order  to 

view  the  spectrum  the  eye  has  to  be  pointed  not  directly 

towards  the  luminous  source,  but  in  some  oblique  direction; 

which  is  sometimes  inconvenient,  especially  in  astrophysi- 

cal  observations.     Accordingly,  various  prism-systems  have 

been  proposed  which  are  designed  so  that  rays  corresponding 

to  some  definite  standard  color  are  finally  bent  back  into 


494  Mirrors,  Prisms  and  Lenses  [§  170 

their  original  direction,  with  the  result  that  there  is  disper- 
sion without  deviation,  which  is  an  effect  precisely  opposite 
to  that  which  is  obtained  with  an  achromatic  prism.  In 
these  so-called  direct  vision  prisms  (prismes  a  vision  directe) 


Fig.  213. — Direct  vision  prism  combination  (dispersion  without  deviation). 

the  spectrum  of  an  illuminated  slit  will  be  seen  in  the  same 
direction  as  the  slit  itself.  The  condition  that  the  light  cor- 
responding, say,  to  the  Fraunhofer  D-line  shall  emerge 
from  the  system  in  the  same  direction  as  it  entered  is  €D  =  0. 
Assuming  that  the  combination  is  composed,  as  before,  of 
two  thin  prisms  juxtaposed  in  the  same  way  (Fig.  213),  and 
employing  the  same  symbols  (§  169),  we  derive  immedi- 
ately the  following  f ormulse : 

jg_nD"-l 

7     nW-l ' 

CO"  €f=Od'-1)    \y--)  ' 

Consider,  for  example,  the  following  combination : 

nD  v 

Light  Phosphate  Crown  1.5159  70.0 

Heavy  Silicate  Flint  1.9626  19.7 

the  difference  of  the  ^-values  here  being  very  great.  If 
we  put  0  =  20°,  we  find:  7  =  10.72°,  €C-  eF  =  22.56'. 

It  will  be  profitable  for  the  student  to  satisfy  himself  by 
several  examples  that  two  kinds  of  glass  which  are  best 
adapted  for  a  direct  vision  prism  combination  are  on  the 


171] 


Direct  Vision  Prism 


495 


contrary  not  very  suitable  for  an  achromatic  prism,  and  vice 
versa;  as  might  naturally  be  expected,  since  the  effects  are 
opposite  in  the  two  cases.  Generally  speaking,  the  two  kinds 
of  glass  used  for  a  direct  vision  prism  should  have  very  dif- 
ferent ^-values,  as  in  the  illustration  given  above. 

In  the  case  of  prisms  of  large  refracting  angles,  the  formula? 
here  and  in  §  169  are  hardly  to  be  considered  as  even  ap- 
proximate. 

171.  Calculation  of  Amici  Prism  with  Finite  Angles. — 
Accurate  formulae  for  the  calculation  of  an  achromatic  or 
direct  vision  prism-system  may  easily  be  derived  when  the 


Fig.  214. — Direct  vision  prism  combination.     Diagram  represents  one-half 
of  so-called  Amici  direct  vision  prism. 


system  consists  of  only  two  prisms.  As  an  illustration  of 
the  method  in  the  case  of  a  direct  vision  prism,  let  us  em- 
ploy here  the  symbols  n\  and  ni  to  denote  the  indices  of  re- 
fraction of  the  crown  glass  prism  and  the  flint  glass  prism, 
respectively,  for  light  of  some  standard  wave-length;  and 
let  j8  and  y  denote  their  refracting  angles.  We  shall  sup- 
pose also  that  the  two  prisms  are  cemented  along  a  common 
face,  as  represented  in  Fig.  214.  A  ray  of  the  given  wave- 
length is  incident  on  the  crown  glass  prism  at  an  angle  d 
and  is  refracted  into  this  medium  at  the  angle  0',  so  that 

tti.sin  0'=sin  6.  (1) 

If  the  angles  of  incidence  and  refraction  at  the  surface  of 


496  Mirrors,  Prisms  and  Lenses  [§171 

separation  of  two  kinds  of  glass  are  denoted  by  ^  and  ^', 
then 

ni.sin^  =  7i2.sin^r,  (2) 

0'  =  $-+;  (3) 

the  angles  here  being  all  reckoned  as  positive.  If,  finally,  it 
is  assumed  that  the  ray  meets  the  second  face  of  the  second 
prism  normally  and  issues  again  into  the  air  in  the  same  di- 
rection as  it  had  originally,  then  also : 

f  =  7,       (4)     and  d=/3-y.  (5) 

The  problem  consists  in  determining  the  angle  of  one  of  the 
prisms  when  the  angle  of  the  other  is  given.  Suppose,  for 
example,  that  an  arbitrary  value  is  assigned  to  the  acute 
angle  7,  and  it  is  required  to  find  the  magnitude  of  the 
angle  fi.  Substituting  in  (1)  the  values  of  6,  d'  as  given 
in  (3)  and  (5),  we  obtain: 

tti.sin(/3-^)=sin(/3- 7), 
whence  we  derive: 

a    fti.sin^  —  sin  7 
tan  p  = -. . 

Eliminating  ^'  from  (2)  and  (4),  we  find: 
ni.sin  ^=712. sin  7, 

and  consequently  also:  

Tii. cos  ^  =  \Zn\— nl.sui'y. 
Accordingly,  the  value  of  /5  in  terms  of  m,  ni  and  7  is  given 
by  the  formula: 

tMnp.         («.-!)  JET . 

-y/nl  —  n^sin2  7  -  cos  7 
If,  on  the  other  hand,  the  value  of  the  angle  /3  has  been 
chosen  arbitrarily,  the  calculation  of  7  will  be  found  to  be 
trigonometrically  a  little  more  difficult.    It  is  left  as  an  ex- 
ercise for  the  student  to  show  that : 

fanT_     ^2-1+ V^(tt2-l)2  +  (n2i-l)  (n22-^)tan2^tsing 
7  (ri2-nDtan2/3+(n2-l)2 

If  it  is  desired  that  the  emergent  ray  shall  not  only  be  par- 
allel to  the  incident  ray  but  that  its  path  shall  be  along  the 


172] 


Direct  Vision  Prism 


497 


same  straight  line,  it  is  necessary  to  add  to  the  above  another 
combination  identical  with  it  and  placed  so  that  the  two 
flint  glass  prisms  constitute  in  reality  one  single  prism  of  re- 
fracting angle  27  inserted  between  two  equal  crown  glass 
prisms  each  of  refracting  angle  j8,  as  shown  in  Fig.  215;  and, 


Fig.  215. — Amici  direct  vision  prism. 

in  fact,  this  is  the  actual  construction  of  the  common  form 
of  the  Amici  prism.  Suppose,  for  example,  that  the  angle 
7=45°  and  that  the  two  kinds  of  glass  are  those  described 
in  the  Jena  catalogue  as  " light  phosphate  crown"  and 
"heavy  silicate  flint"  with  indices  ni  =  1.5159  and  n2  =  1.9626 
corresponding  to  the  D-line;  then  we  find  that  the  angle  /?  = 
98°  7.4'. 

172.  Kessler  Direct  Vision  Quadrilateral  Prism. — One 
of  the  principal  objections  to  a  train  of  prisms  is  the  loss  of 
light  by  reflection  at  the  various  surfaces  and  also  by  ab- 
sorption in  traversing  the  successive  media.  Partly  with  a 
view  to  diminishing  these  losses  and  partly  also  on  account 
of  other  advantages,  many  forms  of  direct  vision  prism  have 
been  proposed  which  are  made  of  one  piece  of  glass  with  four 
or  more  plane  faces;  in  all  of  which,  however,  the  principle 
is  the  same,  namely,  by  means  of  a  series  of  total  internal 
reflections  to  bend  the  rays  corresponding  to  some  standard 
intermediate  color  back  finally  into  their  original  direction. 
The  simplest  of  all  these  devices  is  the  four-faced  prism 
ABCD  (Fig.  216)  proposed  by  Kessler,  a  principal  section 
of  which  has  the  form  of  a  quadrilateral  with  perpendicular 
diagonals.    The  ray  of  standard  wave-length  enters  the  prism 


498  Mirrors,  Prisms  and  Lenses  [§  172 

and  leaves  it  in  a  direction  parallel  to  the  diagonal  BD;  it  is 
totally  reflected  twice,  first  at  the  face  BA  and  again  at  the 
face  AD,  the  path  of  the  ray  between  these  reflections  being 
parallel  to  its  direction  at  entrance  and  emergence.    More- 


A 

Fig.  216. — Kessler  direct  vision  prism. 

over,  in  virtue  of  the  symmetry  of  the  prism,  the  path  of  the 
emergent  ray  will  be  a  continuation  of  the  rectilinear  path  of 
the  incident  ray.  If  the  angles  at  A,  B  and  C  are  denoted  by 
a,  $  and  7,  respectively,  then 

a+2/3+7  =  360o;  (1) 

and  if  the  angles  of  incidence  and  refraction  at  the  face  BC 
are  denoted  by  6,  6',  then 

o-\,     0'=|-<s;       (2) 

and,  finally,  if  the  index  of  refraction  is  denoted  by  n, 

n.sin0'  =  sin0.  (3) 

Consequently,  eliminating  the  angles  6,  6'  by  means  of 
(2)  and  (3),  we  obtain: 

n.sin(-^-/3)=sim|  ;  (4) 

so  that  if  the  value  of  one  of  the  angles  a,  /?  and  7  is  chosen 
arbitrarily,  the  other  two  angles  can  be  determined  by  means 
of  equations  (1)  and  (4). 

If  the  principal  section  of  a  Kessler  prism  has  the  form 
of  a  rhombus  (Fig.  217),  parallel  incident  rays  may  be  re- 


173]  Achromatic  System  of  Lenses  499 

c 


A 

Fig.  217.  — Rhomboidal  form  of  Kessler  prism. 

ceived  on  both  faces  BA  and  BC.     In  this  case  the  angles 
a  and  y  are  equal,  and  hence  /3+7  =  180°,  and  therefore 

0  =  | ,        0'  =  ^-18O°, 


so  that 

7i.sin(^-180°)=sin^, 
whence  we  obtain : 


•    0 
sm^  =  cos 


2     V    \n 


2  2     V     4n 

For  example,  if  n  =  1.64,  we  find  0  =  36°  24', 7  =  143°  36'. 
173.  Achromatic   Combination  of  Two  Thin  Lenses. — 

The  positions  of  the  principal  and  focal  points  of  a  lens- 
sj'stem  vary  for  light  of  different  colors,  and  if  the  system  is 
to  be  used  as  a  magnifying  glass  or  as  the  so-called  ocular 
of  a  microscope  or  telescope,  a  chief  consideration  will  be 
that  the  apparent  sizes  of  the  colored  virtual  images  which 
are  presented  to  the  eye  shall  all  be  the  same,  that  is,  that 
the  red  and  blue  images,  for  example,  shall  subtend  the 
same  angle  at  the  eye,  no  matter  whether  their  actual  sizes 
and  positions  are  different  or  not.  But  the  apparent  size 
of  the  infinitely  distant  image  of  an  object  lying  in  the 
primary  focal  plane  of  the  lens-system  is  measured  by  the 


500  Mirrors,  Prisms  and  Lenses  [§  173 

refracting  power  of  the  system  (§  122) ;  and  hence  the  condi- 
tion of  achromatism  in  this  case  is  that  the  refracting  powers 
(or  focal  lengths)  of  the  system  shall  be  equal  for  the  two 
colors  in  question.  (Achromatism  with  respect  to  the  focal 
length;  see  §  167.) 

Let  us  assume  that  the  system  is  composed  of  two  thin 
lenses  whose  refracting  powers  for  light  of  a  certain  definite 
wave-length  X  are  denoted  by  Fx  and  F2;  then  the  refract- 
ing power  of  the  combination  will  be  F=Fi+F2—c.Fi.F2) 
where  c  denotes  the  air-interval  between  the  two  lenses. 
For  a  second  color  of  wave-length  X+AX  (where  AX  de- 
notes a  small  variation  in  the  value  of  X),  the  refracting 
powers  of  the  lenses  will  be  slightly  different,  and  the  re- 
fracting power  of  the  combination  for  this  color  will  be: 

F+AF=(F1+AF1)+(F2+AF2)-c(F1+AF1)  (F2+AF2). 
Subtracting  these  two  equations,  at  the  same  time  neglect- 
ing the  term  which  involves  the  product  of  the  small  varia- 
tions A,Pi  and   AF2,  we  obtain : 

AF=AFX+AF2-  (F2.AFi+Fi.AF2)c. 
Evidently,  the  condition  that  the  system  shall  be  achromatic 
with  respect  to  its  refracting  power  is  AF=0;  which,  there- 
fore, is  equivalent  to  the  following: 

_  F2.AFi+FlmAF2 
C~       AFi+AF2       ' 
Now  if  n\  denotes  the  index  of  refraction  of  the  first  lens  for 
light  of  wave-length  X,  then 

ft-(m-l)Ki, 

where  Ki  denotes  a  constant  whose  value  depends  simply  on 
the  form  of  the  infinitely  thin  lens,  that  is,  on  the  curvatures 
of  its  surfaces.  Similarly,  for  light  of  wave-length  X+AX, 
we  have: 

Fi+AFi=(m+Ani-l)Ki; 


and  hence 


AFx  =  Ki.Am=Fi-A 


fti—  1 


§173] 


Achromatic  Lens-System 


501 


But  Am/(ni—l)=l/Pi  is  the  expression  for  the  dispersive 
power  of  the  material  of  the  first  lens  (§  165),  and  accordingly 
we  may  write : 

and,  analogously,  for  the  second  lens: 

v2 

Introducing  these  expressions  for  AFi  and  AF2  in  the  equa- 
tion above,  we  find,  therefore,  as  the  condition  that  a  pair  of 
thin  lenses  shall  be  achromatic  with  respect  to  the  refracting 


Fig.  218. — Hutgens's  ocular. 


power  of  the  system,  the  requirement  that  the  distance  be- 
tween the  two  thin  lenses  shall  satisfy  the  following  equa- 
tion: 

v2.Fi+vi.F2 


c  = 


(Vl+V2)Fl-F2    ' 


or 


_^l./l  +  ^2-A 
•  C j , 

where  fi  =  l/Fi  and  f2  =  l/F2  denote  the  focal  lengths  of  the 
lenses. 


502 


Mirrors,  Prisms  and  Lenses 


174 


If  both  lenses  are  made  of  the  same  glass,  then  v1  =  v2,  so 
that  in  this  case  the  condition  of  achromatism  becomes : 

./1+/2 


c  = 


Thus,  for  example,  Huygens's  ocular  (Fig.  218)  is  composed 
of  two  plano-convex  lenses  made  of  the  same  kind  of  glass, 
the  curved  face  of  each  lens  being  turned  away  from  the 
eye  and  towards  the  incident  light.  The  first  lens  is  called 
the  " field-lens"  and  the  second  lens  is  called  the  " eye-lens." 
In  this  combination  /1  =  2/2  (although  in  actual  systems  this 


Fig.  219. — Ramsden's  ocular. 

condition  is  usually  only  approximately  satisfied)  and  c  = 
3/2/2,  or  f2  :c  :/i  =  2  :3  : 4.  Ramsden's  ocular  (Fig.  219) 
consists  likewise  of  two  plano-convex  lenses  of  the  same  kind 
of  glass,  but  with  their  curved  faces  turned  towards  each 
other  and  in  this  combination  f1=f2=f=c.  Both  of  these 
types  satisfy,  therefore,  the  above  condition  of  achromatism 
and  yield  images  that  are  free  from  color-faults  not  only  in 
the  center  but  at  the  border  of  the  field. 

174.  Achromatic  Combination  of  Two  Thin  Lenses  in 
Contact. — If  the  two  lenses  are  in  contact  (c  =  0),  the  con- 
dition of  achromatism,  as  found  in  the  preceding  section, 
becomes : 

Vifi+V2.f2==0, 

or 

^+^  =0. 
The  quotient  of  the  refracting  power  of  a  lens  by  the  dis- 


§  174]  Achromatic  Lens-System  503 

persive  power  of  the  glass  of  which  it  is  made,  namely,  the 
magnitude  F/v,  is  sometimes  called  the  dispersive  strength 
of  the  lens;  so  that  according  to  the  above  equation  we  may 
say  that  the  condition  of  achromatism  of  a  combination  of 
two  thin  lenses  in  contact  is  that  the  algebraic  sum  of  their 
dispersive  strengths  shall  vanish.  Accordingly,  it  appears 
that  such  a  system  can  be  achromatic  only  in  case  the  sub- 
stances of  which  the  two  lenses  are  made  are  different.  More- 
over, while  one  of  the  lenses  must  be  convex  and  the  other 
concave,  their  actual  forms  are  of  no  consequence  so  far  as 
the  mere  correction  of  the  chromatic  aberration  is  concerned. 
It  is  to  be  remarked  also  that  in  an  achromatic  lens  of  neg- 
ligible thickness  achromatism  with  respect  to  the  focal  lengths 
implies  also  achromatism  with  respect  to  the  positions  of 
the  focal  points  and  principal  points,  so  that  such  a  lens  will 
be  achromatic  for  all  distances  of  the  object. 

If  F  denotes  the  prescribed  refracting  power  of  the  com- 
bination then,  since, 

^  =  ^1+^2, 

we  find: 

Fi=^-F,       F2=--^—F. 

Vi-V2  Vi-V2 

The  total  refracting  power  F  will  have  the  same  sign  as 
that  of  the  lens  which  has  the  greater  r-value;  for  example, 
the  combination  will  act  like  a  convex  lens  provided  the 
v-value  of  the  positive  element  exceeds  that  of  the  nega- 
tive element. 

Thus,  being  given  the  values  of  F,  vx  and  v2}  we  can  em- 
ploy the  above  relations  to  determine  the  required  values 
of  F\  and  F2.  Moreover,  if  Ki  denotes  the  algebraic  differ- 
ence of  the  curvatures  of  the  two  faces  of  the  first  lens,  and, 
similarly,  if  K2  denotes  the  corresponding  magnitude  for 
the  second  lens,  then 


ni-1  n2-l 

where  nh  n2  denote  the  indices  of  refraction  of  the  two  kinds 


504 


Mirrors,  Prisms  and  Lenses 


[§174 


of  glass  for  some  standard  wave-length,  as  already  stated, 
which  is  usually  light  corresponding  to  the  Fraunhofer 
D-line.  Thus,  while  the  magnitudes  denoted  by  Ki  and  K2 
may  be  computed,  the  actual  curvatures  or  radii  of  the  lens- 
surfaces  remain  indeterminate;  so  that  there  are  still  two 


Fig.  220. — Dollond's 
telescope  objective. 


Fig.  221.  —  Fraun- 
hofer's  telescope 
objective,  No.  1. 


Fig.  222.  —  Fraun- 
hofer's  telescope 
objective,  No.  2. 


Fig.  223. — Herschel's 
telescope  objective. 


Fig.   224.— Barlow's 
telescope  objective. 


Fig.     225.  —  Gauss's 
telescope  objective. 


other  conditions  which  may  be  imposed  on  an  achromatic 
combination  of  this  kind.  For  example,  in  some  cases  it 
may  be  conventient  to  cement  the  two  components  together, 
and  then  one  of  the  conditions  will  be  that  the  curvatures  of 
the  two  surfaces  in  contact  shall  be  equal.  Usually,  how- 
ever, a  more  important  requirement  will  be  the  abolition  of 
two  of  the  so-called  spherical  errors  due  to  the  fact  that  the 


Ch.  XIV]  Problems  505 

rays  are  not  paraxial,  so  that  the  image  will  be  sharp  and 
distinct,  especially  at  the  center. 

Some  historic  types  of  achromatic  object-glasses  of  a  tel- 
escope are  illustrated  in  the  accompanying  diagrams.  Dol- 
lond's  achromatic  doublet  (Fig.  220)  consisted  of  a  double 
convex  crown  glass  lens  combined  with  a  double  concave  flint 
glass  lens;  whereas  Fraunhofer's  constructions  show  a  com- 
bination of  a  double  convex  and  a  plano-concave  lens  (Fig. 
221)  and  of  a  double  convex  and  a  meniscus  lens  (Fig.  222). 
J.  Herschel's  form  (1821)  is  shown  in  Fig.  223,  Barlow's 
(1827)  in  Fig.  224;  and,  finally,  the  Gauss  type  made  by 
Steinheil  in  1860  is  exhibited  in  Fig.  225.  The  newer  va- 
rieties of  Jena  glass  make  it  possible  to  construct  an  achro- 
matic objective  of  two  lenses  which  is  far  superior  in  achro- 
matism to  any  of  the  older  types  above  mentioned. 

PROBLEMS 

1.  Find  the  values  of  the  reciprocals  of  the  dispersive 
powers  (§  165)  of  alcohol  and  water,  using  data  given  in 
table  in  §  163.  Ans.  Alcohol,  60.5;  water,  55.7. 

2.  The  indices  of  refraction  of  rock  salt  for  the  Fraun- 
hofer  lines  C,  D  and  F  are  1.5404,  1.5441  and  1.5531,  re- 
spectively. Calculate  the  value  of  the  reciprocal  of  the  dis- 
persive power.  Ans.  42.84. 

3.  White  light  is  emitted  from  a  luminous  point  on  the 
axis  of  a  thin  lens.  If  the  yellow  rays  are  brought  to  a  focus 
at  a  point  whose  distance  from  the  lens  is  denoted  by  u', 
show  that  the  distance  between  the  foci  of  the  red  and  blue 
rays  is  approximately  equal  to  2  F.u'/v,  where  F  denotes  the 
refracting  power  of  the  lens  for  yellow  light  and  v  denotes 
the  reciprocal  of  the  dispersive  power  of  the  lens-medium. 

4.  A  lens  is  made  of  borate  flint  glass  for  which  ^  =  55.2. 
The  focal  length  of  the  lens  for  sodium  light  is  30  inches. 
Find  the  distance  between  the  red  and  blue  images  of  the 
sun  formed  by  the  lens.  Ans.  0.54  in. 


506  Mirrors,  Prisms  and  Lenses  [Ch.  XIV 

5.  A  crown  glass  prism  of  refracting  angle  20°  is  to  be 
combined  with  a  flint  glass  prism  so  that  the  combination 
will  be  achromatic  for  the  Fraunhofer  lines  C  and  F.  The 
indices  of  refraction  are  as  follows : 

nc  n-D  nF 

Crown         1.526  849     1.529  587     1.536  052 
Flint  1.629  681     1.635  036     1.648  260 

Using  the  approximate  formulae  for  thin  prisms,  show  that 
the  refracting  angle  of  the  flint  prism  will  be  9°  54'  11",  and 
that  the  deviation  of  the  rays  corresponding  to  the  D-line 
will  be  4°  18'  7". 

6.  A  direct  vision  prism  combination  is  to  be  made  with 
the  same  kinds  of  glass  as  in  the  preceding  problem;  so  that 
rays  corresponding  to  the  D-line  are  to  emerge  without  de- 
viation. If  the  refracting  angle  of  the  crown  glass  prism  is 
20°,  show  that  the  refracting  angle  of  the  flint  glass  prism 
will  be  16°  40'  48",  and  that  the  angular  dispersion  between 
C  and  F  will  be  9'  33". 

7.  An  Amici  direct  vision  prism  (§  171)  is  to  be  made  of 
crown  glass  and  flint  glass  whose  indices  of  refraction  for 
the  D-line  are  1.5159  and  1.9626,  respectively.  If  the  re- 
fracting angles  of  the  two  equal  crown  glass  prisms  are  each 
equal  to  45°,  show  that  the  refracting  angle  of  the  middle 
flint  glass  prism  will  be  98°  7.4'. 

8.  A  Kessler  prism  (§  172)  in  the  form  of  a  rhombus  is 
made  of  glass  of  index  nD  =  1.6138.  Find  the  angles  of  the 
prism.  Ans.  35°  5'  and  144°  55'. 

9.  A  thin  lens  is  made  of  crown  glass  for  which  z>i  =  60.2. 
Another  thin  lens  is  made  of  flint  glass  for  which  *>2  =  36.2. 
When  the  two  lenses  are  placed  in  contact  they  form  an 
achromatic  combination  of  focal  length  10  cm.  Find  the 
focal  length  of  each  lens.      Ans.  /i =3.99  cm. ;  /2  =  —6.63  cm. 

10.  An  achromatic  doublet  is  to  be  made  of  two  thin 
lenses  cemented  together,  and  the  focal  length  of  the  com- 
bination for  the  D-line  is  to  be  25  cm.  The  first  lens  is  a 
symmetric  convex  lens  of  barium  silicate  glass  and  the  other 


Ch.  XIV]  Problems  507 

lens  is  a  concave  lens  of  sodium  lead  glass.  The  indices  of 
refraction  are: 

^D  Up  — Tie 

Barium  silicate         1.6112  0.01747 

Sodium  lead  1.5205  0.01956 

Find  the  radii  of  the  surfaces  on  the  supposition  that  the  rays 
corresponding  to  the  lines  C  and  F  are  united. 

Ans.  The  radii  of  the  first  and  last  surfaces  are  +14.60 
and  —22.65  cm.,  respectively. 

11.  A  symmetric  double  convex  lens  is  made  of  rock  salt 
for  which  nc  =  1.5404  and  nF=  1.5531.  Find  the  thickness 
of  the  lens  if  the  focal  lengths  for  the  colors  C  and  F  are  equal. 

Ans.  d  =  3.4363. r,  where  r  denotes  the  radius  of  the  first 
surface  of  the  lens. 

12.  Two  thin  lenses  of  the  same  kind  of  glass,  one  convex 
of  focal  length  9  inches,  the  other  concave  of  focal  length 
4  inches,  are  separated  by  an  interval  of  20  inches.  A  small 
white  object  is  placed  36  inches  in  front  of  the  convex  lens. 
Show  that  the  various  colored  images  are  all  formed  at  the 
same  place. 

13.  Two  thin  lenses  of  the  same  kind  of  glass,  one  convex 
and  the  other  concave,  and  both  of  focal  length  4  inches,  are 
adjusted  on  the  same  axis  until  the  colored  images  of  a  white 
object  placed  12  inches  in  front  of  the  convex  lens  are  formed 
at  the  same  place.  Show  that  the  interval  between  the  lenses 
must  be  twelve  inches. 

14.  A  lens-system  surrounded  by  air  is  composed  of  m 
spherical  refracting  surfaces.  Assuming  that  the  total  thick- 
ness of  the  system  is  negligible,  show  that  the  condition  of 
achromatism  is 

k=m 

2  (#k_i-i4)  Snk  =  0, 

k=2 

where  Rk  denotes  the  curvature  of  the  kth  surface  and  8nk 
denotes  the  dispersion  of  the  medium  included  between  the 
(k  —  l)th  and  kih  surfaces  for  light  of  the  two  colors  to  be 
compensated. 


CHAPTER  XV 

RAYS  OF  FINITE  SLOPE.         SPHERICAL  ABERRATION, 
ASTIGMATISM  OF  OBLIQUE  BUNDLES,  ETC. 

175.  Introduction. — The  theory  of  the  symmetrical  op- 
tical instrument,  as  it  has  been  developed  in  the  preceding 
chapters,  is  based  on  the  assumption  that  the  rays  concerned 
in  the  formation  of  the  image  are  entirely  confined  to  the 
so-called  paraxial  rays  (§  63)  whose  paths  throughout  the 
system  are  contained  within  an  exceedingly  narrow  cylindri- 
cal region  of  space  immediately  surrounding  the  axis.  With 
this  fundamental  restriction  it  was  found  that  there  was 
perfect  collinear  correspondence  between  object-space  and 
image-space;  so  that  a  train  of  spherical  waves  emanating 
from  an  object-point  was  transformed  by  the  optical  system 
into  another  train  of  spherical  waves  accurately  converging 
to  or  diverging  from  a  corresponding  center  called  the  image- 
point;  and  so  that,  in  general,  a  plane  object  at  right  angles 
to  the  axis  was  reproduced  point  by  point  by  a  similar  plane 
image.  As  a  matter  of  fact,  these  ideal  conditions  are  never 
realized  in  any  actual  optical  system  except  in  the  case  of  a 
plane  mirror  or  combination  of  plane  mirrors.  Moreover, 
according  to  the  wave-theory  of  light,  a  mere  homocentric 
convergence  of  the  rays  is  not  sufficient  for  obtaining  a  point- 
image  of  a  point-source ;  for  this  theory  lays  particular  stress 
on  the  further  essential  requirement  that  the  effective  por- 
tion of  the  wave-surface  which  contributes  to  the  produc- 
tion of  the  image  shall  be  relatively  large  in  comparison  with 
the  radius  of  the  surface,  if  the  light-effect  is  to  be  concen- 
trated as  nearly  as  possible  at  a  single  point  and  not  spread 
over  some  considerable  area  in  the  vicinity  of  the  point.  This 
condition  implies,  therefore,  that  the  aperture  of  the  bundle 

508 


§  176]  Young's  Construction  509 

of  effective  rays  must  not  be  below  a  certain  finite  limit,  in 
other  words  we  are  compelled  by  a  practical  necessity,  wholly 
aside  from  the  principles  at  the  basis  of  geometrical  optics,  to 
employ  more  or  less  wide-angle  bundles  of  rays.  Moreover, 
if  a  wide-angle  bundle  of  rays  is  a  requirement  of  a  distinct, 
clear-cut  image,  it  is  also  equally  essential  for  a  bright  image. 
Thus,  on  both  theoretical  and  practical  grounds,  it  is  found 
necessary  to  extend  the  limits  of  the  effective  rays  beyond 
the  paraxial  region. 

Instead,  therefore,  of  the  ideal  case  of  collinear  correspond- 
ence of  object-space  and  image-space,  the  theory  of  optical 
instruments  is  complicated  by  numerous  practical  and,  for 
the  most  part  irreconcilable  difficulties,  due  chiefly  to  the 
so-called  aberrations  or  failure  of  the  rays  to  arrive  at  the 
places  where  they  might  be  expected  according  to  the 
simple  theory  of  collineation  or  point-to-point  corre- 
spondence (punctual  imagery).  In  the  preceding  chapter 
brief  reference  was  made  to  the  chromatic  aberrations  arising 
from  the  differences  in  the  color  of  the  light;  but  now  we 
have  to  deal  with  the  monochromatic  aberrations  of  rays  of 
light  of  one  definite  wave-length  which  are  caused  by  the  pe- 
culiarities of  the  curved  surfaces  at  which  the  rays  are  re- 
flected and  refracted.  These  surfaces  are  nearly  always 
spherical  in  form,  and  hence  the  aberrations  of  this  latter 
kind  are  usually  called  spherical  aberrations.  A  complete 
treatment  of  this  intricate  subject  lies  wholly  outside  the 
scope  of  this  volume.  In  the  present  chapter  it  must  suffice 
to  point  out  the  general  nature  of  some  of  the  more  important 
of  the  so-called  spherical  errors.  First,  however,  we  must 
see  how  to  trace  the  path  of  a  single  ray  through  a  centered 
system  of  spherical  surfaces  before  we  are  in  a  position  to 
study  a  bundle  of  rays. 

176.  Construction  of  a  Ray  Refracted  at  a  Spherical 
Surface. — In  §  34  a  method  was  explained  for  constructing 
the  path  of  a  ray  refracted  from  one  medium  into  another, 
which  is  always  applicable  to  a  refracting  surface  of  any  form. 


510 


Mirrors,  Prisms  and  Lenses 


[§176 


The  following  elegant  and  useful  construction  of  the  path  of 
a  ray  refracted  at  a  spherical  surface  was  published  in  1807 
by  Thomas  Young  (1773-1829.) 


Fig.  226. — Construction  of  ray  refracted  at  convex  spherical  surface  (n'>ri). 

In  the  accompanying  diagrams  (Figs.  226  to  229)  the 
center  of  the  spherical  refracting  surface  ZZ  is  designated 
by  C.  The  point  R  is  any  point  on  the  path  of  the  incident 
ray  lying  in  the  first  medium  of  refractive  index  n.    The  point 


Fig.  227. — Construction  of  ray  refracted  at  concave  spherical  surface  (w'>n). 

where  the  ray  meets  the  spherical  refracting  surface  is  marked 
B.  The  plane  of  the  paper  which  contains  the  incident  ray 
RB  and  the  incidence-normal  BC  is  the  plane  of  incidence. 


176] 


Young's  Construction 


511 


The  index  of  refraction  of  the  second  medium  is  denoted  by 
n'  and  the  radius  of  the  spherical  refracting  surface  by  r. 
Around  C  as  center  and  with  radii  equal  to  n'.r/n  and  n.rjn' 


Fig.  228. — Construction  of  ray  refracted  at  convex  spherical  surface  (n'<n). 

describe,  in  the  plane  of  incidence,  the  circular  arcs  k  and  k', 
respectively;  and  let  S  designate  the  point  where  the  straight 
line  RB,  produced  if  necessary,  meets  the  arc  k.  Draw  the 
straight  line  CS  ntersecting  the  arc  k'  in  the  point  S'.    Then 


Fig.  229. — Construction  of  ray  refracted  at  concave  spherical  surface  (n'<n). 

the  straight  line  BT  drawn  from  B  through  S'  will  represent 
the  path  of  the  refracted  ray.  In  making  this  construction, 
care  must  be  taken  to  select  for  the  point  S  that  one  of  the 


512  Mirrors,  Prisms  and  Lenses  [§  177 

two  points  in  which  the  straight  line  RB  cuts  the  circle  k 
which  will  make  the  segments  BS  and  BS'  both  fall  on  the 
same  side  of  the  incidence-normal,  since  the  angles  of  in- 
cidence and  refraction  are  described  always  in  the  same 
sense,  both  clockwise  or  both  counter-clockwise. 

The  proof  of  the  construction  is  simple.  Since  the  radius 
r  =  BC  is  a  mean  proportional  between  the  radii  SC=n'.r/n 
and  S'C=n.r/n/,  that  is,  since 

CS  :CB  =  CB  :CS'-n'  :n, 
the  triangles  CBS  and  CBS'  are  similar,  and  hence  Z  CBS  = 
Z  BS'C.    In  the  triangle  CBS : 

sinZCBS  :  sinZBSC  =  CS  :  CB=n'  :  n. 
By    the    law    of    refraction:    n.sma=n'.sma',    where    a  = 
ZCBS.      Consequently,    ZBSC  =  ZCBS'=  a',   so  that  the 
straight  line  BS'  is  the  path  of  the  refracted  ray. 

This  construction  can  be  employed  to  trace  the  path  of 
a  ray  graphically  from  one  surface  to  the  next  through  a 
centered  system  of  spherical  refracting  surfaces. 

177.  The  Aplanatic  Points  of  a  Spherical  Refracting  Sur- 
face.— Incidentally,  in  connection  with  the  preceding  con- 
struction, attention  is  directed  to  the  singular  character  of 
all  pairs  of  points  such  as  S,  S'  determined  by  the  intersec- 
tions of  the  two  concentric  auxiliary  spherical  surfaces  with 
any  straight  line  drawn  from  their  common  center  C.  To 
every  incident  ray  directed  towards  the  point  S  there  will 
correspond  a  refracted  ray  which  will  pass  ("really"  or 
" virtually")  through  the  other  point  S';  so  that  in  this 
special  case  we  obtain  a  homocentric  bundle  of  refracted 
rays  from  a  homocentric  bundle  of  incident  rays,  for  all 
values  of  the  aperture-angle  of  the  bundle.  Thus,  S'  is  a 
point-image  of  the  object-point  S.  The  distances  of  S  and 
S'  from  the  center  C  are  connected  by  the  invariant-relation: 
w'.CS'=w.CS. 

That  pair  of  these  points  which  lies  on  the  optical  axis  is 
especially  distinguished   and   called  the  pair  of  aplanatic 


§178] 


Spherical  Aberration 


513 


points  of  the  spherical  refracting  surface;  they  are  designated 
by  J,  J'  (Fig.  230).    Thus,  we  have: 

CJ  :AC=AC  :CJ'=n'  :n, 
or 

CJ.CJ'  =  r2,       n.CJ=rc'.CJ'. 
The  aplanatic  points,  therefore,  he  always  on  the  same  side 
z 


Fig.  230. — Aplanatic  points  of  spherical  refracting  surface. 

of  the  center  C  so  that  whereas  the  rays  must  pass  "really" 
through  one  of  them,  they  will  pass  "  virtually''  through 
the  other.  In  geometrical  language  the  pdints  J,  J'  are  said 
to  be  harmonically  separated  (§  67)  by  the  extremities  of 
the  axial  diameter  of  the  refracting  sphere. 

178.  Spherical  Aberration  Along  the  Axis. — However,  in 
general,  a  homocentric  bundle  of  rays  incident  on  a  spheri- 
cal refracting  surface  will  not  be  homocentric  after  refrac- 
tion. The  diagram  (Fig.  231)  represents  the  case  of  a  merid- 
ian section  of  a  bundle  of  incident  rays  which  are  all  parallel 
to  the  axis  of  a  convex  spherical  refracting  surface  for  which 
nf>n.  It  will  be  seen  that,  whereas  the  paraxial  rays  after 
refraction  meet  on  the  axis  at  the  second  focal  point  F',  the 
outermost  or  edge  rays  cross  the  axis  at  a  point  L'  between 
the  vertex  A  and  the  focal  point  F';  and  the  intermediate 
rays  cross  the  axis  at  points  lying  between  F'  and  I/.  The 
segment  F'L'  is  the  measure  of  the  spherical  aberration  along 


514 


Mirrors,  Prisms  and  Lenses 


[§17S 


the  axis  or  the  axial  aberration  of  the  edge  ray  of  a  direct 
cylindrical  bundle  of  incident  rays.  (By  a  " direct"  bundle 
of  rays  is  meant  a  bundle  of  rays  emanating  from  a  point  on 
the  axis.)  In  the  figure  this  segment  is  negative,  that  is,  meas- 
ured in  the  sense  opposite  to  that  of  the  incident  light;  and 


Fig.  231. — Spherical  aberration. 

this  effect  is  usually  described  by  saying  that  a  convex  spheri- 
cal refracting  surface  at  which  light  is  refracted  from  air  to 
glass  is  spherically  under-corrected;  whereas,  under  the  same 
circumstances,  a  concave  spherical  refracting  surface  will  be 
found  to  be  spherically  over-corrected,  that  is,  the  segment 
F'L'  in  this  case  will  be  positive.  In  fact,  the  points  of  in- 
tersection of  pairs  of  consecutive  rays  lying  in  the  plane  of 
a  meridian  section  of  a  spherical  refracting  surface  form  a 
curved  line  lying  symmetrically  above  and  below  the  axis, 
if  the  bundle  of  incident  rays  is  symmetric  with  respect  to 
the  axis;  and  this  plane  curve  is  the  so-called  caustic  curve 
of  the  meridian  rays.  The  two  branches  on  opposite  sides 
of  the  axis  unite  in  a  double  point  or  cusp  at  the  point  on 
the  axis  where  the  paraxial  rays  intersect,  so  that  the  axis 
is  tangent  to  both  branches  at  this  point,  which  in  the  figure 


§  179]  Spherical  Zones  515 

is  the  point  F'.  The  system  is  said  to  be  spherically  over- 
corrected  or  under-corrected  according  as  the  cusp  is  turned 
towards  the  incident  light  (<)  or  away  from  it  (>),  respec- 
tively; on  the  supposition  that  the  incident  rays  are  parallel 
to  the  axis.  Each  refracted  ray  in  the  meridian  plane  touches 
the  caustic  curve,  and  hence  this  curve  is  said  to  be  the  geo- 
metrical envelope  of  the  meridian  section  of  the  bundle  of 
refracted  rays. 

If  the  entire  figure  is  revolved  around  the  optical  axis  the 
arc  ZZ  will  generate  a  zone  of  the  spherical  refracting  sur- 
face containing  the  vertex  A;  and  each  incident  ray  pro- 
ceeding parallel  to  the  axis  will  generate  a  cylindrical  sur- 
face, and  all  the  refracted  rays  corresponding  to  the  incident 
rays  which  lie  on  the  surface  of  one  of  these  cylinders  will 
intersect  in  one  point  lying  on  the  axis  between  F'  and  L'. 
The  revolution  of  the  caustic  curve  will  generate  a  caustic 
surface,  which  will  be  the  enveloping  surface  of  the  bundle  of 
refracted  rays  (see  §  187.) 

The  caustic  curve  terminates  at  the  point  H'  where  the 
edge  ray  intersects  the  next  consecutive  ray  in  the  meridian 
section.  If  a  plane  screen  erected  at  right  angles  to  the  axis 
so  as  to  catch  the  light  transmitted  by  the  bundle  of  refracted 
rays  is  placed  initially  in  the  transversal  plane  that  passes 
through  the  extreme  point  H'  and  then  gradually  shifted 
parallel  to  the  axis  towards  the  second  focal  plane,  there  will 
appear  on  the  screen  at  first  a  circular  patch  of  light  sur- 
rounded on  its  outer  edge  by  a  brighter  ring,  which  will  grad-. 
ually  contract  as  the  screen  approaches  L'.  Between  L' 
and  F'  there  will  be  seen  at  the  center  of  the  circular  patch 
of  light  an  increasingly  bright  spot.  For  a  certain  position 
G'  where  the  distance  of  the  screen  from  F'  is  about  three- 
fourths  of  the  length  of  F'L'  the  cross-section  of  the  bundle 
of  refracted  rays  will  have  its  narrowest  contraction.  This 
section  is  sometimes  called  the  least  circle  of  aberration. 

179.  Spherical  Zones. — Since,  in  general,  it  is  not  possible 
to  abolish  the  spherical  aberration  of  a  single  spherical  re- 


516  Mirrors,  Prisms  and  Lenses  [§  180 

fracting  surface,  the  only  means  available  is  to  try  to  ac- 
complish this  result  by  distributing  the  duty  of  refracting 
the  rays  over  a  series  of  surfaces  whose  curvatures  and  dis- 
tances apart  are  so  nicely  adjusted  with  respect  to  each  other 
that  when  the  rays  finally  emerge  they  will  all  unite  in  one 
focus  on  the  axis.  Thus,  for  example,  if  the  incident  rays 
are  supposed  to  be  parallel  to  the  axis  of  the  system,  and  if 
the  system  has  been  designed  so  as  to  be  spherically  corrected 


Fig.  232. — Graphical  representation  of  the  spherical  zones  of  a  lens. 

for  the  edge  ray  which  meets  the  first  surface  at  the  distance 
h  from  the  axis,  it  is  conceivable  that  all  the  intermediate 
rays  of  incidence-heights  z  (where  h  >  z  >  0)  might  perchance 
emerge  from  the  system  along  paths  which  all  likewise  passed 
through  the  focal  point  F';  but  practically  this  never  hap- 
pens. If  the  edge  ray  intersects  the  axis  at  F',  an  intermedi- 
ate ray  of  incidence-height  z  will  cross  the  axis  at  some  other 
point  I/,  and  the  segment  F'L'  is  called  the  spherical  aberra- 
tion of  the  zone  of  radius  z  or  simply  the  spherical  zone  z.  The 
spherical  zones  of  a  lens  may  be  exhibited  graphically  by 
plotting  a  curve  whose  abscissae  are  the  values  of  F'L'  and 
whose  ordinates  are  the  corresponding  values  of  z,  as  repre- 
sented in  Fig.  232. 

180.  Trigonometrical  Calculation  of  a  Ray  Refracted  at 
a  Spherical  Surface. —  The  diagram  (Fig.  233)  represents  a 
meridian  section  ZZ  of  a  spherical  refracting  surface  of  radius 
r  (=AC)  separating  two  media  of  indices  of  refraction  n,  n' . 
A  ray  RB  incident  on  the  surface  at  B  at  an  angle  <x  = 
Z  NBR  =  Z  CBL  crosses  the  axis  at  L  at  a  slope-angle  6  = 
ZALB.      If  the  central  angle  is  denoted  by   </>  =  ZBCA, 


180] 


Calculation  of  Refracted  Ray 


517 


and  if  the  abscissa  of  the  point  L  with  respect  to  the  center  C 
is  denoted  by  c,  that  is,  if  c  =  CL,  then  in  the  triangle  CBL, 
we  have  the  relations: 

a=d  +  4>,        c.sin#  =  —  r.sina. 
The  path  of  the  corresponding  refracted  ray  is  shown  by  the 
straight  line  BT  which  crosses  the  axis  at  the  point  L';  and 


*f 

~~~eT"~- 

A 

C^ 

^N» 

L'^- 

L 

Tl 

\  n» 

-  T 

Fig.  233. — Diagram  for  trigonometrical  calculation  of  refracted  ray. 

if  we  put  a'  =  ZN'BL',  0'  =  ZAL'B  and  c'=CL',  we  ob- 
tain a  similar  pair  of  formulae  from  the  triangle  CBL',  namely: 

a'=d'-\-(j>,  c'.sin#'=  —  r.sina'. 

Accordingly,  being  given  the  constants  denoted  by  n,  n'  and 
r,  and  the  parameters  (c,  8)  of  the  incident  ray,  we  can  find 
the  parameters  (c',  8')  of  the  refracted  ray  by  means  of  the 
following  system  of  equations: 

c   .    a 

sin  a  —  —  -  sin  8, 
r 


sin  a  = —sin  a, 
n 


c'  = 


8'=  8+ a' -a, 


sin  a 
sin  8' 


It  is  easy  to  see  that  if  we  have  given  two  incident  rays 
which  both  cross  the  axis  at  the  same  point  L,  so  that  the 
abscissa  c  has  the  same  value  for  both  rays  while  the  slope- 
angles  8  are  different,  different  values  of  c'  will,  in  general, 
be  obtained  for  the  abscissae  of  the  points  where  the  two  cor- 


518  Mirrors,  Prisms  and  Lenses  [§  180 

responding  refracted  rays  cross  the  axis.    This  is  the  analyti- 
cal statement  of  the  fact  of  spherical  aberration  (§  178). 

The  formulae  for  calculating  the  path  of  a  ray  reflected  at 
a  spherical  mirror  may  be  derived  immediately  by  putting 
n'=—n  (§  75)  in  the  preceding  system  of  equations.  Thus 
we  find: 

sina  =  —  sin  0,      a'=  —  a,     0'=  0— 2a,    c' =r  .   -  -    _      . 

r  sm(0-2a) 

Incidentally,  a  number  of  other  useful  relations  may  be 
obtained  from  Fig.  233.  For  example,  if  the  distances  of  the 
points  L  and  L'  where  the  ray  crosses  the  axis  before  and 
after  refraction  measured  from  the  incidence-point  B  are 
denoted  by  I  and  V,  respectively,  that  is,  if  l  =  BL,  Z'  =  BL', 
where  I  and  V  are  to  be  reckoned  positive  or  negative  ac- 
cording as  these  lengths  are  measured  in  the  same  direction 
as  the  light  traverses  the  ray  or  in  the  opposite  direction, 
respectively;  then 

Z'.sin0'  =  Z.sin0; 
and,  since  by  the  law  of  refraction, 

n'.c'.sin  0'=ft.c.sin  0, 
we  obtain  the  useful  invariant  relation : 

n'.c'  _n.c 

1       T' 

Moreover,  by  projecting  the  two  sides  c  and  I  of  the  triangle 
CBL  on  the  third  side  r,  the  following  formula  is  obtained : 

r  =  Lcosa— c.cos0, 
which  may  be  written : 

c_     r     /cosa_l\ 
I    coscj)  \    r        If 
Similarly,  in  the  triangle  CBL' : 

c'_     r      /cosa/_l\ 

Multiplying  the  first  of  these  equations  by  n  and  the  second 
by  n'  and  equating  the  resulting  expressions,  we  find : 


,/cosa'     1\       /cos  a     1\ 


§  181]       Path  of  Ray  through  Centered  System         519 

which  may  also  be  written : 

n'    n    n'. cos  a'— ft.cosa     ^  ,      N 
V~T r =0(say); 

or  finally: 

L'  =  L+D, 

where  L  =  n/l,  Lr  =  n'llf. 

If  the  ray  is  a  paraxial  ray,  we  may  put  cos  a  =  cos  o!  =  1 
(§  63) ;  and  now  if  we  write  u,  v!  in  place  of  I,  V ,  respectively, 
the  formula  above  will  reduce  to  the  abscissa-equation  for 
the  refraction  of  a  paraxial  ray  at  a  spherical  surface  (§  78). 
Moreover,  if  in  the  last  formula  we  put  ft'  =  ft  (§  75),  we 
find  the  corresponding  relation  for  the  reflection  of  a  ray  at 
a  spherical  mirror,  namely: 

1  1 _  2cos  a 
1+P~~r~' 
181.  Path  of  Ray  through  a  Centered  System  of  Spher- 
ical Refracting  Surfaces.  Numerical  Calculation. — Using 
the  same  system  of  notation  as  in  §  118,  we  may  write  the 
formula  for  the  refraction  of  a  paraxial  ray  at  the  kth  sur- 
face of  a  centered  system  of  spherical  refracting  surfaces, 
as  follows : 

W  =  Uk+Fk, 
where 

Uk  =  nk/uk,      Uk'  =wk+i/%',      and  Fk  =  (nk+i-nk)/rk; 
ftk=AkMk,       V  =  AkMk+i,       7k  =  AkCk. 
And  if  dk  =  AkAk.f  i,  then  also: 

i/^k+i  =  i/t/'k-4K+i. 

According  to  the  relations  given  in  §  180,  we  have  the 

following  system  of  formulae  for  the  refraction  at  the  A;th 

surface  of  a  ray  whose  slope-angles  before  and  after  refraction 

have  the  finite  values  6k  =  Z  AkLkBk  and  0k+i  =  Z  AkLk+iBk, 

respectively : 

ck     .     n  .         .        ftk 

sin  ak  =  —  —  .sin  0k,       sin  ak  =  — —  .sin  ak, 

sin  ak 


#k  +  i=  #k+  0k-  ak,       ck=-rk. 


sin  0k  + 


520  Mirrors,  Prisms  and  Lenses  [§  181 

where  ck  =  CkLk  and  ck'  =  CkLk+i.    Moreover,  if  we  put 

ak  =  CkCk +i  =  dk  +rk  _}_i  —  rk, 
then 

ck_j_i  =  ck  —  ak. 

In  order  to  exhibit  the  methods  of  calculations  by  means  of 
these  formulae,  a  comparatively  simple  numerical  illustra- 
tion is  appended.  The  actual  example  here  chosen  is  one 
given  by  Dr.  Max  Lange  in  his  paper  entitled  "  Vereinfachte 
Formeln  fiir  die  trigonometrische  Durchrechnung  optischer 
Systeme"  (Leipzig,  1909),  pages  24,  foil.  The  optical  sys- 
tem is  a  two-lens  object-glass  of  a  telescope  for  which  the 
data  were  published  by  Dr.  R.  Steinheil  in  the  Zeitschrift 
fiir  Instrumentenkunde,  xvii  (1897),  p.  389,  as  follows: 

Indices  of  refraction  (for  D-line) : 

ni  =  n3  =  Ub=l  (air);  n2  =  1.614  400  (flint);  n4=1.518  564 
(crown) . 

Thicknesses: 

di  =  2;  d2  =  0.01;  d3  =  5. 

Radii: 

ri=+420;  r2= +181.995;  r3=  +178.710;  r4  =  — 40  133.8. 
The  incident  rays  are  parallel  to  the  axis,  so  that 

01  =  O,  ui  =  ci=  oo  (C/i  =  0). 
The  calculation  is  divided  into  two  parts,  namely:  (1)  the 
calculation  of  the  paraxial  ray,  and  (2)  the  trigonometric 
calculation  of  the  edge  ray  which  meets  the  first  surface  of 
the  object-glass  at  the  height  hi  =33  above  the  axis.  When 
C\=  oo ,  we  find  sinai  =  /ii/ri,  which,  according  to  the  above 
data,  gives  lg  sin  a\  =  1.5185139.  This  is  the  starting  point 
of  the  calculation  of  the  edge  ray. 

Each  vertical  column  contains  the  calculation  for  one 
spherical  refracting  surface.  The  sign  written  after  a  log- 
arithm indicates  the  sign  of  the  number  to  which  the  loga- 
rithm belongs.  Generally  the  calculations  do  not  have  to  be 
performed  to  the  degree  of  accuracy  to  which  they  are  carried 
here. 


181] 


Scheme  of  Numerical  Calculation 


521 


1.  PARAXIAL  RAY 


k=l 

k=2 

k=3 

k=4 

lg(ttk+l-ftk) 
clgrk 

9.7884512+ 
7.3767507  + 

9.7884512- 
7.7399406  + 

9.7148D24  + 

7.7478511  + 

9.7148024- 
5.3964898- 

IgFk 

7.1652019  + 

7.5283918- 

7.4626535  + 

5.1112922+ 

Fk 
Uk 

+0.00146286 
0.00000000 

-0.00337592 
+0.00146552 

+0.00290171 
-0.00191036 

+0.00001292 
+0.00099459 

Uk' 

+0.00146286 

-0.00191040 

+0.00099135 

+0.00100751 

clg  Uk' 

2.8347972+ 

2.7188755- 

3.0037761  + 

2.9967506  + 

Igdk 

clg  nk  +i 

0.3010300+ 

9.7919889  + 

8.0000000  + 
0.0000000  + 

0.6989700+ 
9.8185669+ 

lg(dk/nk+i) 

0.0930189  + 

8.0000000  + 

0.5175369  + 

-dk/nk+i 
1/Uk' 

-  1.2389 
+683.5924 

-  0.0100 

-  523.4504 

3.2926 
+  1008.7327 

« 

1/C/k+i 

+682.3535 

-  523.4604 

+  1005.4401 

lg  Uk+i 

7.1659906+ 

7.2811162- 

6.9976438  + 

lg  1*4'  =  clg  UA'  =  2.9967506  +  ;  <  =  A4F,  =  +992.546 

clg  (Ui,.U2,.U3,.Ua,)  =  1.5541994— 
lg  (U2.U3.Ud  =  1. 4447506— 


lg/=  2.9989500  + 
/=  +997.585 


2.  EDGE  RAY 


k=l 

k=2 

A:=3 

k=4 

—  flk-l 
Ck'-l 

8.8952646  + 
9.7919889  + 

+236.0050 

+682.2850 

+  3.2750 
-685.6727 

+40307.51 
+  1353.49 

ck 

+918.2900 

-682.3977 

+41661.00 

IgCk 

lg  sin#k 
clgrk 

2.9629799  + 
8.4765370- 
7.7399405+ 

2.8340376- 
8.8114112+ 

7.7478511  + 

4.6197297  + 
8.3325613- 

5.3964898- 

lg  sinak 
lg  nkfnk  +i 

9.1794574+ 
0.2080111  + 

9.3932999+ 
9.8185669+ 

8.3487808- 
0.1814331  + 

lg  sinak' 
lg^k 

clg  sin0k  +i 

8.6872535  + 
2.6232493+ 
1.5234630- 

9.3874685  + 
2.2600595+ 

1.1885888+ 

9.2118668+ 
2.2521489  + 
1.6674387- 

8.5302139  - 
4.6035102- 
1.4803948- 

lgck' 

2.8339658+ 

2.8361168- 

3.1314544+ 

4.61411^9  + 

522  Mirrors,  Prisms  and  Lenses  [§  182 

_  ai  =  -   4°  30'  23.24"       c4'  =  +41126.23 
ai'  =  +   2°  47'  22.69"       r4  =  -40133.80 
01  =  O,  ai'-ai=  02  =-   1°  43'    0.55"A4L5  =  +     992.43 
-a2=-  8°  41'  40.45"  -i*4'=-     992.55 
- 10°  24'  41.00"  F'L5  =  -         0.12 
a2'=+14°    T  31.28" 
03  =  +  30  42'  50.28" 
-a3=-14°  19'  13.26" 

-10°  36'  22.98" 
a3'=+  9°  22'  26.69" 

04=-  1°  13'  56.29" 
-a4=  +  1°  16'  45.13" 

+  0°    2'  48.84" 
a4'=-   1°  56'  33.95" 

0O  =  -   1°  53' 45.11" 

Thus,  we  see  that  this  object-glass  has  a  slight  spherical 
aberration  of  —0.12,  that  is,  it  is  a  little  under-corrected 
(§  178). 

182.  The  Sine-Condition  or  Condition  of  Aplanatism. — 
Suppose  that  for  a  certain  object-point  M  (Fig.  234)  on  the 
axis  of  a  symmetrical  optical  instrument  the  spherical  aber- 
ration has  been  abolished  for  all  the  zones  of  the  system,  so 
that  rays  proceeding  from  this  point  will  all  be  accurately 
focused  at  the  conjugate  image-point  M'.  On  a  straight 
line  perpendicular  to  the  axis  at  M  take  a  point  Q  very  close 
to  M;  and  let  ?/'  =  M'Q'  denote  the  size  of  the  image  of  the 
object  2/  =  MQ  which  is  produced  by  the  central  zone,  that  is, 
by  the  paraxial  rays.  Now  even  though  .the  system  is  spher- 
ically corrected  with  respect  to  the  pair  of  axial  points  M,  M', 
it  by  no  means  follows  that  rays  emanating  from  Q  will  all 
meet  again  in  Q'.  In  order  that  this  shall  be  the  case,  the 
magnification-ratio  must  be  equal  to  y'/y  for  all  the  zones 
of  the  system.  Draw  the  object-ray  MBi  and  the  corre- 
sponding image-ray  B2M';  if  the  slopes  of  these  rays  are 


§182] 


Sine-Condition 


523 


denoted  by  6  and  0',  it  may  be  shown  that  for  the  zone 
corresponding  to  the  incidence-point  Bi  the  magnification- 
ratio  is  equal  to  n.sin  d/n'.s'n  6';  and  if  this  is  equal  to  y'jy, 
then  the  image  formed  by  rays  belonging  to  this  zone  will 
be  of  the  same  size  as  the  image  y'  made  by  the  paraxial  rays. 


Fig.  234. — Sine-condition. 

Thus,  in  order  that  with  the  employment  of  wide-angle 
bundles  of  rays  a  symmetrical  optical  instrument  may  pro- 
duce a  sharp  image  of  a  little  plane  element  perpendicular 
to  the  axis  of  the  instrument,  not  only  must  the  system  be 
spherically  corrected  for  the  pair  of  conjugate  axial  points 
M,  M',  but  it  must  also  satisfy  the  so-called  Sine-Condition, 
namely, 

n.sin  6  _yf  _ 
n'. sin  d'  y 
This  celebrated  principle  was  clearly  formulated  by  Abbe 
in  1873,  but  it  had  already  been  recognized  by  Seidel,  and 
it  may  be  deduced  from  a  general  law  of  radiant  energy  which 
was  first  given  by  Clausius  (1864).  The  proof  of  it  must  be 
omitted  here.  It  may  be  stated  in  words  as  follows:  The 
necessary  and  sufficient  condition  that  all  the  zones  of  a 
spherically  corrected  system  shall  produce  images  of  equal 
size  at  the  point  M'  conjugate  to  the  axial  point  M  is  that, 
for  all  rays  proceeding  from  M,  the  ratio  of  the  sines  of  the 
slope-angles  6,  6'  of  each  pair  of  corresponding  incident 
and  emergent  rays  shall  be  constant ;  that  is, 

sin  d     n'    wm  ,      , 

- — tt.  =  —  .  y  =  constant, 
sin  0     n 


524  Mirrors,  Prisms  and  Lenses  [§  182 

The  sine-condition 

n.y. sin  6  =  n'.yf. sin  0' 
is  essentially  different  from  the  Smith-Helmholtz  law  for 
paraxial   rays    (see   §88   and    §118),    namely,   n.?/.tan0  = 
n'.i/'.tan  6f,  although  when  the  angles  6,  6'  are  small,  both 
conditions  may  be  expressed  by  the  equation  n.y.  d  =  nf.yr.  6r. 

If  the  optical  system  is  spherically  corrected  for  the  pair 
of  axial  points  M,  M',  and  if  at  the  same  time  the  sine-condi- 
tion is  satisfied,  the  points  M,  M'  are  called  the  aplanatic 
pair  of  points  of  the  system.  It  may  be  demonstrated  that 
no  optical  system  can  have  more  than  one  pair  of  such  apla- 
natic points.  In  the  case  of  a  single  spherical  refracting  sur- 
face the  two  points  J,  J'  (§  177)  whose  distances  from  the 
center  C  are  such  that 

CJ.CJ'  =  r2,  n.CJ=n'.CJ', 
are  a  pair  of  aplanatic  points  as  above  defined;  for  they  are 
free  from  spherical  aberration  and  if  they  are  joined  by 
straight  lines  BJ,  BJ'  with  any  point  B  on  the  spherical  re- 
fracting surface,  and  if  we  put  6  =  Z  CJB,  d'  =  Z  CJ'B,  we 
have  sin  0/sin  d'  =  nfn'  =  constant.  This  property  of  the 
points  J,  J'  of  a  refracting  sphere  has  been  ingeniously  util- 
ized in  the  construction  of  the  objective  of  the  compound 
microscope. 

If  in  Fig.  234  we  put  Z  =  BiM,  then  smd=—h/l,  where  h 
denotes  the  height  of  the  point  Bi  above  the  axis.  Hence, 
the  sine-condition  may  be  written : 


or  since  (§  124) 


where  /,  f  denote  the  focal  lengths  of  the  system  and  x  de- 
notes the  abscissa  of  M  with  respect  to  the  primary  focal 
point  F  (x  =  FM),  we  obtain  also: 

h     _lf 
sin  6'      x 


I. 

h 

sin0' 

n 

-y; 

V 

=/  = 

n 

f 

X 

~~n' 

'  x1 

§  183]  Caustic  Surfaces  525 

Suppose  now  that  the  object-point  M  is  infinitely  distant  so 
that  x  =  l—  oo  ;  then  for  a  ray  parallel  to  the  axis  meeting  the 
first  surface  at  the  height  h,  we  shall  have: 

sin  0'  J  ' 
Thus,  if  the  aplanatic  points  are  the  infinitely  distant  point 
of  the  axis  and  the  second  focal  point  F',  and  if  around  F'  as 
center  we  describe  a  sphere  of  radius  equal  to  /',  the  parallel 
object-rays  will  meet  their  corresponding  image-rays  on  the 
surface  of  this  sphere;  whereas  in  the  case  of  collinear  imagery 
with  paraxial  rays  the  points  of  intersection  of  the  incident 
and  emergent  rays  under  the  same  circumstances  will  all  lie 
in  the  secondary  principal  plane  (§  119),  which  touches  the 
sphere  above  mentioned  at  the  point  where  the  axis  crosses  it. 
If  therefore  we  put  h/sin  6'  =  e,  the  sine-condition  for  an 
infinitely  distant  object  is  e+f=0.  For  example,  in  the 
case  of  the  telescope  objective  calculated  in  §  181: 

lg  hi  =  1.5185139  + 
clg  sin  05=1.4803948- 

lge  =2.9989087-        e=  -997.490 
/=  +997.585 
e+f=  +     0.095 
Accordingly,  the  sine-condition  is  very  nearly  satisfied  in  the 
case  of  this  object-glass. 

183.  Caustic  Surfaces. — The  characteristic  geometrical 
property  of  a  bundle  of  light-rays  emanating  originally  from 
a  point-source  is  expressed  in  a  law  announced  by  Maltjs  in 
1808  (§  39),  which  may  be  stated  in  terms  of  the  undulatory 
theory  of  light  as  follows :  The  rays  of  light  are  always  normal 
to  the  wave-surfaces.  In  fact,  what  is  meant  by  a  wave-sur- 
face is  any  surface  which  cuts  the  rays  orthogonally.  In 
general,  the  curvatures  of  the  normal  sections  at  any  point  of 
a  curved  surface  will  vary  from  one  azimuth  to  another;  but, 
according  to  Euler's  theorem  (§  111),  the  normal  sections  of 
greatest  and  least  curvature,  called  the  principal  sections  of 


526  Mirrors,  Prisms  and  Lenses  [§  184 

the  surface  at  the  point  in  question,  are  always  at  right 
angles  to  each  other.  It  is  well  known  that  the  normal  drawn 
to  any  point  of  a  curved  surface  will  not  meet  the  normal  at 
a  consecutive  point  taken  arbitrarily.  But  if  the  consecutive 
point  is  taken  in  the  direction  of  either  of  the  principal  sec- 
tions, the  two  consecutive  normals  will  intersect.  Thus, 
along  each  normal  to  a  curved  surface  there  are  two  points 
called  the  principal  centers  of  curvature  (§  111),  where  con- 
secutive normals  lying  in  the  two  principal  sections  intersect. 

Accordingly,  if  we  regard  a  bundle  of  rays  of  light  as  a  sys- 
tem of  normals  to  the  wave-surface,  we  may  say  that  each 
ray  determines  two  principal  sections  of  the  bundle,  and  that, 
in  general,  there  will  be  two  points  on  the  ray,  the  so-called 
image-points  (cf.  §  113),  where  contiguous  rays  in  each  of  the 
two  principal  sections  intersect  the  ray  in  question.  The  as- 
semblage of  these  pairs  of  image-points  on  all  the  rays  of  a 
wide-angle  bundle  of  rays  emanating  originally  from  a 
point-source  form  a  surface  of  two  sheets  called  the  caustic 
surface  (cf.  §  42).  Each  ray  of  the  bundle  is  tangent  to  both 
sheets  of  the  caustic  surface.  In  the  special  case  when  the 
bundle  of  rays  is  symmetrical  about  an  axis,  one  sheet  of 
the  caustic  surface  will  be  a  surface  of  revolution,  whereas 
the  other  sheet  will  be  a  portion  of  the  axis  of  symmetry  (see 
§  178). 

184.  Meridian  and  Sagittal  Sections  of  a  Narrow  Bundle 
of  Rays  before  and  after  Refraction  at  a  Spherical  Surface 
— The  apertures  of  the  bundles  of  effective  rays  which  are 
transmitted  through  a  symmetrical  optical  instrument  are 
all  limited  by  the  position  and  dimensions  of  the  aperture- 
stop  (§  134).  For  the  present  it  will  be  assumed  that  the 
diameter  of  the  stop  is  very  small.  Each  point  of  the  object 
lying  in  the  field  of  view  is  the  source  of  a  narrow  bundle  of 
rays  which  contains  one  ray,  called  the  chief  ray  (§  140), 
which  in  traversing  the  medium  where  the  stop  is  placed, 
passes  through  the  center  of  the  stop.  Accordingly,  the  chief 
ray  will  he  in  the  meridian  plane  determined  by  the  object- 


§184] 


Astigmatism  of  Oblique  Bundle 


527 


point  where  the  bundle  of  rays  originates.  The  path  of  this 
chief  ray  may  be  traced  geometrically  by  Young's  construc- 
tion (§  176)  or  it  may  be  calculated  trigonometrically  by 
means  of  the  system  of  formula?  given  in  §  181.  We  have 
now  to  investigate  the  positions  on  this  chief  ray  of  the  two 
image-points  produced  by  the  intersections  of  this  ray  with 
the  rays  immediately  adjacent  to  it  lying  in  the  two  prin- 
cipal sections  of  the  bundle  as  determined  by  its  chief  ray 
(§  183).  Whenever  a  narrow  bundle  of  rays  has  two  such 
image-points,  it  is  said  to  be  a  tigmatic.  Practically,  this  is 
always  the  case  if  the  chief  ray  is  incident  on  a  refracting 
surface  at  an  angle  a  which  is  not  vanishingly  small.  Under 
such  conditions  the  bundle  of  refracted  rays  will  be  astig- 
matic, and  we  have  the  case  which  some  writers  call  "  astig- 
matism by  incidence"  but  which  is  better  described  as  the 
astigmatism  of  an  oblique  bundle  of  rays,  as  distinguished  from 


Fig.  235. — Meridian  section  of  narrow  bundle  of  rays  refracted  at  spherical 

surface. 


the  astigmatism  produced  by  direct  (normal)  incidence  on 
an  astigmatic  refracting  surface  or  surface  of  double  curva- 
ture (Chapter  IX). 

In  the  diagrams  (Figs.  235,  236) ,  which  show  the  meridian 
section  ZZ  of  a  spherical  refracting  surface  whose  center  is  at 
C  and  vertex  at  A  (Fig.  235),  the  point  designated  by  P  (or  Q) 
represents  an  object-point  which  is  the  source  of  a  narrow 


528 


Mirrors,  Prisms  and  Lenses 


184 


homocentric  bundle  of  rays  whose  chief  ray  PB  (or  QB)  is 
incident  on  the  surface  at  the  point  B  at  the  angle  of  inci- 
dence a.  This  ray  crosses  the  axis  at  the  point  marked  L 
in  Fig.  235  and  the  corresponding  refracted  ray  crosses  the 


Fig.  236. — Sagittal  section  of  narrow  bundle  of  rays  refracted  at  spherical 

surface. 

axis  at  I/.  One  of  the  principal  sections  of  the  bundle  of  in- 
cident rays  will  be  the  meridian  section  (§§  112,  113)  made  by 
the  plane  containing  the  optical  axis  and  the  vertex  P  (or  Q) 
of  the  bundle,  that  is  the  plane  of  the  paper;  whereas  the 
other  principal  section,  called  the  sagittal  section  (Fig.  236), 
is  made  by  a  plane  which  intersects  the  meridian  plane 
at  right  angles  along  the  chief  ray  of  the  bundle.  The  point 
G  (Fig.  235)  is  a  point  on  the  spherical  refracting  surface  in 
the  meridian  section,  taken  exceedingly  close  to  the  point  B. 
Likewise,  the  point  D  (Fig.  236)  lies  on  the  spherical  refract- 
ing surface  very  near  to  B ;  but  it  is  contained  in  the  sagittal 
section  and  is  represented  in  the  diagram  as  lying  slightly 
above  the  plane  of  the  paper.  The  ray  PG  (Fig.  235)  after 
refraction  meets  the  chief  refracted  ray  at  the  image-point 
P'  of  the  narrow  pencil  of  refracted  meridian  rays.  Similarly, 
the  ray  QD  (Fig.  236)  after  refraction  will  meet  the  chief 
refracted  ray  at  the  image-point  Q'  where  the  straight  line 
QC  intersects  this  ray,  as  will  be  immediately  obvious  by 


§  185]  Sagittal  Section  of  Narrow  Bundle  529 

supposing  that  the  triangle  QBQ'  is  revolved  around  the 
central  line  QQ'  as  axis  through  a  small  angle  out  from  the 
plane  of  the  paper.  Thus,  whereas  the  meridian  section  of 
the  bundle  of  refracted  rays  is  contained  in  the  same  plane  as 
the  meridian  section  of  the  bundle  of  incident  rays,  the  sagit- 
tal sections  are  in  two  different  planes  BDQ  and  BDQ'  (Fig. 
236)  which  intersect  each  other  in  a  straight  line  perpendic- 
ular to  the  meridian  plane  at  the  point  B,  that  is,  in  the  line 
BD,  which,  since  the  point  D  is  infinitely  near  to  B,  may  be 
regarded  as  a  straight  line. 

185.  Formula  for  Locating  the  Position  of  the  Image- 
Point  Q'  of  a  Pencil  of  Sagittal  Rays  Refracted  at  a  Spher- 
ical Surface. — As  was  explained  (§  184),  the  image-point  Q' 
(Fig.  236)  in  the  sagittal  section  corresponding  to  the  object- 
point  Q  is  at  the  point  of  intersection  of  the  straight  line  QC 
with  the  chief  ray  of  the  bundle  of  refracted  rays.  This  con- 
struction suggests  at  once  a  method  of  obtaining  an  analyt- 
ical relation  connecting  the  points  Q  and  Q';  for  if  the  straight 
line  QQ'  is  regarded  for  the  time  being  as  the  axis  of  the  spher- 
ical refracting  surface,  and  if  we  put  g  =  BQ,  q'  =  BQ'  (where 
the  distances  denoted  by  q,  q'  are  to  be  reckoned  positive  or 
negative  according  as  they  are  measured  from  the  incidence- 
point  B  in  the  same  direction  as  the  light  takes  along  the 
chief  ray  or  in  the  opposite  direction,  respectively),  we  have 
merely  to  write  q,  qr  in  place  of  the  symbols  I,  V  in  the  formula 
derived  in  §  180  in  order  to  obtain  the  desired  relation, 
namely, 

q'     q 

where  the  function  denoted  here  by  D  is  a  constant  for  a 
given  chief  ray  and  is  defined  by  the  following  expression: 
n_w'.cosa/— n.cosa_n.sin(a  —  a') 
r  r.sin  a' 

Thus  having  ascertained  the  path  of  the  chief  ray,  and  know- 
ing the  position  of  the  object-point  Q,  that  is,  being  given 


530  Mirrors,  Prisms  and  Lenses  [§  186 

the  value  of  q,  we  may  calculate  the  value  of  q'  by  means  of 
the  above  formula  and  thus  locate  the  position  of  the  image- 
point  Q'  of  the  sagittal  section  of  the  bundle  of  refracted  rays. 
186.  Position  of  the  Image-Point  P'  of  a  Pencil  of  Me- 
ridian Rays  Refracted  at  a  Spherical  Surface. — The  angles 
of  incidence  and  refraction  of  the  chief  ray  are  denoted  by 

a,  a',  respectively.  Moreover,  let  6,  df  (Fig.  235)  de- 
note the  angles  which  the  chief  ray  makes  with  the  axis  of 
the  spherical  refracting  surface  before  and  after  refraction, 
respectively;  and  also  let  the  central  angle  BCA  be  denoted 
by  4>.  Then  for  a  contiguous  ray  in  the  meridian  section 
which  is  incident  at  the  point  G  very  close  to  the  point  B, 
these  angles  may  be  denoted  by   a+da,    a'+da';    d+dd, 

B'+dd';  and  <j)+d<j),  where  da,  da',  etc.,  denote  the 
little  increments  in  the  magnitudes  of  the  angles  a,  a',  etc., 
in  passing  from  the  chief  ray  to  an  adjacent  ray  in  the  merid- 
ian section.  Now  since  for  the  rays  PB  and  PG  these  angles 
are  connected  by  the  formulae  (§  180); 

a=  d+4>,         a+da=  d+dd+ct>+d<j), 
we  obtain  by  subtraction: 

da  =  dd+d<f). 
Around  P  as  center  and  with  radius  equal  to  PB  describe  the 
small  arc  BU  which  subtends  ZBPG  =  <i0;  so  that  we  may 
write: 

,  n         arc  BU 

a  u  = , 

V 

where  p  =  BP  denotes  the  distance  of  the  object-point  P  from 
the  incidence-point  B,  being  reckoned  positive  or  negative 
exactly  in  the  same  way  as  q  in  §  185.  Now  the  sides  of  the 
little  curvilinear  triangle  BGU  may  be  considered  as  straight 
to  the  degree  of  approximation  with  which  we  are  concerned 
at  present,  and  since  the  sides  of  the  angle  GBU  are  per- 
pendicular to  the  sides  of  the  angle  of  incidence  a,  so  that 
Z  GBU  =  a,  we  obtain : 

arc  BU  =  arc  GB.cosa. 


§  186]         Meridian  Section  of  Narrow  Bundle  531 

Combining  this  relation  with  the  one  above,  we  have  there- 
fore : 

in         arc  GB. cos  a 

dd  =  — . 

V 
Moreover,  since  Z  GCB  =  dcf>, 

,  ,     arc  GB 
d<f>  =  ——-; 

and,  therefore,  by  adding  this  equation  to  the  last  and  taking 
account  of  the  relation  above,  we  find : 

da- g-2!^.  aw  OB.  (1) 

Similarly,  for  the  corresponding  refracted  rays  BP'  and 
GP'  which  intersect  at  the  image-point  P',  for  which  BP'  = 
p',  we  can  derive  the  analogous  relation : 

da'=g-^).awGB.  (2) 

Now  according  to  the  law  of  refraction, 

ft.sina  =  ft'.sina',  n.sin(a+da)  =  n' '.sin(  a' +d  a') , 
and  if  in  the  expansions  of  sin(a+<ia)  and  sm(a'+da') 
we  write  da  and  da'  in  place  of  sinda  and  sinda'  and  put 
cosd  a  =  cosd  a'  =  1,  as  is  permissible  on  account  of  the  small- 
ness  of  these  angles,  we  may  derive  the  following  relation 
between  d  a  and  d  a' : 

nxosa.da  =  n'.cosa'.da'.  (3) 

Hence,  multiplying  equation  (1)  by  n.cosa  and  equation  (2) 
by  n'.cos  a',  and  equating  the  two  expressions  thus  obtained, 
according  to  equation  (3),  we  find,  after  removing  the  com- 
mon factor,  arc  GB,  the  following  formula  connecting  the 
ray-intercepts  p  and  p' : 

,  ,  /cos  a'     \\  /cos  a     1\ 

n.cosa    ( — —  —  -  )  =n.cosa( — -)  : 

V    p         r!  \    p        rl 

which  may  also  be  written  thus: 

n'. cos2  a/_ft.cos2  a  _  ~ 

—    —P     u' 

where  the  symbol  D  has  the  same  meaning  as  before  in  §  185. 
If  we  introduce  Abbe's  differential  notation  and  use  the 


532 


Mirrors,  Prisms  and  Lenses 


186 


operator  A  placed  in  front  of  a  symbol  to  denote  the  differ- 
ence in  the  value  of  the  magnitude  denoted  by  the  symbol 
before  and  after  refraction,  that  is,  for  example,  if  £±z  =  z'  —  z; 
then  we  may  write  the  two  formulae  for  p  and  q  in  the 
following  abbreviated  form: 

\n  _  A^-cos2a_  n 
q~  p 

The  position  of  the  image-point  P'  of  the  meridian  section 
of  a  narrow  bundle  of  rays  refracted  at  a  spherical  surface 
may  also  be  quickly  ascertained  by  a  simple  geometrical 


Fig.  237. — Construction  of  center  of  perspective  (K)  with  respect  to  a  given 
ray  refracted  at  a  spherical  surface. 

construction  which  depends  on  finding  a  point  K  called  the 
center  of  perspective,  which  bears  precisely  the  same  relation 
to  the  pair  of  points  P,  P'  as  the  center  C  of  the  spherical 
surface  bears  to  the  pair  of  points  Q,  Q'  (§  184) ;  that  is,  just 
as  the  straight  line  QQ'  must  pass  through  C,  so  also  the 
straight  line  PP'  must  pass  through  K.  The  existence  of 
this  point  K  was  first  recognized  by  Thomas  Young  (1801). 
In  the  diagram  (Fig.  237)  the  chief  incident  ray  is  represented 
by  the  straight  fine  RB  and  the  chief  refracted  ray,  con- 
structed by  the  method  given  in  §  175,  is  represented  by  the 
straight  fine  BT.  From  the  center  C  draw  CY  and  CY' 
perpendicular  to  RB  and  BT  at  Y  and  Y',  respectively.    The 


§  187]  Astigmatic  Difference  533 

point  K  will  be  found  to  lie  at  the  point  of  intersection  of  the 
straight  lines  YY'  and  SS';  and  hence  if  P  designates  the 
position  of  an  object-point  lying  anywhere  on  the  chief  in- 
cident ray,  the  corresponding  image-point  P'  in  the  meridian 
section  will  lie  at  the  point  where  the  straight  line  PK  meets 
the  chief  refracted  ray.  This  beautiful  construction  is  ex- 
ceedingly useful  in  graphical  methods  of  investigating  the 
imagery  in  the  meridian  section  along  a  particular  ray.  The 
proof  of  the  construction  is  not  at  all  difficult,  but  it  cannot 
be  conveniently  given  here. 

187.  Measure  of  the  Astigmatism  of  a  Narrow  Bundle  of 
Rays. — We  have  seen  that,  in  general,  a  narrow  homocentric 
bundle  of  rays  falling  obliquely  on  a  spherical  refracting 
surface  is  transformed  into  an  astigmatic  bundle  of  refracted 
rays,  so  that  corresponding  to  a  given  object-point  P  (or  Q) 
there  will  be  two  so-called  image-points  P'  and  Q'  lying  on 
the  refracted  chief  ray  at  the  points  of  intersection  of  the 
rays  of  the  meridian  and  sagittal  sections,  respectively.  The 
interval  between  these  image-points,  that  is,  the  segment 
P'Q'  =  q'  —  p'  is  called  the  astigmatic  difference.  However,  it 
is  more  convenient  to  measure  the  astigmatism  by  the  dif- 
ference between  the  reciprocals  of  the  linear  magnitudes  p' 
and  q'.  If,  for  example,  according  to  the  system  of  notation 
introduced  in  §  106,  we  put 

n/p=P,     rt/p'=P',     n/q  =  Q,     n'jq'^Q', 
the  formulae  derived  in  §§  185,  186  may  be  written  as  follows: 

Q'-Q  =  P'.cos2  a'-P.cos2  a  =  D; 
where,  on  the  assumption  that  the  meter  is  taken  as  the  unit 
of  length,  the  magnitudes  denoted  by  the  capital  letters  will 
all  be  expressed  in  terms  of  the  dioptry.  The  astigmatism 
of  the  bundle  of  refracted  rays  is  measured  by  (Pf  —  Qr) .  If 
the  bundle  of  incident  rays  is  homocentric  (Q  =  P),  the  as- 
tigmatism of  the  bundle  of  refracted  rays  will  be : 
P'-Q'=  P'.sin2  a'  -  P.sin2  a. 

Accordingly,  we  see  that  the  astigmatism  of  a  bundle  of 
rays  refracted  at  a  spherical  surface  will  vanish  provided 


534  Mirrors,  Prisms  and  Lenses  [§  188 

Q  =  P   and  P'.sin2a'  —  P.sin2a  =  0;   which  may   happen   in 
two  ways,  as  follows: 

(1)  If  a'  =  a  =  0,  that  is,  if  the  chief  ray  of  the  narrow 
bundle  meets  the  refracting  surface  normally,  as,  for  ex- 
ample, when  it  is  directed  along  the  axis,  then  no  matter 
where  the  object-point  may  lie,  the  two  image-points  will 
coincide.  In  fact,  in  case  of  the  axial  ray  we  may  put  Q  = 
P=U,  Q'  =  Pf=Uf,  D  =  F,  so  that  the  formulae  for  the  me- 
ridian and  sagittal  sections  both  reduce  in  this  case  to  the 
fundamental  equation  for  the  refraction  of  paraxial  rays  at 
a  spherical  surface,  namely,  U'  =  U+F. 

(2)  But  for  any  value  of  a,  we  shall  have  P'  —  Q'  =  0, 
that  is,  P'.sin2a'  =  P.sin2a,  provided  P7n'2  =  P/n2  or 
n'.p'  =  n.p.  In  this  case  the  points  designated  by  P,  P' 
(or  Q,  Q')  are  identical  with  the  points  S,  S'  in  Figs.  226  to 
229.  If  the  vertex  of  the  homocentric  bundle  of  incident  rays 
lies  at  any  point  S  on  the  surface  of  the  sphere  described 
around  C  as  center  with  radius  equal  to  n'.rjn,  the  bundle 
of  refracted  rays  will  likewise  be  homocentric  with  its  ver- 
tex at  the  corresponding  point  S'  on  the  surface  of  the  con- 
centric sphere  of  radius  n.r/n'  (see  §  177). 

188.  Image-Lines  (or  Focal  Lines)  of  a  Narrow  Astig- 
matic Bundle  of  Rays. — In  all  the  preceding  discussion  of 
the  properties  of  an  astigmatic  bundle  of  rays,  it  cannot  have 
escaped  notice  that  only  such  rays  have  been  considered  as 
are  contained  in  the  two  principal  sections  of  the  bundle.  If 
there  were  no  other  rays  to  be  taken  into  account  besides 
these,  we  might  say  that  to  each  point  of  the  object  P  (or  Q) 
there  corresponded  two  image-points  P'  and  Q'.  But  this  is 
by  no  means  a  complete  or  even  approximately  complete 
statement  of  the  image-phenomenon  in  this  case;  for,  indeed, 
the  rays  which  he  in  neither  of  the  two  principal  sections  do, 
as  a  matter  of  fact,  constitute  by  far  the  greater  portion  of 
the  total  number  of  rays  of  the  bundle.  According  to  the 
theorem  of  Sturm  (1803-1855),  the  constitution  of  a  narrow 
bundle  of  rays  is  exhibited  in  the  accompanying  diagram 


188] 


Sturm's  Conoid 


535 


(Fig.  238)  called  Sturm's  conoid  (§  113).  All  the  rays  of  the 
bundle  pass  through  two  very  short  focal  lines  or  image-lines 
XX  and  YY  which  are  both  perpendicular  to  the  chief  ray. 
The  image-line  XX  which  goes  through  the  point  of  intersec- 
tion P'  of  the  meridian  rays  lies  in  the  plane  of  the  sagittal 
section;  and,  similarly,  the  image-line  YY  which  goes  through 
the  point  of  intersection  Q'  of  the  sagittal  rays  lies  in  the 


Fig.  238. — Sturm's  conoid. 

plane  of  the  meridian  section.  Strictly  speaking,  this  theo- 
rem can  be  regarded  as  representing  the  actual  facts  only  on 
the  assumption  that  the  bundle  of  rays  is  infinitely  thin ;  and 
on  this  assumption  the  entire  bundle  may  be  conceived  as 
generated  by  a  slight  rotation  either  of  the  meridian  section 
around  the  image-line  YY  as  axis,  whereby  the  point  P'  will 
trace  the  image-line  XX,  or  of  the  sagittal  section  around 
the  image-line  XX  as  axis,  whereby  the  point  Q'  will  trace 
the  image-line  YY.  Thus,  according  to  Sturm's  theorem, 
with  an  object-point  P  (or  Q)  lying  on  the  chief  ray  of  an 
infinitely  narrow  bundle  of  incident  rays  there  are  associated 
two  exceedingly  tiny  image-lines  lying  in  the  principal  sec- 
tions of  the  bundle  of  refracted  rays  at  right  angles  to  the 
chief  ray.  Not  only  as  to  the  orientation  of  the  image-lines 
of  Sturm,  but  as  to  their  practical,  nay,  even  as  to  their 
mathematical  existence,  there  has  been  much  controversy, 
but  we  cannot  enter  into  this  discussion  here.  In  spite  of 
its  limitations  and  admittedly  imperfect  representation, 
Sturm's  conoid  remains  a  very  useful  preliminary  mode  of 
conception  of  the  character  of  a  narrow  astigmatic  bundle 


536 


Mirrors,  Prisms  and  Lenses 


[§189 


of  rays.  The  only  proper  way  of  arriving  at  a  more  accurate 
knowledge  of  the  constitution  of  a  bundle  of  light-rays  is  by 
the  aid  of  the  powerful  methods  of  the  infinitesimal  geom- 
etry. Mathematical  investigations  of  this  kind  have  been 
pursued  with  great  skill  by  Gullstrand  whose  writings  con- 
tained in  a  series  of  published  papers  and  treatises  dating 
from  about  1890  have  extended  the  domain  of  theoretical 
optics  far  beyond  the  narrow  limits  imposed  upon  it  by 
Gauss  and  the  earlier  writers  on  this  subject. 

189.  The  Astigmatic  Image-Surfaces. — Thus,  the  effect 
of  astigmatism  is  that  the  rays  of  a  narrow  oblique  bundle, 
instead  of  being  brought  to  a  focus  at  a  single  point,  pass 
through  two  small  focal  lines  at  right  angles  to  the  path  of 


ccv 


oo 


oc- 


cc- 


Fig.  239. — Astigmatic  image-surfaces. 


the  chief  ray  in  the  image-space.  If  the  chief  rays  proceeding 
from  the  various  object-points  lying  in  a  meridian  plane  of  a 
symmetrical  optical  instrument  are  constructed,  and  if  along 
each  of  these  rays  the  positions  of  the  image-points  P',  Q'  of 
the  pencils  of  meridian  and  sagittal  rays  are  determined,  the 
loci  of  these  points  will  be  two  curved  lines,  both  symmetri- 
cal with  respect  to  the  axis,  which  touch  each  other  at  their 
common  vertex  on  the  axis.  In  the  diagram  (Fig.  239)  the 
object  is  supposed  to  be  infinitely  distant,  as,  for  example, 
in  the  case  of  a  landscape  photographic  lens.    The  contin- 


§  189]  Astigmatic  Image-Surfaces  537 

uous  curved  line  represents  the  locus  of  the  points  of  inter- 
section of  the  sagittal  rays,  whereas  the  dotted  curve  repre- 
sents the  locus  of  the  points  of  intersection  of  the  meridian 
rays.  These  curved  lines  are  the  traces  in  the  meridian  plane 
of  the  two  astigmatic  image  surfaces  which  are  generated  by 
revolving  the  figure  around  the  axis  of  symmetry.  The  two 
image-surfaces  which  correspond  to  a  definite  transversal 
plane  in  the  object-space,  and  which  are  the  loci  of  the  most 
sharply  defined  images  of  object-points  lying  in  this  plane,  are 
not  to  be  confused  with  the  two  sheets  of  the  caustic  surface 
of  a  wide-angle  bundle  of  rays  emanating  from  a  single  point 
of  an  object  (§  183).  The  focal  lines  of  the  narrow  pencils  of 
meridian  rays  lie  on  one  of  these  surfaces  and  the  focal  lines 
of  the  narrow  pencils  of  sagittal  rays  lie  on  the  other  surface. 
The  positions  and  forms  of  the  image-surfaces  will  depend 
essentially  on  the  place  of  the  stop;  for  it  is  evident  that  if 
the  stop  is  shifted  to  a  different  place,  the  chief  ray  of  each 
bundle  (§§  140,  184)  will  be  a  different  ray,  and  the  points 
P'  and  Q'  will  all  occupy  entirely  different  positions.  If  a 
curved  screen  could  be  exactly  adjusted  to  fit  one  of  the 
image-surfaces,  a  fairly  sharp  image  of  the  object  might  be 
focused  on  it,  but  not  only  would  the  image  be  curved  in- 
stead of  flat,  but  there  would  also  be  a  certain  astigmatic 
deformation  due  to  the  fact  that  each  point  of  the  object 
would  be  reproduced  not  by  a  point  but  by  a  little  focal  line, 
as  has  been  explained.  Between  the  two  image-points  P'  and 
Q'  on  each  chief  ray  there  lies  a  certain  approximately  circu- 
lar cross-section  of  the  narrow  astigmatic  bundle  known 
(§  113)  as  the  "circle  of  least  confusion,"  and  the  locus  of 
the  centers  of  these  circles  will  lie  on  a  third  surface  interme- 
diate between  the  other  two,  which  is  sometimes  taken  as  a 
kind  of  average  or  compromise  image-surface. 

There  can  be  no  doubt  that  astigmatism  of  oblique  bundles 
is  responsible  for  serious  defects  in  the  image  produced  by 
an  optical  instrument,  and  much  pains  has  been  bestowed  on 
trying  to  remedy  this  fault  as  far  as  possible.    Fortunately, 


538 


Mirrors,  Prisms  and  Lenses 


189 


-5 


+2 


the  possibility  of  abolishing  astigmatism  of  this  kind,  that 
is,  of  making  the  two  image-surfaces  coincide  in  a  single 
surface,  is  afforded  by  the  fact  that  the  astigmatic  difference 
(§  187)  is  opposite  in  sign  according  as  the  refracting  surface 
is  convergent  or  divergent.     For  example,  Fig.  240  shows 

graphically  the  opposite 
effects  of  a  convergent 
and  a  divergent  spheri- 
cal refracting  surface 
under  otherwise  equal 
conditions.  The  two 
curves  on  the  left-hand 
side  relate  to  the  con- 
vergent system,  and  the 

Fig.  240. — Astigmatism  of  convergent  two  Curves  On  the  right- 
spherical  refracting  surface  (plotted  ,  ,  •-,  -,  ,  ,  ,,  . 
on  the  left)  and  astigmatism  of  diverg-  nana-  siae  relate  tO  tUe 
ent  spherical  refracting  surface  (plotted  divergent  System  J  and 
on  the  right).                                                                 ,1      ,         ,         -, 

we  see  that  not  only  are 
the  curvatures  opposite  in  the  two  cases,  but  the  relative 
positions  of  the  curves  are  different.  It  will  not  be  difficult 
to  understand  that  it  may  be  possible,  by  suitable  choice  of 
the  radii  of  the  refracting  surfaces  and  of  their  distances 
apart  and  also  of  the  position  of  the  stop,  to  design  a  system 
which  will  be  free  from  astigmatism  at  any  rate  for  a  certain 
zone  of  the  lens;  so  that,  although  we  may  not  be  able  to 
make  the  two  astigmatic  image-curves  coincide  absolutely 
throughout  their  entire  extent,  we  may  contrive  so  that  the 
two  curves  are  nowhere  very  far  apart,  while  at  one  point, 
corresponding  to  the  corrected  zone,  they  actually  intersect 
each  other. 

190.  Curvature  of  the  Image  — Now  let  us  suppose  that 
the  astigmatism  of  oblique  bundles  has  been  completely 
abolished  for  a  certain  angular  extent  of  the  field  of  view,  so 
that  at  last  there  is  strict  point-to-point  correspondence  by 
means  of  narrow  bundles  of  rays  between  object  and  image. 
The  two  image-surfaces  have  thus  been  merged  into  one,  and 


190] 


Curvature  of  Image 


539 


over  this  surface,  within  the  assigned  limits,  the  definition 
of  the  image  is  clear-cut  and  distinct.  There  still  remains, 
however,  another  trouble  due  to  the  fact  that  the  image  is 
curved  and  not  flat;  consequently,  if  the  image  is  received 
on  a  plane  focusing  screen,  only  those  parts  of  the  stigmatic 
image  which  lie  in  the  plane  of  the  screen  will  be  in  focus 
(Fig.  241),  whereas  the.  rest  of  the  image  on  the  screen  will 
be  blurred. 
Now  this  error  of  the  curvature  of  the  image  cannot  be 


Stigmatic 

Surtax 


Focus 
Scrc«U 


Fig.  241. — Curvature  of  stigmatic  image. 

overcome  by  employing  methods  similar  to  those  above  de- 
scribed for  the  abolition  of  astigmatism.  For  the  correction 
of  the  latter  error  the  particular  kinds  of  glass  of  which  the 
lenses  were  made  were  not  essential;  whereas  with  unsuitable 
kinds  of  glass  there  is  no  choice  of  the  radii,  thicknesses,  etc., 
which  will  yield  an  image  which  is  at  the  same  time  stig- 
matic and  flat.  This  fact  was  well  known  to  Petzval  (1807- 
1891).  Petzval's  formula  (published  in  1843)  for  the  abo- 
lition of  the  curvature  of  a  stigmatic  image  produced  by  a 
system  of  infinitely  thin  lenses  in  contact  with  each  other  is 

F 


540  Mirrors,  Prisms  and  Lenses  [§  191 

where  Fj  denotes  the  refracting  power  and  nx  denotes  the 
index  of  refraction  of  the  ith  lens  of  the  system.  This 
formula  is  equivalent  also  to  the  following  statement:  The 
curvature  of  the  stigmatic  image  of  an  infinitely  distant 
object  in  a  system  of  lenses  whose  total  thickness  is  negli- 
gible is  equal  to 

-  2  (refracting  powers  of  all  lenses  of  index  n)  \ 
n  J 

The  general  principal  of  this  equation  was  discovered  by  Airy 
and  was  given  by  Coddington  in  his  treatise  published  in 
1829.  Seidel  pointed  out  that  the  two  faults  of  astigmatism 
and  curvature  could  not  both  be  corrected  at  the  same  time 
unless  some  of  the  convex  lenses  of  the  system  were  made  of 
more  highly  refracting  glass  than  the  con- 
cave lenses.  Now  with  the  varieties  of 
pf  glass  which  were  available  before  the 
production  of  the  modern  Jena  glass, 
this  requirement  was  directly  opposed  to 
the  condition  of  achromatism,  and  as  the 
latter  error  was  considered  more  serious 
than  the  curvature-error,  the  earlier  lens- 


Ff  designers  made  no  attempt  to  obtain  a 

Fig.  242.  —  stigmatic  ga^  stigmatic  image.     But  with  the  new 

image    in     trans-    .  .     _    -  _  ..  .    .,    . 

versai  focal  plane  kinds  of  glass  now  at  our  disposal,  It  IS 
for  a  given  zone  possible  to  design  the  optical  system  so 
that  not  only  is  the  astigmatism  corrected 
for  a  certain  zone,  as  explained  in  §  189,  but  the  point  of  in- 
tersection of  the  two  image-lines  lies  in  the  same  transversal 
plane  as  the  axial  point  where  the  two  image-lines  touch 
each  other  (Fig.  242).  Accordingly,  we  may  say  that  for 
this  zone  the  image  is  both  flat  and  stigmatic.  The  construc- 
tion of  modern  photographic  lenses  which  are  practically 
free  from  these  spherical  errors  is  an  almost  unsurpassed 
triumph  of  human  ingenuity. 

191.  Coma. — Astigmatism  implies  that   the   bundles   of 
rays  concerned  in  producing  the  image  are  very  narrow,  and 


§  191]  Symmetry  in  Sagittal  Section  541 

this  means  that  the  diameter  of  the  stop  is  very  small.  But 
the  validity  of  the  assumptions  which  are  at  the  foundation 
of  geometrical  optics  begins  to  be  caUed  in  question  in  the 
case  of  narrow  bundles  of  rays,  as  was  pointed  out  in  §  175; 
so  that  we  must  be  careful  here  not  to  push  our  conclusions 
too  far.  As  a  matter  of  fact,  in  various  optical  instruments 
and  particularly  in  some  modern  types  of  photographic 
lenses,  the  diameter  of  the  stop  is  by  no  means  small  and  the 


Fig.  243. — Symmetrical  character  of  sagittal  section. 

field  of  view  is  extensive.  The  spherical  aberrations  which 
are  encountered  in  an  optical  system  of  this  kind  are  of  an 
exceedingly  complicated  nature  which  cannot  be  described 
here  in  detail. 

A  bundle  of  rays  of  finite  aperture  emanating  from  a  point 
outside  the  optical  axis  will  show  aberrations  of  a  general 
character  similar  to  the  aberrations  along  the  axis  of  a  direct 
bundle  of  rays  (§  178).  But  the  effects  in  the  two  principal 
sections  of  the  bundle  will  be  very  different  from  each  other; 
because,  whereas  the  rays  in  the  sagittal  section,  being  sym- 
metrically situated  on  opposite  sides  of  the  meridian  plane, 
are  therefore  symmetrical  with  respect  to  the  chief  ray,  as 
represented  in  Fig.  243,  there  will,  in  general,  be  a  complete 
absence  of  symmetry  in  the  meridian  section  (Fig.  244) .  The 
image  (if  indeed  we  may  continue  to  use  this  term)  of  an 
extra-axial  object-point  under  such  circumstances  will  be 
at  best  an  element  of  one  or  other  of  the  two  sheets  of  the 
caustic  surface.    Usually,  however,  what  is  called  the  image 


542 


Mirrors,  Prisms  and  Lenses 


[§191 


is  the  light-effect  as  obtained  on  a  focusing  screen  placed  at 
right  angles  to  the  axis  at  the  place  where  the  central  parts  of 
the  object  are  best  delineated.  The  appearance  on  the  screen 
may  be  described  as  a  kind  of  balloon-shaped  flare  of  light, 


Fig.  244. 


-Unsymmetrical  character  of  meridian  section, 
coma. 


giving  rise  to 


with  a  bright  nucleus  growing  fainter  as  it  expands  in  some 
cases  towards,  in  other  cases  away  from,  the  axis.  This  de- 
fect of  the  image  is  known  to  practical  opticians  as  side-flare 
or  coma  (from  the  Greek  word  meaning  "hair"  from  which 
the  word  "comet"  is  likewise  indirectly  derived).  The  def- 
inition in  the  outer  parts  of  the  field  of  the  object-glass  of  a 
telescope  depends  on  the  removal  of  this  error;  and  this  ap- 
plies also  to  the  case  of  a  wide-angle  photographic  lens.  The 
only  way  to  obtain  a  really  clear  and  accurate  conception  of 
this  important  spherical  aberration  is  to  study  the  forms  of 
the  two  sheets  of  the  caustic  surface.  Generally  speaking, 
we  may  say  that  the  convergence  of  wide-angle  bundles  of 
rays  will  be  better  in  the  case  of  an  optical  system  which  has 
been  corrected  for  astigmatism,  but  even  then  there  will  be 
lack  of  symmetry  in  all  the  sections  of  a  bundle  of  rays  ex- 
cept in  the  sagittal  section.  If  the  slope  of  the  chief  ray  is 
comparatively  slight,  although  not  negligible,  the  condition 
of  a  sharp  focus  is  equivalent  to  Abbe's  sine-condition  (§  182). 
But  for  greater  inclinations  of  the  chief  rays,  it  will  generally 
be  necessary  to  resort  to  the  exact  methods  of  trigonometri- 


192] 


Distortion 


543 


cal  calculation  of  the  ray-paths  in  order  to  determine  the 
nature  and  degree  of  the  convergence. 

192.  Distortion;  Condition  of  Orthoscopy. — Let  us  assume 
that  the  system  has  been  corrected  for  both  astigmatism  and 
curvature  of  the  image,  in  the  sense  explained  in  §  190;  so 
that  by  means  of  narrow  bundles  of  rays  a  flat  stigmatic 
image  is  obtained  of  a  plane  object  placed  at  right  angles  to 
the  axis.  The  next  question  will  be  to  inquire  whether  the 
image  is  a  faithful  reproduction  of  the  object  or  whether  it  is 
distorted.  If  the  image  in  the  "  screen-plane "  (§  134)  is 
geometrically  similar  to  the  object-relief  projected  from  the 
center  of  the  entrance-pupil  on  the  "  focus-plane "  (§  141), 
then  we  may  say  that  the  optical  system  is  orthoscopic  or 
free  from  distortion. 

The  dissimilarity  which  may  exist  between  an  object  and 
its  image  is  a  fault  of  an  essentially  different  kind  from  those 
which  have  been  previously  considered,  and  there  is  no  in- 


Focus  -  Piano  Screen-  Plane 

Fig.  245. — Condition  of  orthoscopy  (freedom  from  distortion) . 

timate  connection  between  this  defect  and  the  others.  Here 
we  are  not  concerned  so  much  with  the  quality  and  defini- 
tion of  the  image  on  the  screen  as  with  the  positions  of  the 
points  where  the  chief  rays  cross  the  screen-plane.  The  po- 
sitions of  these  representative  points  will  not  be  altered  by 
reducing  the  stop-opening  (§§  141,  142);  and  accordingly  the 
image  in  the  screen-plane  is  to  be  regarded  merely  as  a  cen- 
tral projection  on  this  plane  along  the  chief  rays  proceeding 
from  the  center  of  the  exit-pupil. 


544 


Mirrors,  Prisms  and  Lenses 


[§192 


In  the  diagram  (Fig.  245)  the  centers  of  the  entrance-pupil 
and  exit-pupil  of  the  optical  system  are  designated  by  0  and 
0'.  The  straight  lines  PO,  P'O'  represent  the  path  of  a  chief 
ray  which  crosses  the  focus-plane  in  the  object-space  at  the 
point  P  and  the  screen-plane  in  the  image-space  at  the  point 
P'.  If  ?/  =  MP,  s/'  =  M'P'  denote  the  distances  of  P,  P'  from 
the  axis,  then  the  condition  that  the  image  in  the  screen- 
plane  shall  be  similar  to  the  projected  object  in  the  focus- 


t 


a 


Fig.  246. 


-Object   (a)   reproduced  by  image   (b)   barrel-shaped  distortion 
or  by  image  (c)  cushion-shaped  distortion. 


plane,  that  is,  the  condition  of  orthoscopy  (freedom  from 
distortion)  is  that  the  ratio  y'/y  shall  have  a  constant  value 
for  all  values  of  y  within  the  limits  of  the  field  of  view.  If, 
on  the  contrary,  this  is  not  the  case,  and  if  the  ratio  y'/y  is 
variable  for  different  values  of  y,  then  the  image  will  be  dis- 
torted ;  and  this  distortion  will  be  one  of  two  kinds  according 
as  the  ratio  y'/y  increases  or  decreases  with  increase  of  y. 
For  example,  if  the  object  is  in  the  form  of  a  square,  as  shown 
in  Fig.  246,  a,  then,  on  the  supposition  that  y'/y  decreases 
as  y  increases  the  image  of  the  diagonal  will  be  shortened 
relatively  more  than  the  image  of  a  side  of  the  square,  and 
the  square  will  be  reproduced  by  a  curvilinear  figure  with 
convex  sides  as  shown  in  Fig.  246,  b;  this  is  the  case  of  barrel- 
shay  ed  distortion,  as  it  is  called.  On  the  other  hand,  if  the 
ratio  y'/y  increases  in  proportion  as  the  object-point  is  taken 
farther  and  farther  from  the  axis,  we  have  the  opposite  type 
known  as  cushion-shaped  distortion  (Fig.  246,  c). 

If  in  Fig.   245  we  put  OM  =  z,  0'M'  =  z',    ZMOP  =  w, 


§  193]  Airy's  Tangent-Condition  545 

Z  M'O'P'  =  a/,  the  condition  of  orthoscopy  may  be  expressed 

as  follows : 

y'     z'.tano/ 

—  =  =  constant: 

y       2. tana; 

and  if  we  assume,  as  has  been  tacitly  assumed  in  the  pre- 
ceding discussion,  that  the  chief  rays  all  pass  through  the 
pupil-centers  O,  O',  so  that  the  abscissae  denoted  by  z,  z'  have 
the  same  values  for  all  distances  of  the  object-point  P  from 
the  axis,  then  we  derive  at  once  Airy's  tangent-condition  of 
orthoscopy,  namely,  tana/  :  tan  co  =  constant.  But  although 
a  chief  ray  must  pass  through  the  center  of  the  aperture- 
stop  (§  140),  it  will  not  pass  through  the  centers  of  the  pupils 
unless  the  latter  are  free  from  spherical  aberration.  The 
constancy  of  the  tangent-ratio  by  itself  is  not  a  sufficient 
condition  for  orthoscopy;  in  addition,  the  spherical  aberra- 
tion must  be  abolished  with  respect  to  the  centers  of  the 
pupils. 

If  the  optical  system  is  symmetrical  with  respect  to  an  in- 
terior aperture-stop,  the  tangent-condition  will  be  immedi- 
ately satisfied,  because  on  account  of  the  symmetry  of  the 
two  halves  of  the  system,  every  chief  ray  will  issue  in  exactly 
the  same  direction  as  it  had  on  entering,  and  therefore 
tan0:  tan0'  =  l.  Accordingly,  if  a  "symmetric  doublet" 
of  this  kind  is  spherically  corrected  with  respect  to  the  center 
of  the  aperture-stop,  it  will  give  an  image  which  will  be  free 
from  distortion. 

193.  SeidePs  Theory  of  the  Five  Aberrations. — In  the 
theory  of  optical  imagery  which  was  developed  according  to 
general  laws  first  by  Gauss  (§  119)  in  his  famous  Dioptrische 
Untersuchungen  published  in  1841,  the  fundamental  assump- 
tion is  that  the  effective  rays  are  all  comprised  within  a  nar- 
row cylindrical  region  of  space  immediately  surrounding  the 
optical  axis;  this  region  being  more  explicitly  defined  by  the 
condition  that  a  paraxial  ray  is  one  for  which  the  angle  of 
incidence  (a)  and  the  slope-angle  ( 6  ),  in  the  case  of  each 
refraction  or  reflection,  are  both  relatively  so  minute  that  the 


546  Mirrors,  Prisms  and  Lenses  [§  193 

powers  of  these  angles  higher  than  the  first  can  be  neglected 
(§  63).  Evidently,  therefore,  Gauss's  theory  is  applicable 
only  to  optical  systems  of  exceedingly  small  aperture  and 
limited  extent  of  field  of  view.  But  with  the  development 
of  modern  optical  instruments  and  especially  with  the  in- 
crease of  both  aperture  and  field  demanded  for  certain  types 
of  photographic  lenses,  it  became  necessary  to  take  account 
of  rays  which  lie  far  beyond  the  narrow  confines  of  the  central 
or  paraxial  rays.  Long  prior  to  the  time  of  Gauss  important 
contributions  to  the  theory  of  spherical  aberrations  had  been 
made  in  connection  with  certain  more  or  less  special  problems ; 
but  the  first  successful  attempt  to  extend  Gauss's  theory  in 
a  general  way  by  taking  account  of  the  terms  of  higher  orders 
of  smallness  was  made  by  Seidel  (1821-1896)  in  a  re- 
markable series  of  papers  published  between  the  years  1852 
and  1856  in  the  Astronomische  Nachrichten.  Seidel's 
method  consisted  in  tracing  the  path  of  the  ray  through  the 
centered  system  of  spherical  refracting  surfaces  and  in  de- 
veloping the  trigonometrical  expressions  in  series  of  ascend- 
ing powers  which  were  finally  simplified  by  neglecting  all 
terms  above  the  third  order.  If  the  ray-parameters  are  re- 
garded as  magnitudes  of  the  first  order  of  smallness,  it  is 
easy  to  show  that  on  account  of  the  symmetry  around  the 
optical  axis  these  series-developments  can  contain  only  terms 
of  the  odd  orders  of  smallness;  so  that  in  Seidel's  theory 
the  terms  neglected  are  of  the  fifth  and  higher  orders.  It  is 
impossible  to  describe  here  in  detail  the  elegant  mathemati- 
cal treatment  by  which  Seidel  was  enabled  to  arrive  at 
his  final  results;  suffice  it  to  say,  that  he  obtained  a  sys- 
tem of  formulae  from  which  it  was  possible  to  ascertain  the 
influence  both  of  the  aperture  and  the  field  of  view  on  the 
perfection  of  the  image.  In  Seidel's  formulae  the  aber- 
rations of  the  ray,  that  is,  its  deviations  from  the  path  pre- 
scribed by  Gauss's  theory,  are  expressed  by  five  different 
sums,  denoted  by  Si,  $2,  S3,  Si}  and  S5,  which  depend  only  on 
the  constants  of  the  optical  system  and  the  position  of  the 


§  193]  Seidel's  Five  Sums  547 

object-point,  and  which  are,  in  fact,  the  coefficients  of  the 
various  terms  in  the  equations.  The  condition  that  there 
shall  be  no  aberration  demands  that  all  of  these  five  sums 
shall  vanish  simultaneously,  that  is, 

oi  =  02  =  03  =  04  =  05  =  0. 

If,  on  the  other  hand,  these  conditions  are  not  satisfied,  the 
image  yielded  by  the  lens-system  will  not  be  faultless;  and 
therefore  it  will  not  be  without  interest  to  inquire  more  par- 
ticularly into  the  separate  influence  of  each  of  these  five  ex- 
pressions which  occur  in  Seidel's  formulae. 

Thus,  for  example,  if  the  optical  system  is  designed  so  that 
Si  =  0,  then  there  will  be  no  spherical  aberration  at  the  center 
of  the  field  (§  178)  for  the  given  position  of  the  axial  object- 
point.  And  if  not  only  $1  =  0  but  also  $2  =  0,  then  there 
will  be  no  coma  (§  191).  The  condition  S2  =  0  means  also 
that  Abbe's  sine-condition  (§  182)  will  also  be  satisfied,  so 
that  the  image  of  the  parts  of  the  object  in  the  immediate 
vicinity  of  the  axis  is  sharply  defined. 

But  even  when  we  have  0*1  =  0*2  =  0,  the  optical  system 
will,  in  general,  still  be  affected  by  astigmatism  of  oblique 
rays  (§  184),  so  that  an  object-point  lying  at  some  little  dis- 
tance from  the  axis  will  not  be  reproduced  by  an  image-point 
but  at  best  by  two  short  focal  lines  at  different  distances 
from  the  lens-system  and  directed  approximately  at  right 
angles  to  each  other.  Moreover  if  the  distance  of  the  object- 
point  from  the  axis  is  varied,  the  positions  of  these  two  focal 
lines  will  vary  also  both  with  respect  to  their  distance  from 
the  lens-system  and  with  respect  to  their  mutual  distance 
apart.  In  other  words,  when  both  Si  and  $2  vanish,  then,  in 
general,  there  is  no  unique  image  of  a  transversal  object- 
plane,  but  this  latter  may  be  said  to  be  reproduced  by  two 
so-called  image-surfaces  (§  189)  which  are  surfaces  of  revolu- 
tion around  the  optical  axis  and  which  unite  and  touch  each 
other  at  the  point  where  the  axis  crosses  them.  The  ex- 
pressions for  the  curvatures  of  these  surfaces  at  this  common 
point   of  tangency   are   given   by   Seidel's   sums   £3  and 


548 


Mirrors,  Prisms  and  Lenses 


[§193 


St;  so  that  if  also  $3— #4  =  0,  the  two  image  surfaces  will 
coalesce  and  now  the  image  of  the  plane  object  will  be  sharply 
denned,  that  is,  stigmatic,  although  it  will  usually  still  be 
curved.  But  if  also  S3  =  Si  =  0,  the  image  will  be  both  plane 
and  stigmatic.  However,  it  may  still  show  unequal  magnif- 
ications toward  the  margin,  which  means  that  there  is  dis- 
tortion (§  192).  This  last  error  will  be  abolished  provided 
aS5  =  0;  and  now  the  image  may  be  said  to  be  ideal  inasmuch 
as  it  is  flat  and  sharply  defined  not  only  in  the  center  but 
out  .toward  the  edges  and  is  at  the  same  time  a  faithful  re- 
production of  the  plane  object. 

To  attempt  to  derive  Seidel's  actual  formula?  or  even 
to  discuss  the  equations  would  be  entirely  beyond  the  scope 


Fig.  247. — Diagram  representing  the  (i — l)th  and  ith  lenses  of  a  system  of 
infinitely  thin  lenses. 

of  this  volume.  But  it  may  be  convenient  to  insert  here 
without  proof  the  expressions  of  Seidel's  five  sums  for 
the  comparatively  simple  case  of  an  optical  system  considered 
as  composed  of  a  series  of  m  infinitely  thin  lenses  each  sur- 
rounded by  air. 

Let  Ai  (Fig.  247)  designate  the  point  where  the  optical  axis 
crosses  the  ith  lens  of  the  system,  the  symbol  i  being  employed 
to  denote  any  integer  from  1  to  m;  and  let  us  consider  two 
paraxial  rays  which  traverse  the  optical  system,  one  of  which 
emanating  from  the  axial  object-point  Mi  (AiMi=Wi)  and 
meeting  the  first  lens  at  a  point  Bi  such  that  A]Bi  =  /ii, 
crosses  the  axis  after  passing  through  the  (i—  l)th  lens  at  a 
point  Mi  (AiMi  =  wi)  and  meets  the  ith  lens  at  a  point  BA 


§  193]  System  of  Thin  Lenses  549 

such  that  AiBi  =  hY  whereas  the  other  ray,  which  emanates 
from  an  extra-axial  object-point  and  which  in  the  object- 
space  passes  through  the  center  Oi  of  the  entrance-pupil 
(§  139)  of  the  system  (AiOi  =  si)  and  meets  the  first  lens  at 
a  point  Gi  such  that  gfi  =  AiGi,  crosses  the  axis  after  passing 
through  the  (i—  l)th  lens  at  a  point  Oi  AiOi  =  Si)  and  meets 
the  ith  lens  at  a  point  Gj  such  that  AiGi  =  g^    Then  if  we  put 

Ui  =  l/ui,       £i=l/si, 
it  may  easily  be  shown  that 

9i(Si+Fi)=gi+i.Si+1; 

where  F{  denotes  the  refracting  power  of  the  ith  lens.  Now 
if  ft;  denotes  the  index  of  refraction  of  the  ith  lens  and  if  R[ 
denotes  the  curvature  of  the  first  surface  of  this  lens;  and  if, 
further,  for  the  sake  of  brevity,  the  symbols  Ai}  B-lf  Cl}  Di} 
and  E{  are  introduced  to  denote  the  following  functions  of 
nu  Fh  Rv  U\  and  Si,  namely: 

Ai  =  nJ+?FiR\-  (4(^+0^     2^+1      1 

m  {       m  m-l      J 

m  m-l  \m-l' 

m  [   m  m-l      J 

+nj±i  Fim+2^±i  FiUisi+2-^  m 

m  m  m  - 1 

+J*F%+(JH-Yf\; 

m-l  \m-l/ 

Ci  =  3(«i±2)  m_  (6^+1)  ^^3(2^+1)  F\FiRi 

+iFim+2{3n' +2  W+3-^  f& 

m  m  Wi 

m-l  m-l  \m-i/      > 


550  Mirrors,  Prisms  and  Lenses  [§  193 


Di  =  '2-LZ  FiR\-     t^L^  (Ui+Sd  +^~  F{     FiRi+^FiUi 

m  {       rii  ?2i-l       J  ?ii 

ni  ?2i- 1 


+^?Si+(3_)V?; 

m-l  Vm-l/ 


Wi  ?2i—  1  rzi-1  \ni-l/ 

then  Seidel's  formulae  for  the  spherical  errors  of  a  sys- 
tem of  m  infinitely  thin  lenses  may  be  expressed  as  follows: 

i==m  /7>-\4  i=m  /h\sn- 

S3=I(|i.^)ci;  ^(fi.^A; 

i=i  V/ii    fifi/  i=i  \/ii   0i/ 

i=ihi\gi/ 

The  greatest  practical  value  of  these  formulae  is  to  guide 
the  optician  to  a  correct  basis  for  the  design  of  his  instrument 
and  to  supply  him,  so  to  speak,  with  a  starting  point  for  a 
trigonometrical  calculation  of  the  optical  system  which  he 
aims  to  achieve.  But  the  reader  who  wishes  to  pursue  this 
subject  further  will  find  it  necessary  to  consult  the  more  ad- 
vanced treatises  on  applied  optics. 


Ch.  XV]  Problems  551 

PROBLEMS 

1.  If  L,  L'  designate  the  points  where  a  ray  crosses  the 
axis  of  a  spherical  refracting  surface  before  and  after  refrac- 
tion, respectively,  and  if  C  designates  the  center  of  the  sur- 
face, show  that 

0+0' 

,        ,        cos 

n    n     n —n  2 


a -|- a' 
cos- 


2 

where  c  =  CL,  c'  =  CL',  a,  a'  denote  the  angles  of  incidence 
and  refraction,  6,  6r  denote  the  slope-angles  of  the  ray 
before  and  after  refraction,  r  denotes  the  radius  of  the  sur- 
face, and  n,  n'  denote  the  indices  of  refraction.  Also,  show 
that 

a' +6' 


c  +r 


sin  0cos- 


c+r      .    Ql      a  + 
sin  o  cos- 


2 

2.  A  ray  parallel  to  the  axis  meets  the  first  surface  of  a 
glass  lens  (index  1.5)  at  a  height  of  5  cm.  above  the  axis,  and 
after  emerging  from  the  lens  crosses  the  axis  at  a  point  I/. 
The  thickness  of  the  lens  is  1  cm.  Determine  the  aberration 
F'L',  where  F'  designates  the  position  of  the  second  focal 
point,  for  each  of  the  following  cases:  (a)  First  surface  of 
lens  is  plane  and  radius  of  curved  surface  is  50  cm.;  (6) 
Second  surface  of  lens  is  plane  and  radius  of  curved  surface 
is  50  cm.;  and  (c)  Lens  is  symmetric,  radius  of  each  surface 
being  100  cm. 

Ans.  (a)/=±100cm.,F'L'=q=1.13  cm.;  (6)/==*=  100cm., 
W=^f0.29  cm.;  (c) /==»=  100.17  cm.,  F'L'==f0.42  cm.; 
where  in  each  case  the  upper  signs  apply  to  positive  lens  and 
the  lower  signs  apply  to  negative  lens. 

3.  An  incident  ray  crosses  the  axis  of  a  lens  at  an  angle  6\ 
and  meets  the  first  surface  at  a  point  Bi,  the  angle  of  inci- 
dence being  en;  the  slope  of  the  refracted  ray  BiB2j  which 


552  Mirrors,  Prisms  and  Lenses  [Ch.  XV 

meets  the  second  surface  at  the  point  B2  is  02,  and  the  angle 
of  incidence  at  this  surface  is  a2.  If  the  radii  of  the  surfaces 
are  denoted  by  ri  and  r2,  show  that 

r2.sin(ci2— #2)— ri.sin(ai— 00 

-D1-D2  = : — 5 . 

sin  C/2 

4.  The  chief  ray  of  a  narrow  bundle  of  parallel  rays  is  in- 
cident on  a  spherical  mirror  of  radius  32  cm.  at  an  angle  of 
60°.  Find  the  distance  between  the  two  image-points  P' 
and  Q'  of  the  bundle  of  reflected  rays.  Ans.  24  cm. 

5.  The  chief  ray  of  a  narrow  bundle  of  parallel  rays  is  in- 
cident on  a  spherical  mirror  of  radius  r  at  a  point  B,  the  angle 
of  incidence  being  60°.  Determine  the  positions  of  the  image- 
points  P'  and  Q'.  Ans.  BP'  =  r/4,  BQ'  =  r. 

6.  A  narrow  bundle  of  parallel  rays  in  air  is  refracted  at 
a  spherical  surface  of  radius  r  into  a  medium  whose  index  of 
refraction  is  -y/s.  If  the  angle  of  incidence  is  60°,  find  the 
positions  of  the  image-points  P'  and  Q'. 

Ans.  p'  =  3rV3/4,  g'=r\/3. 

7.  A  narrow  bundle  of  parallel  rays  is  incident  on  a  spheri- 
cal refracting  surface  at  an  angle  of  60°.  If  the  meridian  rays 
are  converged  to  a  focus  at  a  point  P'  lying  on  the  surface  of 
the  sphere,  show  that  the  angle  of  refraction  of  the  chief  ray 
is  equal  to  the  complement  of  the  critical  angle  of  the  two 
media. 

8.  The  radius  of  each  of  the  two  surfaces  of  an  infinitely 
thin  double  convex  lens  is  8  inches,  and  the  index  of  refrac- 
tion is  equal  to  \/S.  The  chief  ray  of  a  narrow  bundle  of 
parallel  rays  inclined  to  the  axis  at  an  angle  of  60°  passes 
through  the  optical  center  of  the  lens.  Find  the  positions  of 
the  foci  of  the  meridian  and  sagittal  rays. 

Ans.  The  focal  point  of  the  meridian  rays  is  1  inch  and 
that  of  the  sagittal  rays  is  4  inches  from  the  optical  center. 

9.  If  in  Young's  construction  of  a  ray  refracted  at  a  spher- 
ical surface  (§  176)  a  semi-circle  is  described  on  the  incidence- 
radius  BC  as  diameter  intersecting  the  incident  and  refracted 
rays  in  the  points  Y,  Y',  respectively,  show  that  the  straight 


Ch.  XV]  Problems  553 

line  YY'  is  perpendicular  to  the  straight  line  CS.  The  point 
K  where  the  straight  lines  YY'  and  CS  meet  is  the  center  of 
perspective  of  the  range  of  object-points  lying  on  the  inci- 
dent ray  and  the  corresponding  range  of  meridian  image- 
points  lying  on  the  refracted  ray  (see  §  186).  Show  that 
nTr  _  n.r.sin2  a 

0K w—> 

and  that 

tanZBKC  =  tana+tana'. 

10.  If  the  chief  ray  of  a  narrow  homocentric  bundle  of 
rays  is  incident  on  a  plane  refracting  surface  at  a  point  B, 
and  if  a,  a'  denote  the  angles  of  incidence  and  refraction, 
show  that 

BP'  =  Hl^Jl  .  BP,       BQ'  =  -  .  BQ, 
n  cos2  a  n 

where  P  (or  Q)  designates  the  position  of  the  vertex  of  the 

incident  rays  and  P'  and  Q'  designate  the  positions  of  the 

image-points  of  the  meridian  and  sagittal  rays,  respectively. 

11.  In  the  preceding  problem  show  that  the  straight  line 
QQ'  is  perpendicular  to  the  plane  refracting  surface. 

12.  The  position  of  the  image-point  P'  of  a  pencil  of  me- 
ridian rays  refracted  at  a  plane  surface  may  be  constructed 
as  follows:  Through  the  object-point  P  (or  Q)  draw  PQ'  per- 
pendicular to  the  refracting  plane  and  meeting  the  chief  re- 
fracted ray  in  Q';  and  from  P  and  Q'  draw  PX  and  Q'Y  per- 
pendicular to  the  incidence-normal  at  X  and  Y,  respectively. 
Draw  XG  perpendicular  to  the  chief  incident  ray  at  G  and 
YG'  perpendicular  to  the  corresponding  refracted  ray  at  G'. 
Then  the  straight  line  PP'  drawn  parallel  to  GG'  will  inter- 
sect the  chief  refracted  ray  in  the  required  point  P'.  Using 
the  result  of  No.  10  above,  show  that  this  construction  is 
correct. 

13.  The  chief  ray  RB  of  a  narrow  pencil  of  sagittal  rays 
meets  a  spherical  refracting  surface  at  the  point  B  and  is  re- 
fracted in  the  direction  BT.    Through  the  center  C  draw  CV 


554  Mirrors,  Prisms  and  Lenses  [Ch.  XV 

parallel  to  BT  meeting  BR  in  V  and  CV  parallel  to  BR  meet- 
ing BT  in  V.  If  Q,  Q'  designate  the  positions  of  the  points 
of  intersection  of  the  sagittal  rays  before  and  after  refraction, 
respectively,  and  if  BQ  =  g,  BQ,'  =  q',  show  that 

BV    BV     , 

+— =1, 

Q        2 
and  that 

VQ.V'Q'  =  VB.V'B. 

(Compare  this  last  result  with  the  Newtonian  formula  for 
refraction  of  paraxial  rays  at  a  spherical  surface,  viz.,  x.x'  = 

14.  The  chief  ray  RB  of  a  narrow  pencil  of  meridian  rays 
meets  a  spherical  refracting  surface  at  the  point  B,  and  is  re- 
fracted in  the  direction  BT.  Through  the  center  of  perspec- 
tive K  (see  §  186;  see  also  problem  No.  9  above)  draw  KU  par- 
allel to  BT  meeting  BR  in  U  and  KU'  parallel  to  BR  meeting 
BT  in  U\  If  the  positions  of  the  points  of  intersection  of  the 
meridian  rays  before  and  after  refraction  are  designated  by  P 
and  P',  respectively,  and  if  BP  =  p,  BP'  =  p',  show  that 

BU    BU'     , 

V        V 

and  that 

UP.U'P'  =  UB.U'B. 

(Compare  this  result  with  that  of  the  preceding  problem.) 

15.  If  J,  J'  designate  the  positions  of  the  aplanatic  points 
of  a  spherical  refracting  surface,  and  if  6,  6'  denote  the 
slopes  of  the  incident  and  refracted  rays  BJ,  BJ',  respec- 
tively, show  that 

sin  8  _  n' 
s!nT'~n  'Vi 
where  y  denotes  the  magnification-ratio  for  paraxial  rays. 

16.  A.  Steinheil's  so-called  "periscope"  photographic 
lens  is  composed  of  two  equal  simple  meniscus  lenses,  both 
of  crown  glass,  separated  from  each  other  with  a  small  stop 
midway  between.    The  data  of  the  system,  as  given  in  Von 


Ch.  XV]  Problems  555 

Rohr's  Theorie  und  Geschichte  des  photographischen  Objektivs 
(Berlin,  1899),  p.  288,  are  as  follows: 

Indices:  ni=nz=ns  =  l;  ^2=^4=  1.5233 

Radii:  r\  =  —  r4=  +17.5  mm.;  r2=  —  r3=  +20.8  mm. 

Thicknesses:  di  =  dz—  +1.3  mm.;  d2  =  12.6  mm. 

Distance  of  center  of  stop  from  second  vertex  of  first  lens 
=  +6.3  mm. ;  diameter  of  stop  =  2.38  mm. ;  diameter  of  each 
lens  =  11.32  mm. 

Employing  the  above  data,  determine  (1)  the  position  and 
size  of  the  entrance-pupil,  (2)  the  angular  extent  of  the  field, 
(3)  the  position  of  the  second  focal  point  F';  and  (4)  the 
point  where  an  edge-ray  directed  towards  a  point  in  the 
circumference  of  the  entrance-pupil  and  parallel  to  the  axis 
crosses  the  axis  after  emerging  from  the  system. 

Ans.  (1)  Distance  of  center  of  entrance-pupil  from  second 
vertex  of  first  lens  is  +6.45  mm.;  diameter  of  entrance- 
pupil  is  2.53  mm.  (2)  The  angular  extent  of  the  field  is 
nearly  90° .  (3)  Distance  of  F'  from  last  surface  is  A4F'  = 
+90.946  mm.  (4)  The  edge-ray  crosses  the  axis  at  a  dis- 
tance A4L4=  +90.432  nun. 

17.  The  abscissae  of  the  points  Mk,  Mk+i  where  a  par- 
axial ray  crosses  the  axis  of  a  centered  system  of  m  spherical 
refracting  surfaces  before  and  after  refraction  at  the  &th  sur- 
face are  denoted  by  uk  =  AkMk,  wk'  =  AkMk+i.  If  the  ray 
proceeds  in  the  first  medium  of  index  n\  in  a  direction  par- 
allel to  the  axis,  it  may  be  shown  (cf.  problems  Nos.  16  and 
17,  end  of  Chapter  X)  that  the  primary  focal  length  of  the 
system  is  given  by  the  formula 

f_U2-  Us  •  •  .  Um  (TT  _n\ 

f-w.ut'...um"(Ul-°h 

where  Uk  =  nk/uk,  Uk'  =  nk+i  juk.  Having  calculated  the 
path  of  the  paraxial  ray  in  the  preceding  problem,  em- 
ploy the  above  formula  to  determine  the  focal  length  of 
Steinheil's  "periscope."  Ans.  /= +98.696  mm. 

18.  The  path  of  a  chief  ray  which  in  traversing  the  air- 
space between  the  two  lenses  of  Steinheil's  "  periscope " 


556  Mirrors,  Prisms  and  Lenses  [Ch.  XV 

(see  No.  16)  goes  through  the  center  of  the  stop  will  be  sym- 
metrical with  respect  to  the  two  parts  of  the  optical  system, 
so  that  for  such  a  ray  we  must  have : 

Ca=—  ci,       c4=—  ci,       C3=—c2,       c3=— c2'; 
a4=  a/,       a/=ai,       ct3=  a-2,      a3/=a2; 
0$=  6 1}        6i=  Si. 
Show  that  if   03  =  —  30°  for  a  ray  which  goes  through  the 
stop-center,  the  ray  must  have  been  directed  initially  at  a 
slope-angle   di=  —  28°  2'  54 .43"  towards  a  point  Li  on  the 
axis  whose  distance  from  the  second  vertex  of  the  first  lens  is 
A2Li  =  +6.563  mm. 

19.  The  astigmatism  of  a  narrow  bundle  of  rays  refracted 
through  a  centered  system  of  spherical  surfaces  may  be  com- 
puted logarithmically  by  means  of  the  following  recurrent 
formulae : 

wk.sin(ak— ak') 


Dt  = 


rk.sm  ak 


■/ik  =  rk.sin(ak—  0k),     tk  = 


*k  +1 


wk+i.sin0k+i ' 
Sagittal  Section 

Qk  =Qk+Dk,  Qk+1  = 


l-fc.Qk' ' 


Meridian  Section 

p  ,_Pk.eos2ak+Dk  Pk'      m 

cos2ak  1-ikA 

where  the  symbols  a,  a',  0,  n  and  r  have  their  usual  mean- 
ings and  where  P,  P'  and  Q,  Q'  and  D  are  the  magnitudes  de- 
fined in  §§  186  and  184.  The  calculations  according  to  these 
formulae  will  be  considerably  simplified  in  the  case  of  a  chief 
ray  which  traverses  a  system  like  Steinheil's  " periscope" 
(see  No.  16)  which  is  symmetric  with  respect  to  the  stop- 
center.  For  example,  for  this  particular  system  we  can  write 
for  a  chief  ray: 


Ch.  XV]  Problems  557 

Di  =  D\,    D3  =  D2,    h±=  —  hi,    hz  =  —h2,    h  =  ti; 

0.1—  Oi  =  —  (oi—  0.4),    o2  —  o2 '  =  —  (a.3  —  a3')> 

ai—  B\=  a/—  6b—  a«-  #4,      a2—  62  =  0,3—  Q\=  Oz  —  Oz. 

Apply  the  above  formulae  to  the  optical  system  of  problem 
No.  16  to  calculate  the  astigmatic  difference  (§  186)  of  a  nar- 
row bundle  of  emergent  rays  whose  chief  ray  is  the  ray  whose 
path  was  determined  in  problem  No.  18;  assuming  that  the 
bundle  of  incident  rays  was  cylindrical,  that  is,  Pi  =  Qi  =  0. 

Ans.  p4'— 5/=  +4.849  mm. 

20.  Using  Seidel's  formulae  as  given  in  §  193,  show  that 
the  condition  that  an  infinitely  thin  lens  surrounded  by  air, 
and  provided  with  a  rear  stop,  shall  yield  a  punctual  or 
stigmatic  image  of  a  plane  object  placed  in  the  primary  focal 
plane  of  the  lens,  is  as  follows: 

\n-l  I      n(n-l)2  {n-iy  n{n-l) 

where  F  denotes  the  refracting  power  of  the  lens,  F2  denotes 
the  refracting  power  of  the  second  surface,  1/S  denotes  the 
distance  of  the  stop,  and  n  denotes  the  index  of  refraction 
of  the  lens.  If  the  stop  is  a  rear  stop  at  a  distance  of  30  mm. 
from  the  lens,  and  if  n  =  1.52,  show  that  the  maximum  value 
of  the  refracting  power  of  a  convex  lens  which  will  give  a 
punctual  image  of  a  plane  object  placed  in  the  primary  focal 
plane  is  F  =  + 14.87  dptr. 

21.  Using  Seidel's  formulae  as  given  in  §  193,  show 
that  the  condition  that  an  infinitely  thin  lens  surrounded  by 
air,  and  provided  with  a  rear  stop,  shall  give  a  punctual  or 
stigmatic  image  of  an  infinitely  distant  object,  is: 


fcV+c) 


n  n—  1  n 


where  n  denotes  the  index  of  refraction  of  the  lens,  F  denotes 
its  refracting  power,  Ri  denotes  the  curvature  of  the  first 
surface  of  the  lens,  and  C  =  S—F,  the  magnitude  S  being 
equal  to  the  reciprocal  of  the  stop-distance. 


INDEX 

The  numbers  refer  to  the  pages 


Abbe,  E.:  Porro  prism  system,  50;  refract ometer,  128;  definition  of 
focal  length,  344;  pupils,  401;  magnifying  power,  454;  v-value  of 
optical  medium,  480;  optical  glass,  482,  489;  sine-condition,  523, 
542,  547;  differential  notation,  531. 

Aberration,   Chromatic:  see  Chromatic  Aberration,  Achromatism,  etc. 

Aberration,  Least  circle  of,  515. 

Aberration,  Spherical:  see  Spherical  Aberration. 

Aberrations,  Chromatic  and  monochromatic,  509;  Seidel's  five  sums, 
545-550,  557. 

Abney's  formula  for  diameter  of  aperture  of  pinhole  camera,  5,  26. 

Abscissa  formula  for  plane  refracting  surface,  97,  191,  269;  spherical 
mirror,  154,  155,  191,  276,  285;  spherical  refracting  surface,  191, 
193,  200,  274,  285;  infinitely  thin  lens,  228,  229,  279,  285;  centered 
system,  332,  519.     See  also  Image  Equations. 

Absorption  of  light,  2. 

Accommodation  of  eye,  433-439;  amplitude,  437-439;  range,  438; 
diminishes  with  age,  435,  436;  effected  by  changes  in  crystalline 
lens,  434;  refracting  power  of  eye  in  accommodation,  436,  437. 

Achromatic  combinations:  Prisms,  480,  481,  491-493;  lenses,  480,  481, 
499-505. 

Achromatic  system,  488. 

Achromatic  telescope,  480,  481,  505. 

Achromatism,  480,  481,  487  and  foil.;  optical  and  actinic  or  photo- 
graphic, 489-491. 

Airy,  Sir  G.  B.:  Cylindrical  lens,  315;  tangent-condition  of  orthoscopy, 
545;  curvature  of  image,  540. 

Ametropia,  439  and  foil.;  axial,  curvature  and  indicial  ametropia,  442. 

Ametropic  eye,  440  and  foil.;  distance  of  correction-glass,  445,  446. 

Amici,  G.  B.:  Direct  vision  prism  system,  495,  497,  506. 

Amplitude  of  accommodation,  437-439. 

Anastigmatic  (or  stigmatic)  lenses,  314. 

Angle,  Central,   152,  516. 

559 


560  *  Index 

Angle,  Critical:  see  Critical  Angle,  Total  Reflection. 

Angle,  Slope,  151,  334,  516. 

Angle,  Visual:  see  Visual  Angle,  Apparent  Size. 

Angle  of  deviation,  in  case  of  inclined  mirrors,  43;  in  case  of  refraction, 
78;  in  prism,  50,  51,  125;  in  lens,  293.  See  also  Prism,  Thin  prism, 
Prism-dioptry,   Prismatic  power  of  lens. 

Angles  of  incidence,  reflection  and  refraction,  30,  31,  65. 

Angles,  Measurement  of,  by  mirror  and  scale,  56. 

Angstrom  unit  of  wave-length,  10;  see  also  'i  enth-meter. 

Angular  magnification  (or  convergence-ratio),  351. 

Anterior  chamber  of  eye,  425. 

Anterior  and  posterior  poles  of  eye,  431,  432. 

Aperture-angle,  404. 

Aphakia,  213,  442. 

Aplanatic  points  of  optical  system,  524;  of  spherical  refracting  surface 
(J,  J'),  512,  513,  554. 

Aplanatism,    524.     See  Sine-Condition. 

Apochromatism,  489. 

Apparent  place  and  direction  of  point-source,  15-18. 

Apparent  place  of  object  viewed  through  plate  of  glass,  102,  103,  105, 
106. 

Apparent  size,  20-22,  446  and  foil.;  in  optical  instrument,  449  and  foil. 

Aqueous  humor,  213,  371,  425. 

Astigmatic  bundle  of  rays,  25,  310-314,  526-538,  552  and  foil.;  image- 
lines,  100,  312,  313,  534-536,  547;  image-points,  312,  526,  527, 
529-534;  principal  sections,  311,  528.  See  also  Meridian  rays, 
Sagittal  rays,  Image-points,  Image-lines,  Sturm's  conoid,  Astig- 
matism. 

Astigmatic  difference,  533. 

Astigmatic  image-surfaces,  536-538,   547. 

Astigmatic  lenses,  Chap.  IX,  300  and  foil.;  314. 

Astigmatism  by  incidence,    527. 

Astigmatism,   Measure  of,  533. 

Astigmatism  of  oblique  bundles  of  rays,  527,  547. 

Astigmatism,  Sturm's  theory,  313,  534. 

Astronomical  telescope,  411,  456;  field  of  view,  411,  412;  magnifying 
power,  454-460. 

Axial  ametropia,  442;  static  refraction  and  length  of  eye-ball,  442,  443. 

Axial  (or  depth)  magnification  351. 

Axis  of  collineation,   243. 

Axis  of  lens,  217;  spherical  refracting  surface,  149.  See  also  Optical 
axis. 

Axis,  Visual,  433. 


Index  561 

B 

Back  focus  of  lens,  365. 

Badal's  optometer,   422,   423. 

Barlow's  achromatic  object-glass,  504,  505. 

Barrel-shaped  distortion,    544. 

Bending  of  lens,   284. 

Blind  spot  of  eye,  430,  431. 

Blur-circles,  414-417,  419. 

Brewster,  Sir  D.:  Kaleidoscope,  47. 

Bundle  of  rays,  Character  of,  24,  25,  508,  509,  525;  "direct,"  514; 

homocentric  (or  monocentric),  25,  limitation  by  means  of  stops, 

397-399.     See  also  Astigmatic  bundle  of  rays. 
Bunsen  burner,   66,   473. 
Burnett,  S.  M.:  Prism-dioptry,  135. 


Calculation  of  path  of  ray:  refracted  at  spherical  surface,  516-519; 
reflected  at  spherical  mirror,  518;  refracted  through  prism,  124, 
125;  refracted  through  centered  system,  332,  519-522;  numerical 
example  in  case  of  paraxial  and  edge  rays,  520-522. 

Camera:   see  Pinhole  camera. 

Cardinal  points  of  optical  system,  334-339. 

Cataract:  see  'Aphakia. 

Caustic  curve,  514. 

Caustic  surface,  in  general,  526;  by  refraction  at  plane  surface,  98,  99; 
by  refraction  at  spherical  surface,  515. 

Center:  Of  collineation,  243;  of  curvature,  260,  526;  of  perspective  (K), 
532,  554;  of  rotation  of  eye,  432,  434,  448,  452. 

Centered  system  of  spherical  refracting  surfaces:  Optical  axis,  329; 
construction  of  paraxial  ray,  330,  331;  calculation  of  path  of  parax- 
ial ray,  332;  conjugate  axial  points  (M,  M'),  346,  347;  extra-axial 
conjugate  points  (Q,  Q'),  339-342;  lateral  magnification,  333,  349; 
Smith-Helmholtz  formula,  334;  focal  planes,  333-335;  focal 
points,  332-335;  ray  of  finite  slope,  519-522. 

Centers  of  perspective  of  object-space  and  image-space,  416,  417. 

Centrad,  134,  294. 

Central  angle  (<p),  152,  516. 

Central  collineation,  242-247.  • 

Central  ray,  243. 

Chief  rays,  24,  413,  420,  526. 

Choroid,  425. 

i 


562  Index 

Chromatic  aberration,   487-489,   509. 

Ciliary  body,  427;  mechanism  of  accommodation,  434. 

Circle  of  aberration,  Least,  515. 

Circle  of   curvature,   260. 

Circle  of  least  confusion,  314,  537. 

Circles  of  diffusion:  see  Blur-circles. 

Clausius,    R.:   Sine-condition,    523. 

Coddington,  H. :  Curvature  of  image,  540. 

Collineation :  Central,  242-247;  center  of,  243;  axis  of,  243;  invariant 
of,  246. 

Collinear  correspondence,  242,  508.    See  also  Punctual  imagery. 

Color  and  frequency  of  vibration,  472-476;  and  wave-length,  475. 

Color  of  a  body,  2. 

Colors   of   spectrum,   466. 

Coma,  542,  547. 

Combination  of  three  optical  systems,  374-376. 

Combination  of  two  lenses,  366-370;  achromatic,  499  and  foil. 

Combination  of  two  optical  systems,  356-362;  focal  lengths,  359;  focal 
and  principal  points,  358,  361;  refracting  power,  361. 

Complete  quadrilateral,   162. 

Compound  optical  systems:  Chap.  XI,  356,  foil. 

Concave:  Lens,  221;  surface,  150. 

Concentric  lens,  221,  232,  387,  388. 

Cones  and  rods  of  retina,  428,  429. 

Conjugate  planes,   172,   194,  236. 

Conjugate  points  on  axis  (M,  M'):  Centered  system  of  spherical  re- 
fracting surfaces,  346,  347;  infinitely  thin  lens,  227-229,  232; 
plane  refracting  surface,  97;  plate  with  parallel  faces,  105;  spherical 
mirror,  154,  164;  spherical  refracting  surface,  181,  183. 

Conjugate  points  off  axis  (Q,  Q') :  Centered  system  of  spherical  refract- 
ing surfaces,  339-342;  infinitely  thin  lens,  234-236;  spherical 
mirror,  171-175;  spherical  refracting  surface,  193-196. 

Conoid,  Sturm's,  313,  314,  535. 

Convergence-ratio:  see  Angular  Magnification. 

Convergent  and  divergent  optical  systems,  186,  339,  340. 

Convergent  lens,  221. 

Convex:  Lens,  221;  surface,  150. 

Cornea  of  human  eye,  425;  optical  constants,  371,  372,  401;  vertex, 
431. 

Correction-glass:  Refracting  power  and  vertex-refraction,  443-446; 
distance  from  eye  measured  by  keratometer,  421,  422;  second 
focal  point  of  glass  at  far  point  of  eye,  445. 

Crew,  H.:  ''dioptric,"  287. 


Index  563 

Critical  angle  of  refraction,  80. 

Cross-cylindrical  lens,  315,  317,  319,  320,  325. 

CrystaUine  lens  of  human  eye,  213,  371,  372,  373,  378,  381,  395,  428; 

optical  constants,  371-373,  395,  434;  changes  in  accommodation, 

395,  434;  "total  index,"  436.    See  also  Aphakia. 
Culmann,  P.:  Smith-Helmholtz  formula,  202. 
Curvature  of  arc:  total,  258;  mean,  259;  center  of,  260;  circle  of,  260; 

radius  of,  260;  sign  of,  260;  measure,  260-264. 
Curvature  of  image,  538-540,  547,  548. 
Curvature  of  normal  sections  of  surface,  300-303;  principal  sections, 

302,  303,  525. 
Curvature,  Unit  of,  dioptry,  286-288. 
Curvature  ametropia,  442. 
Curvature-method  in  geometrical  optics,  282. 
Cushion-shaped  distortion,  544. 
Cylindrical  lenses,  217,  310,  314-317;  types,  315-317;  combinations, 

318-326;  transposition,  318-320. 
Cylindrical  surface,  265,  305-308,  310-313;  refracting  power,  307,  308, 


Dennett:  Centrad,  134. 

Depth-magnification,   351. 

Descartes,  R:  Law  of  refraction,  67. 

Deviation  of  ray :  See  Angle  of  deviation,  Minimum  deviation. 

Deviation  without  dispersion,  481,  491-493. 

Diamond,  70,  479. 

Diaphragms  or  stops  for  cutting  out  rays,  397-399. 

Diffraction-effects,  14. 

Dioptry,  286-288;  "dioptrie,"  "dioptre,"  "diopter,"  etc.,  286,"  287; 

millidoptry,  Hectodioptry,  and  Kilodioptry,  287. 
" Direct' '  bundle  of  rays,  514. 
"Direct  vision,"  448. 
Direct  vision  prism-systems,  493-499. 
Direction  of  ray  or  straight  line:  See  Positive  direction. 
Direction  of  source  from  observer's  eye,  15-18. 
Dispersion,  Chromatic:  Chap.  XIV,  465  and  foil.;  anomalous,  477; 

irrationality  of,  477-479;  partial,  479,  483;  relative,  479,  483. 
Dispersion  without  deviation,  481,  493-499. 
Dispersive  power  (or  strength),  479-481;  dispersive  strength  of  lens, 

503. 
Distinct  vision,  Distance  of,  452,  453. 
Distortion,  543-545. 


564  Index 

Divergent  lens,  221;  divergent  and  convergent  optical  systems,  339, 
340. 

Dollond,  J.:  Achromatic  object-glass,  481,  482,  504,  505. 

Donders's  "reduced  eye,"  214;  astigmatism  of  eye  corrected  by  cylin- 
drical glasses,  316;  loss  of  accommodation  with  increasing  age, 
435,  436. 

Double  concave  lens,  219. 

Double  convex  lens,  217. 

Double  ratio  (or  cross  ratio),  156-164. 

Drysdale,  C.  V.,  287. 

Dutch  telescope,  456;  field  of  view,  412,  413;  "eye-ring,"  413,  458; 
magnifying  power,  455-460. 

Dynamic  refraction  of  eye,  438. 

E 

Effective  rays,  23. 

Emergent  rays,  24. 

Emmetropia  and  ametropia,  439-443. 

Emmetropic  eye,  440. 

Entrance-port,   406-409,   410,   413. 

Entrance-pupil,  43,  179,  400  and  foil.,  543;  two  or  more  entrance-pupils, 
405,  406;  entrance-pupil  of  eye,  401,  448. 

Ether,  Light  transmitted  through,  10,  472-476. 

Euler,  L.:  Theory  of  curved  surfaces,  303,  306,  525;  achromatism,481. 

Exit-port,   409,   410,   413. 

Exit-pupil,  400-405,  411-413,  415,  417,  419,  420,  448,  543. 

Eye:  Accommodation,  433-439;  anterior  chamber,  425;  aqueous  hu- 
mor, 371,  425;  bacillary  layer  of  rods  and  cones,  428;  "black  of  the 
eye,"  401;  blind  spot,  430;  center  of  rotation,  432,  434,  448,  452; 
change  of  refracting  power  in  accommodation,  436,  437;  choroid, 
425;  ciliary  body,  427;  cornea,  371,  372,401,425;  cornea- vertex,  431; 
crystalline  lens,  371-373,  428;  decrease  of  power  of  accommodation 
with  age,  435, 436 ;  description  of  human  eye,  425-43 1 ;  entrance-pupil, 
401,  448;  expressions  for  refraction  of  eye,  439;  far  point  and  near 
point,  434,  435;  field  of  fixation,  432,  435;  focal  lengths,  343,  374, 
389,  432;  focal  lengths  in  case  of  maximum  accommodation,  437; 
focal  points,  374,  389,  423,  432;  fovea  centralis,  429,  432,  433,  446; 
iris,  401,  425;  line  of  fixation,  432;  motor  muscles,  431,  432;  nodal 
points,  422,  432;  optical  axis,  431;  optic  nerve,  430;  point  of  fixa- 
tion, 432;  positions  of  cardinal  points  in  state  of  maximum  accom- 
modation, 437;  posterior  pole,  432,  438;  principal  points,  374,  432; 
pupil,  23,  401,   409-413,   421,   425;  refracting  power,  374,  432; 


Index  565 

retina,  428;  static  and  dynamic  refraction,  438  and  foil.;  suspen- 
sory ligament  (zonule  of  Zinn),  428,  434;  variation  of  principal 
points  in  accommodation,  437;  visual  axis,  433;  visual  purple,  430; 
white  of  the  eye,  425,  yellow  spot  {macula  lutea),  428. 

Eye:  see  also  Schematic  eye,  Ametropic  eye,  Emmetropic  eye,  Hyper- 
metropic eye,  Myopic  eye,  "Reduced  eye." 

Eye-axis,  Length  of,  438,  440-443,  448. 

Eye-ring  of  telescope,  413,  458,  459. 

Eye-glasses:  See  Correction-glass,  Astigmatic  lenses,  Cylindrical  lenses, 
Ophthalmic  prisms,  etc. 


Faraday,  M.:  Optical  glass,  482. 

Far  point,  434,  438,  440,  442;  far  point  sphere,  434;  senile  recession, 
436;  coincides  with  second  focal  point  of  correction-glass,  445; 
in  case  of  schematic  eye,  461. 

Far  point  distance,  437,  444. 

Far-sighted  eye,  435.     See  Hypermetropia. 

Fermat,  P.:  Principle  of  least  time,  86. 

Field  of  fixation  of  eye,  432,  435. 

Field  of  view,  18,  19,  406-409,  448;  of  plane  mirror,  40-43;  of  spherical 
mirror,  176-179;  of  infinitely  thin  lens,  247-249,  409-411;  of 
Dutch  telescope,  412,  413;  of  astronomical  telescope,  411,  412; 
"ragged  edge,"  412. 

Field-stop,  19,  178,  249,  406,  410. 

" Fish-eye  camera,"  81. 

Fixation:  field  of,  432,  435;  line  of,  432;  point  of,  432. 

Flat  image,  539,  540,  548. 

Fluorite,  479,  485. 

Focal  lengths  of  schematic  eye,  343,  374,  389,  432 ;  in  case  of  maximum 
accommodation,  437. 

Focal  lengths  of  spherical  mirror,  167;  of  spherical  refracting  surface, 
191,  192,  193,  199,  281;  of  infinitely  thin  lens,  229,  240-242;  of 
compound  system,  359;  of  combination  of  two  lenses,  367;  of  thick 
lens,  363;  of  optical  system  in  general,  342-344. 

Focal  planes  of  spherical  refracting  surface,  197-199;  of  infinitely  thin 
lens,  232;  of  optical  system,  334,  335,  341;  of  centered  system  of 
spherical  refracting  surfaces,  333. 

Focal  point  angle,  447;  as  measure  of  size  of  retinal  image,  449. 

Focal  points  of  spherical  mirror,  166,  189;  of  spherical  refracting  sur- 
face, 186-189;  of  infinitely  thin  lens,  229-232;  of  centered  system 
of  spherical  refracting  surfaces,  332,  333;  of  optical  system,  334, 


566  Index 

335;  of  compound  system,  358,  361;  of  thick  lens,  363;  of  com- 
bination of  two  lenses,  367. 

Focal  points  of  schematic  eye,  374,  389,  423,  432. 

Focus  plane,  400,  402-404,  406-408,  414-417,  543. 

Fovea  centralis,  429,  432,  433,  446. 

Fraunhofer,  J.:  145,  479,  493,  494,  506;  dark  lines  of  solar  spectrum, 
470,  472,  475,  476,  477;  measurement  of  index  of  refraction,  129; 
notation  of  dark  lines,  472 ;  production  of  optical  glass,  482 ;  achro- 
matic object-glass,   504,   505. 

Frequency  of  vibration  and  color,  472-476;  connection  with  wave- 
length, 475. 

Fresnel,  A.  J.:  Principle  of  interference,  14;  use  of  cylindrical  lens,  315. 


Galileo:  Telescope  and  astronomical  discoveries,  456,  462,  463,  464. 

Gauss,  K.  F.:  Reduced  distance,  279,  280;  theory  of  optical  imagery, 
334,  536,  545,  546;  principal  points,  335;  achromatic  object-glass, 
504,  505. 

Glass,  Optical:  see  Optical  Glass. 

Gleichen,  A.:  Lehrbuch  der  geometrischen  Optik,  352. 

Goerz,  P.:  "Double  anastigmat"  photographic  lens,  352. 

Graphical  methods:  Paraxial  ray  diagrams,  168-171;  path  of  paraxial 
ray  through  centered  system,  331;  Young's  construction,  509-511. 

Gregory,    J.,    achromatism,    480. 

Grimsehl,  E.,  Lehrbuch  der  Physik,  363. 

Gullstrand,  A.:  Reduced  distance,  280;  schematic  eye,  343,  370,  371, 
374,  381,  382,  389,  395,  432,  436,  442,  443,  461;  formulae  for  com- 
pound systems,  260,  361;  schematic  eye  in  state  of  maximum 
accommodation,  395,  436,  461;  writings,  536. 

H 

Hadley's  sextant,  58-60. 

Hall,  C.  M.:  Achromatic  telescope,  481. 

Harcourt,  W.  V.:  Optical  glass,  482. 

Harmonic  range  of  points,   161-164. 

Heliostat,  54,   55. 

Helmholtz,     H.    Von:    Ophthalmometer,     103;    Smith-Helmholtz 

equation,  201,  202,  214,  215,  334,  338,  342,  459,  524;  Handbuch  der 

physiologischen  Optik,  371. 
Hero   of  Alexandria,   87. 
Herschel,  Sir  J.  F.  W.:  Achromatic  object-glass  of  telescope,  504,  505. 


Index  567 

Homocentric  bundle  of  rays,  25. 

Houstoun,  R.  A.:  Newton  and  colors  of  spectrum,  466,  469. 

Huygens,  C:  Construction  of  wave-front  in  general,  10-13,  123;  in 
case  of  reflection  at  plane  mirror,  33-37,  61;  in  case  of  refraction 
at  plane  surface,  70-72;  Huygens 's  ocular,  396,  501,  502. 

Hypermetropia,   441,   443,   445. 

Hypermetropic  eye,  441;  correction  glass,  445. 


Image,  5,  17,  18,  25;  ideal,  25,  506,  548;  real  and  virtual,  17,  18. 

Image,  Rectification  of,  by  successive  reflections,  50,  51. 

Image,  Size  of  retinal,  448,  449. 

Images  in  inclined  mirrors,  43-51. 

Image-equations  of  optical  system:  Referred  to  focal  points,  345; 
referred  to  principal  points,  345-347;  referred  to  pair  of  conjugate 
points  in  general,  347,  348;  referred  to  nodal  points,  348;  in  terms 
of  refracting  power  and  reduced  "vergences,"  348. 

Image-equations  of  spherical  refracting  surface,  200,  201. 

Image-lines  of  narrow  astigmatic  bundle  of  rays,  100,  312,  313,  534- 
536,  547. 

Image-lines  of  narrow  astigmatic  bundle  of  rays  refracted  at  plane 
surface,   100. 

Image-point,  25. 

Image-points  of  narrow  astigmatic  bundle  of  rays,  312,  526,  527,  529- 
534. 

Image-rays,  24. 

Image-space  and  object-space,  242,  243. 

Image-surfaces,   Astigmatic,   536-538,   547. 

Incidence:  Angle  of,  30;  height,  151;  normal,  30;  plane  of,  30. 

Incident  rays,   24,   30. 

Inclined  mirrors,  43-51. 

Index  of  refraction:  Absolute,  74;  limiting  value  of,  70;  relative,  66; 
measurement  of,  106,  107,  128,  129;  function  of  wave-length, 
476,  477. 

Indicial  ametropia,  442. 

"Indirect  vision,"  446. 

Infinitely  distant  plane  of  space,  197,  434. 

Infinitely  distant  point  of  straight  line,  158. 

Infinitely  thin  lens,  Paraxial  Rays:  217-257,  276-279,  285;  abscissa- 
formula,  226-229,  285;  character  of  imagery,  237-240;  conjugate 
axial  points,  227-229,  232-234;  construction  of  image,  236;  extra- 
axial  conjugate  points,  234-236;  field  of  view,  247-249,  409-411; 


568  Index 

focal  lengths,  229,  240-242;  focal  planes,  232;  focal  points,  229- 
232;  lateral  magnification,  236,  237;  principal  planes,  239;  pris- 
matic power,  291-295;  refracting  power,  283,  284. 

Infinitely  thin  lens,  Central  Collineation,  246. 

Infinitely  thin  lens,  Conventional  representation,  226. 

Infinitely  thin  lens,  Refraction  of  spherical  wave  through,  276-279. 

Infinitely  thin  lens-system,  289-291;  formulae  for  spherical  aberrations, 
548-550.     See  also  Achromatic  combinations. 

Invariant:  Of  refraction,  76;  of  central  collineation,  246;  in  case  of 
refraction  of  paraxial  rays  at  spherical  surface,  191. 

Iris  of  eye,  401,  425. 

Isotropic  medium,  3,  4. 


Jack  son,  Professor:  New  optical  glass,  484. 
Jansen,  Z.:  Reputed  inventor  of  telescope,  456. 
Jena  glass,  482-485,  540. 


K 

Kaleidoscope,  47. 

Kepler,  J.:  Astronomical  telescope,  455,  456,  sagitta,  202. 

Keratometer,  421,  422. 

Kessler,  F. :  Direct  vision  prism,  497,  498,  499,  506. 

Klingenstierna,  S.:  Achromatic  combination  of  prisms,  481. 

Kohlrausch,  F. :  Measurement  of  index  of  refraction,  128. 


Lagrange,  J.  1^.:  Smith-Helmholtz  formula,  202. 

Landolt,  E.:  Physiological  Optics,  287. 

Lange,    M.:   Calculation-system,    520. 

Lateral  magnification:  Centered  system,  333,  349;  infinitely  thin  lens, 

236,  237;  spherical  mirror,  176;  spherical  refracting  surface,  196. 
Law:  Of  independence  of  rays  of  light,  15;  of  rectilinear  propagation, 

3,  4;  of  reflection,  31;  of  refraction,  66;  of  Malus,  89-91,  525. 
Least  circle  of  aberration,  515. 
Least  confusion,  Circle  of,  314. 
Least  deviation:  see  Prism. 
Least  time,  Principle  of,  86-89. 


Index  569 

Lens:  see  Astigmatic  lens,  Cylindrical  lens,  Infinitely  thin  lens,  Thick 
lens,  Toric   lens,    etc. 

Lens:  Axis,  217;  bending  of,  284;  concentric,  221,  232,  387,  388;  concave 
and  convex,  222;  convergent  or  positive  and  divergent  or  negative, 
223;  definition,  217;  dispersive  strength,  503;  double  convex  and 
double  concave,  217,  219;  meniscus,  219,  226,  385,  386,  387;  of 
zero  curvature,  221,  386;  optical  center,  223-226;  plano-convex 
and  plano-concave,  219;  refracting  power,  283,  363;  symmetric, 
217,  385,  388;  thickness,  219. 

Lens,  Crystalline:  see  Crystalline  lens. 

Lens-gauge,  263-265,  288,  289. 

Lenses,   Forms  of,   217-223. 

Lens-system:  see  Combination  of  two  lenses. 

Lens-system,  Thin:  289-291;  achromatic  combination,  502-505. 

Light:  Rectilinear  propagation,  see  Chap.  I;  wave-theory,  9,  10,  472 
and  foil.;  velocity,  10,  72,  75,  474,  475. 

Line   of  fixation,    432. 

Lippershey,  F.:  Reputed  inventor  of  telescope,  456. 

Listing,  J.  B.:  "  Reduced  eye,"  214;  nodal  points,  337. 

Luminous  bodies,  1. 

Luminous  point,  Direction  and  location,  15-18. 

M 

Macula  lutea  or  yellow  spot,  428. 

Magnification:  see  Angular  magnification,  Axial  magnification,  Lateral 

magnification,  Magnification-ratios,  Magnifying  power. 
Magnification-ratios,  349-351. 
Magnifying  power,   199,  344,  452  and  foil.;  Abbe's  definition,  454; 

absolute,   454;   individual,   454. 
Magnifying  power  of  magnifying  glass,  453;  of  microscope,  454;  of 

telescope,  455-460. 
Malus,  E.  L.:  Law,  89-91,  525. 
Medium:  see  Optical  medium. 
Meniscus  lens,  219,  226,  385,  386,  387. 

Meridian  rays,  311.    See  Meridian  section  of  narrow  bundle  of  rays. 
Meridian  section  of  narrow  bundle  of  rays,  311,  528,  530-533,  535,  552, 

553,  554,  556;  lack  of  symmetry  in,  541. 
Meridian  section  of  surface  of  revolution,  305. 
Michelson,  A.  A.:  Velocity  of  light,  474. 
Minimum  deviation  of  prism:  see  Prism. 
Mirror:  see  Plane  mirror,  Spherical  mirror,   "Thick  mirror,"  (lThin 

mirror,"  Inclined  mirrors,  etc. 


570  Index 

Mirror  and  scale  for  angular  measurement,  56-58. 

Moebius,  A.  F.:  Principal  points,  335. 

Monocentric  bundle  of  rays,  25. 

Monochromatic  aberrations,  509.     See  Spherical  aberration. 

Monochromatic  light,  66,  467,  473-477. 

Monoyer,    F.:    "dioptrie,"   286. 

Moser,  C:  Nodal  points,  337. 

Muscles,  Motor,  of  eye,  431,  432. 

Myopia,  441,  443,  445. 

Myopic  eye,  441;  correction-glass,  445. 


N 

Near  point,  434,  435,  438;  near  point  sphere,  434,  435;  near  point  re- 
cedes from  eye  with  increase  of  age,  435,  436;  in  case  of  schematic 
eye,  436,  461. 

Near  point  distance,  437. 

Near-sighted  eye,  435.    See  Myopic  eye. 

Negative  lens,  223. 

Negative  principal  points,  338. 

Neutralization  of  lenses,  291. 

Newton,  Sir  I.:  11;  prism  experiments  and  dispersion,  66,  465,  466, 
467,  469,  470,  480,  481. 

Newtonian  formula  (x.xf=ff),  168,  201,  237,  345,  554. 

Nodal  planes,  337. 

Nodal  points,  337,  338;  construction,  340;  relation  between  nodal 
points  and  principal  points,  341,  343;  image-equations  referred  to 
348;  of  lens,  226,  363. 

Nodal  points  of  eye,  422,  432. 

Normal  sections  of  curved  surface,  300-305,  525,  526;  cylindrical 
surface,  306. 


Object-point,  25. 
Object-rays,  24. 

Object-space  and  image-space,  242,  243. 
Obliquely   crossed   cylinders,   320-326. 
Oculars  of  Huygens  and  Ramsden,  502. 
Opaque  bodies,  2. 

Ophthalmic  lenses:  See  Astigmatic  lenses,  Cylindrical  lenses,  Correction' 
glass,   Toric  lenses,   etc. 


Index  571 

Ophthalmic  prism:  Base-apex  line,  135;  combination  of  two  ophthal- 
mic prisms,  138-142;  deviation,  133;  power,  134;  rotary  prism,  141. 

Ophthalmometer,   103. 

Optic  nerve,  430. 

Optical  achromatism,  489,  490. 

Optical  axis,  axis  of  symmetry,  23;  of  centered  system,  329;  of  lens,  217. 

Optical  axis  of  eye,  431. 

Optical  center  of  lens,  223-226. 

Optical  disk  for  verifying  law  of  reflection,  32;  refraction,  67,  68;  total 
reflection,   83,   84. 

Optical  glass,  481  and  foil.;  process  of  manufacture,  485-487. 

Optical  image,  5,  17,  18,  25.    See  also  linage. 

Optical  instrument,  23. 

Optical  invariant  of  refraction,  76. 

Optical  length,  89-91,  278,  279. 

Optical  medium,  3;  media  of  different  refractivities,  70. 

Optical  system,  23. 

Optometer  of  Badal,  422,  423. 

Origin  of  coordinates,  149.    See  also  Image-equations. 

Orthoscopy,  Conditions  of,  543-545. 


Paraxial  ray,   Definition,   152. 

Paraxial  rays,  Diagrams  showing  imagery  by  means  of,  168-171. 

Paraxial  rays:  Centered  system,  329-334,  519-521;  infinitely  thin  lens, 
217-257,  276-279,  285;  plane  refracting  surface,  96-98,  191,  265- 
269;  plate  with  parallel  faces,  105-107;  spherical  mirror,  153-179, 
189,  274-276,  285;  spherical  refracting  surface,  179-202,  269-274, 
285,  519,  534;  thin  lens-system,  289-291. 

Paraxial  ray,  Calculation  of,  519-521. 

Pencil  of  rays,  24. 

Pendlebury,  C. :  Lenses  and  systems  of  lenses,  280. 

Penumbra,  7. 

Period  of  vibration,   473. 

Perspective  in  art,   22. 

Perspective,  Center  of,  159;  so-called  center  (K),  532;  pupil-centers  as 
centers  of  perspective,  416,  417. 

Perspective  elongation  of  image,  419,  420. 

Perspective  ranges  of  points,   159-161. 

Perspective  reproduction  in  screen-plane,  417. 

Petzval,  J.:  Curvature  of  image,  539. 

Photograph,  Correct  distance  of  viewing,  417-419. 


572  Index 

Pinhole  camera,  5.    See  also  "Fish-eye"  camera. 

Plane  image,  Conditions  of,  538-540,  548. 

Plane  mirror:  Conjugate  points,  38;  reflection  of  plane  and  spherical 

waves  at,  33-37;  image  of  extended  object  in,  37-40;  uses  of,  52; 

rotation  of,  32,  56;  field  of  view,  40-43;  punctual  imagery,  508; 

reflecting  power,  380,  381.    See  also  Inclined  Mirrors,  Mirror  and 

scale,  "Thick  mirror,"  Sextant,  Heliostat,  etc. 
Plane  Mirrors,  Inclined,  43-51;  rectangular  combinations  for  rectifying 

image,  50,  51. 
Plane  refracting  surface:  Caustic  surface,  98,  99;  narrow  astigmatic 

bundle  of  rays,  98-100,  553;  paraxial  rays,  96-98,  191,  265-269; 

plane  wave,  70-72;  principle  of  least  time,  87-89. 
Plane  wave,  13;  reflection  at  plane  mirror,  33-35;  refraction  at  plane 

surface,  70-72;  refraction  through  prism,   123,   124;  mechanical 

illustration,  72,  73. 
Plano-convex  and  plano-concave  lenses,  219,  225. 
Piano-cylindrical  lenses,  315-317. 
Plate  (or  slab)  with  plane  parallel  faces:  Path  of  ray  through,  101-103; 

refraction  of  paraxial  rays,  105-107;  apparent  position  of  object 

viewed  through  plate  at  right  angles  to  line  of  sight,  102,  103,  and 

inclined  to  line  of  sight,  105-107;  multiple  images  by  reflection  and 

refraction,  107-110;  parallel  plate  micrometer,  103. 
Point  of  fixation,  432. 

Point-source  of  light,  1;  apparent  place  and  direction,  15-18. 
Porro,  I.:  Prism-system  for  rectification  of  image,  50,  51. 
Porta 's  pinhole   camera,    5. 
Porte  lumiere,  53. 

Ports:   See  Entrance-port,   Exit-port. 
Positive  and  negative  directions  along  a  straight  line,   104;  positive 

direction  along  the  axis,  149,  219. 
Positive  lens,  223. 
Posterior  pole  of  eye,  432,  438. 
Power  of  lens  or  prism:  See  Prism,  Prismatic  power  of  lens,  Reflecting 

power,   Refracting  power. 
Power  of  accommodation:  See  Accommodation. 
Prentice,  C.  F.:  Crossed  cylinders,  321;  diagrams,  308,  309,  310;  power 

of  ophthalmic  prism,    135. 
Presbyopia,  435. 

Principal  planes,  335;  of  a  thin  lens,  239;  of  a  spherical  refracting  sur- 
face, 196,  335. 
Principal    point    angle,   447;    as    measure   of   size  of  retinal    image, 

448. 
Principal  points,  334,  335;  relation  to  nodal  points,  341,  343;  image 


Index  573 

equations  referred  to,  345-347;  of  combination  of  two  lenses,  367, 
369,  370;  of  compound  system,  361;  of  compound  system  of  three 
members,  375;  of  infinitely  thin  lens,  239;  of  "thick  mirror,"  377- 
379,  383;  of  thick  lens,  363. 

Principal  points  of  eye,  374,  432;  of  eye  in  state  of  maximum  accom- 
modation, 437;  as  points  of  reference,  437. 

Principal  section  of  prism,  113. 

Principal  sections:  Of  curved  surfaces,  302,  525;  of  surface  of  revolution, 
305;  of  cylindrical  surface,  306;  of  toric  surface,  309;  of  toric  lenses, 
310;  of  a  bundle  of  rays,  304,  311-314,  528,  535. 

Prism,  85,  86,  113  and  foil.;  base-apex  line,  134;  edge,  113;  refracting 
angle,  113,  and  its  measurement,  55;  principal  section,  113.  See 
also  Thin  prism,  Ophthalmic  prism. 

Prism,  Dispersion  by,  465  and  foil. 

Prism,  Path  of  ray  through  a:  Calculation,  124,  125,  and  construction 
of,  113-116;  deviation,  116;  deviation  away  from  edge,  122;  "graz- 
ing" incidence  and  emergence,  117,  118;  limiting  incident  ray,  118; 
minimum  deviation,  119-122,  128-133,  normal  emergence,  129; 
symmetrical   ray,    119-122,    129-133. 

Prism,  Refraction  of  plane  wave  through,  123,  124. 

Prism-dioptry,   135,  294. 

Prism-system:  Achromatic  combination  of  two  thin  prisms,  491-493; 
direct  vision  prism  combinations,  493  and  foil.;  direct  vision 
prism  of  Amici,  495-497,  and  of  Kessler,  497-499. 

Prismatic  power  of  infinitely  thin  lens,  291-295. 

Problems,  25-27,  60-63,  92-94,  110-112,  142-148,  203-216,  249-257, 
295-299,  326-328,  351-355,  384-396,  423-424,  461-464,  505-507, 
551-557. 

Projected  image  and  object,  415,  416. 

Pulfrich,   C:   Refractometer,    128. 

Punctual  imagery,  313,  314,  397,  508,  509;  in  plane  mirror,  508. 

Punctum  ccecum  (blind  spot),  430. 

Punctum  proximum  (near  point),  434,  435. 

Punctum  remotum  (far  point),  434. 

Pupil  of  eye,  23,  401,  409-413,  421,  425. 

Pupils  of  optical  system:  See  Entrance-pupil,  Exit-pupil. 

Purity  of  spectrum,  469-471. 

Purkinje  images  by  reflection  in  the  eye,  378;  calculation  of  equiv- 
alent optical  system,  381,  382. 

Q 

Quartz,  485. 


574  Index 


Radius:  Of  curvature,  260;  of  spherical  reflecting  or  refracting  surface, 
150. 

Ramsden  circle,  458. 

Ramsden  ocular,  463,  502. 

Range  of  accommodation,  438. 

Rays,  Chief:  see  Chief  rays. 

Rays  of  finite  slope,  Chap.  XV,  508,  foil. 

Rays  of  light,  9;  mutual  independence,  15;  meet  wave-surface  nor- 
mally, 13,  14,  89-91.  See  also  Bundle  of  rays,  Effective  rays,  Emer- 
gent rays,  Image  rays,  Incident  rays,  Obiect-rays,  Pencil  of  rays,  etc. 

Ray-coordinates  (or  ray-parameters),  95,  517. 

"Real  and  "virtual,"  17;  images,  17,  18. 

Rectangular  combinations  of  plane  mirrors,  50,  51. 

Rectilinear  propagation  of  light,  3-5. 

Reduced  abscissa,  and  "vergence,"  284-286,  348. 

Reduced  distance,  279-281;  reduced  distance  (c)  between  two  optical 
systems,  360. 

"Reduced  eye,"  214,   437. 

Reduced  focal  lengths,  281;  focal  point  "vergences,"  284-286. 

Reflecting  power  of  mirror,  283;  plane  mirror,  380,  381;  "thick  mirror  " 
379. 

Reflecting  surface,  Quality  of,  29,  30. 

Reflection,  Angle  of,  31,  and  laws  of,  31. 

Reflection,  Regular  and  irregular  (diffuse),  28-30. 

Reflection  as  special  case  of  refraction,  182,  183,  189. 

Reflection  and  refraction,  Generalization  of  laws  of,  86-89. 

Refracted  ray,  Construction  of,  76-78;  deviation,  78.  See  also  Plane 
refracting  surface,  Spherical  refracting  surface,  etc. 

Refracting  angle  of  prism,  113;  measurement  of,  55. 

Refracting  power,  281-284;  in  normal  section  of  refracting  surface,  303; 
of  spherical  refracting  surface,  282,  300;  of  compound  system  of 
two  members,  361,  and  of  three  members,  375;  of  thick  lens,  363;  of 
thin  lens,  283,  284;  of  thin  lens-system,  290;  of  combination  of  two 
lenses,  367. 

Refracting  power  of  correction-glass,  444. 

Refracting  power  of  schematic  eye,  374,  432;  in  state  of  maximum 
accommodation,  437,  438,  439. 

Refraction  of  eye,  438,  439;  dynamic,  438,  and  static  refraction,  438. 

Refraction  of  light,  64,  65;  angle  of,  65;  laws  of,  66,  and  experimental 
basis,  67-69;  mechanical  illustration  of,  72,  73.  See  also  Index  of 
Refraction,  Total  Reflection,  etc. 


Index  575 

Resolving  power  of  eye,  21,  22. 

Resultant  prism  equivalent  to  two  thin  prisms,  138-142. 

Retina,  428. 

Retinal  image,  Size  of,  448,  449. 

Reversibility  of  light-path,  69. 

Rotary  prism,  141. 

S 

Sagitta  of  arc,   262. 

Sagittal  rays,  311.    See  Sagittal  section  of  narrow  bundle  of  rays. 

Sagittal  section  of  narrow  bundle  of  rays,  311-314,  528-530;  symmetry 

in,  541. 
Scheiner,  C:  Astronomical  and  terrestrial  telescopes,  456. 
Schematic  eye:  Far  point,  461;  focal  lengths,  343,  374,  389,  432;  focal 

points,  374,  389,  423,  432;  length  of  eye-axis,  432,  442,  443;  near 

point,  436,  461;  optical  constants,  370-374,  389,  432,  436,  437,  443, 

461;  in  state  of  maximum  accommodation,  395,  436,  437,  461. 
Schott,  O.:  Optical  glass,  482,  489. 
Sclerotic  coat  or  sclera,  425. 
Screen-plane,  400,  402,  414-417,  419,  543. 
Searle,  G.  F.  C:  "Thick  mirror,"  376,  377. 
Secondary  spectrum,  488. 
Segments  of  straight  line,  104,  105. 
Seidel,  L.  Von  :  Theory  of  the  five  spherical  aberrations,  545,  546,  547, 

548,  550,  557;  curvature  of  image,  540;  sine-condition,  523. 
Self-conjugate  point,  243. 
Self-conjugate  ray,  243. 
Sextant,  58-60. 
Shadows,  6-9. 

Sine-condition,  522-525,  547. 
Slab  with  plane  parallel  faces:  See  Plate. 
Slope  of  ray,  151,  334,  516. 
Smith,  R.:  Smith-Helmholtz  formula,  201,  202,  214,  215,  334,  383, 

312,  459,  524. 
Snell  (or  Snellius),  W.:  Law  of  refraction,  67,  72. 
Spectrum,  466  and  foil.;  purity  of,  469-471. 
Spectrum,  Solar,  466  and  foil.;  Newton's  experiments,  465  and  foil. 

Wollaston's  experiments,  469,  470;  Fraunhofer's  experiments, 

472;  dark  lines,  472. 
Spherical  aberration,  Chap.  XV,  509,  513  and  foil.;  along  the  axis, 

513-516,  518,  522,  547. 
"Spherical  lens,"  217. 


576  Index 

Spherical  Mirror,  Ray  reflected  at,  518,  519. 

Spherical  mirror,  Paraxial  Rays:  153-179,  189,  274-276,  285;  abscissa 
formula,  154,  285;  construction  of  conjugate  axial  points,  164-166 
focal  points,  166,  189;  focal  length,  167;  Newtonian  formula,  168 
extra-axial  conjugate  points,  171-173;  construction  of  image,  173 
imagery,  174,  175;  lateral  magnification,  176;  field  of  view,  176- 
179;  reflecting  power,  283;  spherical  wave  reflected  at  spherical 
mirror,  274-276.     See  also  "Thick  Mirror." 

Spherical  over-  and  under-correction,  514,  515. 

Spherical  refracting  (or  reflecting)  surface:  Axis,  149;  convex  and 
concave,  150;  convergent  and  divergent,  186;  magnifying  power, 
199;  radius,  150;  vertex,  149. 

Spherical  refracting  surface:  Aplanatic  points,  512,  513,  524;  calcula- 
tion of  refracted  ray,  516-519;  construction  of  refracted  ray,  509- 
512;  formulae  for  refracted  ray,  517-519. 

Spherical  refracting  surface,  Astigmatism  of  oblique  bundle  of  rays, 
526-534,  553,  554,  556. 

Spherical  refracting  surface,  Paraxial  rays:  179-202,  269-274,  285,  519, 
534;  abscissa  formula,  191,  193,  285;  conjugate  axial  points, 
179-186,  191,  192;  conjugate  planes,  193,  194;  construction  of 
image,  194-196;  construction  of  refracted  ray,  199,  200;  extra- 
axial  conjugate  points,  193-196;  focal  lengths,  191-193,  199; 
focal  planes,  197-199;  focal  points,  186-189;  image-equations,  200, 
201;  lateral  magnification,  196;  refracting  power,  -179-202;  re- 
fraction of  spherical  wave,  269-276. 

Spherical  wave  reflected  at  plane  mirror,  35-37;  at  spherical  mirror, 
27^-276. 

Spherical  wave  refracted  at  plane  surface,  265-269;  at  spherical  surface, 
269-274;  through  infinitely  thin  lens,  276-279. 

Spherical  zones,    515,   516. 

Sphero-cylindrical  lens,  217,  315,  317. 

Spherometer,  263. 

Static  refraction  of  eye,  438,  440,  441,  442,  443;  connection  with  length 
of  eye-ball  in  case  of  axial  ametropia,  442,  443;  relation  with  re- 
fracting power  or  vertex  refraction  of  correction-glass,  444-447. 

Steinheil,  A.:  Data  of  "periscope"  photographic  lens,  554,  555,  556; 
achromatic  object-glass,  505. 

Steinheil,  R.:  Calculation  of  object-glass  of  telescope,  520. 

Stigmatic  (or  anastigmatic)  lenses,  314. 

Stokes,  Sir  G.  G.:  Optical  glass,  482. 

Stop,  Effect  of,  398,  399;  front,  rear  or  interior  stop,  398.  See  also 
Aperture-stop,  Field-stop,  etc. 

Sturm,  J.  C.  F.:  Conoid,  310,  313,  534,  535. 


Index  577 

Surface  of  revolution,  305;  meridian  section,  305;  principal  sections, 

305. 
Surfaces,  Theory  of  curved,  300-303,  525,  526;  normal  sections,  300- 

303,  525,  526;  principal  sections,  302,  525. 
Suspensory  ligament,  428,  434. 
Symmetric  lens,  217,  385,  388. 
Symmetric  points,  339. 


Tangent-condition  of  orthoscopy,  545. 

Telecentric   optical   system,    420-423. 

Telescope:   see  Astronomical   telescope,    Dutch    (or  Galilean)    telescope, 

Terrestrial  telescope. 
Telescope:  Eye-ring  or  Ramsden  circle,   413,  458,  459;  magnifying 

power,  445-460;  invention,  456,  457;  object-glass  and  ocular,  455; 

simple  schematic  telescope,  455. 
Telescopic  imagery,  359 
Telescopic  system,  359. 
Tenth-meter,  10,  475. 
Terrestrial  telescope,  457. 
Thick  lens,  362-366;  focal  points,  nodal  points,  principal  points,  and 

refracting  power,  363;  vertex  refraction,  365,  366. 
"Thick  mirror,"  376-384,  392,  393;  principal  points,  377-379,  383; 

reflecting  power,  379. 
Thin  lens:  see  Infinitely  thin  lens,  Infinitely  thin  lens-system. 
"Thin  mirror,"  377. 
Thin  prism:  combination  of  two  thin  prisms,  138-142;  deviatitn,  133, 

134  and  power,  134-138.    See  also  Ophthalmic  prism. 
Thin  prisms,  Achromatic  combination  of,  491-493;  and  direct-vision 

combination  of,  493-495. 
Thompson,  S.  P.,  38,  135;  axial  (or  depth)  magnification,  351;  obliquely 

crossed  cylindrical  lenses,  321;  symmetric  points  of  optical  system, 

338. 
Toepler,  A.:  Negative  principal  points  optical  system,  338. 
Toric  lens,  310,  314,  316,  317. 
Toric  surface,  265,  305,  306,  308-310,  320. 
Total  reflection,  79-86;  experimental  illustrations,  83-89.     See  also 

Prism. 
Total  reflection  prism,  85,  86,  125,  127. 
Translucent  body,  3. 
Transparent  body,  2. 

Transposing  of  cylindrical  lenses,   318-320. 
Tscherning,  M.:  Physiological  Optics,  287. 


578  Index 

U 

Umbra,  7. 

Undulatory  theory  of  light:  see  Wave  Theory. 

Unit  planes  and  unit  points  of  optical  system,  335. 


Velocity  of  light  in  different  media,  72-75,  475;  varies  with  color,  474; 

in  vacuo,  10,  75,  474,  476. 
Verant,  418. 

Vertex  of  spherical  surface,  149;  of  cornea,  431. 
Vertex  refraction  of  lens,  365,  366;  of  correction-glass,  445,  446. 
"Vertex-depth"  of  concave  surface  of  meniscus  lens,  298. 
Vertices  of  lens,  219. 
Vibration  frequency  and  color,  472  and  foil.;  and  wave-length,  473 

and  foil. 
"Virtual"  and  "real,"  17;  images,  17,  18. 
Virtual  image,  17,  18;  in  case  of  plane  mirror,  38. 
Virtual  object  in  case  of  plane  mirror,  38. 
Vision,  "Direct,"  448;  and  "indirect,"  446. 
Vision,  Distance  of  distinct,  452,  453. 
Visual  angle,  20,  446  and  foil.;  principal  point  angle,  447,  448;  focal 

point  angle,  447,  449. 
Visual  axis,  433. 
Visual  purple,  430. 
Vitreous  humor,  213,  371,  428. 
Von  Rohr,  M.:  Abbreviation  "dptr.,"  287;  verant,  418;  Theorie  und 

Geschichte  d.  photograph.  Objektivs,  555. 

W 

Wave-front,  Plane,  13,  and  spherical,  11.  See  also  Plane  wave,  Spherical 
wave,  Huygens,  Malus. 

Wave-length,  in  vacuo,  5,  475;  wave-length  and  frequency,  475;  wave- 
length and  index  of  refraction,  476,  477;  wave-length  and  color, 
474-477. 

Wave-surface,  Rays  normal  to,  13,  14,  89-91,  525. 

Wave-theory  of  light,  9,  10,  472  and  foil.,  508. 

Wollaston,  W.  H.:  Dark  lines  of  solar  spectrum,  472;  dispersion  ex- 
periments, 469. 

Wood,  R.  W.:  "Fish-eye"  camera,  81;  velocity  of  light  of  different 
colors,  474. 


Index  579 


Yellow  spot  (or  macula  lutea),  428. 

Young,  T.:  center  of  perspective  (K),  532;  construction  of  ray  re- 
fracted at  spherical  surface,  509,  510,  511,  527,  552;  principle  of 
interference,  14. 

Z 

Zinn's  zonule  (or  suspensory  ligament),  428. 


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ASTRO 


.      3  5002  00188  2526 

Southall,  James  Powell  Cocke 

Mirrors,  prisms  and  lenses;  a  text-book 


QC 
385 

57 

AUTHOR 

Southall 

96184 

TITLE 

M-i  T»T»ors . 

■orisms 

and 

lenses 

Astronomy  Library 

QC 

385 

S7  96184