.in
•CD
PRACTICAL OPTICS FOR THE LABORATORY AND WORKSHOP
J. PRACTICAL OPTICS
FOR THE LABORATORY AND WORKSHOP
BY
B. K. JOHNSON
WITH A FOREWORD BY
PROFESSOR CHESHIRE, C.B.E.
DIRECTOR OP THR OPTICAL ENGINEERING DEPARTMENT IMPERIAL COLLEGE OF SCIENCE
LONDON: BENN BROTHERS, LIMITED
8 BOUVERIE STREET, E.C4
1922
Primed in Great Britain by Turnbull & Spears, Edinburgh
PREFACE
THIS little book has been written primarily as a course of instruction for the student in Practical Optics ; and secondly, to deal with the more recent prac- tical applications of Optics for use in the optician's \v< irkshop.
The exercises contuin«l in tin* book are compiled from a series of experiments through which students in the Optical Engineering Department of the Imperial College of Science (South Kensington) usually pass before pro- ceeding to the more advanced optieal work of the de- partinent. It commences with the «juite elementary work and covers a considerable amount of ground, and -hould. I think, prove a useful laboratory course in " Light " • •lieges and Schools of Science.
The experiment- involve as little expensive apparatus a> possible, hut at the present day >at i-faetnry experi- mental optics demands somewhat better apparatus than tli«- rather old-time favoured piece of wood and . ard, and therefore it has heen partly the aim in these pages to suggest means of producing such apparatus in the best possil>l<
Although some of the devices mentioned are not to be found on the market. I have given scale _:s of
-u. h things (as, for example, optical I.enrhe-i. all <»t \\hieh l"-'-n I'Mind thoroughly \> M that those who
a -mall \\..rks hop available ma\ ooDBtntol D606MIJF apparat hem-elves.
That the l.n,,k i- not entirely devoted to the UM ««f the hrought alMuit li\ tin- fad that some of tin- latter chapters— such as Chapter VIII., for example — deal \\iih practical testing of optical instmni* -m-. \\ln«h I
(5 PRACTICAL OPTICS
hope may be of some interest and value to the person engaged on work in the testing department of the optician.
All the diagrams are new and by the author.
My thanks are greatly due to Mr L. C. Martin for very valuable assistance and advice during the process of com- pilation ; also to Prof. Cheshire, C.B.E., for kindness in writing the Foreword.
B. K. JOHNSON
OPTICAL ENG. DEPT.
IMPERIAL COLLEGE OF SCIENCE
CONTENTS
PAGE
PREFACE 5
FOREWORD 11
CHAPTER I
REFLECTION AND REFRACTION OF LIGHT .... 13
Verification of laws of reflection — Formation of an image by a plane mirror — Laws of refraction, and experimental \« n ficat ion— Total internal reflection— Ray plotter— Path of rays through a 60° prism — Minimum deviation — Path of rays through a 45° prism — Constant deviation prisms.
CHAPTER II MIRRORS AND LENSES (OPTICAL BENCH EXPERIMENTS) . 30
Optical l>cn< h (d«-, Tij.ti. m of ne\v. inexpensive, and accurate
bench) — Measurement of the radius of curvature of a concave
mirror or concave lens surface — Radius of curvature of a
convex mirror or « n\. \ lens surface — Focal length of
lenses — Focal length of " thin " concave lenses
— Relation between size of image and focal Im-th of a l«-n-
(graph) — The relation between the conjugate distances an. I
itures for thin positive and nega- - — Simple
t< leM-ojK' : (i) astronomical, (ii) ( ialil.an — The simple compound
niovMoi
CHAPTKK Hi
I'HuiuMl.l l:Y ......... 52
"Richie pri-m photometer — " llmnfonl " photometer — Photoped — " hummer- Brodhui- -Largo photometer
benches (loss of light in a telcM im* nt) — " Nutting "
photometer — " Lummer-Brodhun " MCI t
7
8 PRACTICAL OPTICS
CHAPTER IV
PAGE
SPECTROMETER MEASUREMENTS 66
The spectrometer — Adjustments — Measurement of prism angles — Refractive index and dispersion — Refractive index by immersion — Determination of the wave-length of light by means of diffraction grating — Calibration of the spectrum.
CHAPTER V DETERMINATION OF RADII OF CURVATURE OF SURFACES . 81
Spherometers : 3-legged, ring, Abbe, and Aldis types — Curvature of small diameter surfaces — Curvature by Newton's rings method — Reflection method.
CHAPTER VI MISCELLANEOUS ELEMENTARY EXPERIMENTS ... 93
Use of a measuring microscope — Appearances of " star " image at the focus of a lens : (i) single lens, (ii) achromatic lens — Focal lengths of " eyepiece " systems — Eccentricity of a divided circle — Photographic tests on a lens.
CHAPTER VII
FOCAL LENGTHS OF " THICK " LENSES AND LENS
SYSTEMS . . . . . . . . 105
The " bar " optical bench — Focal length of a thick lens by the magnification method — " Cheshire " focal-length method — Focal collimator — " Lens rotation " method.
CHAPTER VIII
MISCELLANEOUS ADVANCED EXPERIMENTS . . .117
Microscope objectives — Focal length and numerical aperture — Complete measurements of the optical system of the micro- scope for the microscopist — The auto-collimating telescope — Tests on a telescope — The testing of prismatic binoculars.
CONTENTS 9
CHAPTER IX
PAGB
REFRACTOMETERS ...... .141
The " Pulfrich " rt-fnu-tonu>ter — The Abb6 refractometor — Gas interferometric refractometer.
CHAPTER X
APPLICATIONS OF POLARIZED LIGHT . .152
Detection of strain — Microscope polarizer — Sacchari meters.
APPENDIX ......... 161
The cleaning of optical surfaces — .Silvering of glass — Grind- ing and polishing a flat glass surface — Babaming — Developers f-.r photographic work: (i) a frosting Mention for glass, (ii) an optical cement — Table of useful wav» ••!« -n^ths also refractive indices — Tables of : ]n^ ; rc< iprocals ; sines ; cosines ; tangent-.
FOREWORD
THE book which Mr Johnson has written deaU primarily with the experimental side of applied optics. Very little will be found in it about caustics, but a great deal about collimators.
The title of the book has the merit of indicating not only its contents, but at the same time giving information as to the way in which the book differs from other books.
Up to the present time there has been an unfortunate want of co-ordination between the practice of the labora- tories and that of the workshop, to the distinct disadvantage of both. Each has been in a position to assist the other, but for one reason or another has rarely done so.
Laboratory work, on the other hand, has too often ignored everyday wants. The microscopist, however, who requin- simple methods within the compass of his equipment. for the determination of the focal lengths and aj>« -mm-- of lii< lenses, and the magnifying powers of the various combinations of them, will find all the neee>-ar\ informa- tion given in this book. The owner of a telescope, too, who Mispects that only a part of the aperture of his object glass is operative, will now be able to test the matter for himself. He will learn something of the function and importance of that little stop in the erect 01 ol \\hich mav not have ln-m suspected.
F. .1 ^ IIKSHIKK
n
PRACTICAL OPTICS FOR THE LABORATORY AND WORKSHOP
CHAPTER I REFLECTION AND REFRACTION OF LIGHT
(a) VERIFICATION OF LAWS OF REFLECTION
PLACE a piece of cartridge paper on a drawing board. On this place a mirror, preferably silvered on its "front" surface. A microscope "slip," 3 in. x 1 in.. >il\ered, makes an admirable mirror for the purpose ; it -hould be supported at the back so that it will stand with the silvered face at right angles to the paper. Place two pins A and B (Fig. 1) in front of the mirror in the j><»itions shown. Look at the "images" of the pins, A' and I'. . in the mirror, and adjust two more pins C and D so as to appear in (lie same straight lint as these images. Let the lim- 1 1m >ugh AB and CD intersect mi th<- mirror at O. Draw the normal at O and show that it m ikes equal angles with the incident m\ AB and the reflected ray CD. Repeat this experiment three or f« >ur limes, using different positions for the pin A. and -how that in each case the angle of incidence Ifl c.pial to the angle
Section, I li« incident ray, the normal to the mirror at the point
13
I
14
PRACTICAL OPTICS
of incidence, and the reflected ray all necessarily lie in one plane. Tabulate your values of the angles of incidence and reflection for each ray.
(b) FORMATION OF AN IMAGE BY A PLANE MIRROR
Place a pin P (Fig. 2) in front of the mirror as before, and place the eye in such a position that the lower part
of the pin can be seen by A/> reflection. Behind the mirror
adjust another pin so that its upper portion appears to be a continuation of the lower portion 0 Y of the image, for all positions of
• the eye. The second is then at the image position of the first. Let P and Px (Fig. 2) be the positions of the object and image, < >P and let P Px cut the mirror XY in
FIG. 2. 0. Prove by actual measure-
ment that PO=PX0 and that
the angle POX is a right angle. This will show that the image on the normal to the mirror is as far behind the mirror as the object is in front.
In Fig. 1 produce DC back to A' and measure AE, A'E, BF, and B'F.
(c) LAWS OF REFRACTION
First Law. — The incident and refracted rays, and the normal at the point of incidence, all lie in the same plane.
Second Law. — The ratio of the sines of the angles of incidence and refraction for the two media in question is constant. (See Fig. 3.)
| Normal
FIG. 3.
REFLECTION AND REFRACTION OF LIGHT ir»
Explanation of the phenomenon of refraction by the " icave " theory of light.
The " wave theory " is now a fully established fact, and refraction is very easily made clear by considering it on these lines.
Let AB (Fig. 4) be the bounding line between two media, and suppose the lower half to be the denser medium. Let the velocity of light in the upper medium be v, and in the lower vv Let Cc, Dd, and Ee be three rays in an oblique parallel beam of light, and CDE the
Fio. 4.
wave front at any instant. This will advance parallel t" itself until it reaches cde. The ray Cc then enters a different medium, and its velocity is changed from v to Vj. Consequently, whilst the ray Ee is travelling from
e to elt Cc will move through a distance (— xe^J. With
c as centre and ( Vl x eej as radius, describe a semi-circle
in the lo\\cr medium. From el draw a tangent to this semi- • •irele, touching it at cv Join ccv Then cc^ will be the direction of the ray Cc ; cfr will be the new wave h"ht ; and the disturbance at e will travel to e1 in the same period ..t time " t " that the disturbance at c travels
16 PRACTICAL OPTICS
Now FG is a normal to AB at the point c, and the angle the beam was making with this normal was CcF. But having undergone this change of direction (i.e. refraction) in the denser medium the angle is now r2cG.
The ratio of the sines of these two angles is constant whatever incidence is given to Cc, and this ratio is known as the " refractive index " between the two media.
Refractive Index is usually denoted by the letter " n,"
so that the above may be written n = . — .
sin r
It is also easily shown from Fig. 4 that the " refractive index " is also the ratio of the velocities of light in the two media :—
TT. sin i
For n=~^ — .
sin r
Now sin " i "=sin ece-,= — ,
and sin " r " =sin ce1cl= — 1 ;
ee, —
CC
But ee1 = vt. and cc = vt
(c) EXPERIMENTAL WORK FOR — =A CONSTANT.
Place a block of glass (about 4 in. x 3 in. x 1 in.) with parallel sides on a piece of drawing paper, and draw two fine lines along the two edges AB and DC (Fig. 5). Place a pin P in the position shown in contact with the edge of the block, and arrange a series of pins Pf^P^P^ on the circumference of a circle whose centre is P. The radius of this circle should be about 3 in. This gives a series of incident rays PJ?, P2P, P3P, P4P. Determine the paths of these rays through the glass by placing against the side DC pins P5, P6, P7, P8, which appear to be in the
REFLECTION AM) RKFR.V HQN OF LIGHT 17
same straight line as P1P2P3P4 respectively. Remove the block and join PP5, PP6, PP7, PP8. Also draw a normal l'\ to the surface AB.
-ure with a protractor the angle of incidence and
the angle of refraction for each ray and show that ?* <t * ,
sin r is constant.
This ratio is the " refractive index " of the material. It may be determined graphically from the figim 1,\- com- pleting the circle P^PaP,, thus cutting the refracted r»ys, Draw perpendiculars to the normal PN from tli.
points at which an incident ray and a corresponding
ted ray cut the circle. The ratio of these perpendicular lengths gives the required result.
Second Method. — Place the glass block as before on the ;nir paper, and draw fine lines along the edges AH i CD (Fig. 6). Place a pin at P in contact with tin- edge AB. Insert other pin- .\,\2X8 on the other edge DC in the positions shown. On looking through the glass,
further pin^ YjYgYj H<» that thrv appear in tin- sainc
irht lino as Xf, X2P, X3P respectively. Remove th« block, and draw a normal P\ t«. \B at P. Join ^ .in.l produce it back to cut the normal in V.v Produce each other ray back in the saim manner,
B
18 PRACTICAL OPTICS
PX PX
Show by actual measurement that 7 v1 ~7~X2> anc* so
on, is constant. This ratio is the " refractive index " of the glass.
Explanation of foregoing. — Consider Fig. 7. PXX and Y^ are the same rays as lettered thus in Fig. 6. Draw a second normal OR at Xx. Then Y^jR^ angle of incidence, and PXXO= angle of refraction.
Air
FIG. 8.
But PN and OR are parallel.
So
and
• 1C • »
Now n = - — n — r,. sin r
_ „ Therefore n=
(d) TOTAL INTERNAL REFLECTION
Total internal reflection is dependent on refraction. Fig. 8 shows how a series of rays coming from a point " 0 " in a denser medium than air are refracted at the bounding surface XY (e.g. & stone in a pool of water).
Let " n " be the refractive index, which in this case will be less than unity. Now, for any angle of incidence " i " (measured inside the denser medium) the angle of
REFLECTION AND REFRACTION OF LIGHT 19
ivt'rartion ** r " is calculated from the formula n = ~ -.
sin r
So that, in this case, " r " is always greater than
A- / in. -reases the refracted rays get r md
nearer the surface, until a position is reached such as ODY, \vhnv / =90°. Now the sine of 90° is unity, and no angle has a sine greater than unity. >o that for our formula
(-in i\ „ si;
n = — - ) to give any real value for r, - must be
"•i n f*/ w
il to. or less than, unity. Thus, for a refracted ray to be formed, the greatest value of " i " is when sin * This angle is called the " critical angle."
Tin- « j u« -st inn then arises — What happens to tin- in.
rays \\ln-n th< •ml OI), as at K, making
greater angles \Mtli the normal ' In this case no light
?••«!. l.ui all \\hi.-h t.i!U ..n i!,, urface is reflected
10 the first mi*diuin I DMOOO i- t< nned
\ angle, however, a
certain amount <>t mt< mal reflection takes place.
' ' / I' .- .1 n-l.t an-!.
1 rawing paper on a drawing board. The p uld
be a large on,-. |( t|, th.- h\ pot.-nu-..- -urfaoe
about 4 in. IOIIL: l»».i\\ im- Inn - n>un<i the three faces of
tin .-«• ji |,n, I1, an nhown in Y\\i. '». m r«.i,
tlicmirfarr \l%- With I' . oentfl md I'P, an radiiu
20 PRACTICAL OPTICS
(about 4 in.) describe a semi-circle about AB. Place a second Pj in some such position as indicated, and on looking at the hypotenuse face AC, insert further pins R! and Sx so that they appear in the same straight line as PPi- This will give the refracted angle in air N1O1R1 corresponding to the incident angle PO^E^ in the glass. Move the pin Px into another position P2 so that the angle POjEj is increased, and insert the pins R2 and S2. In this way move Px continually towards M until the eye can only just see the two images of P and P1 in line with the perpendicular edge C of the surface AC. This will give the last ray in the glass that is able to get outside the bounding surface AC. Remove the prism, and draw a normal PM. Join PCP. Pc is this last position of Px. Measure the angle MPP with a protractor. With the refractive index of the prism given, calculate the refracted angle, and draw in PO, making this angle with the normal PM. This is the " critical angle " for this particular glass. Experiment II. — Total Internal Reflection. — Replace the prism on the paper as in Fig. 9, and place P in the same
FIG. 10.
position as before. Move Pc further towards M so as to make the angle POE just greater than the critical angle. Then place the eye so as to look in the face BC, and position the pins Rx and Sx (Fig. 10) so as to appear in the same straight line as PPC (Fig. 10). You will now notice that as soon as the angle POE is made greater than the critical angle, the ray is totally reflected at the face AC.
REFLECTION AND REFRACTION OF LIGHT I \
Repeat the experiment for other positions of Pe as indicated at P3 and P4, and show in each case that the ray undergoes "total internal reflection."
X 0 A E |
C |
|
\ v 1 |
1 |
|
^ Unity ^ ri J I |
(e) "SMITH'S" RAY PLOTTER (Trans. OpL Soc., 1919-20, vol. xxi., No. 3).
In the case of all graphic experiments in connection uith refraction, it is continually necessary to draw re- fracted rays at the bounding surfaces of media. These angles have, in an ordinary way, to be calculated from
the formula n= . „ and then drawn out with a pro- sin r
tractor, which, if a number of surfaces are involved, be- comes very laborious and al>n occupies a great deal of time. Therefore it is of great ad- IL:« it these angles can be obtained readily and easily : this " ray plotter " here de- scribed gives a means of doing this, and in a very simple manner.
Procure a piee. ..!' thin sheet celluloid, 6 in. long by 2.J in. \\ide <-<•<• FIL:. 11). On it scratch a fine straight line (H . ujth a marking point; also a line XY at right angles to this at O, as shown in the figure. Mark off -distances OA- - in.. AB = 1J in., and AC = 3 in. The relation between these distances is dependent « n the re- fractive index, hut this is explained later. All glasses, of course, have not the *a x; but for
graphic experiments such as would be done in the la bora- ton the refract x of glass would probably be taken as approximately 1*50. And this is the value on which the above figures are based. If a part i v pe of glass, of known refract m index, is in question, then of course above values will differ l>nt in every case OA must
22 PRACTICAL OPTICS
be equal to unity. AB equal to > and AC equal to " n "
in some convenient unit.
At the points A, B and C drill three very small holes, just sufficient in diameter to take the point of a pin. The " ray plotter "is now complete.
How to use it. — Suppose we wish to determine tilt- direction of the refracted ray EM (Fig. 12) in a piece of
glass corresponding to an in- cident ray VE. Place the " ray plotter " on the paper, so that the line XY lies on the bounding line of the two media PR, and O of the " ray plotter " coincides
with E on the paper.
Insert a pin through the hole
A and revolve the celluloid until Glass the hole C comes directly over
the incident ray VE. Prick the paper through the hole B ; re- move the " plotter/' and join
this point to E. This line produced EM gives the refracted ray.
If the ray in the rarer medium is required to be traced from the ray in the denser medium, the method of pro- cedure is very similar, with the exception that when the celluloid is revolved about the point A,* the point B must be made to coincide with the refracted ray in the denser medium. The paper is then pricked through the hole ( ' and this point joined to E and produced. This gives the ray in the rarer medium.
Proof of Method. — Let VE (Fig. 13 a and b) represent the incident ray in the rarer medium incident at the point E of the denser medium, and EM the corresponding re- fracted ray. EA is the radius of a circle and equal to unity. Draw AC (Fig. 13 a) perpendicular to EA, and in Fig. 13 b draw it obliquely. Where the refracted ray
* In this case the hole A will be in the position A] (Fig. 12).
REFLECTION AND REFRACTION OF LIGHT
KM cuts AC, describe a circle with radius AB ; and where VE produced cuts AC, describe a circle with radius AC.
It is required to show, that for rcrr to be constant,
Mil .M I'. \
AC must equal "n" and AT. i whore n = the refractiv.
H
index).
n). '•).
Now the triangles AEC and AKB an -imilar (from the ratio of sides and a common angl so that </AEC=^EBA and ^ECA=Z.AEB •i. in trianph
\< sin i sin AEC EC AC
_ "
sin
But EA i.s nn i crefore AC Smiil.ulv. in triangle AEB,
KC
KA
-in • sin EBA EB_EA ""sinr^sin AEB "IB "IB"
But I Th« rrfore AB =
24
PRACTICAL OPTICS
(/) PATH OF RAYS THROUGH A 60° PRISM
On a sheet of drawing paper on a drawing board place a 60° glass prism. A large prism should be used for this experiment, preferably about a 3 in. face and refractive index about 1'52. Draw fine lines along the two sides AB and BC (Fig. 14). Remove the prism for a moment, and at the mid-point Px of AB draw a normal NXD. At P! set off a line PiP2 at 40° to the normal with a protractor. Place two pins, one at Px and the other at P2 ; then put the prism back into its former position. On looking in the face BC of the prism arrange two more pins P3 and P4 so that they appear in the same straight line as PiP2. The images of P1 and P2 will be fringed with colour owing to dispersion, but this will not interfere with the position- ing of the pins P3 and P4. Remove the prism, join P4P3 and let it meet the surface BC in F. At this point draw a second normal N2D. Measure the angle of emergence P4FN2 corresponding to the angle of incidence NjP^ (which was 40°). Produce P4F and P2PX and let them meet at " 0." Then the angle P4OM is the deviation produced by the prism.
Increase the angle of incidence NjP^ by 5° and repeat the experiment, and so on until N1P1P2 is as large as possible. In each case measure the angle of emergence from the prism and the deviation, and tabulate the results as follows : —
Angle of Incidence. |
Angle of Emergence. |
Deviation. |
40° |
||
45° |
||
50° |
||
55° |
||
60° |
||
65° |
||
70° |
||
75° |
On a piece of squared paper then plot two curves, one
REFLECTION AND REFRACTION OF LIGHT 25
showing the relationship between the angle of incidence and the angle of emergence, and the other between the angle of incidence and deviation. Plot incidence angles in a horizontal direction and emergence and deviation in
14.
Fig. 15 shows the type of graphs
a vertical direction, obtained.
Observe from the curves you obtain that there is a l»M>iti"ii where the " deviation " is at a minimum. The angle of incidence should be noted for this position ; also
30 40 60 60 70 Anttes in Degrees Incidence,
Fio. 15.
by i '«to the "emergence angle curve " the corre-
sponding emergence angle will be obtained. If
ves are plotted correctly these two angles will be found the same.
This shows that win n • ;• nt .m<l
are equal, the deviat . is at its minimum.
26 PRACTICAL OPTICS
To determine the position of minimum deviation. — Draw a straight line PXP2 (Fig. 14) on a piece of drawing paper, and place two pins in the positions P^. Place the 60° prism as indicated so that Pl touches the face AB. Look into the face BC and place the eye so that the images of Px and P2 appear in the same straight line. Now* rotate the prism slowly, first in one direction and then in the other, moving the eye the whole time so that the two images always appear in the same straight line. A position will be noticed when the two images, moving in one direction, suddenly become stationar}r, and com- mence to move in the opposite direction. This stationary position of the images is the position of " minimum de- viation " for the prism. Insert two pins P3 and P4 so that they appear in the same straight line as P1 and P2. Remove the prism, join P4 P3, and show that the angles of incidence and emergence are equal.
(g) PATH OF RAYS THROUGH A 45° PRISM
(i) Place a 45° prism ABC (Fig. 16) on a piece of drawing paper (the prism should be large, from 3 in.
C
P+
P, P2A B
FIG. 16.
to 4 in. hypotenuse face). Mark fine pencil lines round the three faces. Remove the prism for a moment, and draw five lines to the left of AB parallel to the hypotenuse AC. Number these lines 1 to 5, and replace the prism. On line " 1 " place two pins Px and P2. Look into the face BC, and insert further pins P3 and P4 so that they appear in the same straight line as PXP2. Do this for all the five incident rays, and number each corresponding emergent ray. Remove the prism, and draw in the path of the rays through the prism, remembering the laws of refraction and reflection.
REFLECTION AND REFRACTION OF LIGHT 27
Note that " internal reflection " takes place at the face AB, also that an " up and down " reversal of the object takes place. It will also be seen that there is a limit to the "useful aperture" of the prism when used in this way ; for after No. 5 ray in the figure has got through, the portion FCG is no longer useful, as no more rays above F can get through the face BC. The figure AFGB is called an erecting prism.
(ii) Place the prism on a fresh piece of paper and draw tine lines round the edges as before. Remove the prism and draw a series of parallel lines at right angles to AB
(B)
(4)
(2)
(D
/;
(SX4)(3)(2)(1) FlO. 17.
12345 S4321 Fio. 18.
(Fig. 17). Number these* lin«- an.l replace the pn-m. Insert pin- Px and P2 on the No. 1 line, and looking in the face B< place P8 and P4 so as to be in the same straight lin. with the images of Pl and Pa. Do the same f« i raya Nos. 2, 3, 4 and 5. Note that a- m incident
normal to the face AB, no deviation takes place at tin- acting surfaces, hut that total reflection t ik<- pla.-e he face A< Also, observe that there is a "right and
d ..t the object in this case.
din Determine as befnn l-y means of the pin method of ray-tracing, the pal ^h«-n thc\ are inddenf
on t face AC (see Fig. i^ \«.tc in tin-
ca84. total internal r. tl.-,-iion takes place at l»uth
surfaces AB and !•< md alto that there is again a right
28
PRACTICAL OPTICS
and left reversal of the object. Right-angled prisms are used in this last manner in prismatic binoculars.
(h) CONSTANT DEVIATION PRISMS
(i) There are two special types of prism which should be noted in connection with the work of this chapter.
They are at present in everyday use and involve principles dealt with here. The first of these is illustrated in Fig. 19, and is known as a Pentagonal Prism ; these prisms are used to a very great extent on military and naval " range- finders/' The figure shows the direction and path of the
Thlst surface silvered,
ThhS surrace\~~ silvered. \
Pentagonal Prism FIG. 19.
FIG. 20.
rays through the prism, and, as will be seen, " internal reflection " takes place at the silvered surfaces. This is not the same kind of internal reflection that has been dealt with before in this chapter, as that is dependent on the critical angle ; in this case it is essential that the two surfaces of the pentagonal prism indicated should be silvered. The importance of this prism, however, lies in the fact that the " deviation " between the inci- dent and emergent rays always remains constant, and also that this deviation is 90°. If this type of prism is available in the laboratory, the above points should be proved by ray- tracing with pins.
(ii) The second type of prism is illustrated in Fig. 20, and is used a great deal in connection with spectrometers.
REFLECTION AND REFRACTION OF LIGHT 29
This prism also gives " constant deviation " between the incident and emergent rays. The path of the rays are indicated in the figure, and, as will be seen, they undergo t\v> refractions and one total internal reflection. The prism is all one piece of glass, but the dotted lines indicate ho\\ it may be considered as built up from two 30° prisms and one 45° prism.
If the laboratory has this type of prism, rays should be t raced through it by pin methods.
CHAPTER II
MIRRORS AND LENSES (OPTICAL BENCH
EXPERIMENTS) (a) DESCRIPTION OF OPTICAL BENCH
AN optical bench of the type here described is very convenient in a laboratory. Its combined simplicity and accuracy make it invaluable for both instructional and
Cross Line Object
Lens Holders
Ground-Glass Screen
Eyepiece in Holder
FIG. 21.
commercial work. Fig. 21 shows the general appearance of the bench, and, as will be seen, it consists of a Chesterman steel metre rule supported in a vertical plane, along which all other necessary fittings slide. These fittings are all
FlQ. 2lA.
very simple and inexpensive to construct. A group of these are shown in Fig. 22, such as the cross wire object, ground glass screen, lens holders, mirror, etc. It will be noted that the base of all these fittings is " cut away " in such a manner that readings may be taken direct from the steel
3J
MIRRORS AND LENSES
rule without any appreciable error being introduced. Where more accurate results are necessary a " correction rod " may U employed. The lens holders are designed to
Cross -Wire
Ground- Glass Screen
Correct/on Rod
o
, Mirror m Mount
Achromatic Uns in Mount
tt,
carry len>es from any ordinary spectacle trial case, so for • •xperimental work a large range of lenses may be obtained.
The fittings that >upport the >teel rule in a vertical
Trial Case Lens is
out in here , and clamped by this sliding fitting
,n line with centre of Lens from which resd ing 53 re taken off 5tee/ rule
2. 8.A. Thread
S/ot for Steel Rule,
-IM.UII m I-'IL- L'I \ Th'-se are adaptable M..t .M,! in. -ire rule but to shorter lengt
a foot rule, \\hni ,in ), nzperiineiits ..ul\ jnv«»lve -in. ill ranges. Scale dra\\ the lens holders and ground
32
PRACTICAL OPTICS
glass screen holders are shown in Figs. 23 and 24. From these and Fig. 22 a general idea of all the fittings may be obtained. (See Article by Prof. Cheshire in Trans. Opt. Soc., vol. xxii., No. 2.)
Watch Spring
A Concave Mirror is shown dotted
^Bevelled
Slot for Steel Rule
FIG. 24.
(6) MEASUREMENT OF THE RADIUS OF CURVATURE OF A CONCAVE MIRROR OR CONCAVE LENS SURFACE
The concave mirror * provided for this experiment should be held in one of the optical bench fittings so that the
FIG. 2f>. ,
" pole " of the mirror is in the same plane as the edge of the mount from which readings are taken (see Fig. 24).
Arrange on the optical bench the cross- wire object and the mirror whose curvature is required. Place a plane
* These mirrors may be made very simply by silvering the surfaces of a convex and concave lens from an ordinary spectacle lens " Trial Case," and mounting them with the silvered surface " outwards."
MIRRORS AND LENSES 33
glass reflector G (micro cover slip) diagonally, as shown in Fig. 25, so that light from a lamp L (an electric lamp " frosted " or covered with a piece of tissue paper) illuminates the cross-wire object O. Place the eye in t Im- position shown and an " image " of the cross- wires will be seen near the "real" cross-wires reflected from tin
ace M. It is at once evident that if the " image " and "real" cross- wires are in the same plane the di-- M() \\ill !><• the radius of curvature of the mirror, for all rays diverging from O will return back along ti original paths, and therefore they must strike the mi normally (the normal to a spherical surface at any part icular point is its radius). The method of ensuring that " image " and object are in the same plane is by employing tin- parallax method. By moving the head from side to side ' image " of and "real" cross-wires will appear to move together \\ln-n the mirror is in its correct position ; it however, the "image" does not appear to move as fast as the " real " cross-wires the plane of the image will
}>ehind the plane of the object, and vice rersri. As an alter the image" may be focussed direct!,
on the white surface at the back of the cross-line object.
When the curvature of a concave lens surface is re<|un exactly the same procedure is employed, with the ex< -eption
iface not under t.-t must be covered in H manner in ..nld t«» ; :lccted ha
If tlii-, hack surface of the lens is covered with a thin
• "I •• pi.i-ti. me." this serves the purpose very \\ell.
hlotting ni<k to the back surface with
vaseline does equally \\--ll image" of the cross-
\\irc-, will not he -<> hnuht a- \\hen a -ilven-d surface i-*
used, but sufficiently hriLrht f«r taking measurements.
(c) RADIUS OF CURVATURE OF A CONVEX MIRROR OR A
CONVEX LENS SURFACE
Arrange the apparatu tre "optical IM-IK -h "
as shown m Kig. 26. O is the cross-line object at the
PRA< TICAL OPTICS
end of the steel rule. A is an achromatic lens * (held in
>t the lens holder fittings) which forms an image of
tlu> eross-wiivs on the ground glass screen S. G is a plane
L'lass retire -tor \\hich illuminates the object from a lamp
I.
Determine carefully the reading, on the optical bench, <>f the ground glass screen S when the image is sharply in focus. Interpose the convex mirror f to be tested .M. in the position indicated, and adjust its position until < n viewing the object as in the last experiment, the plane of the image " thrown back " by the mirror M is co- incident with the plane of the "object/' This is done l»y the parallax method as before or by focussing the
S M A O
FIG. 20.
image direct on the white surface at the back of the cross- line object. The reading of the mirror is then taken, and the distance SM is the radius of curvature of the surface. For, in order that the " rays " leaving 0 and A should retrace their paths after reflection from the mirror and form an " image " at 0, they must strike the mirror " normally," and this is only the case when the distance S.M is the radius of curvature of the surface. A number of independent readings should be taken for the position of .M and the mean obtained.
For the determination of the radius of curvature of a ' convex lens surface, the same method is adopted the back surface of the lens being covered by some such method as mentioned in the previous experiment.
* An " achromatic " lens is used to prevent undue dispersion of the light, which would otherwise arise with a " single " Ion-. t See footnote on page 32.
MIRRORS AND LENSES 35
Curvature (Introductory i
Thr curvature of a circle may be defined as being equal t«» the reciprocal of its radius.
CD_ DA DA~CE-CD; and \vhrn the angle DOA is small
j j ^ =-^- (very nearly, r being the radius of circle).
Whence CD <* -• r
Thus the length CD, known as the " sagitta " (trigo- nometrically the versed sine of the angle DOA) is a measure of the curvature of the arc ACB. This fact i- the foundation of the curvature method.
So that, light waves as they reach a lens or mirror from a point source at a distance " u " have a curvature equal
to and t his curvature has a negative sign
M
\\ IK n the waves are "convex-fronted " and thus expanding from a focus ; and a positive sign when they are " concave- fronted " and thus contracting to a focus. Similarly the cur- vature imparted or " impressed " by a positive lens of focal
length / i- « (jual to + ,» whilst in the case of a negative
l« n> it is equal to -
The curvature "impressed" upon a plane-fronted \\.-m- by a mirror or lens is defined as its " focal power."
This power is impressed upon all waves acted upon, DO matter at what distance the object may be. Thus tin « in \ at ure of each wave, as it emerges from a lens, or it may be reflected by a mirror, is equal to the curvature ..i th«- in. id< nt wave added to the curvature impressed by the lens or mirror. In other words, final curvature -equal- initial < urvature-f that impressed.
PRACTICAL OPTICS
It // distance of object to lens
v= „ image and f = focal length of the lens,
V U
(d) FOCAL LENGTH OF A CONVEX LENS (THIN)
(i) Place a 5D lens from the " trial case " in one of the lens holders on the metre optical bench. Direct the optical bench at the furthest bright object that can be seen— for instance, a street lamp, or an electric lamp placed in a long corridor, — the distance should not be less than 50 yards. Place also on the bench a ground glass screen in its holder and receive an " image " of the distant lamp produced by the lens on this. The difference between the readings of the lens holder and ground glass screen holder will give the " focal length " of the lens. Make a number of independent settings and measure the dis- tance in each case. See how nearly any one measurement is likely to be correct.
(ii) After having used a distant object, use an object comparatively near to the lens. This method involves
the use of the formula -,= --- > where "/" is the focal
/ v u
length of the lens, " u " the distance between the object and the lens, and " v " the distance between the " image " and the lens. Due respect must be made to the use of signs when employing this formula, and it should be remembered that divergent light is always reckoned as possessing negative curvature, whilst convergent light is positive. Set up the cross-line object 0 (Fig. 28) at one end of the optical bench and illuminate it with a lamp. Place the 5D lens L (in holder) at a distance of about 45 cms. from the object and receive an image of the cross- lines on the ground glass screen. Take a number of independent readings for the position of this screen. Measure the distance " u " (object to lens), in this case it will be a negative value. Also measure " v " (image
MIRRORS AND LENSES 37
t«» l«ii-». this will be a positive curvature. From these values calculate the result for "/."
Move the lens to another position (say 55 cms. from tin- ohjeet) and repeat the experiment.
. i ) A uto-collimation Method. — It will be seen from Fig. 29 that if light diverging from the object O is rendered
O L G
u 4-*- - - +1; -
Pic;
parallel by the lens L. reflect rd hack by a mirror M, airain brought to a focus by the lens, the di-tan.. <>L will he the focal length of the lens. Set up the object O at the end of the bench as before and illuminate it ; pi c • the lens about 20 cms. from the object, and further along the bench place the mirror M in position. (
O L M
+
fully adjust tin- I'-n- holder until an " image " of the object 1 1 ply focussed "ii tin ulntined back of the object.* Measure the focal length OL. Take a number of in- depend.-nt n-ad - the position ol I. Pnk<
ii i. ult ^ tor «-ach method and compare tin n result
FOCAL LENGTH OF THIN CONCAVE LENSES
up the cross-v iect O (Fig. 30) at one and
<>pn<-;d l.rneh, and form an image of th;
tiltmir
38 PRACTICAL OPTICS
of the achromatic lens A on the ground glass screen ftj. Place a -3D lens from the trial case in one of .the lens holders and insert this in the path of the convergent beam at L. Move the screen until the image is again focussed, as at S2. The image produced at Sj by the lens A serves as the object for the negative lens, so that the distance LSX is " u " and is positive, while the distance LS2 is " v " and is also positive. Using the formula
-: - J as before, the focal length of the negative lens
may be determined. All values of readings taken from the " bench " should be the " mean " of a number of
O
independent settings. Move the negative lens L to a fresh position and repeat the experiment.
(/) RELATION BETWEEN SIZE OF IMAGE AND FOCAL LENGTH OF A LENS
Set up the metre optical bench with a lens holder mounted on it. Arrange at the zero end of the steel rule a piece of ground glass screen (4J in. x 3J in.) in a vertical plane so that the ground surface lies flush with the end of the rule. As far away as it is possible to arrange, set up two light sources at the same height as the optical bench. Make the distance apart of these two lamps about 6 or 8 feet, so that they subtend a small angle at the lens. In the lens holder place, in turn, lenses from the trial case ranging from a +2D to + 12D, varying by ID every time. In every case measure the distance between the centres of the two images produced on the ground glass screen. This is most easily done by laying a short millimetre rule on the ground glass and observing
MIRRORS AND LENSES
with a watchmaker's eyeglass. The position of tin- len- holder mi the optical bench when the images are sharply in focus on the screen must be taken for each individual len-. Tin- will jzive the focus of the lens (appmximai. Tabulate the values fnr the distance apart of the in. ages and the corresponding focal lengths for eaeh lens, and plot these values on squared paper. On the same p of paper plot the reciprocal of the focal length auain-t the distance apart <>f the images, Write do\\n the mean- ing of your graphs thus obtain« d.
2 Metre Steel Rule supported in Wooden Base forming extra long Optical Bench
Stee/ Rule
Wood Clamps
THE RELATION BETWEEN THE CONJUGATE DISTANCES AND CURVATURES FOR THIN POSITIVE AND NEGATIVE IMAGING LENSES
bhfa experiment the nptieal helieh 18 employed.
l.ut in plaee nf the metre steel rule .1 two-metre -leel rule is used, as a larger \\.>ilm_ range is necess t \
tWO -teel rule can be obtained from .Messrs
Chester.! ;l»le. hut it i-
pOSwil'N" to UM- tun o . ; ule- pi
In either OM6 it i- hetter to mntint them wooden base, a por; uhn h > .'U. so
that I Meel t '« d
• periment. — To obtain and j l<.t the <ur\e ^h<. \\m:_ tin-
\e,-II the p. Ill
Positive Lens (< Cose 1. — A ' K.I! object moving a] i«ee
' point <
4i» PRACTICAL OPTICS
of the incident light-waves is negative and varies from O
to - - (where " / " is the focal length of the lens).
In this case the image is always real, and can therefore be focussed on a ground-glass screen.
Place the cross-wire object at the extreme left-hand end of the bench, and illuminate it by means of a lamp placed behind it. Place a 5D lens L (Fig. 32) in one of the optical bench lens holders, and adjust its position on the bench so that the distance 10 (I is the " image plane " and recorded by the ground-glass screen) is the maximum obtainable under the conditions.
Adjust the screen I so that the image is sharply focussed. Then measure the distance L0=w (-) and TL=v ( + ).
FIG. 32.
Move L a short distance (say 5 cms.) nearer to 0 and repeat the experiment. In this way obtain a series of pairs of values for " u " and " v." Plot these values on a piece of squared paper, remembering that when the incident waves are diverging, " u " is plotted negative ; when converging, positive. A graph should be obtained similar to the one shown in the top left hand quadrant of Fig. 38. On a second sheet of squared paper plot the curvatures
and - for the same experimental data (see Fig. 40).
Case II. (see Fig. 33). — A " real " object moving from the first focus to the lens ; i.e. the curvature of the incident
waves is negative and varies from - j to - oc .
In this case the image is virtual and cannot therefore be focussed on a ground-glass screen. So that, for this part of the experiment the optical system is arranged
MIRRORS AND LENSES
41
as shown in Fig. 33. First set up a simple telescope by employing the achromatic lens A and an eyepiece (these should be standard fittings for the optical bench). Focus tin- telescope for "parallel light" on son distant
object, and situate it near the middle of the bench. Place
Telescope
sa,
the "•!) 1« n- L with its holder near the object (i.e. within ins.), and the beam now passing out from L will l»< rgent. By inserting a further lens C (from the trial case) of knu\\n focal length, say a + 2D, in this divergent beam, the light will be rendered parallel, so that looking thnmgh the telescope a virtual image I of O will be seen : tin- image is situated at the principal focus of C. Then LO = M, and v = focal length of C - CL.
< ham:'- the pnHtion of L and repeat. In tin- \\a\ obtain a series of pairs of value- tor " // " and " *-." Diving that
• t th«- (WYe l»et\\ern the limits U= -/ and tt = O.
\\ill be seen in the ]•• \\.-r N tt )i,.nd <(uadrant
;*> \U<> |>l<»t the corresponding em-\ani e curve
• /// («e« l-'i«. 34).— A \irtualobject moving fi
lell- t«> the IlL'llt. /.'. tin- riilAatlire .if (he lll-ldmt
in positive and varies from + « to O.
PRA< TICAL OPTICS
tlu' achromatic lens A to the left of the bench and adjust it so as to give an image O' of the cross-wires mar tlu- riirht-hand end of the bench. Insert the +5D Irns L in the path of the convergent beam and receive the image I on the ground glass screen. Then LO'— u and LI=r. Obtain a series of pairs of values for "u" and as before, commencing with " n " as about 110 cms. and moving L step by step until " u "is about 5 cms. Plot these values as a continuation of the last curve (see Fig. 38, top right-hand quadrant), also the curvature graph as for Cases I. and II. (see Fig. 40).
L C Telescope
FIG. 35.
Curves for Negative Lens (concave)
Case I. (see Fig. 35). — A " real " object moving up to the lens from the left, i.e. the curvature of the incident light is negative, and varies from O to - « .
Place the cross-wire object at the extreme left-hand end of the bench. Arrange the telescope with the achromatic lens and eyepiece (as for Case II. of the posi- tive lens) at the right-hand end. In a lens holder place the -5D lens L, and make its distance from the object (i.e. " u ") 100 cms. Between this lens and the telescope insert an auxiliary positive lens of known focal length (from the trial case), about a + 2D, and adjust it until the object is brought sharply into focus when looking through the telescope. Then OL= -u, and focal length of C-CL = v. Move the position of L and repeat. Make a series of pairs of values for " v " and " u " as before, and plot them on a fresh piece of squared paper (see Fig. 39, both on left-hand quadrant), also the curvature values,
1 and - (see Fig. 41). v
MIRRORS AND LENSES
43
Case 11. (see Fig. 36). — A " virtual " object moving from the len> to the second focus of the lens, i.e. the curvature of the incident light i> positive and varies from
+ cc to . Iii this case the image is real and can be focussed on a ground-glass screen.
A L
O
Pto,
Place the object at the left-hand end of the bench and arrange the achromatic lens A to form an image 0' of O at about the middle of the bench. Place the -5D lens L in the convergent beam about 3 cms. to the left of 0' and adjust the screen until the Image is again sharply in focus. Then 0'L= + »', and IL=+r. Move L a short distance, say 1 cm., and repeat the experi- ment. In this way obtain as before a scries of pairs of values for " u " and " v" Plot these as a continua-
Tclescope
h-
tion of the curve lor the last case (see Fig. 39, top right- hand quadrant). Also plot the corn '-ponding and
" V
curve (see Fig. 41).
Case III. (see Fig. 37). — A "virtual" objen n ,
from the lir-t focus of the leu-, to the ri-jln. /.< the eurvatui-e ot the incident liL'ht i- p..viti\e ,-ind \aries from
' toO.
44
IMJACTK'AL OPTICS
Retain the same positions of the object 0, the achromatic lens A, and consequently the image 0'. The -5D lens L should then be placed a short distance to the right of A, so as to make LO' —u as large as possible. The image I now being virtual, obtain its position by means of the telescope and auxiliary lens as before. Then L0'= +u, and LI= +v.
•20
C/775.1SO 160 HO 120 100 80 60 40 20 0
cms, 160
140
120
+11
cms, FIG. 38.
Move L further to the right by, say, 5 cms., and repeat the experiment. Obtain, as before, a series of pairs of values for " u " and " v " and plot them (see Fig 39, right- hand lower quadrant). Also curvature graphs - and (see Fig. 41).*
* In all the above experiments the sharpness of the "images" may be improved by using a yellow "colour filter" in front of the object, in order to cut out the blue rays.
M I RRORS AND LENSES
45
+u |
|||||||||
1 |
|||||||||
/ |
|||||||||
00 6 |
0 6 |
0 4 |
0 2 |
0 0 |
L |
0 4 |
0 6 |
0 8 |
3 1 |
-w, ^ — ^ |
— — |
• |
- — |
^ |
-f-M, |
||||
/* |
^« — |
• i |
|||||||
/ |
|||||||||
/ |
|||||||||
-V |
+£ |
/ |
||||||
/ |
|||||||
/ |
|||||||
) I |
_ 1 |
5 £ |
X |
i |
> 1 |
3 1 |
S 2 |
-& |
/ |
3 |
|||||
/ |
|||||||
/ |
|||||||
-^ |
These six experiment- give tin full data for plot t MIL M -lm\Mi in I-'IL'-. ."'s. .'{'•». 40 and -I I
46
PRACTICAL OPTICS
(h) SIMPLE TELESCOPE
Thr experiment Consists in setting up a simple astro- nomieul or inverting telescope and taking measurements in ennneetion with the "system/' and then repeating the measurements for a Galilean telescope.
At^trnin, mil-ill. — Use a metre optical bench for the ex- periment. At the left-hand end place a positive lens (from the trial case) of fairly long focal length, e.-. a + 2D, in one of the lens holders. Receive an in age
1 ^ |
|||||||
10 |
/ |
||||||
/ |
|||||||
0 1 |
3 1 |
0 5 |
> 0 |
c |
z |
D 1 |
5 2C |
/ |
|||||||
; |
|||||||
/ |
i r |
||||||
/ |
7.0 |
;. 41.
of some very distant object (such as a lamp), produced by the lens on the ground glass screen (in its holder). Place in a second holder, and on the other side of the ground glass screen, a short focal length positive lens, such as a+12D. Turn the optical bench completely round, and again focus the distant object on to the ground glass screen by adjusting the position of this lens holder. Then remove the ground -glass screen, and look at the distant object through the system of the two lenses. This is a simple form of inverting or astronomical telescope ; the +2D lens would be known as the object glass, while the -f 12D is the eyepiece (see Fig. 42).
MIRRORS AND LENSES
17
that : the image is larger than the object as seen direr tly.
i.e. it subtends a greater angle at the e the imaire i> inverted and reversed. that the edge of the field of view is indefinite and
ill-defined optically.
the distances, off the optical bench, from the glass to the image, and from the eyr l< -us to the and compare these values with the nominal focal
Object Glass
Eye
Plane of
- — J 1 /7T^
- ^^k-4-l
^y~
Focal Length
of O.G.
Focaf Length of Eye Lens
: i .. if.
lengths of the two lenses as given by the focal power en- graved on the lens ring (focal length =p r in cms.^ .
Repeat these measurements with two <>tli< T telescopes in a-!( up from different pairs of lenses, and tabulate resutte, a< follows :
nial |
nee of |
Nominal |
Ob |
ol |
|
Focal |
\*n* |
tad |
mal |
||
from •gfc |
Length <>f |
from Image. |
Length of |
Kv, LMM, |
food Ln|tka |
Observe that the distance apart of the lenses \\lu-n th«- telescope is focussed for parallel light is equal to the MIH,
48 PRACTICAL OPTICS
of the focal lengths. In this condition the telescope is -aid to be in " normal " or " afocal " adjustment.
Find the position with a ground glass screen of the image of the O.G. aperture projected by the eye lens. This image is variously known as the Ramsden circle, the eye- ring, or the exit-pupil. Note that for comfortable vision this image must fall on the pupil of the eye of the observer.
Field of View. — Note that only when the eye is placed in the plane of the " eye-ring " will the whole available field appear fairly well defined.
Measurement of Field of View
Direct Determination. — Place two candles at the far end of the room and adjust their distance apart until the images of the flames as seen in the telescope are just simultaneously visible one in either edge of the field of view. Measure the distance from the O.G. of the telescope to the mid-point between the two candles L, and let the distance apart of the candles be D. Then the field
of view of the telescope in degrees 0 is : tan 0 = -^- (approx.,
as long as the angle is small).
(i) Magnifying Power. — Use the telescope with the + 2D lens as "object glass/' and the +12D lens as eye lens. Observe through it with one eye a distant vertical scale pinned to a wall (the divisions should be about 10 in. apart), whilst with the other eye the scale is seen directly. Note how many divisions of the scale, seen by the unaided eye, are covered by a single division as seen through the telescope. The number of divisions thus seen in the space of one magnified division is equal to the magnifying power of the telescope. Compare this result with the calculated value of the magnifying power obtained by dividing the focal length of the O.G. by that of the eye lens.
(ii) Determine the Magnifying Power from the diameters of the entrance and exit pupils. — Illuminate the O.G. with diffused light, by placing a frosted lamp close to it. Place
MIRRORS AND LENSES 49
a millimetre scale on glass * in one of the optical bench tit tings, and receive an image of the O.G. aperture pro- jected by the eye lens on to it. Measure the M/< thi> image with the scale. Also measure the diameter of the <).<;. .with a pair of dividers). Then, the magnifying _ diameter of entrance pupil diameter of exit pupil
l>raw a sketch to illustrate how the "magnified image is formed in the astronomical telescope.
Galilean Telescope. — Set up, on the metre optical l»en< h as before, a +2D lens in a holder at about the middle of the bench. Receive an image of a distant lamp pro-
Object Eye
Gtess L ens (concave)
duced I »v tl Hi a ground glass screen. Put a - !L'|>
tens in a holder, and place it on the l>< n« h between the O.G. and the ground glass screen. Inn the latter. Ob-
serve the distant object through the telescope and adju-t
thr e\r leli- Illltil thr ..lijrrt is -harplx 111
focus. This i- now a -implr form of Galilean telescope
(see I'll*, n, Otorrc tint
the image is larger than the object as seen i.e. it subtends a greater angle at the eye. the image is erect and <T*ed, as m the oaae
rhr -irnpl.- t. I. ., »p.-
\ i> IIH: ill-
tincd opt • Thwe gUuM »cak» may be obuincd from MCM» Rhcinbrrg * Co., 23 Tho
D
50
PRACTICAL OPTICS
tlic same experiments with the Galilean telescope aa mentioned before with the astronomical telescope, and tabulate all the results.
Draw a sketch to illustrate how the magnified image is formed in a Galilean telescope.
(0 THE SIMPLE COMPOUND MICROSCOPE
Place the cross-wire object at one end of the metre optical bench, and the ground glass screen (in its holder) about three-quarters of the way to the other end of the bench. Place a short focus lens, say a + 10D trial case lens, in one of the lens holders, and adjust its position, not far from the cross-wires, so that a " magnified " image
Objective
Eye Lens
\ i ;. 44.
of the latter is given on the screen. Now, take another fairly strong lens, say a + 12D, and mount it in a holder on the other side of the ground glass. Adjust the position of this lens until a very distant object is focussed sharply on the screen, but do not move the ground glass screen. The screen may new be moved and the " aerial " image of the cross-wires observed by looking through this second lens.
This is now a simple form of con. pound microscope (see Fig. 44).
Measuring the " First Magnification." — This is the ratio of the sizes of the first " real " image produced by the first lens, and the object. Place one of the millimetre scales on glass (paragraph (h) of this chapter) against the cross-wire object, and a second centimetre scale at
M I RRORS AND LENSES 5 1
tin- p. ,-ition where the " aer al " image is forim-d l»y the lir-t lens. See how many divisions of this latter scale i. division of the magnified image; then determine how many eover two magnified divisions, and so on ; thus obtain the " first magnification " of the microscope.
M Compare the image of a definite
number of divi>ions of the millimetre scale against the
-wires, as seen through the microscope, with the
e number of divisions on a second scale as seen directly
with the other eye at a distance of about 10 in. (the
ar point" of the eye). Of course, in making this
:son the microscope must be so focussed that the
image of the first scale seen through it is formed apparently
at a distance of H» in., and not at infinity, as was the case
Again, see how many divisions of the scale seen directly
i one division of the "magnified" scale, and thn- obtain the niatrnifyiiiL: p«»^«-r. Draw a >keteh in illu>trate the formation of the magnified images.
>ips. — Try the effect on the image of euttini: d<»\\n the .1 pert u re of the front lens : >t to half the diameter.
nd to (jiiaiter the diameter. •iill\ d«-eril»e the effects produced.
that tlie eye n.nst be placed at the ••.•\r-ring"
in order that tin- \\ hole a\ailal»|e field >hall he taill\ well
defined.
CHAPTER III PHOTOMETRY
FOR the theory of Photometry, text-books should be consulted ; a good book on this subject is " Illumina- tion and Photometry/' by Wickenden.
Introduction. — The basis of all photometric comparisons between light sources is the law that the intensity of light given out by a source varies inversely as the square of its distance. Suppose a luminous point is giving out light in all directions. It is at once obvious that a sphere, whose centre is the luminous point, will be equally illuminated over its entire interior surface. Let " r " be the radius of any particular sphere, then the area covered
Suppose L to be the amount of light emitted by the source per second, then the illumination per unit area=
Thus, the illumination at a given distance from a source of light is inversely proportional to the square of the distance. This law is known as the " Inverse Square Law/'
A photometer is a means of measuring the relative luminosities of two light sources by the simple expedient of estimating (with the human eye) the quality of two illuminations thrown on a white screen by the two light sources, and by being able to measure accurately the distance between the screen and the lamps, when equality of illumination due to the two lamps is secured. A standard lamp, such as the Vernon-Harcourt Pentane Lamp or the Hefner Alteneck (see text-books), of known candle- power, may be employed as one of the sources of light,
52
PHOTOMETRY 53
so that the other may be determined in candle-pov This is obtained from the distances "r" and measured from the lamps to the screen when the intensities
matched," for :
It L is the amount of light emitted by one source per sec., and L1 is the amount emitted by the other, the
illirninations per unit area are _. and
tiv«-ly, but these intensities are "matched" or equal, so
s-*
If L is the standard of kno\\n cam He-power, by simply measuring "/ and , the candle-power of the lamp L1 under te-t can be obtained. Tin- i- the principle \\hi--li all photometers are based.
There an many types of bench photometers, but they all involve the nee. a darkened room \\ith
walls painted with a "dull Mack " \arni-h. in order to stop any reflections, which would otherwise interfere with the results obtain, d. A -mail n.<.m -hnuld be chosen for the purpose, and a good coat of "dull Mack" sj varnish given to the walls.
(a) " RICHIE " PRISM PHOTOMETER
This photometer consists of two right-angled prisms of about } in. face " balsamed " to the polish i a piece of ussing son-
so that the two edges of i (see Fig. 45) touch one another. The piece of ground glans \\ith the j.
m.\\ attached i> mounted in a vertical
position m a -m.ill u...-d«-n framework (see Fig. 46). -.• nh
Phis can then i..
mteil on ..IK- of the metal tittmifH BO ES U> -
cal bench referred to in I M use, the light sources, i.e. the " standard
54
PRACTICAL OPTICS
and the lamp being tested, are placed in the same straight line as the steel rule, preferably at each end, and on ob- serving through the circular aperture of the photometer the ground glass screen will be seen to be illuminated,
r
2-0'
Right Angled
Prisms GroundGlass
j-
\
v
FIG. 46.
1 — cr
half the aperture from one source of light and the other half from the second source. This is brought about by the manner in which the prisms are arranged (Fig. 45), so that total internal reflection takes place and illuminates the ground glass.
Thus by moving the photometer backwards and forwards along the optical bench, a position will be found where the two halves of the - aperture are of equal intensity, and the distances between the photo- meter and the two lamps obtained. If necessary the " standard " and the lamp under test may be mounted on fittings *to slide on the optical bench, in order to attain more accurate results.
A very good standard lamp for early experiments in photometry is the Hefner- Alteneck. This lamp is shown in Fig. 47 ; the height of the flame can be adjusted and measured ; and this standard may be trusted to within
FIG. 47.
PHOTOMETRY 55
about '2 per cent.. pro vided that correction for pre.— and humidity of the air have been made.
/• , •• riment. — Set up the photometer just described on the "two-metre" optical bench, and place at one end tin- standard t% Hefner " lamp, and at the other
:1\ so, place an electric lamp (preferably "carbon" filament) of about 16 candle-p<>\u -r. Arrange the • d connections for this lamp as shown in Fig. 48, so
Lamp
Voltmeter
Mains
Iwvw/w
Variable ties/stance
that th- • \ariahle re>i>tniK'C * ill the circuit and
also a voltmeter across the la:n|> tern inaU. In tin- the candle-power <»t tin- lamp can be md in •
case determined by the photometer \\hiUt . sponding voltage from the voltmeter may l»e read off in each case. About ten different candle-powers of tin- lamp should be taken, and a graph plotted showing the relationship be'tween the voltage and candle-p, ,\\,T
(b) RUMFORD PHOTOMETER
. ..I thi- t \ pr nt ph»t»m< • ~h«.\\n m
Fig. 49. \ • i-.«| •• \ ; cd a short d
intc Mum D I1-1 the t\\o sources of lii:I • be compared are placed at 1 St so that sha- ll,.- rod tall 00 til'' -T.cn a! d • -l\ .
• A -. - r\ -t tuple and convmimt type of variable mfeUaoe b made by wing a Uige (12 in. x 12 in.) photognphio dervloping dkh which ban it,
um of wat« '.«' wire* fitun t !„• •• main,"
and to the lamp, nhould bare small plate, aoldeml to them, and thru ,..,i
the aohitkm. By varying the dintance apart of tbeM two plate*, immereed. a vrry fine adjuntrornt (,,r , ,r,. -.mny : .l.tl.-rmee in voltage k obtained. The amount of acid hypo put in b umall. l.ut ,- (..und on trial ; the tolution •.houW be well -tirmL
56
PRACTICAL OPTICS
and are coincident at "a." The distances of Sx and S2 are adjusted, usually, by allowing one of them to move
FIG. 49.
along a divided scale, until the two shadows appear equally dark. Then, as before, the intensities of the two lamps will be proportional to the inverse square of their distances from the screen.
Experiment. — Set up a two-metre optical bench as re- ferred to in Fig. 31, in a darkened room, and on one of the sliding fittings for this bench mount an electric lamp (about 16 candle- power carbon filament). At the zero end of the bench place the white screen. This may be constructed in the following manner : cut cut a block of wood to the shape shown in Fig. 50, and on its front face attach a small strip of brass about | in. wide, and then cover the rest of the wood on this face with black velvet. Take a piece of " mag- nesium ribbon " about 6 in. long and ignite one end ; hold the block immediately above the flame and allow the brass strip to be well coated with the oxide thus produced. The velvet should be covered with a piece of card which has an aperture cut in it to allow the brass plate to project through. This gives a very good screen for photometric work.
The standard (Hefner) lamp should then be set up on
FIG. 50.
PHOTOMETRY 57
tin- table in some such position as shown at S2 in Fig. 49, and the electric lamp " wired " as before (see Fig. 48). A -mall circular rod (such as a pencil) should be mounted : the screen and its distance adjusted until the edges he two shadows produced by the lamps are coinridmt. Having set the "standard lamp" at a known distance from the screen (measured with a steel tape), move the • •Irrtrir lamp along the optical bench until the two shadows on the screen appear equally dark. Take tin- distance giv<n 1>\ the optical bench l»rt\\«-i-n the screen and this lamp and obtain its candle-power from tin- formula given before; also note the voltage from tin- voltmeter. Repeat this a number of time-, in < ach case altering the voltage by separating the two (rim in tin hvpo -(.hit ion (see variable n --i-tam-e in last experin - and thus plot a graph showing tin n -lation>hip l>et\\ • IN -power and voltage.
PHOTOPED
The construction of this photon etei i- illu-t rated in 51. It consists of two tubes A and B. about H in diameter, sliding one in-ide tin- other. In-ide the ; B is a metal "stop" with a rectangular aper
n. I in.) «-ut in it. At tin- mil of tin- tulx' A i- attached a tran>lu. •• -nt ion 'u as a piece of oiled or
greased pap< r Ih* tu<. lit to be compared
are placed at 8| and S2 and tin light proceeding through tin- .ijHirure in B illiiinin.iti-H the 8creen attached to \ \Mth two rectangular patchen of light, as nhown 1»
58
PRACTICAL OPTICS
and " ac " in the figure, the edges of which are made to coincide at " a " by the adjustment of the tube B either towards or away from the screen.
B}' arranging the two patches of light to appear equally bright, the intensities of the lamps may be obtained as before.
Experiment. — Precisely the same experiment as per- formed with the other two types of photometer may be done with the " Photoped," by setting it up at the end of the two-meter optical bench and carrying out the same instructions. This type of photometer is used a great deal in actual practice by " gas referees/'
LUMMER-BRODHUN PHOTOMETER
The Lummer-Brodhun type is a rather better and more accurate photometer. The instrument is shown in Fig. 52. Two screens of magnesium oxide (as applied
from
Fm. 52.
before), or zinc oxide C± and C2, are illuminated by the two sources of light Sx and S2. Light from each of these two screens is then brought into the field of view of the telescope T by means of two mirrors Mj and M2 and a
PHOTOMETRY
Lummer-Brodhun cube A. Such a cube is shown in Fig. 53, and consists of two right-angled pri-m> which are put in "optical contact over a small circular ana
in the centre of their hypotenuse faces. The reman «>t tin- face of the prism 1 is ground away a< indi<-.t: this allows light from both mirrors M, and M2 to enter the telescope. The appearance seen i- that <>t t\\<> con- centric circles of light, the centre patch of light
Lummer - Cube
Field as seen in Eyepiece.
from the source S, and the outer ring of light tn.m Sf ; \\itl. arrangenient dr in e.pialitx <>f tin-
intn^ities is v<-r\ ea-ily detected. \Vln-n in us<- tin- light sources an- u-uall\ Urj.t M..I .n.l th.- j»l meter moved until the mtcn tin- \\\» parts of
th«- Held are ecju.il ; \\li.-n in tin- |>n Mil. .11 thr IHUMI.: line between tin- tu«. }>;\n^ \\i\\ not !.«• \i-ihl.-. Tlii^
adapted \' larger t 'I »me in>tnnnents
n. • th. ..)- .,| dl\ idll
60 PRACTICAL OPTICS
LARGE PHOTOMETER BENCHES
The application of the " steel-rule optical bench " for use as a photometer bench is quite suitable for early and introductory experiments in photometry, but for more advanced work and general use a larger type of mounting for photometers is desirable.
For this purpose a double-lined track (similar to that shown in Fig. 54) is usually employed, which supports the carriages for the lamps, screens, etc. Such a track
Divided Rail-
Rails on which (fittings
FIG. 54.
should be straight, level, and firmly supported. The front circular rail of the track should have a scale of equal divisions on it to permit distances apart of the various fittings to be read. The length of the track should be from 10 to 15 ft. long.
With such an apparatus more satisfactory photometric measurements may be made.
Experiment I. — Using this bench and the Lummer- Brodhun photometer, the candle-power of an electric lamp should be obtained. For this purpose it is well to mount the lamp on a suitable fitting (as shown in Fig. 55), in order that the lamp may be rotated and the candle-power measured for various positions of the lamp, from which a " light-flux " diagram can be plotted. It is of the greatest importance that when using electric sources of light in photometric work the state of current passing should be known ; to this end, therefore, either
PHOTOMETRY <>l
an " ammeter " should be put in series in the circuit, or a voltmeter across the lamp terminals.
Tin- candle-power of a lamp determined in this way
Fio. 65.
would be considered as that measured from the centiv <>t rotation of the lamp serving as a reference point.
••periment II. — As a practical application of photomot r\ . the following experiment may be performed. It consists in measuring the loss of light in a telescopic in-tnimrnt
i'/) For this purpose a collimator should be used a- an accessory to the photometer bench in order to produce
a parallel beam of light for passing through the telescopic
*yM«-Mi under tr-t. Tin- m-m-ral arraini'-iiirnt <>t tin- apparatus for the experiment is shown in Fig. 50.
I irst, receive the parallel beam from the collim
i to the photometer I* <>n one nde (i.e. without the telescope in position), and liulit foom an aii\iliar\ l.i!ii|> A
on ' M tll<- |»Mv|t|(,n .,( tlir |»llOtO-
PRACTICAL OPTICS
im-UT until a " balance " is obtained. Take the distance '/! from the photometer to the auxiliary lamp. Focus the telescopic instrument supplied for "infinity/5 and support it in the position T on the bench so that it receives the parallel beam into the eyepiece of the instrument. \M\\. the ratio of the intensity of the emergent beam to
that of the incident beam should be ^2 (neglecting in-
ternal losses), where M is the magnification of the instru- ment ; the student should prove this for himself. The mag- nification should be found as explained in Chapter VIII. With the instrument now in position the position of the photometer should again be adjusted until a second " balance " is obtained ; let the second distance between photometer and auxiliary lamp be d2- Then,
Intensity of final beam (d-^)'2 Intensity of original beam~~(c?2)2'
By the theory above, if instrumental losses were non- existent, we should find
&)*=!
(dtf M" but in practice we shall have
where K is the transmission coefficient of the instrument.
The above description gives the outline of the experi- ment ; the student should suggest and carry out all necessary experimental precautions, such as the deter- mination of the current in the lamps, repetition of readings, etc.
NUTTING PHOTOMETER
This instrument is made as an attachment to ordinary spectroscopes for spectro-photo metric work. It is used for the comparison of light sources as to their intensity
PHOTOMKTHY
of radiation for the various wave-lengths of the spectrum : it may also be used for absorption work. The optieal • •m of the instrument is shown in Fig. 57. Light from the two sources are admitted through the apertim-- A, and A,. (For absorption work it is better to use a " -pi it " beam from one source.) Light through A , passes through a stationary Nicol (or Glan Thompson) pri-m Xj ; that through the aperture A, is brought in
L L
Hypothenusc
Face of
Prism P2
Slit1 of Spectroscope
tin- direction indicated by means of the pri-m 1*,. The inner >urface of the prism P2 (\Uii<li is balsamcd to PI)
Ivered \\ith t\v<> lini-i/niiial -trip-, so that light t'rmn ill l.« it fleeted along the path \ _.!.,. and tlu- light \, \\ill pa-- -traJLrlit thrmiL'li tin- uu-il\ rjvd -tup. li-u- L, irndciv j|,(. |!Lrj,t pai-allrl I .. : !•
ill.- rotating Nir..i N, 'I'hr tani L • usses an
_•• «.t tin- •• in-partite " !i«-ld «-n tin- slit
rotation <«t th. \i,-,,| \ \\\ l>e measured
l>\ .1 di\id<-d curie C and pointer. I in
<u PHOTOMETRY
the eyepiece of the spectroscope should be that of three -'imply defined spectrum " bands/' the centre one of which is varied in intensity by rotation of the Nicol N2. The source or specimen (if for absorption) to be tested is placed in front of the aperture Ax, and the intensity of the centre band varied, until a " match " with the outer bands is obtained. Since only one of the incident beams is polarized, the intensity of the light varies as the square of the cosine of the angular turn of the Nicol N2.
Experiment. — With the Nutting photometer and spec- troscope, the intensities of illumination for various parts of the spectrum, of, say, a piece of cobalt glass or some solution, may be determined, and a graph plotted showing the relationship between intensity and wave- length.
LUMMER-BRODHUN SECTOR
This piece of apparatus is frequently used in connection with photometric measurements. It serves as a means of varying the intensity of a beam of light by a known amount, by inserting a revolving sector (whose apertures are adjustable) in the path of the said beam. Fig. 58
Sector
, ,^_ \ Motor
Arm
FIG. 58.
illustrates the apparatus, and, as will be seen, an electric motor is employed for driving purposes, whilst an ad- justable arm will be noticed for varying the aperture of the sector whilst in motion.
When in use the speed of the sector should be arranged, so that on looking into the instrument with which the sector is being used no flicker of the field is noticeable.
PHOTOMETRY 65
Under this condition, the intni>ity «»f the light trans- mitted by the sector may be taken as being proportional to the angular aperture.
The instrument may be used on the photometer ' in the path of one of the beams of light, and serves as a means of varying it^ intensity without the necessity of rig the source of light n -l.it i\«- t<» tin- photom<
CHAPTER IV SPECTROMETER MEASUREMENTS
(a) THE SPECTROMETER
THE spectrometer is an instrument of fundamental importance for the measurement of refractive index (see Chapter I., section (c)). The essential parts of the instrument comprise a " collimator " SLX (Fig. 59), a rotating prism table T, and a telescope L2E on a movable arm. The collimator consists of a metal tube, at one end of which is an achromatic lens Lx and at the other
V*
FIG. 59.
end a vertical " slit " S in the focal plane of the lens. Light diverging from this slit is rendered parallel (or collimated) by Lx and " parallel light " falls on the prism. The light having passed through the prism, the spectrum thus produced is brought to a focus by means of the lens L2 of the telescope in its focal plane, and this image is viewed by an eyepiece E. The telescope rotates so that it is always directed towards the axis of rotation of the prism table, and is provided with a vernier Vl5 which moves over a divided circle concentric with the prism
66
S1>K( TROMKTKR .MKASfKKMKNTS
67
table. To the latter there is also a vernier V, attached, which moves over the inner edge of the dividing of the circle. It is necessary that the instrument should be thoroughly rigid, and precision must be exercised in the lilting of the bearings, verniers, and circle. It will be found less expensive if such an instrunn -nt is bought out- right rather than to try and construct such an instrument in one's workshop. A selection of numerous "makers" "ill be found in the "Dictionary of British Scientific Instruments."
Dirtied Circie attached Telescope^ to Prism table Lye-piece
ff
Collimator
Slit,
Slow Motion Adjustment
Main Divided Circle
Two opposite Verniers Fio. 60.
Kg. 60 shows a convenient type of spectrometer for laboratory use i ts).
Adjustments.— The following adjustments are necessary before commencing an experim< -m with the spectromct*
1. To adjust the eyepiece. — The eyepiece lens syst« movable in th< ml., \\ln.-h carries the cross-webs. I a piece of white paper in front of the telescope objective so 88 to reflect light into the t< 1,,<,>|>, . then move the eyepiece in or out until the cross-lines are sharply defined
I'o adjust the telescope. — Din < t the telescope towards some distant object, sin- 1, .0 ., < i and move the tube carrying the eyepiece and cross-wires (usually by a rack-motion until tin- m,. •:_••• tant object is seen sharply defined at the same time aa
68 PRACTICAL OPTICS
the cross-lines. To be sure of this see that there is no parallax between the two. The telescope is now in normal adjustment.
3. To adjust the collimator. — Illuminate the "slit" of the collimator. Swing the telescope into such a posi- tion so that it and the collimator tube are in the same straight line ; and then, while looking through the tele- scope, move the slit in or out until there is no parallax between its image and the cross-lines. Set the slit vertical.
4. There are two alternative methods of focussing the collimator for parallel rays which should be taken note of. First, Schuster's method : the prism is set in the position of minimum deviation, and the telescope turned so that the image of the D line or some other convenient line is seen. The telescope is then turned a little to one side of the image ; it is evident that there are now two positions of the image, one on each side of that of minimum deviation, which will bring the image of the line again into the centre of the field of view of the tele- scope. The prism is turned to these two positions in succession, and the line observed in each case ; if the line appears in perfectly good focus at each time, then the telescope and collimator are both adjusted for parallel rays. If, as is more probable, the focus of the line appears better at one position than at the other, the following procedure should be adopted. The prism is first turned to one position, and then the collimator is focussed until the line is seen perfectly sharp ; after turning the prism to the other position the telescope is focussed until the line is again sharp. After one or two repetitions it will be found that the condition will be obtained so that the line remains in perfect focus which- ever way the prism is turned. When this is obtained, telescope and collimator will be in adjustment for parallel light.
The second method is by Lippmann, who employs two strips of " plane parallel " glass, which are set one above
SPECTROMETER MEASUREMENTS
50
th* other and at right angles to one another (see Fig. 61).
- apparatus is set in the path of the beam from the
collimator ; if these rays be truly parallel no effect will be
GttNtattar
Telescope
81,
produced, but if they are convergent or divergent, the upper and lower halves of the image of the slit will appear relatively displaced.
(6) MEASUREMENT OF PRISM ANGLES
First Method. — Let ABC (Fig. 62) be a prism, of which the an-le A is required to be measured. The prism is placed on its table and levelled so that the faces AC and AB are vertical. Adjust its position and that of the telescope so that an image n of the slit is formed on the en>— -uires by means of light reflected from the face AB. Without MIM\ HILT tin- telescope, rotate the prism table until the face AC acquires such a position that light reflected from this face forms an image of the slit on the cross- wires. In order t«r this to be so, it is obvious that \< ,,,,; ,t take up a position parallel to that previously occupied by AB, which is equivalent to rotating the |»n-m through the angle CAD. Therefore the angle of the the supplement of this angle, which
equals 180° - / CAD. The angle ( \ I > obtained by the readings taken from the divide* I he two positions
lie prism table
70 PRACTICAL OPTICS
Second Method. — Arrange the prism with angle to be measured towards the axis of the collimator so that the parallel beam from this falls partly on the face AB (see
Fig. 63) and partly on AC. Move the telescope round until an image of the slit is seen by light reflected from one of these surfaces. Set this image on the intersec- tion of the cross-wires and take the reading of the
~ "~#\. telescope position from the
divided circle. Leaving the prism table stationary, move the telescope round until the image of the slit is again
seen, but this time after reflection from the second surface, and take a second reading from the circle. The difference between these two readings is equal to twice the angle BAC. Proof :
Produce EA to D, GF to M, and PO to M. Then, because HF and ED are parallel
Andzl GFC = ,/AFM (vert, opposite angles). But z.HFA= Z.GFC (by reflection).
Z.GMD= ^FAM+ Z.AFM (two interior and opposite angles).
Similarly,
So that L GMP (the angle moved through by the telescope) = 2 BAC.
Experiment. — Measure the angles of the prism supplied to you by the two methods described. Test the accuracy of your results by seeing if the sum of the three angles added together equal 180°.
SPE< TROMETEB MEASUREMENTS :i
(c) MEASUREMENT OF REFRACTIVE INDEX AND DISPERSION
\Vitli any pri-m thnv is an important relation between
th<- ; \<- Imlrx i/o. the Vertical Angle of the prism
and the Angle of Minimum Deviation (D) (see
Chapter I.). The equation connecting these three quan-
- is written as :
. (A + D)
>'» •>
-in
Before going any further it is well to look at the proof of this formula.
Let the angl. MAC (Fig. 64) be the measured vi
angle of the prism. Call this angle ** A," ami -uppi^r tli«- pri-iii t«. IM- in the position «>t M minimum <lr\iati..n
(see Chapt I i.e. L EGNt = L DF^ = i, and L\\( • I-
HFG=r.
Now. in tin- tiL'in. \nn; the angles \l"M tad AGH are right angles, so that the angles FAG and V \ \ < • n together equal tun right angles, from \vhi< h we see t)>
^lFAG= Z.HFG+ </ HGF = 2^HFG = L
A
i.e. Z.BAC = 2rorr
r DKN, Mil. But IIKLs£GFL+ £(•!•'!!. uhi.h.()ual§
» A 4-ror
72
PRACTICAL OPTICS
(A + D)
sin i
siir
So that n =
Making use of this formula the refractive index of the prism may now be determined. First of all, however, it must be remembered that a prism produces a spectrum and that the various colours or wave-lengths are deviated by different amounts on passing through the glass, the red being the least refrangible and the violet the most refrangible. Therefore " n " will vary, depending on the wave-length of the light ; thus it is that certain de- finite wave-lengths in the spectrum have to be decided on in order that some standard of comparison may be formed for identification of all glasses, also liquids. These wave-lengths are :
Wave-length. |
Produced by |
Notation. |
•00005893 cms. 6563 „ 4861 „ 4102 „ |
Sodium Flame Hydrogen Tube 5 J 5 ) J ) J > |
Dline c „ F „ GI >j |
Experiment. — Place a sodium flame * in front of the slit of the spectrometer — open the slit fairly wide. Put the prism on its table and observe the image of the slit with the telescope after the light has been refracted through the prism. If the telescope and collimator have been carefully set for parallel light previously, the slit-image should be well .defined ; close the slit down gradually until a very narrow image is obtained, and if necessary focus it sharply by means of the rack motion of the telescope.
To set the prism at minimum deviation. — Rotate the
* A very suitable " sodium flame " may be made by employing a " Meker Burner," as supplied by Messrs Baird & Tatlock, Hatton Garden, and by placing common salt on the " grid " at the top of the burner.
SPECTROMETER MEASUREMENTS 73
prism table and observe the movement of the image; if it goes outside the field of view of the telescope, move the latter round the circle so as to keep it in vi- but on continuing to move the prism table in the same direction thr image will reach a limiting position and then commence moving in the opposite diiv«-ti«.n. When the image reaches this position set the intersection of -s-wires on it : this is the position of minimum iation for the sodium line, i It two sodium lines are seen, set the cross-wires midway between the two, for th< -re are actually two lines with six Angstrom units between them.) The reading from the circle should be
Hydrogen Slit of
Tube n n .Spectroscope
Secondary Terminals
taken, thru n-nmvc the pri-m and take the reading when
telescope is set t..i th< -In image as viewed dire< ti\ i.e. in thr same straight line as th« <<>ilimator. The difference in these two readings will give the D. The angle of th< pn,m A has already been deter- mined, so that the refractive index for sodium light (de- noted nD) can h ted.
placing a hydrogen tube* in front of the imiices of the prism for the C (red), F (green), a> tained.
Dispersion noted by the difference in refractive
• \ N voooraient type of" Hydrogen Tube "for thfcwoifcfe the" Ct form. Thk u .hown in Fig. 06. with the electric*! ooOMcUoM wbfn ftccumuUtore and " coil " MB employed
74
PRACTICAL OPTICS
index between the two wave-lengths in question, and is written usually :
C to D = -00481 (for instance)
D to F^« 00970
F to G! = -01741 Dispersive Power of the prism is given by the formula
T>=Ua> ?_?c > where nGl and nc are the values for the re- nD- 1
fractive index for the Gx and C lines respectively.
Refractive Index of Liquids. — The refractive index and* dispersive powei of liquids may be found by the above methods by using a hollow glass prism with the sides of "plane parallel " and optically flat glass, and filling the prism with the liquid under test. Such a prism is shown in Fig. 66. Plaster of Paris makes a good FIG. CG. cement to secure the sides and base.
(d) REFRACTIVE INDEX BY IMMERSION. (See Trans. Opt. Soc., vol. xvii., No. 3, Dec. 1916. Mr L. C. Martin on " Refractometry and Identification of Glass Specimens.")
A very useful means of obtaining the refractive index of specimens of glass in a rough or unpolished state, or of lenses, is by immersing the specimen to be tested in a liquid of the same refractive index contained in one of the hollow prisms shown in Fig. 66. The whole can then be mounted on the spectroscope and the usual necessary measurements taken.
For this purpose, however, it is necessary to have a liquid of variable refractive index. Carbon disulphide and alcohol mixed together provide a readily adjustable solution ; in practice it is found best to start with pure carbon disulphide in the prism, immerse the specimen, and then dilute the solution with the alcohol. The strength of the liquid should be adjusted so that its index is very slightly higher than the value required to focus
SPECTRUM KT I :ii MEASUREMENTS
ply (on looking through the telescope) any particular line <»f the spectrum K.V. the sodium linos), and the evaporation of the carbon disulphide, which usually occurs faster than the alcohol, will presently bring the line into focus very slowly and distinctly. At the moment of sharpest focus, the anirlr of M miniiiunu deviation " is
• n in the usual waj% and the refractive index \\, out iii the usual way from the formula.
One of the most important factors of the \\hole ex- periment is that the liquid in the prism should be kept
niogeneous. To this end it is necessary to have the liquid mechanically stirred ; a small " propeller blade " driven at a suitable speed in the liquid by a small elec- tric motor will secure this condition. The motor must not be mounted on the same table as the spectroscope, as the \il>ration \\ill interfere with the readings. The method is suitable for any rough small pieces of glass, except forms approximating to plane parallel plates.
'periment. — Find the refractive index of the specimen supplied to you by the above method for " D " light (X 5893), and then for the C, F and Gj lines (hydrogen) ; also determine the dispersion and dispersive
(«) DETERMINATION OF THE WAVE-LENGTH OF LIGHT BY MEANS OF A DIFFRACTION GRATING
This experiment again involves measurements \\ith the spectroscope, but instead <>f u-ing it for the d<
i is to be used fW ti- the wave-length of certain lines in the spectrum
purpose a "Grating' d tin- eon-i-u of
a piece of speculum metal uhich has its surface ruled a great numtx tallel lines very close togethei
rulings are about 14,000 lines table
transmission 'grating is made 1> gelatine from a
The full theory of the grating must be revin text-
books (Edser's \.\x\\\ for Students" or Baly'n " Spe
76 PRACTICAL OPTICS
scopy "), and cannot be dealt with in this book, but it will be sufficient to say here that the spectrum produced by a grating is due to the " interference " of waves passing through the spaces in the grating. Let (Fig. 67) AB and CD be two adjacent apertures in the grating,
FIG. 67.
and that parallel light is incident in the direction in- dicated by the arrow, e.g. from a collimator SL, S being a slit parallel to the apertures of the grating. Now the supposed " ether particles " lying in the apertures AB and CD become sources of vibration which proceed chiefly in the direction towards Nl5 N2, N3 and N4, but also, however, in other directions, as towards Ol5 O2, O3 and O4. If the former rays are brought to a focus by means of a lens (i.e. the telescope objective), they will produce a bright image of the slit without any mutual interference taking place, whereas the case with the diffracted rays Olt etc., is rather different. In order to investigate the " interference " among the latter, the straight line BE is drawn perpendicular to D04, when the line DE will represent the difference in path travelled by the two outside rays D04 and BO2 and also between the two outside rays C03 and A0l5 and, therefore, also the difference in path travelled by every pair of corre- sponding rays in the two " pencils/' If now DE is equal to any odd number of half wave-lengths, it follows that for every ray in one pencil there is a corresponding ray in the other " pencil " at opposite " phase/' and, therefore, total interference takes place when the rays are combined at the focus of the lens. The same holds good for every adjacent pair of apertures of the grating.
SPK< TimMKTKi; MEASUREMENTS ::
But if DE be e<|iial to any even number of half \\ length- tli- corresponding ray in the two pei
will be at equal "phase," and. therefore, the rays from these two apertures and every adjacent pair will combine the focus of tin- lens to give a bright image of tin ^\\\. Thus it will be seen that from a "grating" a spectrum will be formed on * 'cither -id. of the direct image of the slit, and the deviation of lines in th« -JM< truin from the direct image is dependent on tin wave-length, i.e. the length DE decides the angle Dl'.K which equals Z.N4DO4. This gives a means of determining experi- mentally the wave-length particular line in the spectrum, for the deviation N4D04 can be measured with spectroscope, and the di-tanee DB can be obtained from knowledge of the number of lines per inch of the rulings:
/. ™'; -m /_DBE.
!• hi: •.\a\r-lenirth the spectrum seen in t In-
direction O4 i- km.\vn as the " first order spectrum." If DE equals "two" wave-lengths a "second order spcetrum \sill be seen, and so on.
Experiment. — Perform all necessary adjustments to th
spectroscope, and then, illuminatim: the slit, take the
reading of the telescope when the image of the slit is
on the cross-wires as seen "directly/' iV. m the same
straight line. Set up the M grating " in a vertical POM
over th- of the prism table It ;- important . tir-t
of all that the grating is set n to the .,\i* of
I > do tin-, move the telescope round
le until it i- exactly 90° from its previous reading,
rotate the prism table with the grating on it until light
ii the collimator is reflected off the plane glass surface
the t« lescope and an image of the slit
tiade to cor ith the mt«-r-«-cti.,n of the cross-
ui!l then be at 46° to ctascope
or eoli Take the vernier readings of the |H
e and then rotate it to a position 45° ft ious
78 PRACTICAL OPTICS
reading. The grating will then be at right angles to tlu- axis of the collimator ; the plane glass side of the grating should be towards the O.G. of the collimator.
For this experiment a very good source of light to use is a mercury vapour lamp, as it has a few prominent and well- spaced lines in its spectrum. These lamps can be obtained from the Cooper, Hewitt Co., and are very suitable for the laboratory. However, if this is not available, the sodium flame and hydrogen tube may be used as before.
Direct the collimator towards the source, and on moving the telescope to about 18° from the direct reading, the spectrum (first order) will be seen in the field of view. Set the cross-wires on some definite line (if the mercury spectrum is used two yellow, one bright green, and one violet line will be seen), and take the reading of the telescope verniers, take also a reading when the telescope is on the other side of the " direct " position ; these two values should be the same, of course. Calculate the wave- length of the line from the formula :
\=d sin 6, where X=the wave-length,
d = the mean distance apart of the rulings, and 0=the angle between the direct and diffracted image of the particular line in question.
Repeat the experiment for the other lines in the spectrum, then move the telescope still further round, when the spectrum will be seen to repeat itself, this being the " second order." Take readings for the same lines in this spectrum and again determine their wave-lengths ; in this case, from what has been said before, the formula will be :
X = -d sin 6,
and X = «d sin 6 for the third order, o
Tabulate all your results.
SPK< TR< ).M KTKK M KASl'H KM KNTS
79
of Prominent Line* in the Sodium, Hydrogen, and Mercury Spectra
Line.
\Vu\«'-l, 'IlL'tll ill (Ml-.
D! Sodium
D2 „
C Hydrogen
F
G!
Mercury lino
6563 red 4861 blue 4102 violet
5461 green- 4359 violet
(/) CALIBRATION OF THE SPECTRUM
The use of the spectrum for the purpose of analysis is now well known, gases and metallic substances each having
a characteristic spectrum when seen through the spectro- scope. Thus it is that the spectrum may be " mapped out "
80 PRACTICAL OPTICS
by simply measuring the various deviations (with one of the previously described spectroscopes) for certain lines of the spectrum of known wave-length, and plotting a curve one against the other. By the aid of this curve we can find the wave-length of any unknown line.
Experiment. — Determine the values and draw out such a curve.
There is a certain type of spectroscope, however, which gives the wave-length of any spectrum line direct, without the necessity of having to make a calibration curve. It is known as the Constant Deviation Spectrometer, and employs a prism of the type shown in the last section of Chapter I. A plan of the instrument is shown in Fig. 68. SLX is a collimator and EL2 a telescope set accurately at 90° to one another. P is the " constant deviation " prism through which the light from the collimator travels as indicated in the figure, and becomes dispersed. This prism rests on a circular table T which is rotated by means of a micrometer screw M ; to this screw is attached the drumhead D, which is engraved in wave- lengths. To use the instrument, all that is necessary is to set the drum to read a known wave-length, such as X5890, then move the prism on its table by hand until that particular line comes coincident with the inter- section of the cross- wires in the telescope. Clamping the prism in this position, the instrument is now adjusted. By bringing any other line of the spectrum on to the cross- wires, its wave-length may be read off direct from the drum, the calibration of which has been carried out once for all by the makers.
CHAPTER V
DETERMINATION OF RADII OF CURVATURE OF SURFACES
(a) *TpH K most usual instrument that is employed for de- JL tennininir the radius of curvature of lens surfaces is a "spherometer." There are numbers of types of this instrument — for example: (i) the "three-legged," (ii) the "ring" type, (iii) the " Aldis type, (iv) A M»e type, etc., but all spherometers are dependent on a certain formula.
This formula is deduced in the follow- ing manner (see Fig. 69) : Let ADB be part of the rinumfer-
of a spherical surface (e.g. a lens) 09.
in section, a require the radm-
< » \ or OD of the surface. Draw AB perpend i» -ular to OD. Then the A OAC is a right-angled triangle, and
OA^OCP + AC* ; alsoOC = OD-CD. Call OA = R, CA = r, and CD-A nR*=R
.•mil R =
Z/I
the spher< is a means of obtaining the dis-
i* CD=A and CA=r, from \\hirh K (the reqn
oaloolatecL Fig. 70 shows a tin
be j" Mnall tripod, in the centre of \\hxh
is mounted a very finely pitched mi. r. -meter sen a divided disc attached to it. T 1 disc reads
r mediate values of di \i-n.n-. <>f the vertical scale v ii
82
PRACTICAL OPTICS
shown in the figure on one of the legs. When using the instrument, it must first be placed on a flat surface (such as an optical " flat "), and the micrometer screw moved
Vertical Scale
Circular Divided Plate
Micrometer Screw
T V
FIG. 70.
up or down until all four " feet " are exactly in contact with the surface at the same moment ; the reading on the scale and divided disc should then be taken ; this should be the zero of the instrument. After this, place the in- strument on the surface whose radius is required, and again move the micrometer screw up or down according as the surface is convex or concave and take a second reading. (All readings should be a mean value of a number of settings.) The difference between the readings taken on the flat and on the curved surface will give the value " h " in the formula. The value " r" which is the radius of the circle on which the three legs lie, is very often engraved on the instrument ; however, for accurate measurements this should always be checked by measure- ment with a travelling microscope. This may be done in two ways * : either by measuring the distance between
* Proof. — In Fig. 71, A, B, and C are the three ''feet" of the spherometer in plan, and the distance AB, BC, and CA are measured. Call their mean value "p."
Then—
: ~p . cos 30C 3
:2P ^!-
This method for obtaining "r" is more espe3ially useful when the "feet " of the spherometer are worn flat.
ItADII OF CURVATURE OF SURFACES 83
the centre leg and each outside leg in turn, and taking the mean value, or by obtaining the mean distance l>et \\een each outside leg and dividing l»y >/3.
Experiment. — Determine the radius of eurvature of the convex and con- cave surfaces supplied to y«»u \\ith the three-legged spherometer. Check the value for " r " by means of a B~~ T
measuring microscope. 71.
"Ring" Type Spherometer.— This
of spherometer is very similar to the three-legged, but it involves the use of a metal " ring " in place of the three legs. The micrometer screw and drumhead are used in a similar manner, but to determine the value "r" in the previously men- tioned formula the maximum in- ternal diameter of the ring must be measured for convex surfaces and tin- maximum external diameter for ive surfaces. The instrument is »ho\\n in Fi^r. 72,
Abbe Type.— A rather better and more accurate type of spheromet* r is the Ahhe t\jn-. The instrument is shoun in KiLr. 7.T It U8C8 a veil- tit ting steel plunger sliding up and down in a vertical direction The surface to be tested is placed on an accurately mrm-d ting situated at the top of the instrument. \\hiUt the spherical nose of tin plunger is kept in contact with the surface by means of a con ^ht suspended over small pulleys.
Attached to tli- plunder is an enuraxed seale divided in
tenths of a millimetre (-1 mms.)t which is observed by a microscope with micrometer eyepiece, and readings may be taken to th of a millimetre. As in the last case,
84
PRACTICAL OPTICS
the internal and external diameters of the particular ring in use must be measured. A series of rings of various
on which Lens rests "-
Steel
£ |
M |
j |
||||
_L |
1 1 |
|||||
1 | |
||||||
£ |
ft- h i/r i/i (i i i i i_i_ij |
0 |
N y |
P |
||
DIL^ Reading Microscop Balar Wei& |
e ice ht |
|||||
f |
•\ |
|||||
( |
||||||
V FIG |
\ . 73. |
diameters is supplied with the instrument for use with corresponding sizes of lenses.
Aldis Type. — A still better and probably the most accurate instrument of its kind is the " Aldis " Sphero- meter (an illustration of it is given in Fig. 74). The surface to be tested is allowed to rest on three small spheres, and the micrometer screw is screwed up to touch the surface. Opposing the screw is a weighted plunger which rests on the other side of the lens ; by this means the instrument is rendered extremely sensitive, for contact between the point of the micrometer screw and the surface is at once detected by touching the edge of the lens with the finger-tips and judging the ease of -rotation. If the lens revolves freely the micrometer screw is too high, and if the lens will not revolve the screw is not touching the surface ; a position will be found when the lens will just and only just revolve, this will be when the point of the screw is in correct contact. The drum attached to the micrometer screw is 2 in. in diameter, and readings
RADII OF CURVATURE OF SURFACES
Bfi
may be taken to -00001 of an inch. In using the sphero- meter formula with this instrument the value " R " is the radius of curvature of the surface + the mean radius of the spheres ; therefore on arriving at the calculated
Pic
value of " R," to obtain tin- true radius of curvature of tin- surface the radius of the spheres must be subtracted. a sketch is drawn this will become evident : it is equivalent to working on a sphere of radius R + x, where x is the radius <>f the spheres.)
URVATURE OF " SMALL DIAMETER " SURFACES It is obvious that the use of the spherometer is limited
the diameters of lenses ul.. n N uses are from 1 m.
liamcter downwards, and more especially microscope objective lenses, some other method than the sphero- meter has to be employed. The following method gives a good and very accurate way of determining the radii of CM; 1 diameter lens surfaces.
86
PRACTICAL OPTICS
jbfifo
Fig. 75 gives a diagrammatic explanation of the method. Light from a distant lamp is reflected into the eyepiece of a microscope by means of a plane glass reflector G. (For this experiment it is better to remove the field lens of the eyepiece.) Then an image of the lamp will be
formed in the focal plane of the eye lens at I1? and also a second image by the micro, objective O at I2. Now, if the surface to be tested is placed at I2, light will be reflected from it, and, returning
along its original path, will form
another image at Il5 so that an eye placed at E will see this
Concave image ; the first image will, of Surface , ,
course, not be seen, as the light
is travelling in the wrong direc- tion. Similarly, by placing the surface in a second position, as at I3 (when all the rays from 0 strike the surface normally), another image will again be seen at Ij.
The distance between these two positions of the surface, namely, at I2 and I3, will give the radius of curvature of the surface. Refer to method of determining the curvature of a convex mirror (Chapter II., section (c)).
For measuring this distance I2I3 accurately, either the microscope must remain fixed and the lens move in a vertical direction, or the lens remain stationary and the microscope move on a vertical axis. In the latter case the experiment is simplified by employing a measuring microscope with a special adapter (made by the Cambridge & Paul Scientific Instrument Co., Cambridge), described in Chapter VI., section (a), as this instrument can be used very conveniently in a vertical direction, and measurements taken to a thousandth of a millimetre. In the former case a simple piece of apparatus may be made up by adapting a Brown & Sharpe micrometer
75.
IlADII OF CURVATURE OF SURFACES
87
1 to the stage of a student's microscope, as depicted in Fig. 76; in this case the' lens would be attached to the movable head and moved up and down with it. readings being taken from the micrometer drum for the two positions of the lm>
ice I2 and I3 in Fig. 65. Experin - Determine the
radius of curvature of the convex and concave surfaces supplied to you by one of the above methods. (The method applies equally well to concave surfaces as well as < -on vex.)
CURVATURE : NEWTON'S RINGS METHOD
Micrometer Head.
Thoroughly clean a long focus convex lens and a piece of plate glass (flat), press them together
and examine the reflection of th« ^k\ n< n tin point of con- tact. A dark spot surrounded by a series of colour. -d rings "ill be seen. By using monochromatic light, such as a sodium flame or im-miry vapour lamp, many more rings, alternately light and dark, may be seen It will he found igs are closer together as they are larger, also it \\ill IM- noticed that the rings are closer for \rll..\\ than d light, and still closer for green or him- light. Th< n formation is due to the interference between the light in tit* ti'.nt and back surface* of the air film .••••n the lens and glass plate. The i be seen \>\ tted light . m this case, however, they
are mn<-l.
Let ua con*id< i thr theory; and to Himj.hfy this n i> »• are cone< t \\ith ih.
<• the \*> the
i. -tm- <>n a j.imr Hurfucc OMN (Fig. 77). In the figure, O is the point of contact of the lens and surface, to
88
PRACTICAL OPTICS
the complete figure is symmetrical about the point O. Consider light coming in the direction LO normal to the surface OMN. At a given point, A, the thickness of the air film between the two surfaces is AM. Part of the light incident at A passes straight through the film at this point without reflection ; another part is reflected at M,
r,
N
FIG. 77
and again at A, and finally passes out at M in the direction MM.^ It therefore suffers a retardation in path equal to 2AM. If 2AM is equal to half a wave- length of the light considered, or any multiple of half a wave-length, the two portions of light differ in phase by half a period and " interfere/' producing a dark band at A. If, however, 2AM = a whole wave-length (h.) or any multiple of h, the two portions of light combine at M in the same phase, and A is the middle point of a bright band. At 0, where there is no difference of phase, there is a bright spot. On passing outward from 0 the thick- ness of the air film increases until it becomes equal to
— At this point there is a dark ring : still further out
the thickness has increased to ~> and at this point there
3h is a bright ring : then when the thickness is -r there is
a second dark ring : and so on.
If M is the position of a dark ring and R is the radius of curvature of the surface OAB, then by a property of the circle
RADII OF CURVATURE OF SURFACES 89
I'll x AM = rl2 nearly ; or -AM = ?R'
If B is the position of the next dark ring, 2BN = -=|~ Hence
r2 , X 3X 5X (2n+l)X t ,. must equal > 0 > 0 , etc., or generally r2 = ^ =— - R,
I . Z Z Z -
where " n " is any integral number or zero.
The radii of successive dark rings, therefore, increases as the square roots of the odd natural numbers, and the
areas of the annul! hetween successive rin^s are the same.
Also iMiN i'AM=X:
therefore X = gW - rt*) .
If B is not the next but the nth dark ring after A, we have
1
In this experiment the wave-length of the light being used would be known (that of sodium light being •0000589 cms. or that of a mercury vapour lamp passing through a green filtn 1>« -ing •000054 <> ems.), so that the expression may be made, with a km.\\l«-dge of the radii « 't th»- rings, to give the value of R.
When the rings are viewed by reflected light dark bands are seen \\h«-n tin retardation within the film is X or any multiple of X.
This is due to the t\\«> reflections not taking place under MM- same condition-, lii the transmitted light both re- fl<( tions are from -urfaceR of the glass, but for reflected light on< ntl«,n«>n i. it A i i- from a surface of air and one (at M) from a surface of glass. IY«m tin- MOM th« -n-
i- produced a retardation of phase ', \\ln.-h must be add< d
to that due to t in paths.
With th< |, -us and glass "flat" supplied to \ nigs by placin- tin- t\\«» m oontaot, th«-
flat surface resting on the curved one (see Fig. 78). The
90
PRACTICAL OPTICS
system may be made <]iiiu* stable by small pieces of soft
,it A ami l».
The measi: of the rings formed by reflected light
is effected by means of a measuring microscope. The Cambridge & Paul Scientific Instrument Co/s type, as de- scribed in Chapter VI., section (a), is very suitable.
The point of contact of the two sur- faces is viewed with the microscope, and is illuminated by means of a " vertical illuminator " in the micro- scope. This piece of apparatus is shown in Fig. 79, and consists of a small plane glass plate placed diagonally between the objective and the microscope body tube ; in this way light from the monochromatic source is reflected down normally upon the " flat " and lens. As a mono- chromatic source, light from a mercury vapour lamp filtered tli rough a green gelatine filter gives best results for this experiment, although, of course, a sodium flame may be oaecL
By means of the microscope measure the diameter of the 3rd, 5th, 6th, and 7th dark rings— also the 15th, 16th, and 17th — or even of three rings further from the centre, say the 25th, 26th, and 27th if possible.
Calculate an approximate value of the radius from the 3rd and 7th rings say —correcting this value from calculations made from the radii of the most widely separated pairs measured, say the 5th and 25th, the 6th and 26th, etc.
The determination from the 3rd and 7th riiiL^ will prevent mistakes being
FIG. 79.
made if a wrong number of rings is counted in the further work.
In this way, applying the formula, the radius of curvature of the surface R may be obtained.
RADII OF CURVATURE OF SURFACES 91
(d) RADII OF CURVATURE : REFLECTION METHOD (KOHLRAUSCH)
This experiment gives another convenient method of determining the radius of curvature of lens and mirror surfaces ; moreover, the method is applicable to both large and small surfaces.
Fig. 80 shows the method employed. Two light sources,
.--" i &*™~
FIG. 80.
such as candle flames, or preferably two illuminated vertical slits, are placed at Lt and L2. At the mid-point between these two is situated a telescope T, so that the object-glass lies in the same straight line as the two lights. The surface to be tested, either convex or concave, is placed at S, at a distance not less than 3 metres, so that on looking through the telescope two images of the light sources will be seen by reflection from the surface under test. If, now, a glass scale G is placed in contact with the surface, the separation of the t\v<> iriiai^-s may !><• nic.i-invd. l<Y<>m this and a knowledge of the distances ST and LjL2 (these can be measured with a steel tape), the radius of curvature of ih«- surface may be obtained from the following formula :
r- "' for a convex surface and r- ~' (/ for a concave surface
PRACTICAL OPTICS
\\ here r = the radius of curvature of the surface, d = the distance ST, / = the measured separation of the images on the
scale, L = the distance apart of Lj and L2.
The student should prove these formulae for himself from previous knowledge ; however, the proof for a convex surface is given below :
The line L gives an image behind the spherical surface
1121 at a distance xy by the rule - = -,+ - (-r is the focal length).
X u> T £
The length y of this image is also given by
y _x L~d
From these two formulae we find
dr Lr
The length between the two images seen in the surface and measured with the glass scale is " I " and equals y • -* - , from which, by substituting the above values of x and y,
1 rL
~2dT?
2dl r = L-U
In exactly a similar way is deduced the formula for concave surfaces.
CHAPTER VI MISCELLANEOUS ELEMENTARY EXPERIMENTS
(a) THE MEASURING MICROSCOPE
THE measuring microscope is an instrument of funda- mental importance, and therefore its use should be familiar to all students. A very good type of instrument, especially for laboratory work, is made by the Cambridge & Paul Scientific Instrument Co., and is shown in Fig. 81. As
81.
will be seen, it consists essentially of a microscope M, which i- made to travel by means of an accurate, finely-pitched nii< rometer screw S. The microscope is attached to the tube T, along which it may be adjusted at will, hut can be ii\<d rigidly when measurements are being taken. The tube T slides in two V's at Vt and V2, in which it is held by two opposing springs ; at the end of the tube is situated the micrometer drum D, by means of which intermediate values of the whole divisions on the scale C are read off. Usually C is divided into millimetres and the drum D into one hundred parts, so that with careful estimation readings
93
94 PRACTICAL OPTICS
may be taken to one-thousandth of a millimetre. F is the stage on which the object to be measured is placed. An advantage of this type of instrument is that it may be used cither in a horizontal (as shown in the figure) or a vertical position.
Experiment. — Examine carefully the measuring microscope supplied to you, noting its mechanical construction, the arrangement of the optical parts, and the adjustments, etc., and draw a sketch of the instrument.
Adjust the eyepiece of the microscope to view the " cross- wires " clearly when the eye is " at rest/' i.e. so that the " accommodation " is not strained. Place the object (a "graticule" or "spectrogram") to be measured on the stage, and carefully focus it by means of the milled head O until the " image " is seen sharply defined at the same time as the cross- wires. Arrange the cross- wires diagonally so that a line of the object may be set accurately on their intersection. In this way measure the distance between consecutive lines of the object by readings obtained from the scale C and drum D. Care should be taken in making a " setting " always to rotate the milled head R in one direction for each independent reading, in order to overcome any error due to " backlash " of the micrometer screw.
As an additional experiment, the " pitch " of a screw may be measured in a similar manner. Measurement of (say) three threads will give the interval very nearly ; a large number may then be measured without counting, the actual number of threads being found by the first approximate result. The length divided by the number of threads then gives a value for the pitch.
(b) APPEARANCES OF " STAR " IMAGE AT THE FOCUS OF A LENS
One of the best ways of testing the performance of a lens or lens system is by viewing the image of a distant star produced by the lens under a high power, such as a microscope or high-power eyepiece.
M I SCELLANEOUS ELEMENTARY EXPERIMENTS 95
As actual stars are not always available, a very good artificial star may be made by allowing light from a circular aperture to fall on to a small steel ball about J in. in diameter (one from a " Hoffmann " ball-bearing acts extremely well) at right angles to the direction in which the tests are to be made. The extremely small image seen in this spherical surface affords a very satisfactory " point " source. The distance of the lens under test from the artificial star should not be less than 50 feet ; it is advisable also to have a black non-reflecting background immediately behind and in the neighbourhood of the steel ball.
Experiment. — Mount a single lens of about 25 cms. focal length in one of the optical bench lens holders (see Chapter II.), and place it on a one -foot steel rule made up as an optical bench, as described in that chapter. Place a high-power eyepiece (in its fitting) also on the steel rule, direct the latter towards some distant object, and arrange the position of lens and eyepiece until the object is clearly seen. Now direct the optical bench towards the artificial star, carefully centring the system so that the image of the star as seen in the eyepiece appears perfectly central.
Focus the image until it appears at its best focus and make a coloured sketch of what is seen. Then move the eyepiece about 2 cms. inside the " best focus " position, observe the appearance, and again draw and colour the rings seen. Do the same when the eyepiece is moved - < IMS. outside the " best focus " position.
Explain with a sketch the reason why "inside" the best focus a red ring is seen on the edge and blue in tin- centre, and why " outside " the best focus a blue ring is seen on the edge and red in t he centre.
Fig. 82 shows what actually happens — light from the star on reaching the lens L is refracted, and exactly as in the case of a pri-m i> >|>lit up into its various eom- ponents, blue it will In <1 \yeing deviated more
than red ; so that \\h«-n the rays are brought to a focus, l>Iue rays will focus at a point nearer the lens than the
96 PRACTICAL OPTICS
red, as shown in the diagram. Consequently when the appearance is viewed inside the focus, as at I, a red ring will be seen on the edge and blue in the centre ; and conversely for outside the focus.
position FIG. 82.
This appearance of colours at the focus of a lens or lens system is known as " chromatic aberration/'
An " achromatic " lens should now be substituted in place of the " single " lens and the difference in appear- ance noted. Unless the achromatic lens is an extremely good one the coloured rings will still be detected inside and outside the focus, only on a very much smaller scale, and from these it will be possible to tell whether the lens is "over-" or "under-corrected." In connection with spherical aberration, the most noticeable effect seen with an achromatic lens, more especially when a microscope is used to view the star image, is the appearance of a series of concentric dark and light rings ; these are due to diffraction. With an " over-corrected " objective the ring system outside the focus will be clearer and better defined than that inside the focus. With an under- corrected objective the reverse will be the case. If the lens is satisfactorily corrected the appearance will be the same both inside and outside the focus.
(c) DETERMINATION OF THE " FOCAL LENGTH " OF EYEPIECE SYSTEMS
The equivalent focal length of an eyepiece system may be determined very conveniently in the following manner :
MISCELLANEOUS ELEMENTARY EXPERIMENTS 97
Tin i -\v piece to be tested should be held in some suitable mount (a retort stand) on the table at a distance of about 15 to 20 feet from the wall. To the wall >hould be attached a piece of paper or cardboard on which are painted two bold Indian ink lines about 2 metres apart. If now the eyepiece is directed toward- the mid-point of these two lines, images of them will be formed by the eyepiece, as at Ix and I2 (Fig. 83), and whose distance
i i
apart can be measured either \\ith a dynameter " * or measuring microscope. In the case of "negative" •
M n i^ hetter la n>e the latter \\ith about a 2 in. objective.
In the figure (83) ASB i- the cardboard with the t\\«. hnrs at A and B. Pl and P2 represent the t\\<» prinrip.il plain-s of the < \^i.in ,md IJ, the images of B
and A. It is at < <irnt that tin- triangles ABPj
and IiIsPs an- -imilar. BO that :
SP =fT^* (FP«i8tllr '•'•'!""rd tncal length.)
_I1I8xM P«" Ah
The distances AB and SP, can be measured \\iti
or | itad t..p. . it these distances are large,
a small error in thnr measuremt>nt \\ill n<>t cause any
• /'>/""""'". A Mimll j.Hirc.f ftjijiarntusconniiitingof a R*mJHlrn or jx
eyepi.. • . ;n tl» f... .1 i.l.i mounted A finely divided gl*M Male,
usually I . in. ih it*. It prove* very tweful in many experiment*. O
98 PRACTICAL OPTICS
appreciable error in the focal length of the eyepiece. The three values on the right of the equation having been obtained, the equivalent focal length of the eyepiece may thus be found.
Second Method. — Another very satisfactory method of determining eyepiece focal lengths is by using a colli- mator (see Fig. 84) having two lines A and B subtending
Microscope
FIG. 84.
a certain angle 0 at the object glass C. (This angle is carefully obtained beforehand.) The eyepiece E to be tested is placed in the path of these two parallel beams, and two images are formed at Ix and I2 and their separation measured.
Then if I = this separation and "/" the required focal length of the eyepiece,
-, = 0 (in angular measure, when 0 is small, as it is).
So that when once I is measured it need only be multiplied by a constant (i.e. the reciprocal of " 0 " in angular measure), and the focal length of the eyepiece is obtained.
Sometimes a microscope is used to view the image I and which has a scale in its eyepiece. In this case
where M is the " first " magnification of the microscope, and I3 is the separation of the two images measured by the scale in the eyepiece.
A " f ocometer " of this kind may be very easily made by attaching such a collimator as shown in Fig. 84 to the underside of the stage of any ordinary microscope.
MISCELLANEOUS ELEMENTARY EXPERIMENTS 99
'11i«- lens C should be achromatic and about 1| in. to 2 in. focal length : the twn lin. •- A and B should be about 1 mm. apart.
See also Chapter VII., section (d).
Searle's Goniometer. — In connection with these experi- ments a piece of apparatus known as Searle's Gonio- meter will be found useful. It consists, as will be seen from Fig. 85, of an arm A, on which are mounted a len^ L and a single vertical line object O, the latter
Fio. 85.
being at the focus of this lens. This arm swings about
the centre L, and the amount of rotation is read off a scale
S by means of a fine wire W. A strip of mirror M is
situated at the side and slightly below the scale, in order
to ensure a directly vertical observation of the reading
being made. This is done by moving the eye until the
urn- and it- iniaL'e I'nmi the mirror appear coincident.
So that )>y the use of this appaiai uigular subtense
«-t the object 0 may be obtained at will.
As an example, this goniomct- r may l>< n place
lie scale on the wall, mentioned in the first method
for determinim: the focal lengths of eyepieces in this
•ter.
(d) ECCENTRICITY OF A " DIVIDED CIRCLE " The testing of the ecceni a di\i ,-le is
iys a necessary experiment in order to obtain a \vledge of the error of readings taken from such a
circle \\hen in use. More especially is this im-
100
PRACTICAL OPTICS
portant when only one vernier is employed on the circle. In the case of more accurate instruments, where micro- meter microscopes are used instead of verniers, besides the systematic error brought about by eccentricity, the individual error of each division of the circle must be taken into account. For such a circle a " calibration curve " is made out, so that error for any part of the circle may be read off from the graph.
Fig. 86 will illustrate effects on the readings of the circle due to eccentricity. Let D be the " dividing centre " of the graduations, C the centre of the alidade
(i.e. the arm on which the verniers are carried), and Vx and V2 the zeros of the verniers. Suppose, in this case, that the circle remains stationary and the verniers move round the circle. Evidently, when VxCVg coincides with the diameter through C and D, the readings of the two opposite verniers will differ by exactly 180° (this assumes that the zeros of the two verniers a re- in one and the same straight line as C), and when at right angles to that diameter the difference will be a maxi- mum. In the figure, "V^CVg represents this position, and the " angular eccentricity " will be half the difference in the readings, that is, the angle VjDA.
Experiment. — The student should be supplied with some instrument fitted with a divided circle with two opposite verniers fitted, such as a spectrometer or theodolite.
Take the readings of the two opposite verniers at twelve or more points round the circle and obtain their differ- ences, care being taken to subtract these values always in the same direction. Then plot the differences on squared paper against the angle ; from this the position of zero or minimum departure from the ideal difference
MISCELLANEOUS ELK.M i:\TARY EXPERIMENTS 101
of 180° may be found. In i\\'\> way a diagram may be <lra\\n >1 n>\ving the relative eccentricity, and a table of values drawn up from the graph, giving the angular error •eiitrieity at any point on the circle.
If the difference of the vernier readings is never exactly 180°, the zeros of the verniers and the point of rotation C are not in the same straight line (such as at V3 instead of V They -hould be adjusted to be so. This error
can be obtained from the graph by the difference between the minimum eeeentrieity >hown and 180° exactly.
Fig. 87 show- typical eccentricity curves. Angular
readings of the circle are plotted laterally, and the dif-
!ice -f or between the two vernier readings are
Difference in Seconds of the ty»o Verniers from true I8O° reacting, I i r i i + +<* + * 8. S. S. g. 5. o 5. 8. S. 8. §. |
||||||||
*/ |
x^ |
N |
V |
|||||
/ , |
/^ |
N |
A |
|||||
// |
\\ |
|||||||
2 |
V |
V |
||||||
y |
> |
\ |
/ |
|||||
v\ |
// |
|||||||
\\ |
/ |
7 |
||||||
\ |
^ |
H/ |
/ |
|||||
A |
X. |
s |
||||||
90*
160
th
270"
plotted tlx>ve and below th« /.cro posit i"i
•pcctivi-ly. ( urvr A iiulicates that the alidade V^V,
t \\iih th«- «li.i Kig. 86)
at 0° on i' ' the greatest ece<
''
was at 90° nn«l 'I WM «-'i";'l to ,, • • • -i \I--.
M tin- error wa« the same at each <>f these last two
102 PRACTICAL OPTICS
mentioned positions, the zeros of the verniers must have been set at exactly 180°.
Curve B shows that the greatest angular eccentricity is
again -^ seconds, but that as the two exact 180° differences
of the verniers occur at 200° and 340° on the circle (i.e. not at 180° apart), it indicates that the verniers are not set exactly opposite one another, as illustrated by an alidade VjVg in Fig. 86.
(e) PHOTOGRAPHIC TESTS ON A LENS
Apart from the tests for spherical and chromatic aberrations of a lens or lens system, as mentioned in section (d) of this chapter, it is sometimes necessary to test the performance of a lens by the actual results given on a photographic plate when a photograph is taken with the lens.
For this purpose the lens should be mounted in some type of camera which has a fairly large " rack adjust -
FIG. 88.
ment " for movement of the focussing screen. Two " test-charts " should be made similar to the one shown in Fig. 88, one small and one large. This type of chart is extremely good, as it is designed to bring out every
MISCELLANEOUS ELEMKV1 A K Y EXPERIMENTS 103
effect of error that the lens can produce. The size of the two charts depend somewhat on the focal length of the lens under test ; the small one can be drawn on a piece of white card with Indian ink. of such dimensions so that when it is placed at the same distance in front of the lens as the image is behind (i.e. when u =v) the image of the chart will cover the whole of the focussing screen or photographic plate. The second chart will have to be very much larger, as the distance from the chart t<» tin- lens in the second case is made about ten times that from lens to image (i.e. u = lQv). It is better if this chart is painted with Indian ink on a flat white wall or board. Card is not advisable, as it is very liable to bend when of large dimensions ; and " flatness " is essential.
When tln-M- two charts have been prepared. they should be illuminated either with daylight or by a carbon arc and photographs of them taken with the lens. As mentioned before, one photograph should be taken \\lnii u=v and one when u = \Qv. The point of making " u " equal to " v " is that defects due to the lens will be more pronounced and are an aid for judging the other ivMilt. Of course, the images must be focussed carefully on the ground glass screen of the camera before any photograph is taken; it is best to use a fairly high pow< i eyep for this purpose. Exposure should be found by trial— lltord " ordinn \ plates are good for such a test.
\\lirn tin- plates are developed. ti.\«-d. washed and dried they should be examim-d « u« tull\ and th< following points looked for :
i • ' ntral Definition.
(2) AstigiiMii in
(3) Distort) Coma.
(5) Flatness of I'M M.
As regards No i < Ultra] iMimtinn u ^. sharp-
rie«* of th< lm< -s (supposing tlm th< plate is at its best
PRA< TICAL OPTICS
ia position) would indicate tliat aberrations, either chromatic or spherical, are presented by the lens. It is a good thing to use a yellow screen in front of the lens and so cut out the blue rays, which will to a great extent do away with chromatic aberration. A-tiirmatism would be detected by the lack of definition on certain of the "radial" lines at right Bfi to those on which the definition appeared good.
i'>) Distortion, if present, would be most evident at the edge of the plate, where the straight lines of the square would appear curved (a straight-edge should be laid along them). Distortion would be either " barrel-shape " or " pin-cushion/' I "Coma" would be indicated by the appearance of the small white circles in the large central cross as being blurred or diffused on one side. Having the effect of a " tail of a comet/'
• ) If the definition is equally bad at all four edges of the plate, and if by taking a photograph slightly in-ide or outside the best focus position for the centre of the plate, the definition at the edges im- proves, roundness of the field would be indicated. In this way the photographic test on a lens may be carried out ; this, combined with the " visual star test " already described for spherical and chromatic aberrations, will give a very good idea as to the performance of the lens.
CHAPTER VII
FOCAL LENGTHS OF " THICK " LENSES AND LENS SYSTEMS
(a) THE " BAR " OPTICAL BENCH
IX connection with experiments dealt with in this chapter it is important that a good type of optical bench be available. The one described in Chapter II. is extremely good for early and more preliminary ex-
Lerrs Holder
\
Eyepiece wth a
Fio. 89.
, but for first-class work it is essential to have i larger and somewhat more serviceable type. Therefore
Rectangular Steel Bar. '( Hnch * '* inch section)
Supports.
'
Divisions engrdved on top of Rod. Circular Steel Rod.
Length • ISO Cms. divided into Millimetres. (//nsfi diam.)
it will l.«- \\rll here to describe an « \; good bench
which, although not on tin- m.irkn. \\ill l»o found suitable
KM;
1 TACTICAL OPTICS
for the experiments suggested ; therefore scale drawings
are given for those who may have the opportunity of making this type of bench for them- selves.
Fig. 89 shows the general appearance of the optical bench with the holders and various fittings. Figs. 90 and 91 illustrate rather more clearly the construction of the " bed " of the bench. It consists of a steel rod and vertical bar mounted side by side and parallel to one another, the former being divided in milli- metres. Along these two slide the " holders/' one of which is shown in Fig. 92 ; the design of these holders makes them quite rigid and free from any
FOCAL LENGTHS OF " THICK " LENSES 107
tendency to turn on a vertical axis when placed on the " bed " of the bench. The cylinder underneath is of lead,
!•'!•:.
in order to make the holder steady, engraved on a knife-edge on j. & the holder with which read- ings are taken from the di vided rod (constituting part of the " bed " of the l>ench). The milled head at the top of the hollow "pillar"
t he holder is for clamping stems of the various fitting which fit into tin-" pillars.
Almost any fitting can. in tin- way, be adapted to the optical bench ; the more e».-ntial ones, hn\\r\rr, are (i) l"iis carriers, (ii) object
FIG. 94. An index line is
l"sep.Ramscten eyepiece „
,. 1 1
108
PRACTICAL OPTICS
or scale holders, and (iii) a scaled eyepiece (/..tli millimetre scale mounted in the focal plane of an eyepiece). Drawings are shown of these fittings in Figs. 93, 94, and 95. Other iiM't'ul fittings can be made up as desired.
Such a " bench " as this will be found an invaluable piece of apparatus for almost every type of experiment.
(b) FOCAL LENGTH OF A " THICK LENS " BY THE MAGNIFICA- TION METHOD
Revise the theory of the method and prove the following formula :
/=2^«j
ml m2 where / = required focal length,
u1 and u2 = the readings taken from the optical bench for the two positions of the object scale,
and ml and m2 — the two corresponding magnifications measured with the micrometer eye- piece.
A glance at Fig. 96 explains the formula. The well- known Gauss construction is used, and if this be remem-
FIG. <w.
bered the formula may be re-derived. Pj and P2 represent the two " principal planes " of a thick lens, and AB an object on the left-hand side of the axis. Draw a ray from A passing through the first principal focus F and cutting the first principal plane at M ; then this particular ray will emerge from the lens parallel to the axis, and
FOCAL LENGTHS OF " THICK " LENSES 109
the image of A must fall somewhere along MN. Therefore, the size of the image of AB must be MO ; so that the triangles ABF and MOF are similar, and thus
OM FO FO m, (the magnificat* = = =
Similarly m2 (the magnification when the object is moved to some other =
j. -*r\
position, as at X)
These two ('({nations may be written :
i JBO-FO
///, FO 1 XO-FO
and m,
Subtracting, J. m
and therefore
in
Experiment. — Place a photographic lens (to be tested) in one of the lens holders on the optical bench. At a distance of about 50 cms. set up a millimetre scale on glass in one of the scale carriers, and then focus the image of this scale on the other side of the lens with a micrometer eyepiece. Measure the size of a number of di\i>ions of the scale as seen through the eyepiece, and thus get tin- magnification for this position of the object. Make a note of the reading of the " object holder " on the optical bench scale. Move the glass object scale to a fresh lion on the bench, focus up the image again and determine the second maLnifi'-ation. From the value-, nl.t.iiiMd .ind using the formula, the focal length of the \- be obtain. -«l. Tin- experiment should be repeated for a number of posit inn- ,,t the object and tin- '• in. MII " rc-ult ohtainnl.
Negative Lens.— In the case of a "thick" negative
. or l.-ns system, the same formula hold- r.jually well.
110
1'liACTICAL OPTICS
but an auxiliary positive lens has to be used, in the same way as in Chapter II., section (e). Form an image Ix of the scale by means of the positive lens (see Fig. 97) — this serves as the " object " for the negative lens- measure the size of a number of divisions of the scale with the micrometer eyepiece, this value is then the " object/'
Scale
Pos.Lens
FIG. 97.
Insert the negative lens between the positive lens and this last image, and move the micrometer eyepiece until an image of the scale is again seen (say at I2, Fig. 97). Measure the size of the same number of divisions of the scale ; this value divided by the last will give the first magnification. Then move the negative lens to another position and repeat the procedure. In this way and using the formula the focal length may be determined.
(c) " CHESHIRE " FOCAL LENGTH METHOD
A simpler and perhaps more accurate method of deter- mining focal lengths of lens systems has been developed by Professor Cheshire recently. A and B (Fig. 98) are
FIG. 98.
two lines of known separation or a millimetre scale on glass. L is the lens to be tested and E the micrometer eyepiece. S is a piece of metal with a narrow (1 mm.) vertical slit cut in it ; this piece of apparatus is known as a telecentric
FOCAL LENGTHS OF " THICK " LENSES J 1 1
-top and increases the exact ness with which A^ may be focussed. The slit S is set at the first principal focus ol the lens under test by placing a mirror M behind the lens and adjusting the latter until a sharp image of the -lit i- seen reflected back near the "real" slit, When thi- i- the case S will be at the principal focus of L. As the ray- AS and BS pass through the first principal focus of the lens, the images Al and Bx must lie on parallel lines and EAX, so that the triangles ABS and EDS are
/" can be
x f -imilar, and therefore r~ = -.- -r , f rom which
found, for A1Bl is measured with the micrometer eyepiece, AB is known, and the distance x is obtained by a " measuring rod." A metre or half-metre steel scale set up on the optical bench serves admirably.
This method can be performed very satisfactorily on the " Bar " optical bench described previously.
(d) " FOCO-COLLIMATOR " METHOD (Trans. Opt. Soc., vol. xxii.. No. 1, 1920-21).
Thi< method of determining "focal lengths** is accurate (to -2 per cent.), extremcl y nmple, but chiefly a quick method. It is this last point which makes the " foco-collimator " very suitable as a " workshop tool."
The principle of the method will be seen from Fig. 99. A and B are two fine diamond lines on glass, situated
Fio. 00.
accurately in the focal plane of an achromatic lens " C," in<l -nl >tending a certain definite angle at the first prim-ipal plane of thi- len-. Tims, two parallel beams emerge from the 1m- inclined at the -aid angle to I
II:.' PRA< IK AL OPTICS
so that the len> \<> be tested L placed in the path of the t\\o beams will form an image of the two lines at A1 and B! ; and their separation is measured with a micrometer eyepiece E, or " dynameter."
It i^ (jiiite obvious that the two triangles ABC and AjBtL are similar, so that the angle A1LE1 = the angle
Now the angle ACB is previously determined accurately l»y a method described later, and as A^ is found by the micrometer e}repiece, it follows, therefore, that the distance " / " (i.e. the focal length) may be obtained. For
l. l = 0 (in angular measure)
or/=A1B1xg; but T\ is a constant, so that all that is necessary to deter-
\7
mine the focal length of a lens is to measure the distance A^! accurately and multiply it by the previously worked- out " factor." Thus the operation becomes a very quick one and is ideal for the workshop or testing department.
The graticule AB and the lens C, constituting the " foco -collimator," are mounted in metal cells at the ends of a suitable tube and fixed permanently with " set- screws " when finally adjusted. The " multiplying factor " should be engraved on the tube. The lens C should be about 8 in. focal length, and the distance between A and B about 4 mms.
Focussing and Measurement of Angle. — The accurate setting of the two lines A and B in the focal plane of the lens C, and the measurement of the angular subtense of these two lines at the lens, are both of extreme importance. These two settings can be done very completely using the same apparatus in each case. Set up the foco-collimator in a horizontal position and illuminate the graticule from a lamp by means of a microscope cover slip or a piece of mica, as shown in Fig. 100. A mirror M is then placed
FOCAL LENGTHS OF "THICK" LENSES 113
as shown, and a microscope (using a 2 in. objective) is arranged to view the graticule. A back reflected " image " of lines of the graticule will thus be produced by the mirror M. It at once becomes evident that when the " image "
Spectrometer or Theodolite Circle
FIG. 100.
of the lines and the " real " lines themselves are in focus simultaneously, as seen on observation with the micro- scope, the graticule lines must be in the focal plane of the lens C. The distance between the graticule and the lens should be adjusted until correct.
In order to determine the angular subtense of the two lines at C, the apparatus can be used exactly as it is, with the exception that the mirror M should be mounted on the centre of the prism table of a spectrometer or some m-trument on which angular rotation of the mirror may be measured. All that is necessary then is, on observing through the microscope, to adjust the mirror until the images of the two lines are exactly coincident with the " real " lines ; take a reading of the vernier from the circle on which the mirror rotates, then rotate the prism table i \\ith mirror on it, of course) until the "first" line of the image has become coincident \\ith the "second" "real " line, and read the circle again. This value will give just half the angular subtense.
A Workshop Tool. — A very convenient and useful "tool" for use commercially or in a testing department may be made by a simple adaptation of the principle of the " foCD-collimator." It i- an ni-iiuuent for deter- mining the focal length- of .-u< Imses or leu-
114
I'UACTK'AL OPTICS
systems (such as eyepieces) <|iii< -kl\ -. All that is necessary is to attach a " foco-collimator on a much smaller scale " to the stage of a simple upright microscope. Fig. 101 shows such an instrument in side elevation. C is the small foco - collimator, employing an achromatic lens of about 1J in. to 2 in. focus and a graticule with the separa-
Graticule
FIG. 101.
tion of the two lines equal to about 1 mm. This is mounted to a metal case which carries a right-angled prism P. The lens to be tested L is rested on the microscope stage, and the images of the two lines formed by this lens are viewed by means of the microscope (which has a tenth- millimetre scale in its eyepiece), with which the separation of the images are measured. Therefore, taking into account the " first " magnification of the microscope (which must be determined beforehand and called here 44 M "), the separation of the two images will now be : MxA1B1 = (say) A2B2,
so that
AlBl~ M2
Substituting in the previous formula at the beginning of this section, namely,
FOCAL LENGTHS OF "THICK" LENSES 115
AB
(in angular measure);
0 x J is constant and will be the multiplying factor.
So that on measuring the separation of the two lines \vith the scale in the eyepiece of the microscope it is only necessary to multiply this separation by the " factor " and the focal length of the lens under test is
obtained.
(e) " LENS ROTATION " METHOD
This method employs the rotating of the lens system about a vertical axis and can be performed very suitably on the " bar " optical bench. The theory of the method will be seen from Fig. 102. Let Nx and N2 be the nodal points of the lens system, which we will suppose has been rotated through an angle 0. Now, a ray ANj entering the system and passing through the first " nodal " point will emerge from the lens parallel to it- original direction from N2. If, then, tin- lens be rotated about any point other than .V. the ray N2B will shift from side to side. It is using this fact that the following method is based : A collimator /-^. i \\itli «it li< r a slit or small circular aperture as ohjeet) is set up on one of the "V1 >upports on the optical bench. The lens (((>
to be tested is held in one of the lens holders nniilar to that >h<>\\n in KiiT. '.'.'I. \\ith the exception of a " rack motion " being fitted for movement back \\ar«l- or forwards of tin- upper portion of the holder. The image produced by the l«-n- ifl \ ie\\ <l \\ith a mi. MI-MIL: I in. objective). Tic- leu- li..|<lrr i- thru rotated through a small annle and hark again to the Other \\hrn the image of the slit or aperture <as the case may
IK; PRACTICAL OPTICS
be) will be seen to move across or perhaps right out of tlu' field of view of the microscope. The lens should be moved either backwards or forwards by means of the iimt inn on the lens holder and again rotated. When the image remains stationary the second nodal point of the lens will he over the centre of rotation of the lens holder. This will be recorded on the optical bench by the index line on the holder. The focal length of the lens will be the distance between this last position and the focal plane of the microscope. This focal plane may be recorded on the bench by resting a set-square on the dividing and moving it backwards or forwards until its edge is sharply focussed when observing through the microscope.
CHAPTER VIII MISCELLANEOUS ADVANCED EXPERIMENTS
(a) FOCAL LENGTH AND NUMERICAL APERTURE OF MICRO- SCOPE OBJECTIVE
THE focal length of a microscope objective may be determined in a very simple manner with no apparatus other than a microscope itself, and by the adaptation of an alternative formula used in the " mag-
'n;. lo:i.
nification method " of finding the focal length of any ordinary objective (as in the last chapter). The formula is deduced as follows (see Fig. 103) : —
A1B1 is the size of an " image " of AB produced by a lens at a distance vl from the second principal plane of the lens, so that the (first) magnification
Similarly, if the image is made to fall at a second position A2B2 at a distance v2 from the second principal plane, thr magnification (w ,) in this caae \\ill be
Subtracting,
117
"
w2 - ml = a"T 1 •
118 PRAiTlrAL OPTICS
It is using this formula and the fact that the distance between the two images (i.e. the distance v2 - vj is required, that the microscope itself can be used for the determina- tion of the focal length, for (v2~vi) can be measured from readings taken on the side of the draw-tube of the microscope.
i; i periment. — Screw the micro-objective to be tested in position on the microscope : if a Huygenian eyepiece is fitted, the field lens should be removed as it introduces a slight error in the magnifications. With a Ramsden eyepiece, which should be used if possible, this is not neee--ar\ . Whichever type is used, a " tenth-millimetre " glass scale should be fitted in its focal plane for the experiment.
Place a second " tenth-millimetre " scale on the stage of the microscope, draw out the " draw-tube " of the microscope to its full extent and focus this scale. De- termine the magnification by estimating the number of divisions in the eyepiece scale covered by one or a number of divisions of the " image/'
Reduce the tube length by a known amount (say 4 cms. either by taking a reading from a " divided " draw- tube or with a pair of calipers, and measure the second magnification.
Having thus obtained the value for (t^-t^), also m2 and w1, " / " may be determined from the formula.
Repeat the experiment for various tube-lengths and take a mean of the values calculated.
(a) Numerical Aperture (N.A.) of a microscope objective.
Numerical Aperture (usually written " N.A/') is de- fined as Ix-ing equal to the product of the refractive index, " n," of the medium immediately outside the objective, and the sine of half the apical angle of the cone of light taken up, i.e.
w sin "a/J
Numerical Aperture, in connection with the " resolving
MISCELLANEOUS ADVANCED EXPERIMENTS 119
power " of a microscope, is even of more importance than the magnification.
Determination of N.A. — This experiment may be per- formed very conveniently on the " bar optical bench " described in the last chapter. A microscope mounted on a horizontal axis should be placed in one of the holders on the bench. At the extreme end of the bench should be mounted a metre steel rule held in. one of the clips on the optical bench. Two pieces of white paper with straight edges should be cut and folded so that they slide con-
Ramsden Circle
FIG. 104.
vrniently along the edge of the rule. The principle of the method will be seen from Fig. 104. M is the microscope, whose working distance is at A.* CBD is the steel scale \\itli the pieces of paper at C and D. The Huygenian eyepiece of the microscope should be of low power (about 50 m/m sep.), and in place of the ordinary stop a 2 m/m diameter stop should be inserted. The Ramsden circle produced by this eyepiece should be viewed by a positive Ramsden eyepiece placed behind it. The pir.-o <>t paper on the steel scale should then be moved outwards from the centre until their edges can only just be seen in the extreme edges of the Ramsden circle. We then have
* 'I'!..- point \ may be fixed relative to tin- <li\i.ling of the o|>tiral l><-n. h l.y resting a set-square - .n- <>n th« ,1* or
forwards until t I edge is seen sharply in f
tin mi, i.... ..-.-... it,,, reading of the bottom edge of the square may then be
•; fn.ni tli. .,ns.
L20 PRACTICAL OPTICS
a means of determining the angle " a," for CB or DB (which -In >ul<l be the same) can be obtained from the steel scale and AB from the optical bench. So that
Various distances of AB should be taken for the same objective, the experiment repeated, and a mean value of " a " obtained.
Various quick methods whereby the numerical aperture may be read off " direct " have been devised by Prof. < hc-shire, which give very good results. One of these methods consists in placing on the stage of the microscope a piece of card on which is painted the design shown in Fig. 105. This design, when seen in the plane of the Ramsden circle of the microscope, projects as a number
FIG. 105.
of straight lines of equal thicknesses. The distance of the card from the front of the objective is of importance ; to obtain this correct distance, a small metal or hard wood block is made of the right length * ; this is rested on the card and the top surface of the block focussed with the microscope. The block is then removed, the positive Ramsden eyepiece placed so as to view the Ramsden circle as before, when the number of lines corresponding to the N.A. will be seen just to fill the diameter of the circle.
(b) COMPLETE MEASUREMENTS OF THE OPTICAL SYSTEM OF THE MICROSCOPE FOR THE MICROSCOPIST
This section is written for the microscopist who wishes to take measurements on the optical system of his own
* This length should be the distance between the " working distance " of the objective and the position at which the card was calibrated ; in this case 25 mms. These cards are obtainable from Messrs Baker, 244 High Holborn.
MISCELLANEOUS ADVANCED EXPERIMENTS HM
instrument, and, therefore, naturally does not want to go to the expense of having to obtain much auxiliary apparatus for the purpose.
(a) Considering first, then, the Numerical Aperture of his objective ; this is best done by using the Cheshire Apertometer * shown in Fig. 105, the method of using being described in the preceding section. (6) The focal length of the objective may be determined by the magnification method mentioned in sec- tion (a) of this chapter, i.e. using the draw-tube extension.
(c) The focal length of the eyepiece can be obtained in a similar way by making up a simple adaptor (see Fig. 105A) to carry the eyepiece, and which can
Microscope
Adaptor which screws
into Microscope
Small Stop
yepiece in its Adaptor
^$ca/e on Microscope Stage
Fio. 105*.
be screwed in position in place of the objective. This is, in « (T< ct, using the eyepiece as an objec- tive, \\liicli incidentally must be stopped <l»\\n. Another eyepiece (which has a tentli-millimrt n> scale in its focal plane) is then used at the eye- piece end of tli- -cope, and by tin- maLrnifira-
timi method described previously the focal length may be obtained exactly as before.
'/) Magnifications. — The fir t " magnification may be • See footnote on page 1
122
I'll ACTICAL OPTICS
determined by placing a tenth-millimetre scale on the stage of the microscope and comparing the size of the image of a certain number of divisions of this scale, projected by the objective, on a second scale situated in the focal plane of the eyepiece.
The total " magnifying power " may be very conveni- ently obtained by the method shown in Fig. 105B. A piece of neutral tint glass G (if this is not available, a piece of ruby or cobalt glass will do, or plane glass)
FIG. 105s.
is placed at about 45° to the axis of the microscope. If now the eye is placed at E, the magnified image of the scale S as seen through the microscope can be viewed so that it appears on a piece of card S1S2 at about 10 in. away (i.e. at the " near point " of the eye). Whilst thus observing, two lines can be drawn with a pencil at the positions where two particular lines of the magnified scale are seen, and then this distance S^g measured, from which the magnifying power of the microscope may be obtained.
(c) THE AUTO-COLLIMATING TELESCOPE
This instrument is, as its name implies, a combination of a collimator and telescope, and plays an important part in the testing department of the optician. Its ap- plications are many, but it is used chiefly in connection
MISCELLANEOUS A I >\ A \< ED EXPERIMENTS 123
with the measurement of prism angles and the testing of parallelism of glass plates. It will be well, first of all, to look at the optical system of the instrument ; this is shown in Fig. 106. O is the object glass (usually about 1- in. focal length), in the focal plane of which is mounted a graticule G. One of the best types of graticules is that shown in the figure, the horizontal line on the left being
Graticule Bloc* l/nes on ctoargla**
Gratiqule
Ramsden Eyepiece
L
Object Glass
45°Prism
FIG. 106.
covered by a small 45° prism as indicated by the dotted lines, and the spaced lines on the right correspond to a definite angular subtense at the object glass. If a " tenth- millimetre " scale is used, each division may be made to <ni respond to 1 min. angular subtense by choosing an object glass of suitable focal length, so that by estimation reading may lie taken to 6 sec. of arc.
F and E are the field lens and eye lens respectively of a Ramsden type eyepiece. By means of an aperture in the side of the telescope tube light is admitted from a lamp, and thus the previously mentioned horizontal line becomes illuminated. Tin- line serves as the object for the collimator. Rays from thi^ collimator go out " parallel," and it a mirror or plane glass surface is placed in the path of the beam normal to the axis, the rays will return ;d«»ni: their ori-mal path and come to a focus again in the plane of G, when an image of the hnri/ontal line
u viewed l>\ meant ot the eyepiece. In this way an\
di-plaerment of the image from the centre line «.i" the scale may he mea-ured in aii'jular amount
124
PRACTICAL OPTICS
Sometime^ an eyepiece with a plane glass reflector in it is used (see Fig. 107), and a graticule of the design
G F D
OifT
FIG. 107.
shown in Fig. 108 instead of the graticule and 45° prism, but owing to scattered light from the plane glass reflector
FIG. 108.
FIG. 109.
it is not nearly so successful as the type of auto-collimating telescope shown in Fig. 106. A very suitable mount and stand for the auto-collimating telescope is shown in Fig. 109 ; the arm A can be swung into any position within
the 180° and can be clamped at will by a " winged " nut at the back of the instrument.
Parallelism of a Glass Plate.— If, now, a glass plate is placed in front of the objective, in most cases (unless the two faces are absolutely parallel) two images of the horizontal line will be seen.
FIG. 110.
These are due to reflection from the two surfaces of the I .late, the brighter of these two images being the reflection from the first surface. The angular separation of the two images can then be measured on the graticule.
MISCELLANEOUS ADVANCED EXPERIMENTS 125
Fig. 110 illustrates the path of the rays in the plate; the angle NOP is the one measured with the auto- collimating telescope, from which, with a knowledge of the refractive index of the glass (it is near enough to take n = l-5) the inclination of the two surfaces may be obtained.
For^iSAR (the required angle) = 1 80° - zASR -
= 180° -90° -(90 -
But L SRO - L ROM = l x L NOP.
ft
NQP /. z. SAB = 180 -90 -90 + n
NOP = _ // .
2 If n is taken as 1-5
Testing the Angles of a Right-angled Prism. — The auto- colli mating telescope may be used to great advantage for the testing of the angles of right-angled prisms, and becomes an extremely simple and quick method when the observer is once acquainted with his instrument.
It is general to determine the error of the 90° angle fir-t. a- this aids the determination of the 45° angles. I MI tli is purpose the auto-collimating telescope may be n-ed in two ways: one as shown in Fig. H!A and the oth< T as in Fig. Ills. In the first case, if the angle between tin prism face and the "flat i- exactly 90°, only one im.iL'f of the hori/.ontal line would be seen, and that (••.incident \\ith the "zero" line of the graticule. Tin-. houever. i- not u-ual : more often two images \\ill l>e seen equally displaced each >ide of the /.< -n». Thi> in- dicates that tli- rn i in the !m . and tin- error «.f
say «) will be represented by an angular di-|.la< •ement of t he t \\o imagM ot " 4a " on the graticule.
In the second case, \\hen the liL'ht travel- m-ide the
l'RA< TICAL OPTICS
prism, the <1< -\iation is increased to w(4ct), where n is the . «• index df the prism. The student should prove these for himself. To test the 4.1° angles, the auto-collimating telescope
Prism
Flat
Prism
(a)
Fio. 111.
should be adjusted until its axis is " normal " to the face A I! (see Fig. 112), when the face BC is put carefully in
contact with the flat. The prism should then be carefully taken off and the face AC put in contact with the flat. On looking into the telescope, but without altering its position in any way, it will be observed (in all probability, unless the 45° angles are exactly equal) that the horizontal line image has moved a certain number of divisions. This angular move- ment will be just twice the difference in angle between the two 45° angles. Let this difference be /3. Then the < AB (supposing that it is the greater of the two 45° angles)
Optical 'Flat'
in-.
MISCELLANEOUS ADVANCED EXPERIMENTS 127
/180°-C
and the
2
u here < ' is the actual value of the 90° angle.
«/) TESTS ON A TELESCOPE
Tests on the performance of a complete telescope are of the greatest importance. They may be divided into t u n sections :
(i) Geometrical Tests (such as angular field of view,
magnification, etc.), and (ii) Definition Tests.
In dealing with the first section, the focal lengths of the object glass and eyepiece may be determined by methods described in previous chapters. The magnifica- ma\ l>e determined very accurately by the following method: Focus the telescope on some very distant object (parallel light). Then support it in a vertical position on the table, with a frosted lamp immediately beneath the object glass. In front of the object glass place one of the millimetre glass scales, and over the piece place a " dynameter " (see page 97), and focus -harply the 1 Jam-den circle, when the divisions of the glass scale in front of the object glass should also be in tncii-. In tin- u.iy the -i/.e of both scale and image
n ay l<« measured m nltaneously, and the magnification obtained ;
M_ Size of Scale ~~ Size of Image*
/ >fld of View. — The angular extent of the field of view
n i\ I.e l.r-t nhtaimd l.y observing two distant objects appear at the ext rem« edge of tin field as seen \\ ln-n
thnniL'li the trloenpe, .ind atteruai'ds measuring
[28 I'KA< TICAL OPTICS
the angular >ul>ten-e to the naked eye of these two objects l.v means of a theodolite or sextant
If a permanent seale can be set up at some distance (such as might be done in connection with any optical testing dt •}' which has been previously divided
according to known angular subtenses, it is possible to read off the angular field of any telescope directly from the scale.
Owing to the (possible) finite distance of the scale, however, it i> necessary to place the instrument under test, so that the front or anterior focus of the object glass coincides with the point from which the angular subtense of the scale divisions were previously measured.
Effective Aperture of Object Glass of a Terrestrial Telescope. — The determination of the position and size of the stop
L£ L3 L,
in the ''erecting eyepiece" of a terrestrial telescope is of considerable importance, as this is frequently found to be incorrect, with the consequence that the " effective aperture " of the object glass is reduced.
O in Fig. 112A is the object glass, and L! and L2 are the lenses of the "erector." From the paths of the rays proceeding from the object glass shown in the figure, it becomes evident that the stop S must have a definite size and position between the lenses Lx and L2 in order to ensure that all the light regularly transmitted by the object glass passes to the image and ultimately to the eye. At the same ti i e any stray light reflected by the sides of the telescope tube are prevented by the stop u passing to and thus confusing the image. Makers frequently take advantage of this point and place tin's stop in some position such that the definition of their
MISCELLANEOUS ADVANCED EXPERIMENTS 129
instrument is increased, but which in effect decreases the aperture of the object glass. This is unfortunate for the customer, who always has to pay for aperture !
To test the Position of the Stop. — Focus the telescope for infinity. Illuminate the aperture of the object glass with a piece of paper and measure the size of the " exit pupil," as mentioned in Chapter VIII., section (d). Then take out the complete eyepiece and remove the stop S altogether. Replace the eyepiece and again measure the size of the "exit pupil." If this latter exit pupil (which is the true one) is found to be larger in diameter than the previous one, it is obvious that the stop is cutting off some of the aperture of the object glass. The position of the stop should then be adjusted until the true diameter of the exit pupil is obtained.
(ii) Definition Tests (Test Objects). — For these tests it is essential to have certain definite " test objects." They should preferably be illuminated by daylight, and should be situated at not less than 150 feet from the
• • •• •
• +
113.
position at which observations are to be taken. The most important of these objects is an "artificial star"; this in i\ be made up very easily, as explained in section (6), ( h.iptrr VI.. by employing a small steel ball, on to \\liidi light from a < ircular aperture is allowed to fall, at right angles to the direction in which observations are
n: v TICAL OPTICS
to be taken Such a device gives quite a >ati>i'actnry " point
tot ol.jrct can he made by painting with Indian ink a sketch of a tree (without foliage), showing branches and tuiirs. upon a piece of "opal" glass, and illuininatinLT it fmni l)chind. This serves admirably, as the "degree of blackness" of the branches and twigs, as D through the telescope, serves as an all-important test tor tlic presence of " spherical aberration."
Tin- third object should be one of some such design as
-hown in Fig. 113. It consists of a metal plate with
squares and circles of varying size cut in it. It should be
illuminated behind either by artificial daylight or real
iiirht. in the latter case by means of a mirror at 45°,
Circular aperture.
Steel
.*.-)'
-"•••I
=- — 4
I, I , I i I i I i I , I i I , I, I i I
Kiu. 114.
and in the former by using a " Chance " artificial daylight •aeon in trout of a MVatt electric lanip.
Fig. 114 shows a useful set of test objects which may be mounted together in some form of wooden casing. They should each have a hinged door which can be swung in front at will, in order that any one object may be used without interference from any of the others. Such a set of test objects as illustrated in Fig. 106 is very simple to make, and introduces everything that is essential for telescope testing.
Performance Sheet for a Telescope. — The procedure for i '-ting a telescope will be as follows :
Determine —
MISCELLANEOUS ADVANcKD EXPERIMENTS 131
Magnification (including size of "Exit Pupil"). Angular field of view.
Set the telescope on the " artificial star " and observe the appearances of the image at the centre of the field :
First. At the best focus. Second. Inside the best focus. Third. Outside the best focus.
A properly corrected instrument should show a clearly defined diffraction " ring system " on each side of the best focus. If the rings are " harder " on one side than on the other, " spherical aberration " is indicated. If they are more clearly defined inside the focus, "under- correction " of the system is indicated ; and if more clearly defined outside the focus, " over-correction."
If the rings seen are " elliptical," as shown in Fig. 115, " astigmatism " is present (due probably to some cj-lin- « 'ririty of one of the refracting surfaces). "Coma" is identified by the appearance of the ring system shown in Fig. 116.
Colour Correction. — Observations should then be taken on the " colour correction " of the instrument. This is
115. Lift
best seen by directing the telescope towards the edge
of one of the bright larp iquaree -li«»\\n in Fig. 114. The
appearance of this edge inside and outside the focus \\ill
be that it i- fringed with colour : it it has a red fringe
<Ie the focus and a blue fringe outside the focus, " under-
• •ction " will be indicated. If " red " outside the focus
;ind hlue inside, "over-correction" \\ill be indicated.
PRACTICAL OPTICS
A further test for spherical aberration may be given
using the telescope on the black tree test object.
Spherical aberration would be indicated by any lack of
ickness" of the image seen. For any considerable
difference in focus between the marginal and par-axial
- will cause a considerable " scattering " of light, and
consequent 1\ tin- imuirc will appear " greyish " instead
of black.
The angular extent of " good " field should then be determined for each of the above tests, i.e. " the star/' the bright edge, and the black tree.
Therefore the " Performance Sheet " may be tabulated in the following manner :
PERFORMANCE SHEET FOR A TELESCOPE
Form to be filled up for each instrument tested
1. Description of Instrument. -. Mairnifyinjr Power.
3. Angular Field of View.
(Including size of " Exit Pupil.")
4. Angular extent of " good " field.
First. For " star " object = Second. For " bright edge " (i.e. colour) = Third. For " black tree " object = Effective Aperture (stop).
' Artificial Star " Tests.
(a) State whether the system is " over-corrected " or " under-corrected " as regards " spherical aberration."
(6) Is " astigmatism " present ? (c) Is " coma " present ?
6. Colour Correction.
State whether the system is " over ' corrected as regards " colour."
7. Centring.
8. General Remarks.
MISCELLANEOUS ADVANCED EXPERIMENTS 133
A'lto-cottimation Test for testing Telescope Objectives.— Another very convenient way of testing the object glass of a telescope is by using the following auto-collimation method. It is in effect a " star " test, using an artificial star by means of light reflected from a small steel ball.
Light from a lamp (Fig. 117) is reflected by means of a steel ball (situated at the focus of the lens), which travels
Lamp
GoodMirror
^'diameter Cycle
Lens being tested
KM;. 117.
F
High Power Ram set en Eyepiece
towards the lens in the direction indicated. A " good " * mirror is placed as shown and the reflected beam brought to a focus F in the same plane as the steel ball, but slightly to one side of it. The appearances of the " star " image may then be viewed by a high-power eyepiece, and the performance of the lens judged therefrom.
It is of great importance also in this test that the lens be properly " centred " before the star-images may be
«
Ms.
fairly judged. For this purpose a piece of apparatus known as a " self -centring eyepiece" should be used.
* It i-i ..f the greatest importance that this *ilvered mirror w 11 i>< •rf.vtly good "flat."
PI: \< TICAL OPTICS
-Imuii in Fig. 118, and was devi>«l
hv PP> h consists of an outer tube A, into
\\hirh sli<lr< tin- portion B. B consists of a tube, at one ••ml of \\hic! Jily-pnlislird >i Ivor or german silver plate
P sweated on at l~> with a hole (about J in. in diameter) bored mit rally in it. whilst at the other end is a knurled ring II \\hich has a pin-hole H drilled in it. S is a stop
he aperture shown.
In use tin- "eyepiece " is set up approximately on the axis of the object glass to be tested. A lamp is then arranged so as to illuminate the reflector P through the ( -nt -a way portion of the tube A, so that an annulus of light will be sent towards the object glass, and to an observer's placed at H an effect such as shown on the left of the figure will be seen reflected in the first surface of the object glass. When the object glass is satisfactorily " centred " the annuli thus seen should all appear concentric.
. TESTS ON PRISMATIC BINOCULARS
The testing of the prism binocular is inevitably an all- important subject, and it is the aim of this " section " to give a complete description of how this should be done.
The tests may be divided up under various headings.
Treating each half of the binocular as a telescope :
1. iMinition,
2. Magnification,
3. Field of View,
may be determined in exactly the same manner as de- scribed in the previous section (c). Other tests are :
4. Parallelism of axes in all positions.
5. Strain in prisms.
6. Inversion produced by the prisms. 7 Stray light.
8. Angular subtense of graticule (if fitted). Parallelism of Axes. — This is the most important test of all in connection with binoculars. The apparatus
M IS( KLI ANEOUS ADVANCED EXPER I M KXTS 135
needed for this test is essentially of a somewhat special nature, but as it also serves as a means for adjusting binoculars, it is well worth while having such a device constructed. A diagram of the apparatus is shown in Fig. 119. It consists of two collimators parallel to one another at a distance apart equal to the average separation of the binocular object glasses, an adjustable table on
Mounts with 3" 'Capstdn' screws for adjustment of cross- fines
Supports on which table swivels
Milled Head for raising^ or lowering the table
Adjustable /table
MilledHead /for lateral adjustment of table
, Slide for Telescope
Telescope (F-6"dbout)
II!'.
\\hich to rest the binoculars, and a small telescope which travels along a geometric slide at right angles to the axes of thr collimators (see Fig. 120).
I'irst of all, the axes of the collimators are adjusted parallel to one another l>\ -luiing the telescope in front of each collimator object glass in mm. and adju- the " adjustable " cross-lines of the collimators to coincide with the cross-lint m th«- eyepiece <>t the telescope. \\ hm tin- ha- hrrn (l<»nr tin- hinoculars to be tested are placed nn th« table and adjusted until tin image" \i<\M<l
LSI
PRA< TIGAL OPTICS
•!i the telescope) through one-half of the binocular made coiiu id. nt \\ith the cross-line in the telescope. On sliding the telescope so as to view the " image " through the other half of the binocular, any deficiency in coin- cidence of the image will at once be seen, and this is a measure of the want of parallelism of the axes of the two
halves of the binocular. The actual displacement of the image may be determined in angular amount by the graticule in the focal plane of the eyepiece (a suitable type of graticule is shown in Fig. 108). In this way both the vertical and horizontal angles between the image forming rays from the two halves of the instrument may be determined. Below is given the maximum allowance in angle between the axes of the two halves and the corresponding magnification :
Magnifying Power. |
Horizontal Allowance. |
Vertical Allowance. |
3x |
30min. |
lOmin. |
6x |
12min. |
4min. |
10 x |
6min.40sec. |
2min. 12 sec. |
12 x |
5min. 30 sec. |
1 min. 50 sec. |
This " binocular testing bench " is also convenient for the ordinary adjusting of binoculars, for the quickly ad- justable table allows the binocular to be placed in position and tested with the least amount of trouble possible. Adjustments in binoculars are effected, either by move-
MISCELLANEOUS ADVANCED EXPERIMENTS 137
ment of the prisms or by rotation of the object glasses.
Strain in Prisms. — Owing to the method by which the prisms are held in the binocular, excessive pressure is sometimes exerted on them. This is an extremely bad fault, as the double-refracting effect thus produced will appreciably affect the definition of the instrument, and sometimes if the binocular is accidentally given a sharp " jar " a piece of the prism will chip off owing to the strain.
Strain may be quite easily detected by holding the binoculars in a clamp stand and allowing light reflected from a " blacked glass " polarizer (at the polarizing angle, see Fig. 121) to enter them. The "exit pupil" may
Diffusing Screen
V
'Prism
Binoculdr
Lamp
Blackened Glass Plate
Vl\.
then be examined with a Xieol prism, whieh has been rotated to give "extinction" before tin binoculars are inserted in the path of the polarized beam Any strain in the instrument will be shown up* by the appearance of " light patches" among the darkened field. IV and lenses of all kinds should be held sufficiently tightly \\ 1 1 hout any undue strain being imposed on them.
Inversion produced by Prisms. — For this test the binoculars are supported horizontally and focussed on a vertical lint \\hich is not less than Inn teot away. A theodolite \\hieli has been previously made to "transit" over this vertical line satisfactorily (after levelling, etc.) then views
li; \. TK'AL OPTICS
Pun vie* of 2 right-angled Prisms K used in a Binocular showing an error O 'which produces bad inversion effect.
the lump' «'t the lin«- through each half of the binocular in turn. In this \\ay the perpendicularity of the image
the line may be estimated, and hence the perfection of the inversion produced by the prisms. Any lack of " inversion " is due to error in the setting of the two
l)ris!iw at riSht an^le to one another (see Fig. 122).
Stray Light. — A square frame should be made up, with tissue paper stretched across it and having a black circular disc of
r in the middle, such as is shown in Fig. 123.
tram* should be brightly illuminated from behind,
one-half of the binocular directed towards the black
disc. The distance of the binocular from the disc should
be such that the disc rather more than fills the field of view.
Tissue Paper
Wooden Frame
Circular Disc of BlackPaper (V/i'diam.)
FIG. 123.
On examining the exit pupil, either with an eyepiece or l.y taking a photograph, any stray light in the instrument will be made manifest. Bright reflections are distinctly detrimental to the action of the binocular, especially in " night glasses."
Test of the Graticule. — A graticule is very often fitted
MISCELLANEOUS ADVANCED EXPERIMENTS 139
in the focal plane of one ocular of a binocular, for purposes of " range-taking/' and it is necessary that the angular subtense of the graticule divisions (usually 30 min>.) should be tested.
For tin's purpose the binocular should be supported in a horizontal position, with some source of illumination (preferably diffused) placed in front of the eyepiece. At tin1 object glass end a theodolite should be arranged so
Mirror
L
Fi. . li'l.
as to view the linage of the graticule lines. The angular separation may then be measured by setting the cross- win^ in the theodolite on the images of the lines in turn and taking readings from the theodolite circle.
Second Method. — Another method may be adopted, \\ hieh involves the use of a " mirror mounted on a theodolite table"* (a piece of apparatus invaluable to the testing room), and is sho\\ n in Fig. 124. Light from a lamp is reflected into the eyepiece of the l.inocular l»\ UK an- «•!' a plane glass reflector. The theodolite table, with a good
•Tlii- »>•]><• <>f tli<Mxlolit«- i.ii.li- with"-:1 unting may be
obtu-i ,l.«-r\\.-ll.
140 PRACTICAL OPTICS
mirror mounted on it, is then placed as shown in the figure, and u(Iju>u-d until an image of the graticule is seen in the same plane as the " real " graticule on observing thiMiiirh the eyepiece. On rotating the theodolite table the image of the central graticule line may be made to travel across the divisions of the real graticule, when readings from the theodolite table may be taken which will give just half the value of the actual angular subtense.
CHAPTER IX REFRACTOMETERS
THE subject of determination of refractive index by the spectrometer was dealt with in a previous chapter, and this instrument in reality is the fundamental instrument for such determinations. There are, how- ever, other instruments designed solely for refractomctiv which either give " refractive index " direct or by the simple determination of one angle and the use of tables. As these instruments are in considerable use at the present day, this chapter has been devoted to the explanation of the more important types.
(a) PULFRICH REFRACTOMETER
The principle of this type of refractometer will be seen from Fig. 125. The substance or liquid whose refractive
index is to be measured is placed on the top of a glass block of known refractiv m<l. -\. In tin- MM «>t a solid, a thin layer of liquid of high refractive index is placed
between the two surfaces. The angle A iK'twrm tin- in
I i_» PRA< TICAL OPTICS
vertical and hori/ontal Miriace- of the prism is usually
accurately !«n .
It. then, li^rht niters the substance or liquid of unknown 'idex from a position L, that entering above the normal LO will enter the Pulfrich prism and pass out again as indicated along the path NP ; a telescope placed at P would then see a band of light with a sharp bounding line on the upper side. The rays which enter normally along LO will graze the two surfaces in contact and will be the limiting rays of the band of light observed at P. Any rays entering below the normal LO will not be able to enter the Pulfrich prism at all.
So that, the sharp line observed in the telescope of the refractometer represents the rays which have just been able to enter the prism ; the angle through which these " grazing " rays have been refracted is the com- plements of (90° - r) .... [see Fig. 125], which is the " critical angle " of the Pulfrich block with respect to the substance above it, and depends solely on the refractive indices of the two materials. (An inter- mediate medium if of greater refractive index than the one above it has no appreciable effect.)
The angle " i " at which the beam emerges into the air depends on the magnitude of the angle " r," and is measured with the refractometer.
Considering the refraction at the two prism faces in turn, we have
sin <:o ^ sin (90° - r) ) and sin " 1"=^ sin "r" I
where n^ and n2 are the refractive indices of the Pulfrich block and substance or liquid to be tested respectively.
Combining these equations, the unknown refractive n2 is calculated from the expression
n2 = Jnj2 - sin 2i. The instrument is usually supplied, however, with a
RKFHA( TH.MKTKRS
143
tal>l<- \\hirh iv prepared for all values of the angle " * " from tlie above formula, so that refractive index may be
ni lined directly from the table.
Fig. 126 gives a general illustration of the instrument. Light (usually from a hydrogen tube) is sent into the substance or liquid being tested, through the condenser " C," which renders the light convergent. The beam on
Thermometer
Clamp for
Hydrogen
Tube
iil'i.
< merging from the Pulfrich prism face F is received by tin- telescope T, \\lii* -h is attached to the rotating circle S. With tliis circle, by meanfl of a vernier and tangent screw tin- angle i- MM -a-im -d. The telescope is auto-
collimating, in order that the normal to the face F of the Pull rich Unrk may be obtained by back reflection. Kin ins:* W for a water circulation are provided, so that Mil»-tan<<> may be investigated at raised trni|»« • atures ; also it is particularly useful in the case of substances, -IK h as fats and waxes, vlm-h mily become liquid and transparent at these temperatures. D is a right-angled
144 PRA< IK AL OPTICS
I'MMii \\hirh can be swung in and out of the path of light
•IK I» -user so that sodium light may be used when
!vd without having to remove the hydrogen tube.
K i- -imply a device for limiting the aperture of the
incident beam.
The Instrument in Use. — When testing a solid it is essential that the specimen has two surfaces nearly at right angles, and the one which is placed in contact with the Pulfrich prism should be well polished and reasonably flat, while the other need only be sufficiently polished to allow light to enter. It is important, however, that the edge at which the two surfaces join should be very sharp. For measurements on liquids a small glass cell is cemented on the top of the Pulfrich block,* into which a small amount (a layer of about 3 mms. deep) of the liquid can be held (see Fig. 125 (6)).
First of all, the reading of the circle when the telescope is " normal " to the face of the " block " should be obtained (i.e. the zero setting checked). For this purpose a lamp should be arranged to illuminate the small prism near the eyepiece of the telescope, and the circle rotated until an image of two small lines will be seen in the field of view. When they are in such a position that the one line of the real graticule is midway between them, this should give the zero setting of the instrument. The type of graticule generally used is shown in Fig. 127.
This being done, the specimen to be measured should be placed on top of the " block." This has to be done with great care. The two surfaces to be put in contact should first be thoroughly cleaned. A small quantity of liquid f (of higher refractive index than either the specimen or the Pulfrich prism) is then placed between the two
* Two Pulfrich prisms are supplied with the instrument, a "light" and 44 dense," the former for use with liquids and the latter for solids.
f A suitable liquid to use, and of high refractive index, is " Monobrom- naphthaline."
REFRACTOMETERS 145
surfaces, and the specimen pressed firmly on to the prism. The surface of contact should then be examined by re- flecting from it monochromatic light (a sodium flame), \v I it'ii alternate light and dark interference bands will be seen. The bands should be made as broad as possible 1>\ pressing on the specimen, as the surfaces are then most nearly parallel. The number of bands seen should not be more than six, and with " well-worked " specimens having flat surfaces it will be found possible to bring the >urfaces so parallel that one band fills the whole surface of contact.
The instrument is now ready for taking readings. Sending " sodium light " first, therefore, through the specimen, the telescope should be moved round until the graticule is brought on to the sharp bounding edge between the sodium coloured and dark part of the field. The difference in readings taken at this position and that of the "normal" or "zero" reading will give the value "»'" (Fig. 125).
By a simple reference to the tables supplied with the instrument, the refractive index (for D light) of the specimen can be obtained corresponding to the value of " »."
Similarly, by using the hydrogen tube* with the in- Mrumrnt the values of "t" may be obtained for the C, F, and Gt lines, and with the use of the tables the refractive index for each line. Also mean and partial dispersions may be obtained.
Tables are also supplied for temperature variation \\lini such are needed.
(b) THE ABBE REFRACTOMETER
The principle of this instrument again depends on the use of a stan <l.ii<l prism and the border line between the
hi m.-t -uit;il.lr i tube to use is the type nn-iiti..nr.l in
< h i|»ter IV., as the large side bull* all..wH a much heavi • to be put
th- tul>< u Mil.. ut the great rise in pressure, and thus increases the
146
l'i:.\< TICAL OPTICS
light and dark parts of the field, due to "grazing inci- dence " illumination.
Its general arrangement will be seen from Fig. 128. It consists of two 30° prisms A and B, mounted in a metal casing which can be rotated on a horizontal axis immediately beneath a telescope T. To this metal case
Graduated Scale
Object 6/ass Amid Prisms.
FIG. 128.
is attached an arm R, at the end of which is a graticule line ; this moves over a scale graduated directly in terms of " refractive index."
The general principle of the use of the standard prism is the same as in the case of the Pulfrich refractometer, but it should be noted its angle is no longer 90°. B in Fig. 117 is the standard prism and is usually of "dense flint " ; the auxiliary prism A is solely for the purpose of leading light at " grazing* emergence " into the liquid film, when of course it will fall on the main prism face at " grazing incidence."
Fig. 129 (a) shows the path of "grazing incidence " light on the face of the standard prism ; Fig. 129 (b) shows the
REFRACTOMETKRS
147
use of a prism when testing a solid or when using a test prism; and Fig. 129 (c) shows the prisms as generally used when the liquid being measured is spread out as a thin film between the flat glass surfaces.
Light is admitted into the prism system by means of a mirror M (Fig. 128). This may be either light from the
(b)
sky or from a lamp ; monochromatic light is not necessary, as the colour of the " bounding line," as seen in the tele- scope, is annulled by the use of two " direct-vision " prisms (known as Ainici prisms), situated in front of the object glass, and wliieh can be rotated in opposite directions by means of a rack and pinion. Only \\lnn the "bounding line " is properly achromatized can readings be taken.
Theae Amid prisms (see Fig. 130) are so constructed that they have no deviation for "D" light, but \\ill produce deviation for all 'other colours ; so that two such pi i itmly inverted will be achromatic, but
similarly plaeed \\ill produce approximately double the di-persion due to one alone. For details of construction a text-book on Geometrical Optics should be consult. « I
148 PRACTICAL OPTICS
The exact calculation "t tlic dispersion due to two
such prisms when placed at any relative angle is a very awkward one, and it is probably better to calibrate the instrument ex- perimentally. It should be remembered, however, that the Crown Crown dispersion value furnished by this test is only approximate and is only meant for rough identification purposes.
The ln*trtinn at in Use. — The two surfaces of the prisms betwt < n \\hich the test liquid is to be put should be thoroughly cleaned. The instrument is then swung into such a position so that the hypotenuse face of the standard PHMII is horizontal ; a few drops of the liquid are then put on, and the other prism in the other half of the metal case swung over and clamped. The telescope of the in- strument should then be brought into its most convenient position, and on looking in through the eyepiece the mirror M (Fig. 128) should be adjusted until good illumination is present in the field. The arm R should then be moved round until the " bounding line " between the light and dark part of the field comes into view. This must then be made quite free from any colour fringes by rotating tin- Amici prisms by means of the milled head provided for that purpose. The cross- wires in the eyepiece should be sharply focussed and the bounding line set accurately "ii to the intersection of the former. The reading given by the graticule index line on the graduated scale will give the " refractive index " of the liquid. The scale is graduated from 1'3 to 1'7 and is divided to the third decimal place of refractive index, the fourth place being obtained by estimation. As with the Pulfrich refractometer, temperature precautions are of the greatest importance with liquids, and therefore the prisms are surrounded with a water-jacket to secure constancy in this respect. If available, a " thermostat " should be used to ensure uniform circulation of the water.
REFRACTOMETERS 149
(c) REFRACTOMETER FOR GASES
The determination of the refractive indices of gases is obviously a more delicate operation than that of liquids and solids. To obtain the required sensitiveness of the instrument a method employing "interference" of two beams of light is used. The original principle was from Lord Rayleigh, but the instrument described here is a modification of this principle. A diagrammatic sketch of the optical system is shown in Fig. 131. Light from a Miiall electric lamp L illuminates a slit " S," which is in
PLAN
TJ
c
3
tin- tneal plane of an achromatic lens O (about Gin. focal length) The parallel beam emerging from this objective travels on until it reaches a mirror M, which has two parallel slits of km>\\n separation in front of it. The light then returning along the same path will come to a focus again in the plain- of the slit S, where a bright imaire <>t the slit will be seen with a number of diffraction bands on either side of it. Two gas cells G (of km»\\n length) side by side are situated in one halt <>t the parallel In -am. a- illustrated hrtueen i he object-glass O and the mirror M, so that half the IMMHI thrmiL'h the «-elU and the
<M her half over the top of them. C1 and Ca are two " com- pensators," one of Avhieh is adjustable by means of a slow motion with micrometer screw. P8 is a block of glass >nuated in the top part of the beam and equal in thick-
160
PRA( TICAL OPTICS
t«» tin- combined thicknesses of the plates Pl and P2 and one thickness of a compensator C\ ; this is done in order that both halves of the beam shall travel in equal amounts of glass. The appearance, therefore, as seen \\ith the eyepiece E,* will be that of two sets of diffrac- tion bands, one set from the lower half of the beam, the light of which will have passed through the gas cells and back again ; whilst the other set will be formed by light passing over the top of the cells and back again.
When the instrument is adjusted correctly the central bright band of light of the lower set of bands should be coincident with the central bright band of the upper set. If, now, the gas to be tested is allowed to flow through one of the gas cells, and air is still kept in the other, any
FIG. 132.
in refractive index will be represented by a re- tardation of the beam through the gas cell. This retarda- tinn in one of the halves of the lower beam will result in a lateral displacement of the lower set of diffraction bands. The upper set of bands, of course, will not move at all, and thus they serve as a reference mark. By bringing the central bright band of the lower set of bands back to its original position by tilting the compensator Ct with the micrometer slow motion, the retardation produced by the gas cell may be calculated for (see Fig. 132).
Let AC be the second principal plane of the object glass and also the distance apart of the two slits, DE the focal length of the objective ; and let B be the position of the
* The eyepiece is made up from a piece of cylindrical glass rod about 3 mms. in dininrtcr ; this gives a very high magnification, which is necessary in order to be able to see the diffraction bands at all.
REFRACTOMETERS 1 5 1
central bright band of the lower set of bands, after being displaced a distance EB.
Then for a bright band to be formed at B, the difference in path between AB and BC must be an even number of half wave-lengths. This difference in path is obtained from the following :
Call AD =b, FB =d, and EB =x, then and
Subtracting AB2 - BC2 =
or (AB + BC)(AB - BC) = 4bx : but (AB + BC)^= 2d (sufficiently near).
So that
a
From this the actual retardation of the beam passing through the gas cell may be obtained ; with this and a knowledge of the length of the gas cells,
2Z (AB-BC) "=X + X 21 X where w = the refractive index,
l = the length of the gas cells, X =the wave-length of light.
AB and BC — the distances referred to in Fig. 132.
Pressure and drying precautions of both air and gas should, of course, be taken.
CHAPTER X APPLICATIONS OF POLARIZED LIGHT
TIIK theory of the subject of polarization should be revised from other text-books, as this chapter deals with useful applications of polarized light.
(a) DETECTION OF STRAIN
One of the most convenient ways of producing a beam of " polarized " light is by reflection. If skylight is re- flected from a blackened glass plate so that the reflected
Extraordinary Kay
Ordinary Ray
'ilm of Canada Balsam
\\\:\.
beam leaves the plate at the correct angle (viz., 56 J° with
the normal), the light thus reflected will be plane polarized.
Another very usual method is by employing a " Nicol "
in. Such a prism is shown in Fig. 133; it consists
of a rhomb of Iceland spar cut and cemented together
APPLICATIONS OF POLARIZED LIGHT 153
along the face AB with Canada balsam. A ray entering the face AC will be split up into its two components, the " ordinary " and " extraordinary " rays, the former of which has a greater refrangibility. Canada balsam having a refractive index between that of the ordinary and ex- traordinary rays, and the length of the prism ABC being suitable, the " ordinary " ray is " totally reflected " at the face AB, whilst the extraordinary ray passes almost straight on and leaves the prism with its original direction. So that there will no longer be two beams coming out of the spar with vibrations at right angles to one another, but one beam with vibrations only in one direction or plane. Thus the Nicol prism is a very suitable means of obtaining plane polarized light.
The double refracting effect produced by Iceland spar is also present when ordinary glass is under any stress, due either to applied pressure or to bad annealing of the
Fm.
glass setting up internal stresses. Such strain becomes very evident when a suspected specimen is examined by |M.|,ui/<d light. It is this fact that makes the use of polarized light of such importance. Suppose two Nicols Nj and N2 (Fig. 134) to be " crossed " so that all light is extinguished, and let the specimen to be tested be placed in het \\ern the two. If any strain is present it will be n -presented by the appearance of patches of light in the previously dark field, and colour effects will be seen \\lu-n great pressure is present .
A con vr nil -lit piece of apparatus for detecting sti may In- arranged as shown in Fig L3fl \.\n\\\ from the sky or from a diffused an \\\<\i\ source strikes a blackened * glass reflector B. The reflected beam is then viewed \\ .th
vamiith.
UU PRA< IK AL OPTICS
a Nienl in hont of the eye. On rotating the Nicol and
l>\ moving the head up or down a position will be
.(1 when the retleeted beam is almost entirely ex-
dflhed, The Nienl >hnuld tlu-n be rigidly held in this
position with a i lamp of some kind. The object to be
tested is thru placed between the reflector and the Nicol
as shmvn. when any strain in the specimen will at once be detected. If a piece of glass is held in a small vice, placed in the beam as before and the vice gradually tightened, the effects due to increased pressure will at once become obvious.
Almost any optical work, both mounted and unmounted, may be examined for strain in this way. More especially is this test essential to ascertain whether object glasses are held too tightly by their counter-cells, the over-clamping of prisms, and numerous other cases.
(6) MICROSCOPE POLARIZER
A simple application of the blackened glass reflector is to form a polarizer for the microscope, for use in con- nection with petrological work. For observation of rock sections, etc., polarized light is greatly advantageous in the microscope. In place of the somewhat expensive I pri<m usually used as the polarizer, a 3 in. x 1 in. cover slip may be blackened with varnish and stuck with soft wax to the tilting mirror beneath the microscope
APPLICATIONS OF POLARIZED LIGHT 155
stage. On observing through the analyser and rotating same the tilt of the slip polarizer can be adjusted until
40-
a 16 32
NUMBER OF PLATES Fio. 136.
the best position of " extinction " is obtained. This makes quite an efficient polarizer. The polarization of
*
Showing adaption of 'Pola riser ' to a fixed vertical Microscope.
Glass 311 ff
Block of wood
tin- beam, however, may be increased if necessary l>\ superposing a second :; in. 1 in -lip on the face of
156 PH\< TK'AL OPTICS
tin- first. Fig. 136 shows a graph by Stokes giving the relat inn-hip between the percentage of polarized light re- flected from a number of plates from 1 to 32, the light U-mg incident at the polarizing angle (i.e. 56J°). From tli is curve it will be seen that practically the maximum amount of polarized light which can be obtained by re- flection is from eight plates. In practice, however, one, or at most two, will be sufficient for most work. Fig. 137 shows a " slip-polarizer " used with a simple vertical microscope, where it is necessary to use an auxiliary mirror lying flat on the table in order to get light into the instrument.
C SACCHARIMETERS
One of the most important applications of polarized light at the present day is saccharimetry.
Certain transparent substances possess the property that when plane polarized light is passed through them it emerges plane polarized, but in a different plane to tli.tt of polarization at incidence. These substances are said to rotate the plane of polarization ; such a substance is quartz. The effect is also produced by solutions of certain substances ; for instance, a solution of sugar in water rotates the plane of polarization. The rotation which a substance produces is the key to the determina- tion of the degree of concentration of that substance in -"lution. The instrument for measuring this rotation is kimun as a saccharimeter or polarimeter ; they are used to a very great extent commercially in testing " sugar " solutions.
It has been determined that the rotation produced is proportional to the " mass " of substance in a given volume of the solution.
Now, suppose a mass, " w" of a substance to be con- tained in each cubic centimetre of an inactive solvent (i.e. one that does not rotate the plane of polarization), and let plane polarized light of a definite wave-length
APPLICATIONS OF POLARIZED LIGHT 157
(say sodium) traverse a length " / " of the solution. Then the rotation R is proportional to " lw."
So that R=Klw . (i)
w IK« re K is known as the " specific rotation " of the substance. K is dependent to some extent on the wave- length of light, also temperature and concentration. To determine K by experiment, suppose " x " grammes of the substance to be dissolved in " y " grammes of the solvent, and let the density of the solution be " d." Then, the volume of the solution is
y + x
d ccs-
Therefore, the mass of the substance contained in a cubic centimetre is :
d xd
Thus (from (i)) R Klxd
.•, . . . (ii)
So that, from formula (i) it is clear that if for any sub- stance the value of K is known, and the rotation R, pro- duced by a known length "I" of the solution, can be determined, the concentration " w " of the solution may be obtained.
The optical system of a very usual type of saccharimeter is shown in Fig. 138. The source of light S is placed at the focus of the lens Llf so that a parallel beam of light enters the polarizing Nicol Nt. Two auxiliary Ni I A and B (known as Lippich prisms) are situated im- ni'diately behind Nr Thus the field as seen with the telescope consists of three parts : the central part Nj corresponds to light which has passed through the analyser
L68
I TACTICAL OPTICS
milv. \\hile the two outer parts A and B correspond to light which has passed in addition through the two auxiliary Nicols. It is found that dividing the field into three parts in this way facilitates the accuracy with which tin* Nicol N2 can be set. N2 is the second Nicol which together with its mount rotates with the divided circle C, from whirl i readings of rotation are taken. The solution to be investigated is placed between the polarizer and analyser ; it is enclosed in a tube of known length,* at the ends of which are plates of optically worked glass.
Vernier
FIG. 138.
These plates are held against the ends of the tube with metal caps, so that the plates may be removed for cleaning and filling the tube. T is a low-power telescope which focusses on the sharp edges Ex and E2 of the auxiliary Nicols, thus giving a sharp dividing line to the three parts of the field as seen through the
telescope.
lid " Type. — In some forms of saccharimeter the angle through which the plane of polarization is rotated i- measured by interposing a certain thickness of some substance which rotates the plane in the opposite direction, and thus neutralizes the rotation produced by the solution under investigation, instead of measuring the rotation with a divided circle.
* They are usually 10, 20 or 30 cms. in length.
APPLICATIONS OF POLARIZED LIGHT 159
Soliel devised a means involving the use of two quartz wedges ABC and DEF (Fig. 139), which by means of a rack and pinion are caused to move in opposite direc- tions, thus enabling varying thicknesses of quartz to be obtained. The wedges have equal angles and are cut with the optical axis of the quartz perpendicular to the faces BC and DF.
When the wedges are immediately behind one another, a scale mounted above should read zero. The solution to be tested is then placed in the instrument, and the wedges moved so as either to increase or decrease the thickness of quartz until the two halves of the field (i.e.
K
ED
when using a bi-quartz) again appear equally dark. The reading from the scale can then again be taken. The scale is calibrated beforehand with solutions of kim\\n rotation, so that any reading on the scale may at once be converted into angular rotation from the graph. The optical system of the Soliel saccharimeter is shown in Fig. 139.
N! is a polarizing Nicol ; Q a bi-quartz prism; T the tube emit Dining the solution ; Qj is a right-handed qua it/ plate cut at right angles to the axis ; ABC and DEF are the two wedges of left-handed quartz. N2 is the k ' ana 1 yser," and L a lower power Galilean telescope \\hich can be focussed on tin l»i-«juartz Q.
\\hen th< wedges are set at zero, they with (^ produce no effect, and the tint of passage is seen on both sides of the bi-quart/. When an "active" -olution is inter-
LOO PRACTICAL OPTICS
posed in T, the change in tints is neutralized by bringing the wedges into play. N and Q2 are a Nicol and quartz plate to be used if the solution in T is coloured. By their means the light emerging from N\ is made to be com- plfUH-ntury in colour to the solution, and then the appear- ance is as if the solution were colourless.
APPENDIX
(a) THE CLEANING OF OPTICAL SURFACES
THE cleaning of surfaces of optical glass is a subject which cannot be too fully emphasized. Not only is it of importance in the laboratory, but still more so in the optician's assembling or testing room.
One of the best methods of " thoroughly cleaning " an optical surface is to wash it well with soap and hot water, using a perfectly clean linen cloth, then rub it well with a cloth dipped in alcohol, finally rinsing it in distilled water and drying with a piece of " grease-free " chamois leather. Great care should be taken not to let the hands or finger-tips come into contact with any surface ; it will be found advisable to wear a pair of chamois leather gloves when cleaning.
When mounting optical work into instruments it will be found advantageous to immerse the glass in a 20 per «'nt. solution of nitric acid for about two hours before the cleaning (as mentioned above) is begun, as this prevents to some extent the very objectionable " filming " that occurs on the optical surfaces when optical work remains in an instrument for some considerable period. In in- struments that are finally sealed and made air-tight it is advisable to do all mounting in a perfectly dry atmo- sphere. All particles of dust should be removed with a small camel-hair brush. Such brushes should be con- tinually washed out in distilled water to prevent grease clinging to the small hairs.
For surfaces of ordinary glass (i.e. non-optical) a paste, made up of " rouge and ammonia," serves extremely \\cll for cleaning purposes, and should be applied \\ith apiece of chain.. i, I, , it her or a " Selvyt " cloth.
161
PRA< TICAL OPTICS
•• pith " in Mirk- of "elder" an- very useful for re- moving " tarni>h " tn>m surfaces of the denser flint glasses.
(b) SILVERING OF GLASS
In >il\vriiiLr. rlcanliiK — Is again the all-important factor for success.
KiiM ot all prepare two solutions :
1. Di— <>l\e silver nitrate in distilled water, and add ammonia till the precipate first thrown down is almost entirely redissolved. Filter the solution, and dilute it so that 100 c.cs. contain 1 gramme of silver nitrate.
-. hi--olve 2 grammes of silver nitrate in a little dis- tilled water and pour it into a litre of boiling •illed water. Add 1-6 grammes of Rochelle -alt. and boil the mixture for a short time, till the precipitate contained in it becomes grey ; filter the solution whilst it is still hot.
The glass should then be " thoroughly " cleaned, with tin- same precautions taken as mentioned in the previous section, and whilst still wet from the lastly applied dis- t illed water, should be placed in a clean glass vessel (e.g. a t;i]'i/m<4 dish), with the surface to be silvered placed uppermost.
Kijual (jiiantities of the solutions 1 and 2 should then be mixed together and poured into the vessel so as to cover the glass, — the solutions should be cold. After about an hour the silvering will be completed. The liquid can t hen be poured off and the glass removed ; any of the silver d< -posit can be rubbed off where it is not required, and that which is required may be coated with some black \ ;i rnish for preserving purposes when the silver has dried.
(c) GRINDING AND POLISHING A FLAT GLASS SURFACE
The fact of being able to grind and polish a flat surface on a piece of glass is of great importance both for instruc-
APPENDIX 163
tional purposes in the laboratory and for commercial purposes in the workshop.
Such a subject is of too large a scope to deal with very fully in these pages, as practical experience is the chief key to success ; but a general outline of the methods employed will no doubt be of use.
It will be presumed that some sort of machine for re- volving the tools is available, either the treadle type of " grinder " or the type fitted with a small power unit.
First of all screw the " roughing tool " to the spindle of the machine, and take a little emery (about grade 90 *), mix it with water, and use a little at a time on the tool. Hold the piece of glass in the fingers of both hands firmly, and revolving the " rougher," press the glass down on the tool, giving it a backward and forward motion. In due time all the prominent irregularities of the glass surface will be removed and a smooth ground surface will be left. Another tool for finer grinding is now used. This tool should have already been made a fairly correct Hat surface, and therefore may be used for the more exact work. " Fine grinding " can then be done by using 10-minute,* 15-minute, 20-minute, and 60-minute emery in succession in the same way, the grinding being continued with each grade until all bits and scratches left from the coarser grades are removed. The surface must be continually viewed with a fairly high-power eyepiece in order to detect such scratches.
As each grade of emery is used care must be taken to remove any particles of a previous grade ; this is best done with a small soft sponge, and by rubbing a rough piece of flat glass known as a " bruiser " on the tool prior to using the actual glass surface on the tool.
When the surface has been successfully brought to the finest condition, the polisher may then be prepared.
* 111' numbered grades of emery, such as 90, refer to the number of meshes INT in> h of a sieve through which ul.tr emery has passed. The
10, 20, 60, etc., minute emery rrf- r t<> tin; particles that an 1. ft m -us|., in irafea Mowed to stand for the respective number of
minutes.
n; I PRACTICAL OPTICS
Preparing the Polisher.— The polisher is made up of a •r of pitch melted on to the surface of one of the iron tools. The pitch, which may be softened by the addition of tallow or lard, is freed from grit by straining it through a piece of fine muslin on to the tool while the pitch is molten and hot. The tool is heated sufficiently to keep the pitch plastic, and its surface is then flattened Im- pressing the pitch down on a cold iron plate. Before the pitch is quite hard a number of grooves may be cut in it, in order to give places which will accumulate the polishing medium.
The polishing can then be commenced. The " polisher " should be screwed to the spindle of the machine, warmed slightly, and moistened with a little " rouge and water." The glass surface should be rubbed over the " polisher " as during the grinding process, but in this case the speed of relative movement between glass surface and polisher should be very much slower.
After about twenty minutes' polishing the glass surface will be ready for the " test plate/' This is placed on the uice and the interference fringes viewed by reflected light from a Mercury Vapour Lamp ; from this is ascer- tained the relative roundness of the surface and its form, u hcther convex or concave.
If the surface is convex, the best procedure to try and correct this tendency is to increase the " stroke " and to press harder on the polisher. The result will be to increase the wearing of the surface in the centre and thus give a tendency towards concavity. If the surface is con- cave, however, the stroke should be shortened and some of the pressure on the tool relaxed. There are various ways of varying the relative amounts of wear in different regions of the surface, such as cutting grooves in certain parts of the polisher to alter the glass surface in the same part ; but experience is the only master which can teach all the devices used in practice for the correction of surfaces in such a manner.
The period necessary to complete the polishing will,
APPENDIX 165
of course, depend on the time taken entirely to remove all trace of " grey " from the surface and to produce the best flat possible.
(d) BALSAMING
When balsaming it is of first importance that the surfaces to be put in contact are absolutely clean and " dust-free." The surfaces should be cleaned as mentioned in section (a) and carefully dusted with a soft camel-hair brush. All 44 balsam " should be carefully filtered before use.
The two optical parts which are to be cemented together should first be slightly warmed in the balsaming oven. A very suitable oven for this purpose is the small (9 in. cube) copper oven supplied by Messrs Baird & Tatlock of Hatton Garden, and is fitted with gas heating. Failing this, an ordinary biscuit tin may be converted into an oven, the heating being provided by a carbon filament electric lamp in the circuit of which is arranged a vari- able resistance. The lamp should be placed inside the tin, and means for fitting a thermometer and adjustable air regulation provided in the lid. Such a device works extremely well.
A small amount of balsam should then be placed in the centre of one of the surfaces which is to be balsamed, and the other surface pressed carefully but firmly (with a piece of cork) on to the first until the balsam spreads out as a thin film over the entire surface. Any small bubbles should be removed by pressure with the cork. The parts being balsamed should then be placed on a glass plate covered with paper and supports placed at the sides to prevent any sliding movement of one surface relative to the other. The parts are then put into the oven and the temperature slowly raised until it reaches 77° C., where it -houN I..- k«-pt for four Imurs, and then slowly redu< « • I until the temperature of the room is again attained. The parts can then be removed I mm tin- oven and all super- fluous balsam cleaned off with benzol. The operation is thru complete.
1,,,, PRA< TICAL OPTICS
There are various grades of Canada balsam, known as
td "soft" balsam, but all except the "very
soft" should be taken to 77° C. The "very soft" will
be sufficiently mobile to be put on without any heat and
3-jaw adjustable mount similar to that shown in i. 86. M
Light from lamp or —
'Groove for mount to slide in
Mirror i i 140.
will set when left exposed to the atmosphere for an hour or two.
When achromatic lenses are being balsamed it is necessary to " centre " the two lenses while the balsam is still " plastic." For this purpose a piece of apparatus similar to that shown in Fig. 140 will be found of great convenience. It consists of a cross-line object 0, an ad- justable mount M for the lens L, and a telescope T, all
APPENDIX 167
mounted on the same rigid base and supported in a vertical position.* The lens is rested in the recessed mount M, which is adjusted so that O is in the focal plane of the lens. Observing through the telescope the lens is then rotated, when any centring defect will be shown up by movement of the image. The lens and lens mount can then be heated while in this position until the balsam becomes sufficiently plastic to move one lens relative to the other, when the test can again be repeated until the centring is correct.
(e) DEVELOPERS FOR PHOTOGRAPHIC WORK Hydroquinone Developer FOR PLATES
Solutions A and B to be mixed in equal quantities when required for use. They should be kept in separate bottles.
Solution A .
Hydroquinone ..... 25 gms.
Potassium Metabisulphih . . 25 gms.
Potassium Bromide . . . . 12 gms.
Water . . . 1000 c.cs.
Solution B.
Potassium Hydrate .... 150 gms. Wai. , . . . 1000 c.cs.
Fixing />'///>
Hypo . . . 150 gms.
Water . . . 1000 c.cs.
f cross-line and positive component <>f tho achromatic lens is first - r with the axis of the telescope, before the negative or flint
component is put on to the first.
his PRACTICAL OPTICS
Pyro Developer
FOR PLATES
Solutions A and B to be mixed in equal quantities when required for use. They should be kept in separate bottles.
Solution A.
I'vn.LMllir Acid . . . . 10 gms.
Potassium Metabisulphite ... 2-4 gms.
Water 1000 c.cs.
Solution B.
Sodium Carbonate . . . .100 gms. Sodium Sulphite ... 100 gms.
Potassium Bromide ... 12 gms.
Water 1000 c.cs.
Developer
FOR GASLIGHT PAPER Sodium Carbonate . . . .170 gms.
Sodium Sulphite . . . . 30 gms.
II vdroquinone ..... 8 gms.
Mrtol ...... 2-5 gms.
Potassium Bromide . . . . 1 gm.
Water 1000 c.cs.
(i) A FROSTING SOLUTION FOR GLASS
Such a solution is very convenient for frosting electric lamp bulbs, instead of using tissue paper over a bulb, a much practised method in opticians' workshops.
Dissolve :
25 grammes of (leaf) gelatine and 120 grammes of either • ;il< ium carbonate or magnesium oxide in 250 c.cs. of hot distilled water.
Let the solution cool to 34° C. and dip the glass into
APPENDIX
169
it. Allow to dry and then immerse the glass a second time.
Two coats will in general be enough, but more may be Driven if required.
(ii) A CEMENT FOR OPTICAL PURPOSES
For cementing glass cells, glass windows to metal cells, etc., etc., one of the best cements will be found by mixing equal quantities of " beeswax " and " rosin " (whilst molten), and on cooling make it into thin " sticks." It should be applied with a small heated rod, and then, placing all parts to be cemented into a hot-air oven, should be left until the cement becomes " plastic." At this stage the re- <|iiired surfaces should be put in contact, and then allowed to cool.
This cement will resist the action of aqueous solutions and organic solvents for a very considerable time.
(/) TABLE OF USEFUL WAVE-LENGTHS
Snl stance. |
How emitted. |
Wave-length in 10 -8 cms. |
Colour. |
Sodium |
Bunsen Flame |
5890-2 |
Orange |
» |
» |
5896-2 |
» |
Lithium |
On pole of "Arc" |
r.Tns.L' |
Red |
Rubidium |
if |
7947-0 |
Far red |
» |
» |
7806-1 |
it |
Hydrogen |
Vacuum Tube |
li.Vi.'Mi |
Red |
» |
ii |
4861-5 |
Blue-green |
ii |
4340-7 |
Violet |
|
Mercury |
Mercury Lamp |
5790-7 |
Yellow |
» |
it |
5769-6 |
ii |
'» |
5460-7 |
Green |
|
ii |
ii |
4078-1 |
Violet |
Cadmium |
Vacuum Tul>< |
6438-5 |
Red |
ii |
5085-8 |
Green |
|
« « |
>» |
!7'.»9-9 |
Wur |
Strontium |
Bunsen Flame |
4607-5 |
Blue |
PRACTICAL OPTICS REFRACTIVE INDICES FOR SODIUM LIGHT (A=589
Refractive Index. |
|
Fluorspar |
1-4339 |
Quartz |
1-5442 ordinary |
» |
1-5533 extraordinary |
Rocksalt |
1-5443 |
Water |
1-3329 |
Carbon Bisulphide |
1-6277 |
Benzene |
1-5004 |
Iceland Spar |
1-6584 ordinary |
» |
1-4864 extraordinary |
TABLES
LOGARITHMS
Mean DilTctvi |
ires. |
||||||||||||
1 2 3 |
456 |
7 8 9 |
|||||||||||
10 |
0000 |
0043 |
0086 |
0874 |
4 8 12 |
17 21 2.1 |
29 33 37 |
||||||
11 |
0414 |
...V, |
0607 |
0648 |
<n;*2 n7l;i .17.1.1 |
4 8 11 |
M 111 2." |
•j.; :;n :u |
|||||
12 |
0793 |
'*2* |
1004 |
IOM M72 lino |
3 7 10 |
It 17 21 |
|||||||
13 |
1139 |
1 1 73 |
ISM |
1271 |
L808 |
1MB |
i:;o7 i",:i'.i 1 i:;n |
3 6 10 |
13 16 19 |
2", 20 29 |
|||
14 |
1461 |
1614 |
1'iil |
]i;7:; I7n:i 17:12 |
.", (i li |
12 15 18 |
21 21 27 |
||||||
15 |
1761 |
1790 |
L841 |
1*71 |
L808 |
L981 |
I'.'.v.' 1987 |
9014 |
:; o s |
11 11 17 |
._.,, -,._, ._,- |
||
16 |
3041 |
:•,„> |
MM |
2122 |
8148 |
8178 |
2 2n I |
2227 22-1." |
2279 |
358 |
11 13 10 |
Is 21 21 |
|
17 |
n 'i |
•::.:;- |
MM |
8880 |
8408 |
9458 |
1 2.121. |
2 .1 7 |
in 12 1.1 |
17 2i • 22 |
|||
18 |
2553 |
2601 |
- |
M79 |
20 -.1.1 |
27 IS 2712 270.1 |
257 |
9 12 14 |
n; in 21 |
||||
19 |
2788 |
2810 |
8888 |
• |
2900 |
2:i2:i |
211 1.1 -J'107 2'.IS1» |
2 7 |
9 11 13 |
16 18 20 |
|||
20 |
3010 |
3032 |
3096 |
3118 |
3139 |
SlOn 3181 |
3201 |
2 6 |
8 11 13 |
1.1 17 19 |
|||
21 |
3243 |
M68 |
3304 |
889 i |
8846 |
3365 |
8886 |
8404 |
2 6 |
8 W 12 |
1 1 10 IS |
||
22 |
3444 |
- |
8809 |
.".122 |
8841 |
3560 |
M79 |
3598 |
2 6 |
8 10 12 |
14 15 17 |
||
23 |
3636 |
M74 |
3092 |
",711 |
3729 |
3747 |
8766 |
37SI |
2 6 |
7 9 11 |
l.'i M 17 |
||
24 |
3802 |
3820 |
- |
J874 |
:{'.iu'.» |
3927 |
3945 |
3962 |
2 5 |
7 9 11 |
12 14 16 |
||
25 |
-.-:, |
3997 |
!••! 1 |
1081 |
4048 |
1089 |
4099 |
4116 |
4133 |
235 |
7 9 10 |
12 14 15 |
|
28 |
4150 |
4166 |
1800 |
4 2 Hi |
1989 |
1249 |
4205 |
198] |
4298 |
235 |
7 8 10 |
11 13 15 |
|
27 |
4314 |
4330 |
1869 |
4378 |
4393 |
1409 |
4496 |
(440 |
1 I.K; |
235 |
689 |
11 13 14 |
|
28 |
1471 |
4487 |
4518 |
1648 |
1864 |
4579 |
i.v.M |
4609 |
235 |
689 |
11 12 11 |
||
29 |
16*4 |
ir.r.'.i |
4683 |
4728 |
4742 |
4757 |
134 |
679 |
in 12 i:; |
||||
30 |
1771 |
17--; |
4800 |
IMi |
1848 1857 |
4871 |
4886 |
4900 |
134 |
6 7 U |
10 11 13 |
||
. |
|||||||||||||
31 |
4914 |
4928 |
1941 |
4983 |
1997 |
6011 |
5024 |
5038 |
134 |
678 |
10 11 12 |
||
32 |
MM |
5079 |
5099 |
5105 |
5119 |
5189 |
6145 |
5159 |
5172 |
134 |
578 |
9 11 12 |
|
33 |
6198 |
.1211 |
.122 1 |
5250 |
5263 |
59 76 |
5289 |
5302 |
134 |
568 |
9 10 12 |
||
34 |
5315 |
5378 |
5391 |
5408 |
5416 |
5428 |
134 |
508 |
» 10 11 |
||||
35 |
.'-111 |
- |
MM |
5502 |
.1.11 1 |
,1.127 .1.i:Hl |
5551 |
124 |
.1 0 7 |
9 10 11 |
|||
38 |
1.1.;:; |
5599 |
Mil |
M98 |
-,635 |
r,oi7 5668 5670 |
] 2 1 |
5 0 7 |
8 10 11 |
||||
37 |
•1'>2 |
5717 |
172:1 |
5740 |
5769 |
:.703 1775 |
5786 |
1 2 :; |
.1 0 7 |
s 11 in |
|||
HS |
5798 |
5809 |
MS] |
- |
.ISM; |
-<8 5899 |
1 2 3 |
507 |
s 9 10 |
||||
3f» |
5911 |
59M |
5977 |
l 2 :$ |
1 .1 7 |
S !l 111 |
|||||||
40 |
0...13 |
1064 |
6075 |
• ins;, |
iniifi »;in7 HI 17 |
l 2 :; |
450 |
8 9 10 |
|||||
41 |
6128 |
8149 |
6160 |
H70 |
0191 |
!2'il .1212 0222 |
l 2 :; |
1 .1 o |
789 |
||||
42 |
021; :t |
1271 |
i'.--".i 1 |
;:;«n i;:;i i i;:i2.i |
123 |
l .1 i; |
7 s :• |
||||||
43 |
' .:.:, |
o:n;.i |
(876 |
M96 |
i|i i.l 0 11.1 012.1 |
1 2 3 |
1 .1 0 |
789 |
|||||
44 |
.III |
<; n;i |
; i: i |
8484 |
8498 |
1508 6518 0.122 |
1 2 .", |
l .1 r, |
789 |
||||
45 |
IM9 |
M61 |
6561 |
«m |
6590 |
(099 0609 |
0018 |
123 |
l :, i; |
789 |
|||
46 |
...... |
1665 |
.;.,->! |
6702 |
0712 |
123 |
1 .1 i; |
7 7 .s |
|||||
47 |
r,7lti |
8767 |
6776 |
•-7S.1 07SM |
6808 |
123 |
455 |
0 7 8 |
|||||
48 |
.is.;.; |
is 7.1 (iSS| |
OK 93 |
123 |
1 1 .1 |
078 |
|||||||
49 |
mi |
092* |
6955 |
;:'!•. I 8979 |
0981 |
1 2 .'! |
445 |
ii 7 s |
|||||
50 |
, ,;, , |
7016 |
7049 |
7050 |
7n.V.i |
7007 |
] 2 :; |
345 |
r, 7 s |
||||
51 |
7'.;o |
7093 |
7101 |
nio |
7118 |
7120 |
71:1.1 7li:i |
71.12 |
123 |
345 |
078 |
||
52 |
710-. |
7177 |
71;.:; |
72D2 |
72 in |
721.* 7220 723.1 |
122 |
3 4 5 |
0 7 7 |
||||
53 |
. • |
7292 |
~:;nn 731 >s 7:110 |
122 |
:i l .1 |
0 0 7 |
|||||||
54 |
....•I |
7340 |
7848 |
7864 |
7372 |
1 2 2 |
3 4 5 |
r, r, 7 |
LOGARITHMS
173
55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 8J 83 84 85 86 87 88 8!) 90 91 98 0<» |
0 |
1 |
2 7419 7497 7574 7649 7796 7868 7938 8007 8075 8142 8209 1374 8338 8401 8463 8525 8585 8645 8704 8762 8820 8876 8932 8987 9043 9096 9149 M01 9253 9304 |
3 |
4 7435 7513 7589 7664 7738 7810 7882 IBM 8021 8089 8156 8222 8287 8351 8414 8476 8537 8597 8657 8716 8774 8831 8887 8943 8998 9053 9159 :'•_• 1 •_• MM 9315 MM MM Mil MM |
5 7li:. 7745 7818 7889 7959 8028 8096 8162 8357 8482 8543 8603 8663 8779 8837 8893 8949 9004 MM 9112 ;'!»,:, 9217 MM MSO M70 MM MM MU MM 9614 Mtl MOO M4I MM Ml » M7I |
6 7451 7528 7604 7679 77.VJ 7825 7896 7966 8035 8102 8169 8235 8299 MM 8426 8488 U48 8609 8669 8727 8785 8842 SSJ.JI SU.-.l 1 '.' •«»;:{ 9117 9170 9222 9274 9325 •.»:;;:. •.'»•.'.•- •.'171 ;..vj:t '.'.-.71 9619 ;i«;i;r, i»7.vj ;,„,.-, I.V..I •.•vt :.'..:; :i •r.'v; |
7 |
8 7:,):; 7619 7.:;' 1 7 7. .7 71M" TWO M4I sin; 8181 >-J|s s:;r_- BSTfl -»::•.» BMO SMi BM1 8681 B7M B7t7 8854 M10 8965 MM 9074 9S85 94S5 9581 9628 9675 9722 9768 9899 9903 9991 |
9 |
Diflerenon. |
71"! 74S3 7. ••:.•.• . - - 7993 8062 8129 - 8388 MM - 8633 8692 8751 8808 MM 8976 9031 9085 9138 IMIU '.'.".' 1 MM |
74U 7566 7 ••,!•_• 771G 7860 8000 8069 8136 8331 8395 MM Ifll 8639 8698 -M I -71 MSI MM MM •.-1 c; M4I |
71.7 T.'.s-j 7.;:. 7 7731 7803 7-7--. 7945 8014 8082 8149 S^'15 8280 -:;il MO? 8470 8531 8591 8651 8768 8882 8938 8993 9047 9101 9154 9206 9258 9309 9360 • 1'." |
MM 7536 761'2 IfU 77f." ran rwM 7;- 7:; 8041 not 8176 U 1 1 MM 8370 843'J 8494 8555 8615 8675 B7M B7f] 8848 M04 MM 1018 MM 9122 9175 '.••."-•7 9279 9330 9380 9430 9479 9528 9576 MM '.'.;71 9717 •..;..:. MM MM MM M4I 9987 |
7171 7551 76S7 7701 7771 7846 7917 7987 8055 Blf] 8189 3264 8319 8382 8445 8506 8567 8627 8686 8745 8802 8859 8915 8971 9025 9079 - 9238 9289 •i:;in 9390 9440 '.Ms'J •i.VIS y.iso 963S •M;X,I 9727 9818 -,,;i ..,.„, n •..-.. :.j •r.....j |
:> •; 7 567 567 567 .567 566 |
||||||
l -J -.- 1 1 2 112 1 1 2 |
345 345 344 344 344 |
||||||||||
112 1 1 2 112 1 1 2 |
|||||||||||
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 L' 1 2 1 2 |
334 334 234 |
556 1 :, i; 4 5 C |
|||||||||
234 234 |
j :, :, l :, :, |
||||||||||
233 |
i :, :, |
||||||||||
445 |
|||||||||||
1 2 |
|||||||||||
1 . 1 •_' |
|||||||||||
0 1 1 0 1 1 0 1 1 -1 1 1 |
|||||||||||
<) 1 1 Oil |
223 I | | |
344 a i i |
|||||||||
• ' 1 1 " 1 1 •t 1 1 |
238 |
344 |
|||||||||
338 |
3 4 4 |
||||||||||
171
AMMLOGARITHMS
.Mean DilVei-eiu-e-. |
|||||||||||
00 1000 |
1002 |
1005 1007 |
1009 |
1014 |
1016 |
1019 |
|||||
1021 |
001 |
111 |
2 2 2 |
||||||||
•01 1023 |
1026 |
1033 |
1038 |
l"l" 1"I2 |
1045 |
001 |
111 |
2 2 2 |
|||
•02 1047 |
1050 |
1057 |
1".H H..17 |
L069 |
001 |
111 |
2 2 2 |
||||
-08 1072 |
1076 1079 |
1081 |
L089 L09] |
L094 |
001 |
111 |
2 2 2 |
||||
04 1096 |
11"2 IK.'I |
1107 |
li"n 1112 |
1111 |
1117 |
L119 |
Oil |
112 |
222 |
||
-06 1122 |
1127 11."" |
1132 |
1135 |
111" 1113 |
1146 |
Oil |
1 1 2 |
2 2 2 |
|||
•08 1148 |
1151 |
1153 1156 |
1159 |
nc.i in;i |
1H17 lit;'.' 1172 |
Oil |
1 1 2 |
2 2 2 |
|||
•07 1175 |
- |
1186 |
n-.' n '.'1 |
ll'.'l 11-.I7 11U9 |
Oil |
112 |
222 |
||||
08 1202 |
1213 |
1210 |
1219 |
1-22 1226 1227 |
Oil |
112 |
2 2 :; |
||||
09 1230 |
1236 1239 |
1212 |
1246 |
1217 |
12:," 12.-.:; ii'.-n; |
Oil |
112 |
223 |
|||
•10 125'.. |
:_• . |
1271 |
1271 |
1276 |
1271. 1281 L286 |
Oil |
112 |
223 |
|||
11 12-- |
1300 |
L809 1:112 |
1315 |
Oil |
122 |
223 |
|||||
•12 1318 |
1321 |
1330 |
i:;:;i i:;:;7 |
i:il" 1343 134G |
Oil |
122 |
223 |
||||
•18 1349 |
- |
1361 |
l. •;.;:, L86S |
1371 |
i:i 71 i:i77 |
Oil |
122 |
233 |
|||
•14 1380 |
1881 1 •'••'.'" |
1393 |
l :.'."'. 1 l"" |
1403 |
1 lor, 11"'.) |
Oil |
122 |
233 |
|||
•16 1413 |
in.; |
M1H 1122 |
1426 |
1I2H Ii:i2 |
1435 |
1439 1442 |
Oil |
122 |
233 |
||
16 1445 |
11. V2 ll.V. |
1459 |
11(12 lltltl |
1469 |
1 172 |
1-17(1 |
0 1 |
122 |
233 |
||
•17 11 71* |
1486 1489 |
1 in:; |
1496 1500 |
1503 |
15ti7 151" |
0 1 |
122 |
233 |
|||
•18 1514 |
1811 |
1 52 1 1 52 1 |
1528 |
1531 1535 |
1538 |
1512 1515 |
0 1 |
122 |
233 |
||
•19 1549 |
1882 |
15G3 |
L687 |
1570 |
1574 |
1578 1581 |
0 1 |
122 |
333 |
||
•20 1585 |
1888 |
1600 |
n ;..,:; io"7 |
1611 |
n; n IGIS |
0 1 |
122 |
333 |
|||
•21 1622 |
1637 |
Hill Hill |
1648 |
L662 |
1656 |
Oil |
222 |
333 |
|||
•22 1660 |
L66J |
1675 |
1679 1GK3 |
L687 |
1690 |
1694 |
Oil |
222 |
333 |
||
•23 1698 |
1706 1710 |
1714 |
1718 |
1722 |
172G |
1730 |
1734 |
Oil |
222 |
334 |
|
•24 1738 |
17.2 |
1746 1750 |
1 7:. 1 |
1762 |
17GG |
1770 |
1774 |
Oil |
222 |
334 |
|
•28 1778 |
1786 1791 |
1795 |
1 7n it |
1803 |
1807 |
1811 |
1816 |
Oil |
222 |
334 |
|
•28 1820 |
1837 |
L84J |
1845 |
1849 |
1854 |
1858 |
Oil |
223 |
334 |
||
•27 1862 |
1871 1875 |
1879 |
1884 |
1888 |
1892 |
1897 |
1901 |
Oil |
223 |
334 |
|
•28 1905 |
L910 |
inn nun |
L982 |
1928 |
1932 |
1936 |
1941 |
1945 |
0 1 |
223 |
344 |
•29 1950 |
1977 |
1982 |
1986 |
1991 |
0 1 |
223 |
344 |
||||
•80 1995 |
2014 |
2018 |
2023 |
2028 |
2032 |
2037 |
0 1 |
223 |
344 |
||
2041 |
• |
2061 |
2065 |
2<>7ii |
2075 |
2080 |
2084 |
0 1 |
223 |
244 |
|
•82 2089 |
2094 |
2109 |
2113 |
2118 |
2123 |
2128 |
2133 |
0 1 |
223 |
344 |
|
•88 2138 |
2158 |
2163 |
2168 |
2173 |
2178 |
2183 |
0 1 |
223 |
344 |
||
•84 2188 |
2208 |
2218 |
2223 |
2228 |
2234 |
112 |
233 |
445 |
|||
2844 |
2259 |
2265 |
2270 |
2275 |
2280 |
2286 |
112 |
233 |
445 |
||
2:1 17 -j:;2:; |
2328 |
2333 |
2339 |
112 |
233 |
445 |
|||||
37 ...II |
2366 2371 2:1 7 7 |
2889 |
2388 |
2393 |
112 |
233 |
445 |
||||
•88 2399 |
211" I'll:, |
2421 |
2127 |
2482 |
2438 |
2443 |
2449 |
112 |
233 |
445 |
|
•89 2455 |
Ml • |
2 Ilt5 |
2600 |
2506 |
1 1 2 |
233 |
455 |
||||
2 Ml 2517 |
2553 |
2559 |
2564 |
112 |
234 |
455 |
|||||
2694 |
2612 |
2618 |
2624 |
112 |
234 |
455 |
|||||
•l.'i |
MM 2 7 1 u |
2M-.I 2MI7 •_'7L':i 272H |
2871 2735 |
2679 2742 |
2685 2748 |
112 1 1 2 |
234 |
4 5 G |
|||
44 2751 |
2780 |
- i l - 2806 |
2812 |
L X z 112 |
334 |
456 |
|||||
45 -'-I- |
2844 |
28« |
2-5- |
2884 |
2.S71 |
2877 |
112 |
334 |
5 5 G |
||
•48 2884 |
2891 |
2881 2804 |
21U7 2H21 |
2931 |
2988 |
2944 |
112 |
334 |
556 |
||
2968 |
2992 |
21)1111 |
3006 |
3013 |
112 |
334 |
B B •; |
||||
•48 3020 |
3027 |
2084 204] |
8066 |
3062 |
;"«;:. |
3076 |
:;">:; |
112 |
344 |
566 |
|
•49 3090 |
. 181 |
1119 |
312(1 |
nil :;ns |
3155 |
112 |
344 |
5 G 6 |
|||
\\PILOGARITHMS
175
0 |
123 |
Differences. |
||||||||||
•50 31C-J •53 MM -54 3167 •56 3631 •58 MOJ •61 «"7« •62 41«y • 63 Jl'GG •64 «65 •85 no; •66 »:''l •68 »7SO •69 »-'•"> •70 •''"!- •71 .'.l-J'.t '-.' •si '-i"'; .go M01 •83 '"''-I •86 72441 *88 ' • * " ' 1 '89 77'>'-'l |
:;i7" 3243 3319 :;:;•...; 3556 3811 3899 MM urn UTS r_'7«; 4477 tin 1688 4797 5768 MM 6180 6324 6471 |
.". 1 7 7 un 3327 3404 3483 3565 3648 3733 3819 3908 3999 4093 4188 4285 4385 L487 1091 MM 4808 IMQ MM |
3192 3266 3342 3420 :;i n'j 3581 3664 3750 3837 3926 4018 4111 MM 1KM 4721 4831 I'.M:; 5058 ii?< IM1 MM '.:. ic 'it; 7."p 5808 5943 6081 6223 6368 6516 ,,;,> >•_•:; 6982 7146 7311 7482 KM v.i ."l |
3199 3273 MM 3428 IMC MM 1671 3758 3846 3936 4027 US1 1217 iin; 4624 4842 MM 5070 5188 not MM MM MM 5821 5957 ION 6237 IM1 6531 MM 1881 IMI 71-.1 rest 7499 -'•.71 7862 MM Ml 8414 |
3206 3281 3357 lift UK :;:.•.. 7 M81 B767 UN 4036 4130 4529 4634 4742 4853 L964 5082 5200 5445 5572 5702 >r.l 5970 ;],,., ;•_•:.•_• 6397 ; ;,|,; ;.;•..•.. >:,:, 7015 7178 ;:;,.•, 7516 ;<,'.n -•.;«. - •.,! v: 1 1 si:;:i V.I |
::-JMI :;:;.;:, :;u:; 3524 3606 3690 3776 3864 :;:•.-, I 4046 4140 4236 4335 4436 4539 4645 4753 4864 4977 5093 5212 5333 5458 5585 5715 U4I MM fill l MM •; 1 1 •-• IM1 1711 6871 7031 7194 7362 7634 7709 7889 8072 MO IN MO -.-.I Oil Ml i*i |
3221 :;-.".M :;:;7: 3451 3532 1614 MM B784 3873 3963 IOM I 1 :,. ' LS4J IMC AMO MM J7.U ^7;. ISM 5105 5470 5598 5728 5861 6998 6138 M8I '.I-J 7 6577 6730 6887 7047 7211 7379 7551 7727 7907 8091 8279 8472 8670 8878 9078 9290 MM 9727 MM |
MM MO |
567 6 7 1 7 6 7 6 7 |
|||
3258 3334 :: 1 1 1 MM U7I MM :; 7 1 1 3917 4009 4102 4198 4498 4603 4710 4819 :.MI 7 .V.>1 |
1 2 2 |
345 |
||||||||||
MM 3540 3622 3707 3793 3882 3972 4064 4159 4256 4355 4457 4560 1667 1771 4887 5000 5117 5236 5358 5483 5610 5741 5875 6012 6152 6295 ;n-j •,:.'.••_- ;;i:. ;•...<_• 7063 7228 -:,.M; -:„> •;i:. ••.'•::. 8110 ^•yj -r.«: ,;-.,„ -:.: „,-.,-., :.ll :.•-•- |
||||||||||||
1 2 2 |
345 |
|||||||||||
1 2 3 |
345 |
7 8 |
||||||||||
7 8 |
||||||||||||
7 8 7 8 7 8 |
||||||||||||
1 2 3 |
456 |
|||||||||||
123 I 2 3 |
456 457 |
7 9 10 8 9 10 |
||||||||||
1 -J :: 124 124 124 134 134 1 3 4 134 1 3 4 134 1 3 4 |
5 7 ~ 5 7 5 7 5 8 5 8 5 8 578 578 678 679 |
8 9 10 8 9 11 8 10 11 9 10 11 9 10 11 9 10 12 9 10 12 9 11 12 10 11 12 10 11 1." 10 11 13 10 ]•_• i:; 11 12 14 11 12 14 11 13 14 11 13 15 12 13 16 12 13 15 12 14 16 12 14 16 13 14 16 13 16 17 13 15 17 14 15 17 i LI LI 14 16 18 16 17 19 16 17 19 U LI N 16 18 20 16 18 20 |
||||||||||
5781 Mil MM SIM MM MM |
5794 :.•.••_".' .;..,;: •;•_•..•.. .;:;:,:; .;:.., 1 • ;• ,:. |
|||||||||||
235 |
689 |
|||||||||||
235 -• :; 1 236 • | :, 246 246 246 1 •'. 4 6 4 6 4 6 1 I 4 r, 4 7 4 7 :. 7 |
6 8 10 7 8 10 7 8 10 7 9 10 7 9 11 7 9 11 7 9 11 8 9 11 8 10 12 8 10 12 8 10 12 8 10 12 8 11 13 9 11 13 9 11 13 9 11 14 |
176
NATURAL SINES
Mean l>irtViviuv>. |
||||||||||||
0' |
6 |
12' |
18' |
24 30' 36' |
42' |
48' |
54' |
|
||||
r 2' 3' |
4' 5' |
|||||||||||
0 |
•0000 |
0017 |
0052 |
0070 |
0087 |
0105 |
0122 |
0140 |
0157 |
3 f. y |
l-j ir, |
|
•0175 |
0192 |
0887 |
"•j|| |
0262 |
0279 |
< i-_".. 7 |
0314 |
0332 |
3 (1 '.i |
12 I.'. |
||
•0141 |
0401 |
0419 |
0436 |
0454 |
0471 |
0488 |
0506 |
369 |
1 •_• 1 :. |
|||
•0523 |
0541 |
MM |
0576 |
0593 |
0610 |
0628 |
0645 |
0663 |
0680 |
3 6 9 12 Ifi |
||
•MM |
0715 |
0750 |
0767 |
0785 |
0801 |
0819 |
0837 |
0854 |
369 |
12 11 |
||
•0872 |
0889 |
Q994 |
0941 |
0958 |
0976 |
0993 |
1011 |
1028 |
369 |
I-.' 11 |
||
1063 |
1080 |
1097 |
1115 |
1132 |
1119 |
1167 |
1184 |
1201 |
369 |
12 1! |
||
•l.'l'.t |
1236 |
1851 |
1-J71 |
1288 |
1305 |
1323 |
1340 |
1357 |
1*74 |
369 |
1 •_' 11 |
|
•1392 |
L409 |
1426 |
1444 |
1461 |
1478 |
1495 |
1513 |
1530 |
1547 |
369 |
12 ] 1 |
|
•1564 |
1582 |
1599 |
1616 |
1633 |
1650 |
1668 |
1685 |
1702 |
1719 |
369 |
12 11 |
|
10° |
•1736 |
1754 |
1771 |
1788 |
1805 |
1822 |
1840 |
1857 |
1874 |
1891 |
369 |
11 14 |
11 |
•1908 |
1925 |
1942 |
1959 |
1977 |
1994 |
2011 |
3088 |
2045 |
2062 |
369 |
11 14 |
12 |
•2079 |
2096 |
2113 |
2130 |
2147 |
2164 |
2181 |
2198 |
2215 |
2233 |
369 |
11 14 |
13 |
•SSM |
2267 |
2284 |
2300 |
2317 |
2334 |
2351 |
2368 |
2385 |
2402 |
368 |
11 11 |
14 |
•2419 |
2436 |
2453' |
2470 |
2487 |
2504 |
2521 |
2538 |
2554 |
2571 |
368 |
11 11 |
15 |
•2588 |
2605 |
2622 |
2639 |
2656 |
2672 |
2689 |
2706 |
2723 |
2740 |
368 |
11 14 |
16 |
•2756 |
2773 |
2790 |
2807 |
1888 |
•JS10 |
2857 |
2S7I |
2890 |
2907 |
368 |
11 14 |
17 |
•2924 |
2940 |
2957 |
2974 |
2990 |
3007 |
3024 |
3040 |
3057 |
3074 |
368 |
11 14 |
18 |
•3090 |
3107 |
3123 |
3140 |
3156 |
3173 |
3190 |
3206 |
3223 |
3239 |
368 |
It 11 |
19 |
•3256 |
3272 |
3289 |
3305 |
3322 |
3338 |
3355 |
3371 |
3387 |
3404 |
358 |
11 14 |
20° |
•3420 |
3437 |
3453 |
3469 |
3486 |
3502 |
3518 |
3535 |
3551 |
3567 |
358 |
11 11 |
21 |
•3584 |
3600 |
3616 |
3633 |
3649 |
3665 |
3681 |
3697 |
3714 |
3730 |
358 |
11 14 |
22 |
•3746 |
3762 |
3778 |
3795 |
:isn |
3827 |
3843 |
3S59 |
3875 |
3891 |
358 |
11 1 1 |
28 ; '3907 |
3923 |
3939 |
3955 |
3971 |
3987 |
4003 |
4019 |
4035 |
4051 |
358 |
11 14 |
|
24 |
•4067 |
ins:; |
4099 |
4115 |
4131 |
4147 |
4163 |
4179 |
4195 |
4210 |
358 |
11 13 |
25 |
•i-j.i; |
4242 |
4258 |
4274 |
ll'S'.l |
4305 |
4321 |
4337 |
4352 |
4368 |
358 |
11 13 |
26 |
•4384 |
ISM |
4415 |
4431 |
1116 |
4462 |
4478 |
4493 |
4509 |
4524 |
358 |
10 13 |
27 |
• r.l'i |
l :,:>:> |
4571 |
4586 |
16(12 |
4617 |
4633 |
4648 |
4664 |
4679 |
358 |
10 13 |
28 |
•4695 |
4710 |
4726 |
4741 |
4756 |
4772 |
4787 |
4802 |
4818 |
4833 |
358 |
10 13 |
29 |
•4848 |
4863 |
4879 |
4894 |
4909 |
4924 |
4939 |
4955 |
4970 |
4985 |
358 |
in i:; |
30° |
5015 |
5030 |
5045 |
5060 |
5075 |
5090 |
5105 |
5120 |
5135 |
358 |
in i:; |
|
31 |
•5150 |
5165 |
5180 |
5195 |
5210 |
5225 |
5240 |
5255 |
5270 |
5284 |
257 |
10 12 |
32 |
-. "• •-".)'.• |
5314 |
5329 |
5344 |
5358 |
5373 |
5388 |
5402 |
5417 |
5432 |
257 |
10 12 |
33 |
•MM |
5461 |
5476 |
5490 |
5505 |
5519 |
5534 |
5548 |
5563 |
5577 |
257 |
lit ]•-• |
34 |
•MM |
MM |
5621 |
5635 |
5650 |
5664 |
5678 |
5693 |
5707 |
5721 |
257 |
10 12 |
35 |
•5736 |
B7M |
5764 |
5779 |
B7M |
5807 |
5821 |
5835 |
5850 |
5864 |
257 |
it 12 |
36 |
•5878 |
MM |
5906 |
.V.rjo |
5934 |
5948 |
5962 |
5976 |
.V.I'.IO |
6004 |
257 |
'.1 I L- |
37 |
•6018 |
6032 |
6046 |
•5060 |
6074 |
6088 |
6101 |
6115 |
6129 |
6143 |
257 |
II 1 •-' |
38 |
•6157 |
6170 |
6184 |
6198 |
6211 |
6225 |
6239 |
6252 |
6266 |
6280 |
257 |
9 11 |
39 |
•6JM |
6307 |
6320 |
6334 |
6347 |
6361 |
6374 |
6388 |
6401 |
6414 |
247 |
•t I! |
40 |
•6428 |
0111 |
6455 |
6468 |
6481 |
6494 |
6508 |
6521 |
6534 |
6547 |
247 |
9 11 |
41 |
9674 |
6587 |
6600 |
6613 |
•;r,2<; |
6639 |
6652 |
6665 |
0678 |
247 |
9 11 |
|
42 |
•6691 |
6704 |
6717 |
6730 |
6743 |
6756 |
6769 |
6782 |
6794 |
6807 |
246 |
9 11 |
43 |
6833 |
6845 |
6858 |
6871 |
I5S.X1 |
6896 |
6909 |
6921 |
6934 |
246 |
8 11 |
|
44 |
•6947 |
6959 |
6972 |
6984 |
6997 |
7009 |
7022 |
7034 |
7046 |
7059 |
246 |
8 Kt |
NATURAL SINES
177
A |
81 .1 1Q- |
OA' fit}' |
00' |
49" |
*»• |
Mean Differences. |
||||||
u |
U lo |
— ^ «>U |
uO |
4 . 1 -3 |
o* |
1 2 3 4 5 |
||||||
45 |
7083 |
not |
7i:.7 |
7169 |
7181 |
246 |
8 10 |
|||||
46 |
•7193 |
LI 7 •_•:;•• |
7266 |
246 |
||||||||
47 |
7385 |
- |
246 |
|||||||||
48 |
UU 71 'if, |
7501 |
7.-.:..; |
246 |
8 10 |
|||||||
49 |
-7.-.I7 |
~ •> >.<*n 1 |
7593 7604 . i" i -1 |
7615 |
mi |
246 |
8 9 |
|||||
,».!.' l |
7**OO |
IOOJ — -(\n |
ion WOA 1 |
1 tOo |
< t 10 |
7727 |
~Q 1 U TOKO |
77''1 ' -fc ., . |
||||
62 |
till •7880 |
7701 7891 |
, Ml J 7912 |
folO |
7S.">7 7'.. 1 » |
, •» i - Jon |
, Mi'.l 7976 |
245 |
||||
53 |
•TtM |
7997 |
8018 |
8028 |
8039 |
SllJil |
8080 |
•j :• |
7 » |
|||
54 rr |
•MM • v 1 '*•' |
8100 |
Bill |
8121 |
8131 |
Mil |
H5i |
8181 |
•j 1 :• |
7 |
||
§• 56 |
s 1 .*- |
8300 |
8211 |
8221 - |
8251 v;i> |
BZ/J |
8281 8377 |
i; - |
||||
57 |
•8887 |
8396 |
8406 |
8415 |
1441 |
8471 |
235 |
6 8 |
||||
58 CQ |
•8480 • «^7*» |
aroi |
8499 QKCIA |
8508 |
v.-,:;,; |
BM| |
235 |
6 8 |
||||
•9 60 |
•8660 |
8669 |
»OW 8678 |
•>">'.''.' 8686 |
»l',l'."i 8712 |
»i;.v_' 8738 |
134 |
6 7 |
||||
61 |
•8746 |
8780 8788 |
-7'...; |
8821 |
134 |
f. 7 |
||||||
62 |
•8829 |
8864 |
8878 |
8902 |
1 3 4 |
:. 7 |
||||||
6'J |
•8910 |
8934 |
- |
134 |
5 6 |
|||||||
64 |
.0901 |
8996 |
Mil |
1»M|1 <«II|S |
MM |
1 3 4 |
;, «. |
|||||
65 |
•9063 |
9070 |
9085 |
9092 |
9107 |
9114 |
III'JI |
9128 |
124 |
;, .. |
||
66 |
9157 |
'.- 1 1: l |
'.•171 |
9178 |
919S |
i •_• r, |
5 6 |
|||||
67 |
'.'•.•.•:. |
MM |
1 2 3 |
4 6 |
||||||||
68 |
•an |
9323 |
MM |
i •_• :i |
4 5 |
|||||||
69 |
•9336 |
9354 |
9373 |
9379 |
9385 |
i •_• :; |
» |
|||||
70 M*1 |
•MM |
•.. 1 1 :. |
9438 |
'..III |
9449 |
1 2 3 |
4 5 |
|||||
71 72 73 * |
.' 1 • > > •9511 •Ma 'it i " |
MCI Mil MM , , • . - |
'..-,7* |
9583 |
'.' I *«;; MM |
'.. I1' I PMJ MM |
9600 9563 |
'.i.". ' i.'i 9558 9608 ... - • |
i •-• :; i •_• -j |
3 4 8 4 |
||
/* 75 |
."-... •MM |
Po] i MM |
'•'. . |
'."'•-' 7 9673 |
MM ••.,77 |
MM |
9686 |
'.., |i. MM |
'.ll'i.'d l 9694 |
.tt,,,., MM |
1 I 2 |
3 4 |
76 |
•9703 |
9707 |
'..711 |
9715 |
9720 |
mi |
9736 |
1 1 2 |
1 | |
|||
77 |
9748 |
tnt |
'7.V |
9763 |
9770 |
9774 |
9778 |
1 1 2 |
8 8 |
|||
78 |
•9781 |
9785 |
9789 |
MM |
9799 |
9803 |
MM |
1 1 |
:. |
|||
79 |
'Mil |
9820 |
MM |
MM |
MM |
9838 |
MM |
MM |
M4I |
1 1 •„• |
1 |
|
80 |
•M4I |
9851 |
9854 |
MM |
9860 |
9863 |
MM |
MM |
" 1 1 |
|||
81 |
•9877 |
9882 |
••HH:, |
9888 |
9890 |
9895 |
9900 |
H 1 1 |
I |
|||
82 |
••..:...: |
MM |
9907 |
•.••.•In |
9912 |
9914 |
9917 |
1 1 |
2 2 |
|||
83 |
•MM |
MM |
9930 |
...•:;•: |
9934 |
9936 |
993H |
B940 |
9942 |
9943 |
0 1 1 |
1 | |
84 |
•9946 |
9947 |
9949 |
,,-,- |
9964 |
MM |
9957 |
II 1 1 |
1 3 |
|||
85 |
•MM |
..... |
9966 |
.•..,.. |
.„.. |
MM |
9971 |
(1 .. 1 |
I I |
|||
86 |
>M7fl |
9978 |
9979 |
, ... |
M II | |
I 1 |
||||||
87 |
•MM |
MM |
.... |
1 1 |
||||||||
88 |
•MM |
9096 999A |
,,„ |
9998 |
«, , |
0 0 |
||||||
89 |
•MM |
,-.,..•:, |
i i i i |
1 ,,.,., |
" 0 |
0 0 |
178
NATURAL COSINES
0 |
6 |
12- |
18- |
24 |
30 |
38 |
42' |
48' |
54' |
Mean Differences. |
||
1' 2' 3' |
4 5 |
|||||||||||
Q |
1-000 |
1*000 |
1-000 |
1-000 |
1-000 |
-MM |
MM |
9999 |
9999 |
000 |
0 0 |
|
•MM |
9998 |
9998 |
9997 |
9996 |
MM |
9995 |
9995 |
000 |
0 0 |
|||
•MM |
•r....:: |
9993 |
'.'•.•'.'_' |
9991 |
MM |
MM |
M89 |
9988 |
9987 |
000 |
1 1 |
|
....-•. |
'.'•-.-. |
9984 |
•.'•-:; |
M8S |
9981 |
9980 |
M79 |
9978 |
9977 |
001 |
1 1 |
|
•H7I |
9974 |
9973 |
9972 |
9971 |
9969 |
MM |
9966 |
9965 |
9963 |
001 |
1 1 |
|
•MM |
;,.„;,, |
9959 |
9957 |
MM |
9954 |
H'.i.Vj |
9951 |
M49 |
9947 |
Oil |
1 2 |
|
g |
••Ml |
r.'i:; |
9942 |
MM |
9936 |
;i;i." \ |
9932 |
9930 |
9928 |
Oil |
1 2 |
|
7 |
•MSI |
9923 |
9921 |
nn in |
9917 |
9914 |
9912 |
9910 |
9907 |
9905 |
Oil |
•_' - |
g |
•MOI |
9900 |
MM |
-.«-.«:, |
9893 |
MM |
9888 |
M86 |
9882 |
9880 |
Oil |
•_.' |
9 |
•9877 |
9874 |
9871 |
9869 |
9866 |
9863 |
9860 |
MS? |
9854 |
9851 |
Oil |
•_' L' |
10 |
•9848 |
9845 |
Mti |
9839 |
9836 |
9833 |
9829 |
9826 |
9823 |
9820 |
112 |
•2 3 |
11 |
•9816 |
9813 |
9810 |
9806 |
9803 |
9799 |
9790 |
9792 |
9789 |
9785 |
112 |
2 3 |
12 |
•9781 |
9778 |
9774 |
9770 |
»767 |
9763 |
9759 |
9755 |
9751 |
9748 |
112 |
3 3 |
13 |
•9744 |
9740 |
9736 |
9732 |
9728 |
9724 |
9720 |
9715 |
9711 |
9707 |
112 |
:; :; |
14 |
•9703 |
9699 |
'.";;' 1 |
9690 |
9686 |
9681 |
9677 |
9673 |
9668 |
9664 |
112 |
:; l |
15 |
•9659 |
9655 |
9650 |
9646 |
9641 |
9636 |
9632 |
M27 |
9622 |
9617 |
122 |
3 4 |
16 |
•9613 |
9608 |
9603 |
9598 |
9593 |
9588 |
9583 |
9578 |
9573 |
9568 |
122 |
3 4 |
17 |
•9563 |
9558 |
9553 |
9548 |
9542 |
9537 |
9532 |
9527 |
9521 |
9516 |
123 |
3 4 |
18 |
•9511 |
9505 |
9500 |
9494 |
9489 |
9483 |
9478 |
9472 |
9466 |
9461 |
123 |
4 5 |
19 |
•9455 |
9449 |
'.-III |
9438 |
MM |
9426 |
9421 |
9415 |
9409 |
9403 |
123 |
|
20 |
•9397 |
9391 |
9385 |
:»:»7'.i |
9373 |
9367 |
9361 |
9354 |
9348 |
9342 |
123 |
4 5 |
21 |
•9336 |
9330 |
9323 |
9317 |
9311 |
9304 |
9298 |
9291 |
9285 |
9278 |
123 |
4 5 |
22 |
•9272 |
9265 |
MM |
9252 |
M48 |
9239 |
9232 |
9225 |
9219 |
9212 |
123 |
4 6 |
23 |
•9205 |
9198 |
9191 |
9184 |
9178 |
9171 |
9164 |
9157 |
9150 |
9143 |
123 |
r, r, |
24 |
•9135 |
9128 |
9121 |
9114 |
'.M 07 |
9100 |
9092 |
9085 |
9078 |
9070 |
124 |
5 6 |
25 |
•9063 |
9056 |
9048 |
9041 |
9033 |
9026 |
9018 |
9011 |
9003 |
8996 |
134 |
5 6 |
f)n |
•8988 |
8980 |
8973 |
8965 |
8957 |
8949 |
8942 |
8934 |
8926 |
8918 |
134 |
r, •; |
:.'7 |
•8910 |
8902 |
MM |
8886 |
8878 |
8870 |
8862 |
ssr.i |
8846 |
8838 |
134 |
5 7 |
:.'8 |
•8829 |
8821 |
8813 |
8805 |
8796 |
8788 |
8780 |
8771 |
8763 |
8755 |
134 |
G 7 |
29 |
•8746 |
8738 |
8729 |
8721 |
8712 |
8704 |
8695 |
Hf.Sfi |
8678 |
8669 |
134 |
6 7 |
t>0 |
•8660 |
MM |
8643 |
8634 |
8625 |
8616 |
8607 |
8599 |
8590 |
8581 |
134 |
G 7 |
31 |
•8572 |
8563 |
8554 |
8545 |
8536 |
86J6 |
8517 |
8508 |
8499 |
8490 |
235 |
<> s |
.'{:.' |
•8480 |
8471 |
8462 |
8453 |
8443 |
84S4 |
si •_'.-, |
8415 |
8406 |
8396 |
235 |
6 8 |
33 |
•8387 |
8377 |
8368 |
S.'S.-.s |
BUS |
8339 |
8329 |
8320 |
8310 |
8300 |
235 |
6 8 |
'' l |
•8290 |
8281 |
8271 |
si'«;i |
886] |
8241 |
8231 |
8221 |
8211 |
8202 |
L' :i 5 |
7 8 |
35 |
•8192 |
8181 |
8171 |
8161 |
8151 |
8141 |
8131 |
8121 |
8111 |
8100 |
235 |
7 8 |
36 |
•8090 |
8080 |
8070 |
* i.V.i |
Si i J'l |
8039 |
8028 |
8018 |
8007 |
7997 |
235 |
7 '.» |
37 |
•79M |
7976 |
7965 |
7'..:,:, |
7944 |
7923 |
7912 |
7902 |
7891 |
2 5 |
7 9 |
|
38 |
•7880 |
7869 |
7859 |
7-1* |
7837 |
7826 |
7815 |
7804 |
7793 |
77SL' |
2 5 |
7 9 |
w |
•7771 |
7760 |
7749 |
7738 |
7727 |
7716 |
7705 |
76M |
7683 |
7672 |
2 6 |
7 9 |
40 |
•;.,.,,, |
7M8 |
7638 |
7627 |
76U |
7604 |
7593 |
7581 |
7570 |
7559 |
2 6 |
8 9 |
41 |
•7547 |
7536 |
T.VJI |
::.!:; |
7501 |
7490 |
7478 |
7406 |
7455 |
7443 |
2 6 |
8 10 |
42 |
7420 |
74M |
7:;'."'. |
7385 |
7373 |
73G1 |
7349 |
7337 |
7325 |
2 6 |
8 10 |
|
43 |
•7314 |
7302 |
7290 |
7278 |
7266 |
7254 |
7242 |
7SM |
7218 |
7206 |
2 6 |
8 10 |
44 |
•7193 |
7181 |
7169 |
7157 |
7145 |
7133 |
7120 |
7108 |
7096 |
7083 |
246 |
8 10 |
NATURAL COSINES
179
M.'Mll I>i!V.
Mean DJfft ; |
||||||||||||
A' |
r-i .•)• 10 |
.11 on Ofi |
4O' AO' K.A' |
|||||||||
0 |
IX lo |
•I |
M |
ow |
Vb |
VOJ |
«•* |
r v r |
4' 5 |
|||
45 -7071 |
\>\ 7"."l |
7m.'L> 7,Hi:i |
6984 |
6971 |
246 |
8 10 |
||||||
1 |
6858 |
M4B |
i' i 6 |
8 11 |
||||||||
47 '6820 |
>:756 |
6730 |
R717 |
6701 |
246 |
11 |
||||||
48 '6691 |
Mil |
MOO |
M07 |
.' 1 7 |
11 |
|||||||
49 -6561 |
1547 U |
6481 |
544J |
•J 1 7 |
11 |
|||||||
50 '6428 |
f.lll I'T.l r,:;ss |
M47 |
6334 |
6307 |
•2 -i 7 |
11 |
||||||
51 |
•6293 |
••.•_•*.• .;-••,.; .;•_>:,•_• |
61M |
6184 |
»;i7«' |
l' :> 7 |
11 |
|||||
52 -6157 |
••.ii:> f,i-.".t tin:, |
c.lul i;n88 6074 |
BOM |
BOM |
l' 5 7 |
12 |
||||||
53 -6018 |
•,!)48 |
:>•.•:; I |
5920 |
5906 |
5R92 |
i' 5 7 |
I] |
|||||
54 -5878 |
:>M; i |
5850 |
5821 |
:,7'.':; |
6779 |
257 |
'.' ] •_' |
|||||
56 -5736 |
vm |
5707 |
MM |
5678 |
:,.;.; i |
MM |
MM |
-' 5 7 |
10 12 |
|||
56 "5592 |
.'..•.77 |
MOI |
:,:,ls |
5534 |
.-•.-> 19 |
5505 |
5490 |
M7< ">i'''i |
267 |
10 12 |
||
57 '5446 |
5432 |
5417 |
MM |
5388 |
5358 |
:.:; ) i |
267 |
in 12 |
||||
58 |
•MM |
5284 |
5270 |
5240 |
5210 |
5195 |
:,lsu .-,1.;:, |
257 |
10 12 |
|||
59 |
•5150 |
5135 |
r>r.'<i |
.-.Him |
M7B |
5060 |
5045 |
5030 |
r,oi ;, |
368 |
10 13 |
|
60° |
IMI |
1070 |
i;.:;;. |
I'.'L'I |
4909 |
ism |
IS T;I |
1801 |
358 |
10 13 |
||
61 |
•4848 |
4833 |
IMS |
1001 |
4787 |
4772 |
I7M |
1711 |
I7M |
171" |
358 |
10 13 |
62 |
• l'il'0 |
4679 |
IM;| |
MM |
M8J |
4617 |
4602 |
4586 |
LOT] |
IMI |
o 5 8 |
10 13 |
63 |
•I.-.IM |
4524 |
IBM |
1471 |
1 1 ir, |
4431 |
1415 |
UM |
358 |
10 13 |
||
64 |
•4384 |
I:;IM 1808 |
4289 |
I'.' 71 |
LSM |
4MJ |
:; :• •» |
11 13 |
||||
65 |
•4226 |
ino |
ir.i.i H7'.» |
4163 |
1117 |
1 1 :; i |
4115 |
IOM |
IOM |
358 |
11 13 |
|
66 '4067 |
4051 |
IM:;:, |
1011 |
4003 |
M07 |
M71 |
3955 |
IMI |
MM |
358 |
11 14 |
|
67 || '3907 |
3891 |
M7I |
MM |
3843 |
18*7 |
3SM |
I7M |
•Ml- |
1701 |
358 |
11 14 |
|
68 |i -3746 |
3730 |
::7i I |
M07 |
M61 |
MM |
M4I |
3633 |
Ml« |
MOO |
:; ;. s |
11 14 |
|
69 "3584 |
3567 |
Iff] |
MM |
3518 |
MOI |
MM |
MM |
MM |
M07 |
358 |
11 14 |
|
70 i- |
MM |
1069 |
M71 |
MM |
MSI |
MOB |
3289 |
3272 |
368 |
11 14 |
||
71 |
•DM |
3239 |
MSI |
MM |
noo |
3156 |
3140 |
3107 |
:; •; s |
11 14 |
||
7:2 |
•3090 |
3074 |
1007 |
1040 |
MM |
3007 |
MOO |
•-".'71 |
•J'.'lH |
:: .; s |
11 14 |
|
73 |
•MM |
MOT |
ION |
2857 |
2807 |
2773 |
:: .; ^ |
11 14 |
||||
74 |
•_•:.-.••, |
2740 |
ITU |
•.'7".; |
2089 |
2656 |
2639 |
MOI |
2605 |
368 |
11 14 |
|
75 |
•flOOfl |
MM |
•-•:.:> |
Mil |
2487 |
2453 |
MM |
S 6 8 |
11 14 |
|||
78 |
•2419 |
2402 |
2S85 |
2368 |
2351 |
MOO |
MM |
M07 |
3 6 |
11 14 |
||
77 |
•SSM |
2233 |
•Jin- |
2147 |
noo |
mi |
MM |
:; •; |
11 14 |
|||
78 |
•1071 |
H61 |
_,,_.. |
MU |
1994 |
1077 |
LOM |
I'.'U |
1 :•:•-, |
:; .; |
11 14 |
|
79 |
•1SOC |
1857 |
1822 |
1805 |
L7M |
1771 |
1754 |
:; .; |
11 14 |
|||
80' |
•17M |
1685 |
LOM |
LOM |
1.;:;:; |
i',|., |
1 Ml |
I :.--.• |
| .; |
12 14 |
||
81 |
• i M i |
1530 |
1513 |
1495 |
1478 |
IP,] |
mi |
MM |
1409 |
8 6 |
IS 14 |
|
82 |
•1 :::.:• |
1340 |
LOM |
LOM |
IV- |
LOM |
i •.•:•..: |
8 6 |
||||
83 |
•1219 |
ISO] |
1149 |
M.-.J |
Ull |
Ins,, |
in,;;; |
8 6 |
18 14 |
|||
84 |
•1046 |
LOM |
lull |
..:.<.,;; |
1071 |
OOM |
• ••.Hi |
MM |
,,ss;. |
3 6 |
IS 14 |
|
85 |
•0671 |
iM |
oosi |
..sin |
Ml |
OfM |
0767 |
1700 |
Ofn |
0715 |
8 6 |
IS 14 |
86 |
•06M |
I06Q |
<,.!.-, |
)6S8 |
0010 |
• ..V.i.1 |
,:,;,-, |
MM |
-Ml |
.1 r, |
IS 15 |
|
87 |
•out |
I6M |
MM |
n||'.' |
MOI |
MM |
..;;,,,, |
.1 r, |
IS 1ft |
|||
88 |
•0349 |
ion |
0014 |
•'•.".'7 |
171 |
OOM |
.••.•II |
001 |
MOO |
.. 1 •..'.- |
| •: |
1 1 1 1 |
89 |
•0175 |
mi |
0140 |
•il'.-J |
001 |
0007 |
1111711 |
IOM |
OOM |
,,,,17 |
3 0 |
13 15 |
ISO
NATURAL TANGENTS
Menu DillViviii'i M. |
||||||||||||
0' |
6' |
11 |
18 |
24' 30' 36' |
42' 48' 54' |
r 2- 3' |
4' 5' |
|||||
0 |
o-ooo |
mi |
MM |
,.,,:,•_ |
070 |
0087 "in:, |
1188 "1 lo nir, 7 |
:; c, it |
1 •_• 1 :, |
|||
0227 |
••_• 1 1 |
1891 ".".11 ":;:;•_• |
c, n |
l-j i:, |
||||||||
•0349 |
• iin 041 |
. 17'J (Us1.. 11.107 |
:i t; it |
l •_• l .-, |
||||||||
•MM |
•41 |
•2 or.-jit |
n;.|7 <.i;i;i i.r.si' |
:; c, j» |
i •_• 1 :, |
|||||||
MM |
• 717 |
• s •_••_• 084 |
:; r, 9 |
!•_' 1.1 |
||||||||
•• 171 |
0910 |
,,,-JS |
0945 |
0981 |
098 L016 i":;:! |
:; r, 9 |
1 -J 1 .1 |
|||||
•1061 |
1104 |
11 •_'.' |
L189 |
1157 |
1 7--. 1 1 icj |
L210 |
:i c. '.. |
12 15 |
||||
•;•_••.> |
l •-".''.< i::i7 mi |
:;.vj r,7" i::ss |
:; (i 9 |
12 15 |
||||||||
•1405 |
11 H.V.I |
1177 mi:, i:,l-j |
:,.-," i.-.is i:,t;i; |
a r. 9 |
12 15 |
|||||||
•1584 |
.0 1638 |
L6M n;::-, ir.sn |
7o;i 17:.' 7 17i:> |
369 |
1 2 1 1 |
|||||||
icr |
•1763 |
1731 1 |
1817 |
J68 1871 |
isiio 1908 |
1926 |
3 r, it |
1 •_• 1 .1 |
||||
11 |
•]'.'!» |
Ht'.tS |
jolt; |
2071 |
8089 |
2107 |
369 |
1 1' 1 -> |
||||
12 |
•2126 |
Jill 2 K.-1 |
2180 |
J 1 1'H |
•_'•_' 17 •_'-•:',:. |
2254 |
2272 |
2290 |
3 K li |
1 L' 1 r, |
||
13 |
•not |
2364 |
1 'Jill* |
2438 |
2456 |
2475 |
3 6 9 |
U' i:, |
||||
14 |
•MM |
2549 |
M M05 |
8688 |
•_M;|-J |
2661 |
3 6 9 |
12 16 |
||||
15 |
•1671 |
2736 |
27.11 -J77:1, 27112 |
2811 |
8880 |
2849 |
369 |
13 16 |
||||
16 |
•M61 |
I •_••••_• | |
-."..1:1 |
2962 |
2981 |
3000 |
3019 |
3038 |
:; c, g |
13 16 |
||
17 |
•3057 |
:;u96 |
3115 |
3134 |
3153 3172 |
3191 |
3211 |
3230 |
:; r, 10 |
13 16 |
||
18 |
•3249 |
3269 |
3288 |
3307 |
3327 |
3346 3365 |
8888 |
3404 |
3424 |
3 6 10 |
1 :; 1 r, |
|
19 |
•3443 |
3463 |
3482 |
3522 |
:;:.n :;:,.;! |
3581 |
3600 |
3620 |
3 7 10 |
13 16 |
||
20° |
•3640 |
3659 |
3679 |
3699 |
3719 |
3739 |
3759 |
3779 |
3799 |
3819 |
3 7 10 |
13 17 |
21 |
•MM |
3859 |
3879 |
3899 |
3919 |
:;:.:;;. |
3959 |
3979 |
4000 |
4020 |
3 7 10 |
i :; 17 |
L':.' |
•4040 |
4081 |
4101 |
4122 |
11-12 in;:; |
4183 |
4204 |
4224 |
:! 7 10 |
1 1 17 |
||
23 |
•4245 |
4286 |
4307 |
4327 |
1848 1869 |
4390 |
4411 |
4431 |
3 7 10 |
1 1 17 |
||
24 |
•4452 |
4494 |
\:>\:> |
4536 |
1.1.17 |
4578 |
Ifilili |
4621 |
4642 |
1 7 11 |
14 18 |
|
25 |
•4663 |
4684 |
4706 |
4727 |
1748 |
177«» |
4791 |
4813 |
4834 |
4856 |
4 7 11 |
14 18 |
:.'.; |
I '4877 |
4899 |
r.'-ji |
11U2 |
4964 |
4986 |
5008 |
5029 |
5051 |
5073 |
4 7 11 |
15 18 |
27 |
•MM |
r-117 |
5139 |
r.ir.i |
5184 |
5206 |
5228 |
5250 |
5272 |
5295 |
4711 |
1.1 is |
28 |
•5317 |
:,:;i" |
5362 |
:.:;s i |
:.t"7 .vi:;n |
5452 |
5475 |
5498 |
5520 |
4 8 11 |
1 .1 1 H |
|
2!« |
•5543 |
MM |
U89 |
B6U |
5635 |
5658 |
5681 |
5704 |
5727 |
5750 |
4 8 lli |
1 :, 1 it |
30 |
5797 |
5820 |
.isr,7 |
5914 |
5938 |
5961 |
.M.sr, |
4 8 12 |
16 20 |
|||
31 |
•«00| |
BOM |
ION |
aow |
6104 |
6128 |
6159 |
6176 |
6200 |
6224 |
4 S 1 2 |
16 20 |
32 |
•'. „•!•.' |
6273 |
.',•.".'7 |
6420 |
6445 |
r.ic.ii |
4 8 12 |
it; 211 |
||||
:>,:>, |
.-..Ml. |
6544 |
.;:,.;;. |
6594 |
Of! Ill |
6644 |
6669 |
(it;:, i |
6720 |
4 8 13 |
17 21 |
|
34 |
6771 |
6794 |
i!M7 |
6878 |
6899 |
6924 |
6950 |
6976 |
4 li 1 :; |
17 _ 1 |
||
35 |
7028 |
7".1I |
7080 |
7107 |
7188 |
7159 |
7186 |
7212 |
7L':w |
4 9 13 |
IS -2-1 |
|
:;*, |
7292 |
7373 |
7KKI |
7127 |
7454 |
7481 |
7r,o.x |
r. 9 u |
IK W |
|||
37 |
>7§M |
7.-,.,:; |
7690 |
7618 |
7646 |
7r, 7:1 |
7701 |
7789 |
7787 |
7785 |
r, i. i 1 |
is -j:; |
:;s |
•7813 |
7841 |
7869 |
7898 |
7926 |
7'.»:,» 7ns:; |
8012 |
SOIO |
8069 |
5 It 1 1 |
19 24 |
|
:;•. |
8127 |
SIM |
8186 |
8214 |
8243 |
8878 |
8302 |
8332 8361 |
5 10 15 |
•jo -j| |
||
40 |
•8391 |
8421 |
8461 |
148] |
8511 |
8541 |
S571 |
8601 |
8632 |
8662 |
5 10 15 |
•jo 2.1 |
41 |
•MM |
8786 |
-i.; |
8847 |
8878 |
S'.MO s'.MI |
8973 |
5 10 16 |
21 26 |
|||
42 -9004 |
BOM |
11 1 :; 1 |
0168 919! |
HL'L'S li'jc.ii '..•_".»: |
5 11 16 |
•Jl 27 |
||||||
43 -MSI |
m:, 7 in:". '.»:,•_•: |
9556 9590 9623 |
6 11 17 |
22 2S |
||||||||
44 '9657 |
HS-.M; HUSO 9965 |
6 11 17 |
j:; -".. |
\ ATI' HAL TANGENTS
181
0 |
6' 12' 18' |
24 |
30 36 |
42' 48' 54' |
Mean Differences. |
|||||||
1' 2* 8- 4' 5' |
||||||||||||
45 I'OOOO |
0035 |
0070 |
0105 |
0141 |
0176 |
0212 |
0247 |
6 12 18 24 30 |
||||
46 1*0355 |
0392 |
0428 |
0104 |
1611 |
0686 |
6 12 18 25 31 |
||||||
47 l-"72i |
0761 |
0799 |
iiss; |
>*"> .IVU3 '0951 |
0990 1028 |
1067 |
6 13 19 |
25 32 |
||||
48 1-1 IM.; |
1145 |
1184 |
12 24 |
. |
1383 142:; 1 i<;:; |
7 13 I'" |
27 33 |
|||||
49 |
1544 |
1585 |
1626 |
IM1 |
1750 |
17'.«2 1833 |
1875 |
7 It 21 |
28 34 |
|||
IH |
1-1918 |
1960 |
2002 |
2045 |
2088 |
2131 |
2174 |
2305 |
7 11 |
29 36 |
||
51 |
1-2349 |
2393 |
2437 |
2482 |
2527 |
2572 |
2617 |
_'.;•;-.' 27'is |
2753 |
8 15 23 |
30 38 |
|
52 |
2846 |
2892 |
2938 |
MM |
3032 |
3079 |
3127 3175 |
MM |
8 16 2 i |
31 39 |
||
53 |
;:;i'.' |
3367 |
3416 |
3465 |
3514 |
3564 |
3613 |
3663 |
3713 |
8 10 •_>:, |
33 41 |
|
54 |
1-3764 |
3814 |
3865 |
3916 |
3968 |
4019 |
4071 |
1124 |
4176 |
ISM |
9 17 26 |
34 43 |
55 |
1*418] |
4335 |
4388 |
1112 |
4496 |
4550 |
4605 |
MM |
4715 |
4770 |
9 18 27 |
:;.; «:, |
56 |
1-4826 |
4882 |
4938 |
4994 |
5051 |
5108 |
5166 |
5224 |
5282 |
5340 |
10 19 2 ii |
38 48 |
57 |
1-5399 |
5458 |
5517 |
5577 |
MSI |
5697 |
5757 |
5818 |
5880 |
5941 |
10 20 30 |
40 50 |
58 |
1-6003 |
;• .»;.; |
6128 |
6191 |
6255 |
6319 |
6383 |
6447 |
6512 |
6577 |
11 21 32 |
43 53 |
59 |
6709 |
6775 |
6842 |
6909 |
6977 |
7045 |
7113 |
7182 |
7251 |
11 23 34 |
i.-) :..; |
|
60 |
l-ToiIl |
7391 |
7461 |
7532 |
7603 |
7675 |
7747 |
7820 |
7893 |
7'.";.; |
12 24 36 |
48 60 |
61 |
1-8040 |
8115 |
8190 |
s •.'»;:, |
8341 |
8418 |
8495 |
8572 |
8650 |
8728 |
13 26 38 |
51 64 |
62 |
1-8807 |
8887 |
SM7 |
9047 |
9128 |
9210 |
'.'•_••.'•: |
8375 |
9458 |
M4I |
11 27 41 |
;,:, ••.* |
63 |
i •'.»;•_•«; |
9711 |
•797 |
•.•--:; |
9970 |
0057 |
ill 45 |
0233 |
0323 |
0413 |
15 29 44 |
68 73 |
64 |
0594 |
MM |
0778 |
0872 |
0965 |
1000 |
1155 |
1251 |
1348 |
10 31 17 |
•-.:; 7< |
|
65 |
1543 |
1642 |
L74J |
1842 |
1943 |
2045 |
2148 |
2251 |
I |
17 34 51 |
68 85 |
|
66 |
M4M |
HM |
M7I |
2781 |
2889 |
2998 |
:uo-.i |
ISM |
MM |
1448 |
18 37 55 |
73 92 |
67 |
- MM |
;••,::; |
1780 |
:;'..»,; |
ion |
4142 |
t 2H2 |
4383 |
4504 |
I.-.-J7 |
10 60 |
7'.' i.y |
• |
2 I7.M |
4876 |
5002 |
list |
mi |
5386 |
5517 |
5649 |
5782 |
.VI.; |
, 65 |
87 108 |
60 |
6187 |
HM |
f.lf.l |
MM |
6746 |
»;x-j |
7034 |
7179 |
I |
21 17 71 |
'.'.'. 11'.' |
|
70 |
7625 |
7776 |
7 '.'•.•;. |
8083 |
UM |
8397 |
8556 |
8716 |
8878 |
^ |
104 131 |
|
71 |
rt04i |
9208 |
9376 |
9544 |
ITU |
9887 |
0061 |
0415 |
0596 |
29 58 87 |
11.', I);, |
|
72 |
OM1 |
1 1 If. |
1334 |
1524 |
i7i.; |
1910 |
uoa |
12 H I'.l |
||||
73 |
2914 |
MM |
3759 |
:r..77 |
11 '.'7 |
4420 |
4646 |
36 72 108 |
111 1*0 |
|||
74 |
3-4874 |
5106 |
5339 |
6576 |
5S16 |
6059 |
.;:;..:, |
6554 |
MM |
IOM |
11 -1 122 |
i«'.:; 104 |
75 |
:;•;:; 21 |
7583 |
7848 |
nu |
8391 |
8667 |
-'..17 |
9232 |
MM |
Mil |
46 93 139 |
l»r. •_•:;•: |
78 77 |
4-0108 43316 |
0408 8662 |
0718 4015 |
LOS1 I.-.7I |
1335 4787 |
LMI 5107 |
L976 5483 |
1801 6864 |
MM 8MS |
M7I MM |
||
78 |
4-7046 |
8716 |
9152 |
•...'.•..I |
0041 |
0604 |
||||||
79 |
5-l44« |
:;»:..-. |
MM |
MM |
:,'.•„•.; |
|||||||
80 |
6-6713 |
7894 |
8602 |
9124 |
MM |
ftM |
1066 |
17»'.' 2I3-. |
||||
81 |
6-3128 |
>.v.« |
4506 |
6860 |
6122 |
Mil |
7720 |
M41 |
MM |
J/«rt differences no |
||
82 7-1164 83 8-1443 |
:">, ..•,;;,, |
8002 8962 3863 6126 |
4 -.it 7 «,i;7 |
MM |
MM 9161 |
HM-,2 (.:,;•.. |
IU:,H i06l |
fen |
longer fufflcUutlj accurate. |
|||
84 |
•••.,77 |
LOW |
LO-M |
IM ;, |
10-99 |
|||||||
85 |
11-43 |
12-71 |
i :;••••• |
1 :;•:;.. |
1 .V..;' |
18-96 |
||||||
86 |
14-80 |
14-67 |
15-06 16-46 |
I.V-. |
16-Sfi |
\ ,.•*:: |
17 :.i |
17-89 |
18-46 |
|||
87 |
I-..-7I |
.'.••'." |
tr« |
,, ,„, |
:.;-•. |
27-27 |
||||||
88 |
28-64 |
... 11 |
31-82 33-69 |
86-80 |
47-74 |
MDI |
||||||
89 |
67-29 |
.,:,-..,. |
71-62 81-86 |
•J :.•!•.' 1 |
r..i .' |
182
IJM, \KITHMIC SINES
0 |
6' |
12' |
18* |
24' |
150 |
36' |
42' |
48 |
54' |
Mean DilTriviH-t-.-. |
||
r 2- 3- |
4' 5' |
|||||||||||
0 |
- 00 |
7190 |
MS9 |
9408 |
5so |
0870 |
1450 |
LM |
||||
- •: 1 1 '.- |
3880 |
4179 |
1409 |
4723 |
4971 |
5206 |
||||||
- MM |
5640 |
5842 |
6035 |
6567 |
6731 |
8889 |
7041 |
|||||
8-7188 |
7330 |
7468 |
r< - |
7731 |
7857 |
7979 |
8098 |
8213 |
8326 |
|||
- MM |
8543 |
8647 |
-71'.' |
SSl'l |
8946 |
9041 |
9188 |
•.»:; i : |
16 32 48 |
64 80 |
||
8-9103 |
9489 |
9573 |
9655 |
9736 |
9816 |
9894 |
9970 |
0046 |
01* |
13 26 39 |
52 65 |
|
9-0192 |
0264 |
0334 |
0403 |
0472 |
0539 |
0605 |
0670 |
0734 |
0797 |
11 22 33 |
1 1 :,:, |
|
9-0859 |
0920 |
0981 |
1040 |
L099 |
1157 |
lL'1 1 |
1171 |
1326 |
1381 |
10 19 29 |
38 48 |
|
9-1436 |
1489 |
1542 |
1594 |
1646 |
1697 |
1717 |
1797 |
1847 |
1895 |
8 17 25 |
34 42 |
|
9-1943 |
1991 |
2038 |
20S5 |
2131 |
2176 |
211] |
2266 |
2310 |
2353 |
8 15 23 |
30 38 |
|
10' |
9-2397 |
2482 |
..VJI |
2563 |
2606 |
1647 |
2687 |
2727 |
2767 |
7 14 20 |
IT :;t |
|
11 |
9-2806 |
2845 |
2883 |
•-".'•Jl |
2959 |
2997 |
3034 |
3070 |
3107 |
3143 |
6 12 19 |
25 31 |
12 |
9-3179 |
3214 |
3250 |
1384 |
3319 |
88M |
3387 |
3421 |
3455 |
3488 |
6 11 17 |
23 28 |
13 |
9-3521 |
3554 |
3586 |
3618 |
3650 |
3682 |
3713 |
3745 |
3775 |
3806 |
5 11 16 |
21 i'i; |
14 |
9-3837 |
3867 |
3897 |
3927 |
3957 |
3986 |
4015 |
4044 |
4073 |
4102 |
5 10 15 |
20 24 |
15 |
9-4130 |
4158 |
4186 |
I-J11 |
[243 |
4269 |
4296 |
4323 |
4350 |
4377 |
5 9 14 |
18 23 |
16 |
9-4403 |
4430 |
4456 |
L481 |
4508 |
4533 |
4559 |
4584 |
4609 |
4634 |
4 9 13 |
17 21 |
17 |
9-4659 |
4684 |
4709 |
4733 |
4757 |
4781 |
4805 |
4829 |
4853 |
4876 |
4 8 12 |
16 20 |
18 |
•>4800 |
4923 |
I'.Mii |
4969 |
4992 |
5015 |
5037 |
5060 |
5082 |
5104 |
4 8 11 |
15 19 |
19 |
5148 |
5170 |
5192 |
5213 |
5235 |
5256 |
5278 |
5299 |
5320 |
4 7 11 |
14 18 |
|
20' |
9-5341 |
5361 |
5382 |
5402 |
3i •_>.-! |
5443 |
5463 |
5484 |
5504 |
5523 |
3 7 10 |
14 17 |
21 |
9-5543 |
5563 |
5583 |
5602 |
5621 |
5641 |
5660 |
5679 |
5698 |
5717 |
3 6 10 |
13 16 |
22 |
9-5736 |
:.7--.i |
5773 |
5792 |
5810 |
5828 |
5847 |
5865 |
5883 |
5901 |
369 |
12 15 |
23 |
PHU |
5937 |
5954 |
5972 |
5990 |
6007 |
6024 |
6042 |
6059 |
6076 |
369 |
1 2 1 r. |
24 |
9-60M |
6110 |
6127 |
6144 |
61G1 |
6177 |
6194 |
6210 |
6227 |
6243 |
368 |
11 14 |
25 |
•...;•_'.-,'.. |
6276 |
8191 |
6308 |
6324 |
6340 |
6356 |
6371 |
6387 |
6403 |
358 |
11 13 |
26 |
9-6418 |
6434 |
6449 |
8465 |
6480 |
6495 |
fi3 10 |
6526 |
6541 |
6556 |
358 |
10 13 |
27 |
9-6S70 |
6585 |
6600 |
6615 |
6629 |
6644 |
6639 |
6673 |
6687 |
6702 |
257 |
10 12 |
28 |
•.'••; : i«; |
6730 |
6744 |
6759 |
6773 |
6787 |
0801 |
6814 |
6828 |
6842 |
257 |
11 1 2 |
29 |
'.••,>.-,,; |
6869 |
6883 |
6896 |
6910 |
•;:»•_':; |
6937 |
6950 |
6963 |
0977 |
247 |
9 11 |
80° |
»-6t90 |
7003 |
7016 |
7029 |
7042 |
7055 |
7008 |
7080 |
7093 |
7100 |
246 |
9 11 |
81 |
9-7118 |
7i:;i |
7144 |
7156 |
7168 |
7181 |
7193 |
7205 |
7218 |
7230 |
246 |
8 10 |
32 |
9-7242 |
7254 |
7266 |
7278 |
7190 |
7302 |
7314 |
7326 |
7338 |
7349 |
2 4 c, |
8 10 |
38 |
'.- ::,'•. i |
7373 |
7384 |
7396 |
7407 |
7419 |
7430 |
7442 |
7453 |
7464 |
246 |
8 10 |
84 |
9-7471 |
7487 |
7498 |
7509 |
7 3 -jo |
7531 |
7542 |
7553 |
7564 |
7575 |
:• -1 (i |
7 9 |
85 |
•j •;.>.; |
7597 |
7607 |
7618 |
7629 |
7640 |
7050 |
7661 |
7671 |
7082 |
245 |
7 9 |
86 |
9*ntl |
7703 |
7713 |
7723 |
7734 |
7744 |
7731 |
7764 |
7774 |
7785 |
235 |
7 9 |
87 |
9-7795 |
7805 |
7815 |
7825 |
7835 |
7844 |
7854 |
7864 |
7874 |
7884 |
235 |
7 8 |
38 |
9-7893 |
7903 |
7913 |
7922 |
7932 |
7941 |
7951 |
7960 |
7970 |
7979 |
235 |
0 8 |
88 |
••IBM |
7998 |
8007 |
8017 |
BOM |
8035 |
8044 |
053 |
8068 |
8072 |
235 |
6 8 |
40° |
9-8081 |
8090 |
8099 |
8108 |
8117 |
8125 |
8134 |
143 |
8152 |
8101 |
134 |
6 7 |
41 |
:• 8169 |
8178 |
8187 |
8195 |
8204 |
8213 |
8221 |
230 |
8238 |
8247 |
134 |
0 7 |
42 |
9-8255 |
8264 |
8272 |
s-.-so |
8389 |
8397 |
8305 |
313 |
8322 |
8330 |
134 |
6 7 |
48 |
9-8338 |
8346 |
8354 |
8362 |
8370 |
8378 |
8386 |
394 |
8402 |
8410 |
134 |
5 7 |
44 |
9-8418 |
8426 |
8433 |
sill |
8449 |
8457 |
8464 |
472 |
8480 |
8487 |
134 |
5 G |
LOGARITHMIC SINES
183
0 |
6' |
12- |
IP |
24' |
30 |
36 |
42' |
43 |
54 |
Mean Differences. |
||
r 2 s |
4 5 |
|||||||||||
45 |
9-8495 |
8502 |
8510 |
8517 |
8525 |
8532 |
8540 |
8547 |
-.-,:,:, |
8562 |
124 |
.; |
46 |
! 9-8569 |
-.-.77 |
MM |
8591 |
8598 |
8606 |
8613 |
8620 |
MSI |
8634 |
1 2 4 |
| .. |
47 |
9-8641 |
8648 |
MM |
MtJ |
8669 |
MTfl |
8683 |
^,;:,,, |
-•.'..7 |
8704 |
1 2 3 |
5 6 |
48 |
9-8711 |
8718 |
8724 |
8731 |
s;.-{s |
8741 |
8751 |
8758 |
8765 |
8771 |
1 2 3 |
4 6 |
49 |
| 9-8778 |
-7-1 |
8791 |
8797 |
ss,,| |
8810 |
--17 |
-• |
8830 |
8836 |
1 1 |
4 5 |
50' |
9-884J |
--!'.' |
8855 |
8862 |
8868 |
8874 |
8880 |
8887 |
8893 |
8899 |
1 2 3 |
4 5 |
51 |
9-8905 |
8911 |
8917 |
8929 |
8935 |
8941 |
8947 |
8953 |
MM |
1 2 3 |
4 5 |
|
52 |
9-8965 |
8971 |
8977 |
M8I |
s-iix-i |
8995 |
: |
'. ; |
9012 |
9018 |
123 |
4 5 |
53 |
9-9023 |
9029 |
9035 |
9041 |
9046 |
9057 |
9063 |
9069 |
M74 |
1 2 3 |
1 |
|
54 |
|>908Q |
Mfcl |
9091 |
9101 |
'.HI-.' |
9118 |
lltt |
1 2 3 |
4 5 |
|||
55 |
9-9134 |
1119 |
9144 |
9149 |
9155 |
9160 |
9165 |
9170 |
9175 |
9181 |
1 2 3 |
3 4 |
56 |
•.'••..I-.'. |
9191 |
9196 |
Hl'nl |
9206 |
9211 |
9216 |
9221 |
1 2 3 |
3 4 |
||
57 |
M4] |
M44 |
9251 |
MM |
M6Q |
9265 |
9270 |
1 2 2 |
I I |
|||
58 |
M8f |
•..-.••.•I |
9298 |
MOI |
MM |
9317 |
1 I |
:; i |
||||
59 |
:•••..:;:;! |
MM |
M40 |
9344 |
M49 |
•..:;:,:; |
'..:;:,- |
M<I |
..:;.; 7 |
9371 |
1 1 •-' |
3 4 |
60 |
9-9375 |
9380 |
9384 |
•.'.-;-- |
9393 |
•.•:;:• 7 |
9401 |
9406 |
9410 |
'.>lll |
1 1 2 |
:; i |
61 |
9-9418 |
• I-.".' |
Mil |
1 |
Mtl |
MM |
'.MI:; |
9447 |
9451 |
MM |
1 1 2 |
I :; |
62 |
'.'•'.'I.V.. |
M«| |
MM |
'-•171 |
•17.-, |
.'17'.' |
•.-I-:; |
'.'1-7 |
• I'.M |
MM |
112 |
3 3 |
63 |
,,.,,,,,,, |
9503 |
9507 |
...'.In |
9514 |
9518 |
9522 |
9525 |
1619 |
9533 |
1 1 I' |
I :; |
64 |
•.••:•:.::: |
9540 |
9544 |
.•.'.is |
9551 |
MM |
1..V.S |
9562 |
1161 |
•.'.-..:•.. |
1 1 2 |
•j :; |
65 |
'.••'.'.-,::; |
9576 |
9580 |
..:>-:; |
9587 |
9590 |
9594 |
9597 |
MU |
9604 |
112 |
2 3 |
66 , • - |
9-9607 |
9611 |
9614 |
.".17 |
9621 |
9624 |
9627 |
9631 |
9634 |
...',:; 7 |
1 1 2 |
2 3 |
o7 68 |
'.'• '."'. I1 ' -..- '...;:•_• |
"'. i .. 9675 |
."'> 1 7 M71 |
.»':',' ' .'•;-i |
MM 9684 |
.it',.",r, 9687 |
."'..".'.' H-.-...I |
H'.t;-.' 9693 |
u'.i'.i; 9696 |
."'•.'.. ...;;.;. |
Oil |
•J . |
69 |
> m |
9704 |
9707 |
.•71" |
9713 |
9716 |
9719 |
I7SI |
9724 |
n |
0 1 1 |
-.' i' |
70° |
9-0730 |
9733 |
9735 |
.'7:;- |
9741 |
9743 |
.'71'. |
'71'.' |
9751 |
9754 |
0 1 1 |
- i' |
71 |
•.."..7:,; |
• 7.-.'.' |
9762 |
-7--.I |
9767 |
9770 |
•77-J |
9775 |
9777 |
,7-.. |
0 1 1 |
•J I' |
72 |
'.••'.• : •_• |
9785 |
9787 |
• 7s-.i |
9792 |
9794 |
':•'! |
9799 |
9801 |
,SM, |
0 1 1 |
•J •_• |
78 |
,„„ |
9811 |
•si:: |
9815 |
9817 |
,,-..,, |
Ml |
MM |
is-.v. |
0 1 1 |
•_• - |
|
74 |
MM |
9831 |
9833 |
-:;:, |
9837 |
MM |
.-11 |
9843 |
M4I |
• si; |
0 1 1 |
1 9 |
75 |
'.'•'.'-r.. |
9851 |
• -:,:; |
,.:,:, |
9857 |
9869 |
.-.,i |
9863 |
9865 |
.-..; |
0 1 1 |
1 2 |
76 |
9871 |
9873 |
.-7.'. |
9876 |
9878 |
<-- |
9882 |
9884 |
• HS.-, |
0 1 1 |
1 2 |
|
77 |
9-9887 |
9889 |
9891 |
• -•.'.• |
9894 |
MM |
.-..7 |
9899 |
9901 |
.-.»..•: |
0 I 1 |
1 1 |
78 |
:. MM |
.>..; |
9907 |
,;,.,.., |
9910 |
9918 |
.:M:: |
9915 |
9916 |
••.MS |
0 1 1 |
1 1 |
79 |
9-9919 |
Mtl |
,,_.., |
-. 1 |
Mi |
'•.-•: 7 |
.:.;- |
Ml |
9931 |
.:...: |
0 0 1 |
1 1 |
801 |
-.. MM |
9935 |
9936 |
.,... |
9939 |
9940 |
j.'ll |
9943 |
9944 |
••.•i:. |
001 |
1 l |
81 |
:• MM |
9947 |
9949 |
.•.,.-.., |
9951 |
9952 |
... ,; |
'.'M |
9966 |
..... |
M Q 1 |
1 1 |
82 |
Ml |
MM |
• •.'•: 1 |
.'M,; |
.•...,; |
, „ , |
Ml |
MM |
..... |
0 0 1 |
1 1 |
|
83 |
MM1 |
...„,, |
MM |
,,,-,, |
9971 |
M7I |
9974 |
9976 |
...;:, |
000 |
1 1 |
|
84 |
M71 |
9978 |
• ••-," |
9979 |
MQ |
...-] |
9981 |
MM |
000 |
•. 1 |
||
85 |
..-i |
9986 |
r..-:, |
9986 |
9987 |
r..-r |
,,,.- |
9988 |
..,-, |
000 |
0 0 |
|
86 |
, , , |
,..,.., 1 |
mi |
9993 |
..•'•.. 1 |
|
,, ,, |
|||||
87 |
I MM |
... .1 |
MM |
.••••... |
...... |
....... |
........ |
....... |
9997 |
.,.,.,; |
000 |
0 0 |
88 |
9-9997 |
,..„ |
MM |
,.,.„ |
•.,.,.,. |
MM |
,,,, |
;...... |
MM |
.,., |
000 |
0 0 |
89 |
Mm |
....... |
too |
|
oooo |
UOOO |
(j( |
MOO |
000 |
0 t |
||
LOGARITHMIC COSINES
•r,ict M«TUi DilTorn
0 |
6 |
12- |
18 |
24' |
30 |
36' |
42' |
48 |
54 |
Mean DitTi'i. |
||
r 2- v |
4' 5' |
|||||||||||
0 |
10-0000 |
0000 |
0000 |
0000 |
oooo |
0000 |
• |
oooo |
OOOO |
9.9999 |
000 |
0 0 |
1 |
'.'•'.".''.'•.' |
9999 |
9999 |
ww |
MM |
9999 |
9998 |
9998 |
9998 |
9998 |
0 0 0 |
0 0 |
2 |
9»Mti |
9997 |
9997 |
mmr. |
9996 |
9996 |
9996 |
9995 |
9995 |
:''.<'.' 1 |
0 0 0 |
II .1 |
3 |
,,.,,,,,,, |
M9 i |
•.'•.".'.-; |
vw:\ |
I''.".'-.' |
MM |
9991 |
9991 |
9990 |
9990 |
000 |
II (1 |
4 |
9-9989 |
9999 |
9988 |
.,«i>S |
9987 |
9987 |
M86 |
9985 |
9985 |
9984 |
000 |
0 0 |
5 |
Ftttl |
9981 |
9982 |
9981 |
9981 |
9980 |
9979 |
9978 |
9978 |
9977 |
000 |
0 1 |
6 |
'.'".'•.' 7'-. |
9975 |
9975 |
9974 |
9973 |
9972 |
9971 |
9970 |
9969 |
9968 |
000 |
1 |
7 |
.,;,.„;> |
M61 |
9966 |
9965 |
9964 |
9963 |
9962 |
9961 |
9960 |
9959 |
0 0 |
|
8 |
9-9958 |
MM |
MM |
9954 |
9953 |
9952 |
9951 |
9950 |
9949 |
9947 |
0 0 |
|
9 |
9-9946 |
9945 |
9943 |
9941 |
9940 |
9939 |
9937 |
9936 |
9935 |
0 0 |
||
10 |
9-9934 |
9932 |
9931 |
9929 |
9928 |
9927 |
9925 |
9924 |
9922 |
9921 |
0 0 |
|
11 |
9-9919 |
MU |
9916 |
9915 |
9913 |
9912 |
9910 |
9909 |
9907 |
9906 |
0 1 |
|
12 |
9-9904 |
9902 |
9901 |
9899 |
9897 |
9896 |
9894 |
9892 |
9891 |
9889 |
0 1 |
|
13 |
9-9887 |
9885 |
9884 |
9882 |
9880 |
9878 |
9876 |
9875 |
9873 |
9871 |
0 1 |
2 |
14 |
9-9869 |
9867 |
9865 |
9863 |
9861 |
9859 |
9857 |
9855 |
9853 |
9851 |
0 1 |
2 |
15 |
9-9849 |
9847 |
9845 |
9843 |
9841 |
9839 |
9837 |
9835 |
9833 |
9831 |
0 1 |
2 |
16 |
9-9828 |
9826 |
9824 |
9822 |
9820 |
9817 |
9815 |
9813 |
9811 |
9808 |
0 1 |
, ,, |
17 |
9-9806 |
9804 |
9801 |
9799 |
9797 |
9794 |
9792 |
9789 |
9787 |
9785 |
0 1 |
2 _' |
18 |
9-9782 |
9780 |
9777 |
9775 |
9772 |
9770 |
9767 |
9764 |
9762 |
9759 |
0 1 |
2 2 |
19 |
9-9757 |
9754 |
9751 |
9749 |
9746 |
9743 |
9741 |
9738 |
9735 |
9733 |
Oil |
2 2 |
20 |
9-9730 |
9727 |
9724 |
9722 |
9719 |
9716 |
9713 |
9710 |
9707 |
9704 |
Oil |
2 2 |
21 |
9-9702 |
'.'.;;.;. |
B6M |
9693 |
9690 |
9687 |
9684 |
9681 |
9678 |
9675 |
Oil |
2 2 |
22 |
9-9672 |
9669 |
966S |
9659 |
9656 |
9653 |
9650 |
9647 |
9643 |
i i • i |
2 3 |
|
23 |
9-9640 |
9637 |
9634 |
9631 |
9627 |
9624 |
9621 |
9617 |
9614 |
9611 |
1 1 J |
2 3 |
24 |
9-9609 |
9604 |
9601 |
9597 |
9594 |
9590 |
9587 |
9583 |
9580 |
9576 |
112 |
2 3 |
25 |
9*M7I |
9569 |
9566 |
9562 |
9558 |
9555 |
9551 |
9548 |
9544 |
9540 |
11-2 |
2 3 |
26 |
9-9537 |
9533 |
9529 |
9525 |
9522 |
9518 |
9514 |
9510 |
9507 |
9503 |
112 |
3 3 |
27 |
,,.,,,.,,, |
9495 |
9491 |
9487 |
9483 |
9479 |
9475 |
9471 |
9467 |
9463 |
112 |
3 3 |
28 |
9-9459 |
9455 |
9451 |
9447 |
9443 |
9439 |
'.U.V. |
9431 |
9427 |
9422 |
112 |
3 3 |
29 |
9-9418 |
'.Mil |
9410 |
9406 |
9401 |
9397 |
9393 |
9388 |
9384 |
9380 |
112 |
3 |
80 |
9-9375 |
9371 |
9367 |
9362 |
MM |
9353 |
9349 |
9344 |
9340 |
9335 |
112 |
3 |
31 |
9-9331 |
'.•:;•_'•; |
9322 |
9317 |
9312 |
9308 |
9303 |
9298 |
9294 |
9289 |
122 |
3 |
32 |
9-9*84 |
9279 |
9275 |
9270 |
M68 |
9260 |
9255 |
9251 |
9246 |
9241 |
122 |
3 |
33 |
9-9236 |
9231 |
9226 |
9221 |
9216 |
981] |
9206 |
930] |
9196 |
9191 |
123 |
8 |
34 |
9-9186 |
9181 |
9175 |
9170 |
9165 |
9160 |
9155 |
'.'! I'.t |
9144 |
9139 |
1 2 3 |
3 |
35 |
9-9134 |
9128 |
9123 |
9118 |
9112 |
9107 |
9101 |
9096 |
9091 |
9085 |
123 |
4 5 |
36 |
9-9080 |
'.•"71 |
9069 |
9063 |
M57 |
9052 |
9046 |
9041 |
9035 |
9029 |
123 |
4 5 |
87 |
9-9023 |
in • 1 - |
9012 |
9006 |
9000 |
8996 |
8989 |
8983 |
8977 |
8971 |
123 |
4 5 |
38 |
9-8965 |
*'..:,;. |
MM |
8947 |
8941 |
MM |
8929 |
8923 |
8917 |
8911 |
123 |
4 5 |
M |
9-8905 |
«-:.:. |
8893 |
sss7 |
8880 |
8874 |
HXI5S |
8862 |
8855 |
8849 |
123 |
4 5 |
40 |
9-8843 |
8836 |
8830 |
8823 |
8817 |
8810 |
ssoi |
8797 |
8791 |
8784 |
123 |
4 5 |
41 |
9-8778 |
8771 |
8765 |
8758 |
8751 |
8745 |
8738 |
8731 |
8724 |
8718 |
123 |
5 6 |
42 |
9-8711 |
8704 |
8697 |
8690 |
8688 |
8676 |
8669 |
8662 |
MM |
8648 |
123 |
5 6 |
43 |
9-8641 |
8634 |
8627 |
8620 |
8618 |
8606 |
8598 |
8591 |
8584 |
8577 |
124 |
5 6 |
44 |
9-8569 |
8562 |
8555 |
8547 |
8540 |
8532 |
8525 |
8517 |
8510 |
8502 |
124 |
5 6 |
LOGARITHMIC COSINES
Subtract Mean Differences.
185
0' |
6' |
\9> |
18' |
04 |
TO' |
Oft' |
4O' |
AQ |
r 1 |
Mean L)iff< |
:rences. |
|
MM |
AO |
£>* |
ov |
oo |
•Ji |
VJB |
ov |
V V V |
4' 5' |
|||
«D |
0407 |
<*. 1 VI I |
x 1 "• |
V 1 .' \ |
V 1 - - |
g j IQ |
Q 4 4 1 |
o f Q« |
0 «.)/* |
|||
46 |
'.»•* i '.'"> 9-8418 |
- * < 8410 |
s lx" 8402 |
is 1 i - 8394 |
> 11. 1 8386 |
x 1 «» i 8378 |
o44» 8370 |
0441 8362 |
MM 8354 |
x I -* 8346 |
134 |
5 7 |
47 |
9-8338 |
8330 |
8322 |
8313 |
8305 |
8297 |
3*81 |
8280 |
8'.' 72 |
8264 |
1 3 4 |
6 7 |
48 |
9-8255 |
8247 |
8238 |
>.•:;< |
- |
8213 |
8204 |
sm:, |
8187 |
8178 |
134 |
6 7 |
49 |
9-8169 |
8161 |
8152 |
U4I |
8134 |
8125 |
BUI |
8108 |
S099 |
8090 |
1 3 4 |
6 7 |
50 |
9-8081 |
8072 |
MM |
BOM |
- |
8035 |
8026 |
8017 |
8007 |
7998 |
i' 3 5 |
6 8 |
51 |
'.'•7 •.<--.' |
7979 |
7970 |
;'.M;M |
7951 |
7941 |
7932 |
7922 |
7913 |
7903 |
•2 3 5 |
.; s |
b'2 |
9-7893 |
7-M |
7874 |
7-r,| |
7s:.l |
7844 |
7835 |
7825 |
Tsi.-i |
7805 |
•2 :\ :, |
7 8 |
53 |
9-7795 |
1 - |
7774 |
77.; i |
77.M |
7744 |
7734 |
7723 |
7713 |
7703 |
:! 3 5 |
7 y |
54 |
9-7692 |
7682 |
7671 |
7661 |
7650 |
7640 |
7629 |
78U |
7607 |
7597 |
245 |
7 9 |
55 |
9-7586 |
7575 |
7564 |
7553 |
7542 |
7531 |
7520 |
TIM |
7498 |
7487 |
246 |
7 9 |
56 |
9-7476 |
7444 |
7453 |
7442 |
7419 |
7409 |
r.:\»: |
7384 |
7373 |
1 6 |
8 10 |
|
57 |
9-7361 |
7349 |
7338 |
7326 |
7314 |
7302 |
7'-".", |
7J78 |
7266 |
7254 |
246 |
8 10 |
58 |
7218 |
7.".-. |
7193 |
7181 |
7168 |
7156 |
7144 |
7131 |
t 6 |
8 10 |
||
59 |
9-7118 |
7106 |
7093 |
7080 |
7061 |
7055 |
704J |
7029 |
7016 |
7003 |
a 4 Q |
9 11 |
60 |
M9M |
6977 |
6963 |
MM |
6937 |
6923 |
6910 |
6896 |
6883 |
6869 |
247 |
9 11 |
61 |
M8M |
6842 |
6828 |
•814 |
6801 |
6787 |
6773 |
B7M |
•-,711 |
6730 |
257 |
9 12 |
62 |
9-6716 |
M81 |
6673 |
MM |
6644 |
6629 |
Mil |
6600 |
6585 |
257 |
10 12 |
|
63 |
9-6570 |
,;.-,.-„; |
6541 |
6526 |
6510 |
6495 |
6480 |
6465 |
6449 |
MM |
358 |
10 13 |
64 |
9-6418 |
MM |
6387 |
6371 |
MM |
6340 |
6324 |
6308 |
6292 |
358 |
11 13 |
|
65 |
MSM |
•Ml |
6210 |
•194 |
6177 |
6161 |
6144 |
6127 |
6110 |
368 |
11 14 |
|
66 |
9-6093 |
6076 |
6059 |
6042 |
6007 |
:,'.''.»> |
M7I |
MM |
5937 |
6 9 |
12 15 |
|
67 |
9-5919 |
5901 |
5883 |
.>,;:, |
5847 |
5828 |
5810 |
5792 |
5773 |
B7M |
6 9 |
12 15 |
68 |
:••.-,;:..; |
5717 |
5698 |
5679 |
5660 |
M 1 1 |
:.»•.•_• i |
5602 |
5583 |
5563 |
6 10 |
13 16 |
69 |
MMI |
5523 |
5504 |
:.IM |
5463 |
M4J |
5423 |
1401 |
5382 |
5361 |
7 10 |
14 17 |
70 |
•.-:.:. il |
1880 |
5299 |
:..-> |
5256 |
MM |
5213 |
5192 |
5170 |
5148 |
7 11 |
14 18 |
71 |
9-5126 |
5104 |
6082 |
:,..,;,, |
BM1 |
1011 |
I '.".••_• |
UM |
I'.'l', |
4923 |
8 11 |
15 19 |
72 |
Ml |
4876 |
I8M |
1 •.-.••• |
MM |
4781 |
4757 |
4733 |
L7W |
4684 |
S 12 |
!•; M |
73 |
MMI |
MM |
MM |
l.-.-l |
MM |
4533 |
4508 |
MM |
MM |
4430 |
9 13 |
17 1M |
74 |
M40I |
4377 |
1880 |
\:.-:: |
4296 |
4269 |
4214 |
4186 |
4158 |
5 9 14 |
i- M |
|
76 |
9-4130 |
4102 |
4073 |
4044 |
4015 |
3986 |
;•.•:,; |
8927 |
3897 |
;sr.7 |
5 10 15 |
20 24 |
78 |
9-3837 |
1771 |
:;7i:, |
8713 |
3682 |
n;:,. • |
8618 |
3586 |
C..1I |
5 11 16 |
21 26 |
|
77 |
9-3521 |
MM |
MM |
.TI-JI |
8387 |
3363 |
;:;!•.' |
8284 |
3260 |
3214 |
6 11 17 |
23 28 |
78 |
B*817I |
U4I |
3107 |
:;..;.. |
8034 |
2997 |
:•.'.-.'.. |
2921 |
MM |
j»t:. |
6 12 19 |
25 31 |
79 |
:,..•„„; |
2767 |
2727 |
:••>; |
2647 |
MM |
:;,.;;, |
BM « |
2482 |
.'I.V.I |
7 14 20 |
27 34 |
80 |
•.••:,. .7 |
2363 |
2310 |
-M; |
2221 |
2176 |
: i :; i |
2086 |
2038 |
i '.-•., i |
8 16 28 |
30 38 |
81 |
•••r.'i:; |
1895 |
1847 |
1797 |
1747 |
1697 |
!• 1' |
1694 |
1542 |
!!-, |
8 17 26 |
34 42 |
«:.' |
9-1436 |
1381 |
1826 |
1-J71 |
1214 |
1167 |
i •'-.«•.• |
1040 |
0981 |
'•.'I' • |
10 19 29 |
38 48 |
83 |
M8M |
0797 |
0784 |
«M |
UM |
0403 |
0384 |
'..,, |
11 22 88 |
44 65 |
||
84 |
H)1M |
0120 |
1041 |
B81Q |
i8M |
Mil |
9736 |
too |
Mil |
.,s., |
18 26 89 |
M M |
85 |
9-9403 |
9816 |
'-.'•; |
.-: . • |
1041 |
MM |
--I-.. |
8749 |
8647 |
v,i:, |
64 80 |
|
S«i |
B*84M |
8326 |
8213 |
7979 |
7867 |
. |
-',,; |
7468 |
;:;:,< |
|||
87 |
|«718I |
7041 |
.--'.. |
....i |
8881 |
6397 |
.-"-•" |
DM |
6842 |
,r,|.i |
||
88 |
B-MM |
5206 |
4971 |
ITM |
4459 |
4179 |
>KM |
IMI |
3210 |
•v.: |
||
89 |
8-2419 |
1961 |
1460 |
OS70 |
M8 |
9408 |
MM |
7190 |
6429 |
2419 |
ISti
LOGARITHMIC TANGENTS
Mr. HI DillVi. |
||||||||||||
0 |
6 |
12* |
18 |
24' |
30 |
38 |
42' |
48 |
54' |
|||
r 2- 3' |
4 5' |
|||||||||||
r |
— 00 |
5429 |
7190 |
8439 |
9409 |
0200 |
0370 |
1450 |
ins:; |
|||
i |
2833 |
3211 |
1881 |
4181 |
IK'.l |
LTSfi |
1071 |
5208 |
||||
8-5431 |
5643 |
5845 |
BOM |
6223 |
6401 |
6571 |
6786 |
6894 |
7046 |
|||
7475 |
7f."'.' |
7739 |
7865 |
7988 |
8107 |
8223 |
8336 |
|||||
8-8446 |
8554 |
MM |
8762 |
8862 |
8960 |
M)5€ |
9150 |
9241 |
9331 |
16 32 48 |
64 81 |
|
t-MJO |
9506 |
PHI |
9674 |
9756 |
9836 |
9915 |
9992 |
0068 |
0143 |
13 26 40 |
53 66 |
|
9-0216 |
0289 |
0360 |
0430 |
0499 |
0567 |
0633 |
OG99 |
0764 |
0828 |
11 22 34 |
||
9-0891 |
0954 |
1015 |
1076 |
1135 |
1194 |
1252 |
1310 |
1367 |
1423 |
10 20 29 |
."'.i I'.i |
|
9-1478 |
1533 |
1587 |
1640 |
1693 |
1745 |
1797 |
1848 |
L898 |
1948 |
9 17 26 |
35 43 |
|
9.1997 |
2094 |
•Jll-J |
2189 |
2236 |
2282 |
2328 |
2374 |
2419 |
8 16 23 |
31 39 |
||
HP |
9-2463 |
2507 |
2551 |
MM |
2637 |
2680 |
•-'7l'2 |
2761 |
2805 |
2846 |
7 14 21 |
28 35 |
11 |
9-2887 |
2927 |
2967 |
3006 |
3046 |
3085 |
3123 |
31G2 |
3200 |
3237 |
6 13 19 |
26 32 |
12 |
9-3275 |
3312 |
3349 |
3385 |
3422 |
M58 |
3493 |
3529 |
3564 |
3599 |
6 12 18 |
24 30 |
is |
9-3634 |
3668 |
3702 |
3736 |
3770 |
3804 |
3837 |
3870 |
3903 |
3935 |
G 11 17 |
22 28 |
14 |
9-3968 |
4000 |
4032 |
4064 |
4095 |
4127 |
4158 |
4189 |
4220 |
4250 |
5 10 16 |
21 26 |
15 |
9-4281 |
4311 |
4341 |
4371 |
4400 |
4430 |
4459 |
4488 |
4517 |
4546 |
5 10 15 |
20 25 |
16 |
9-4575 |
4603 |
4632 |
4660 |
4688 |
4716 |
4744 |
4771 |
4799 |
4826 |
5 9 14 |
19 23 |
17 |
9-4853 |
4880 |
4907 |
4934 |
4961 |
4987 |
5014 |
5040 |
5066 |
5092 |
4 9 13 |
18 22 |
18 |
9-5118 |
5143 |
5169 |
5195 |
5220 |
5245 |
5270 |
5295 |
5320 |
5345 |
4 8 13 |
17 21 |
19 |
9-5370 |
5394 |
5419 |
5443 |
5467 |
5491 |
5516 |
5539 |
5563 |
5587 |
4 8 12 |
16 20 |
20' |
9-5611 |
5634 |
5658 |
5681 |
5704 |
5727 |
5750 |
5773 |
5796 |
5819 |
4 8 12 |
15 19 |
21 |
9-5842 |
5864 |
5887 |
5909 |
5932 |
5954 |
5976 |
5998 |
6020 |
6042 |
4 7 11 |
15 19 |
22 |
1*064 |
6086 |
6108 |
6129 |
6151 |
6172 |
6194 |
6215 |
<;:.':>(; |
G257 |
4 7 li |
14 18 |
28 |
9-6279 |
6300 |
6321 |
6341 |
6362 |
6383 |
6404 |
6424 |
6445 |
6465 |
3 7 10 |
14 17 |
24 |
9-6486 |
6506 |
6527 |
6547 |
6567 |
6587 |
6607 |
6627 |
6647 |
6667 |
3 7 10 |
13 17 |
25 |
9-6687 |
6706 |
6726 |
6746 |
6765 |
8788 |
6804 |
6824 |
6843 |
6863 |
3 7 10 |
13 16 |
26 |
9-6882 |
6901 |
6920 |
6939 |
6958 |
6977 |
6996 |
7015 |
7034 |
7053 |
369 |
13 16 |
27 |
9-7072 |
7090 |
7109 |
7128 |
7146 |
7165 |
7183 |
7202 |
7220 |
7238 |
369 |
12 15 |
28 |
'.'•71' :.7 |
7275 |
7293 |
7311 |
7330 |
7348 |
7366 |
7384 |
7402 |
7420 |
369 |
12 15 |
29 |
9*7438 |
7455 |
7473 |
7491 |
7509 |
7526 |
7544 |
7562 |
7579 |
7597 |
369 |
12 15 |
80° |
9-7614 |
7632 |
7649 |
7667 |
7684 |
7701 |
7719 |
7736 |
7753 |
7771 |
3 G 9 |
12 14 |
81 |
9-7788 |
7805 |
7822 |
7839 |
7856 |
7873 |
7890 |
7907 |
7924 |
7941 |
369 |
11 14 |
82 |
'.'•7'.<:.* |
7976 |
7992 |
Si MIS |
8025 |
9041 |
8059 |
8075 |
8092 |
8109 |
368 |
11 14 |
88 |
9-8125 |
8142 |
8158 |
8175 |
8191 |
MM |
8224 |
8241 |
8257 |
8274 |
358 |
11 14 |
84 |
1*8990 |
8306 |
8323 |
8339 |
B855 |
8371 |
8388 |
8404 |
8420 |
8436 |
358 |
11 14 |
85 |
1*84*1 |
8468 |
8484 |
8501 |
-.-.17 |
8533 |
8549 |
8565 |
8581 |
8597 |
358 |
11 13 |
36 |
'.•••.: I. ", |
8629 |
-'•,11 |
8660 |
8676 |
8692 |
8708 |
8724 |
8740 |
8755 |
358 |
11 13 |
87 |
9-8771 |
8787 |
8803 |
8818 |
8834 |
8850 |
SSI5.1 |
8881 |
ss'.i? |
8912 |
358 |
10 13 |
38 |
9-8928 |
8944 |
-...v.i |
8975 |
MM |
9006 |
9022 |
9037 |
9053 |
9068 |
358 |
10 13 |
39 |
MOM |
9099 |
'..li:- |
9130 |
9146 |
9161 |
9176 |
9192 |
9207 |
9223 |
358 |
10 13 |
40 |
9-9238 |
9254 |
M6S |
MM |
MOO |
9315 |
9330 |
9346 |
9361 |
937G |
358 |
10 13 |
41 |
'.'•:.:;'.'. |
9407 |
9422 |
Mia |
9453 |
9468 |
9483 |
9499 |
9514 |
9529 |
358 |
10 13 |
42 |
'.Kir, 11 |
9560 |
'...-,7.-. |
9590 |
9605 |
9621 |
9G36 |
9651 |
9666 |
9681 |
:; r, 8 |
10 13 |
43 |
••Mtl |
9712 |
tni |
9742 |
9757 |
9773 |
9788 |
9803 |
9818 |
9833 |
358 |
10 13 |
44 |
9-9848 |
9864 |
9619 |
'.<-.. 1 |
•.Hi. in |
9924 |
9939 |
9955 |
9970 |
9985 |
358 |
10 13 |
LOGARITHMIC1 TANGENTS
187
Mean Differences. |
||||||||||||
ft' |
12' |
18* |
24' 30' 36' |
42' |
48' |
54' |
||||||
u |
*l* |
1' 2 3" 4' V |
||||||||||
45 |
10-0000 |
0015 |
0030 |
0045 |
'076 |
0091 |
0106 |
0121 |
0136 |
358 |
10 13 |
|
46 |
10-0152 |
0167 |
0182 |
0197 |
Mil |
0243 |
0258 |
OT1 |
MM |
358 |
10 13 |
|
47 |
10-0303 |
0319 |
0334 |
0349 |
0396 |
0410 |
04JI |
0440 |
358 |
10 13 |
||
48 |
10-0456 |
"171 |
0486 |
0501 |
n.-.i; M |
0562 |
0578 |
MM |
358 |
10 13 |
||
49 |
10-0608 |
0639 |
0654 |
0670 |
0716 |
0731 |
0748 |
:; :, > |
10 13 |
|||
50 |
10-0762 |
0793 |
0808 |
0824 |
OM9 |
0854 |
0870 |
0885 |
0901 |
358 |
10 13 |
|
51 |
10-0916 |
0932 |
0963 |
0978 |
loin |
LOM |
l"ll |
1056 |
:; :, ^ |
10 13 |
||
52 |
10-1072 |
1088 |
1103 |
1119 |
1135 |
1150 |
1166 |
1182 |
1197 |
1213 |
358 |
10 13 |
53 |
LO*1SM |
LS4I |
1260 |
1-J7.1 |
mi |
1340 |
1356 |
1371 |
358 |
11 13 |
||
54 |
10-1387 |
1 1 1 ;• |
1 l.'i.-i |
ll.-.l II.;; 11-:; |
i ism |
1516 |
i. -•:;•-• |
358 |
11 13 |
|||
55 |
1564 |
1580 |
1596 |
Mil |
L6M |
it;i:, |
1661 |
1677 |
1694 |
358 |
11 14 |
|
56 |
10-1710 |
1726 |
1743 |
1759 |
1776 |
1 ::••_• |
IS.l'l |
1825 |
1858 |
358 |
11 1J |
|
57 |
10-1875 |
1891 |
1908 |
1925 |
194J |
1992 |
2008 |
2025 |
:; r. s |
11 14 |
||
58 |
10-2042 |
2059 |
2076 |
2093 |
2110 |
•J1-J7 -.'Ml |
2161 |
2178 |
I1M |
1 |
11 11 |
|
59 |
2229 |
•-'•.'••, 1 |
2281 |
. |
2333 |
M*1 |
2368 |
3 6 |
1 •_• 11 |
|||
60 |
i 10-2386 |
2403 |
2421 |
2438 |
2456 |
1474 |
jiyi |
MM |
2527 |
M4B |
3 6 |
1 L1 1 :. |
61 |
10-2562 |
BMQ |
2598 |
2616 |
2634 |
2670 |
1689 |
2707 |
I7M |
3 6 |
12 15 |
|
62 |
2762 |
2780 |
2798 |
2817 |
•js:,l |
2872 |
2891 |
2910 |
3 6 |
1 •_• 1 :, |
||
63 |
10-2928 |
2947 |
MM |
MM |
1004 |
., •!_ |
B061 |
3080 |
;{• msi |
3 6 |
13 16 |
|
64 |
10-3118 |
3137 |
3157 |
3176 |
3196 |
MM |
MM |
3274 |
MM |
3 6 10 |
13 16 |
|
65 |
10-3313 |
:;:;:;:; |
3353 |
3373 |
MM |
3473 |
3494 |
3 7 10 |
LI 17 |
|||
66 |
1MB |
:;.-,:,:, |
lift |
3596 3f.I |
MM |
3679 |
3700 |
3 7 10 |
11 17 |
|||
67 |
3743 |
3764 |
:;7-:, |
3871 |
3892 |
1914 |
4 7 11 |
14 18 |
||||
68 |
10-3936 |
:;•..:,- |
3980 |
1001 |
4091 |
in:; |
4136 |
4711 |
15 19 |
|||
69 |
11-1 |
4204 |
Itfl |
4319 |
tMI |
4366 |
4 8 11 |
15 19 |
||||
70 |
10-4389 |
ill:; |
4437 |
IP, i |
MM |
4509 |
i:,:.:; |
4557 |
4581 |
4606 |
4 8 12 |
16 20 |
71 |
LO-4MQ |
MM |
4680 |
4705 |
1780 |
1801 |
IMI |
4857 |
4 8 13 |
17 21 |
||
12 |
10-4882 |
MM |
4934 |
I960 |
:,'.:;•., |
1064 |
MM |
5120 |
4 9 13 |
i- M |
||
78 |
:.17J |
.'-.'-I |
.-.•_'•_".• |
Mil |
5340 |
MM |
:,::•.- : |
6 9 14 |
LI M |
|||
74 |
MM |
:,\*:: |
:.:,i-j |
:,r,.m |
5629 |
MM |
:,.;-.:. |
5 10 15 |
20 25 |
|||
75 |
10-5719 |
6750 |
5780 |
5811 |
5873 |
5905 |
6936 |
-,;.,> |
i: |
6 10 16 |
:•! |
|
76 |
10-40*1 |
6065 |
6097 |
KM |
net |
r.-.Ti.. |
Mfl I |
MM |
.;:;:;•: |
6 11 17 |
M M |
|
77 |
1<>. ,.;;,;,; |
M01 |
MM |
r.iTI |
6507 |
6578 |
6615 |
MS] |
I',.;HS |
6 12 18 |
24 30 |
|
78 |
10-0725 |
ITU |
MOO |
.>:;- |
•;>77 |
6954 |
(Ml |
7033 |
7"::; |
6 13 19 |
M M |
|
79 |
i"-7in |
7154 |
7196 |
MM |
::;.;:; |
7406 |
rut |
7I»3 |
7 11 21 |
28 35 |
||
80 |
10-7537 |
7681 |
7626 |
7672 |
7-11 |
nu |
not |
79.', 1 |
8 16 23 |
31 39 |
||
81 |
10-8003 |
mi |
8102 |
8162 |
MM |
8266 |
8807 |
MM |
Mil |
-n ; |
9 17 26 |
36 43 |
8L> |
LO*MM |
-:,;; |
MM |
.,;;„, |
8748 |
8806 |
ss,,, |
MM |
.„,,, |
10 SO 29 |
39 49 |
|
HI! |
10-9109 |
9172 |
MM |
•.'.-.Ml |
••...; |
9570 |
...i , |
'.'711 |
11 12 34 |
i , |
||
84 |
LOfTM |
Mil |
Mil |
,H)U, |
MM |
(•114 |
. |
4M |
MM |
I 'i |
13 26 40 |
H , | |
85 |
ii • • > |
.„•,,,, |
0769 |
..,:,.. |
IMI |
i ..11 |
1 1 h. |
16 32 48 |
||||
H |
11-1504 |
1664 |
1777 |
1893 |
2012 |
S261 |
ftM |
:.-.•.•:. |
,-.,.-. |
|||
87 |
L1>MM |
MM |
8106 |
::v. i |
MM |
IM1 |
. " ' |
I! .. |
I . .. |
|||
88 |
UN |
&027 |
5275 |
II! |
• .>:• |
|||||||
89 |
11-7681 |
MM |
8660 |
9130 |
:-i • |
4571 |
7.-.-1 |
i:i:< IPROCALS OF NUMBERS
20 |
4 R ft |
70 g |
|||||||||||
o |
Tt u D |
o u |
|||||||||||
10 |
Ml |
M04 |
•>!".' |
815 |
'.vj i |
9434 |
9346 |
9259 |
•171 |
9 18 27 |
;.; j:, :,:, |
||
11 |
MM, |
*>:„, |
772 |
MM |
8621 |
8547 |
8475 |
8403 |
8 15 23 |
30 38 45 |
53 61 68 |
||
1-2 |
> |
8197 |
• |
8065 |
7987 |
7874 |
7813 |
7752 |
6 13 19 |
26 32 38 |
i:, ;,i :,s |
||
13 |
1 • |
463 |
7407 |
7:;:,:; |
"299 |
7246 |
7194 |
5 11 16 |
22 27 33 |
:;s 11 in |
|||
14 |
<::'•.':. |
944 |
1897 |
6849 |
6803 |
6757 |
-.711 |
5 10 14 |
19 24 29 |
33 38 43 |
|||
15 |
. . . : |
6536 |
6494 |
6452 |
6410 |
6369 |
6329 |
6289 |
4 8 13 |
17 21 25 |
29 33 38 |
||
16 |
.,;:;:, |
6098 |
6061 |
6024 |
5988 |
5952 |
5917 |
4 7 11 |
15 18 22 |
||||
17 |
5747 |
5714 |
5682 |
(650 |
5618 |
5587 |
3 6 10 |
13 16 20 |
•-'." L'C, -JH |
||||
18 |
5435 |
5405 |
5376 |
5348 |
5319 |
5291 |
369 |
12 15 17 |
20 23 26 |
||||
19 |
1181 |
5155 |
5128 |
(101 |
5076 |
5051 |
5025 |
358 |
11 13 16 |
is L'l •_'! |
|||
20 |
• |
4902 |
4878 |
4854 |
4831 |
4808 |
4785 |
257 |
10 12 14 |
17 19 21 |
|||
21 |
4739 |
4717 |
if,-..:. |
4673 |
4651 |
4630 |
4608 |
4587 |
4566 |
247 |
9 11 13 |
i:. 17 •_'!' |
|
22 |
1 :. I :. |
4525 |
uoi |
MM |
4464 |
HI) |
4425 |
4405 |
4386 |
4367 |
246 |
8 10 12 |
14 16 18 |
23 |
i :•.!>• |
4329 |
4310 |
ISM |
4274 |
4255 |
US 7 |
4219 |
4202 |
4184 |
245 |
7 9 11 |
13 14 16 |
24 |
4149 |
4132 |
an |
4098 |
4082 |
4065 |
4049 |
1039 |
4016 |
235 |
7 8 10 |
12 13 15 |
|
25 |
i • '•• |
3984 |
3968 |
3953 |
no, : |
3922 |
3906 |
3891 |
3876 |
3861 |
235 |
689 |
11 12 14 |
26 |
:;-|i |
3831 |
3817 |
3802 |
3788 |
3774 |
3759 |
3745 |
3731 |
3717 |
134 |
678 |
10 11 13 |
27 |
3690 |
3676 |
3663 |
B660 |
3636 |
3623 |
3610 |
3597 |
3584 |
134 |
578 |
9 11 12 |
|
28 |
,.-,:,'.. |
3546 |
3534 |
3521 |
3509 |
3497 |
3484 |
3472 |
3460 |
124 |
567 |
9 10 11 |
|
L>'t |
- |
3436 |
3401 |
3390 |
3378 |
3367 |
3356 |
3344 |
i i- :; |
567 |
8 9 10 |
||
30 |
3333 |
usi |
3311 |
3300 |
:;•_'-;' |
3279 |
3268 |
3257 |
3247 |
3236 |
123 |
456 |
7 9 10 |
31 |
M |
mi |
3205 |
3195 |
3185 |
3175 |
8165 |
3155 |
3145 |
3135 |
123 |
456 |
789 |
32 |
3125 |
mi |
UOfl |
3096 |
3086 |
3077 |
3067 |
3058 |
3049 |
3040 |
123 |
456 |
789 |
83 |
:;..:;.. |
Mm |
3012 |
:;•">:; |
. |
2985 |
2976 |
2967 |
2959 |
2950 |
123 |
445 |
678 |
34 |
'.'941 |
•...:.:, |
2915 |
2899 |
2890 |
2882 |
2874 |
2.S65 |
123 |
345 |
678 |
||
35 |
2857 |
1841 |
2841 |
2833 |
2825 |
2817 |
•JSII'.I |
2801 |
2798 |
2786 |
123 |
345 |
667 |
36 |
2778 |
2770 |
2762 |
•J7:.:. |
.'717 |
2740 |
2732 |
2725 |
2717 |
2710 |
122 |
345 |
567 |
37 |
270S |
MM |
2688 |
2681 |
2674 |
2667 |
IT.IHI |
2653 |
2646 |
2639 |
112 |
344 |
566 |
38 |
2632 |
2625 |
2618 |
2611 |
1604 |
2597 |
L'.V.tl |
2584 |
2577 |
2571 |
1 1 2 |
334 |
556 |
:•.'. |
2564 |
2558 |
2545 |
2538 |
2532 |
L-.VJ: |
2519 |
2513 |
2506 |
112 |
334 |
456 |
|
40 |
2494 |
2488 |
2481 |
2475 |
1468 |
2463 |
2457 |
2451 |
2445 |
1 1 2 |
234 |
455 |
|
41 |
MM |
2433 |
2427 |
2421 |
2415 |
'.'tin |
2404 |
2398 |
2392 |
2387 |
1 1 2 |
233 |
1 ~> :, |
42 |
n*i |
•.:.;:. |
2370 |
2364 |
•-•:;:,-> |
•j:i:,:; |
2347 |
2342 |
•j:;:n; |
2331 |
112 |
233 |
445 |
43 |
.... |
:•:::•'• |
mi |
2309 |
2304 |
•_••_".•'.' |
2294 |
2288 |
8988 |
2278 |
112 |
233 |
445 |
44 |
..•;: |
:•.. - |
MM |
2257 |
8S58 |
2247 |
2242, |
2237 |
2232 |
2227 |
1 2 |
233 |
445 |
45 |
.... |
..'17 |
1 1 1 1 |
not |
2198 |
2193 |
2188 |
2183 |
2179 |
0 1 |
L' :.' :; |
344 |
|
46 |
.:; |
. : - :< |
nei |
2160 |
2155 |
2151 |
2146 |
2141 |
2137 |
2132 |
0 1 |
223 |
344 |
47 |
.:. |
.:... |
2119 |
2114 |
2110 |
2106 |
2101 |
2096 |
•Jiilrj |
2088 |
0 1 |
223 |
344 |
48 |
. -. |
: . .• |
I .;:. |
2070 |
».„;, |
2062 |
2058 |
2053 |
•JO 111 |
•.'()!.- |
0 1 |
223 |
334 |
49 |
. i |
. . |
: ,:;:; |
2028 |
. |
2020 |
2016 |
2012 |
2008 |
20H4 |
0 1 |
222 |
334 |
60 |
. |
; .... |
1 •..:._< |
1988 |
1984 |
!'."•< |
1976 |
1972 |
1 :'•;:. |
1965 |
0 1 1 |
222 |
334 |
51 |
1961 |
1 '.- -, 7 |
1958 |
1949 |
1946 |
1942 |
1938 |
1934 |
1931 |
1927 |
Oil |
222 |
333 |
.v.' |
: •. |
l-.-l'.. |
L914 |
1912 |
].,,,, |
1905 |
1901 |
1898 |
IS:M |
IHW |
Oil |
122 |
3 3 3 |
53 |
:-- |
1--:, |
!--• |
1876 |
1873 |
1869 |
1866 |
1862 |
1859 |
1855 |
Oil |
122 |
233 |
54 |
:-••• |
1-1- |
1845 |
1842 |
1V> |
L8M |
1832 |
1828 |
1825 |
1821 |
Oil |
122 |
233 |
RECIPROCALS OF Nf.MUKKs
189
7' |
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9 |
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55 |
1818 |
1815 |
1812 |
lsn> |
is,,.', |
1802 |
1799 |
1 7'.':. |
1792 |
1789 |
0 1 1 |
2 :; I |
|
56 |
IT-: |
1783 |
1779 |
1776 |
1773 |
177" |
17-; 7 |
1 70 1 |
1761 |
1757 |
(i 1 1 |
2 2 |
2 1 I |
57 |
17M |
1751 |
1748 |
1745 |
1742 |
1739 |
1736 |
1733 |
1730 |
175.' 7 |
0 1 1 |
2 2 1 |
|
58 |
i :•_• i |
1721 |
1718 |
171* |
1712 |
1709 |
1706 |
1704 |
1701 |
ir.'.i.S |
(111 |
1 -2 |
223 |
59 |
1695 |
1692 |
H>'.' |
l»;sr, |
1684 |
1681 |
1678 |
1675 |
L671 |
1669 |
0 1 1 |
1 2 |
2 i' :. |
6C |
1GG7 |
1 i:.; l |
1661 |
1658 |
1656 |
1653 |
1650 |
1647 |
L64J |
1642 |
II 1 1 |
1 2 |
223 |
61 |
1639 |
1637 |
1634 |
1631 |
1629 |
1626 |
1623 |
Itl-Jl |
1618 |
1616 |
(1 1 1 |
1 1 2 |
|
62 |
1613 |
1610 |
1608 |
1605 |
1603 |
1 .;• .. i |
1597 |
1595 |
LMI |
1590 |
<» 1 1 |
112 |
2 2 2 |
63 |
1587 |
1585 |
1582 |
1580 |
1577 |
1575 |
1071 |
1570 |
LM7 |
1565 |
.1 II 1 |
1 1 1 |
|
64 |
1563 |
1560 |
1558 |
IMI |
1553 |
LMO |
i:.i> |
1546 |
LMI |
1 :. u |
1) II 1 |
1 1 1 |
|
65 |
1538 |
1586 |
1534 |
i:.:;i |
I :.-.".' |
1.VJ7 |
!524 |
LMI |
1.-.17 |
0 o 1 |
1 1 1 |
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66 |
K.i:. |
1513 |
1511 |
1508 |
L5M |
1504 |
1502 |
1 in-.* |
1 l'«7 |
1495 |
(I 0 1 |
111 |
2 •-' 2 |
67 |
i »'.':; |
1490 |
1488 |
1 1-1; |
1 1-1 |
1481 |
1 1 IV |
1 177 |
1 17:. |
1473 |
I) 0 1 |
1 1 1 |
2 2 2 |
68 |
1471 |
1468 |
MM |
Itr.i |
1462 |
1460 |
1 «:.> |
1 l.ir, |
1453 |
1451 |
u n i |
1 1 1 |
|
69 |
111'.' |
1447 |
1445 |
1443 |
1111 |
1439 |
1437 |
i »:;:. |
1431 |
u n i |
1 1 1 |
2 2 •_' |
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70 |
1427 |
1425 |
1 » 22 |
1 12" |
1418 |
1416 |
L414 |
1112 |
1410 |
0 0 1 |
1 1 1 |
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71 |
1408 |
1406 |
11"! |
MM:; |
1401 |
1399 |
LSM |
1393 |
1391 |
0 (1 1 |
1 1 1 |
1 2 2 |
|
72 |
1389 |
i:;-7 |
1385 |
- |
1S81 |
1379 |
1377 |
L876 |
1 :; 7 1 |
1372 |
.1 (1 1 |
1 1 1 |
|
73 |
1368 |
1366 |
LMI |
1362 |
1361 |
LS57 |
1355 |
1353 |
001 |
1 1 1 |
1 2 |
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74 |
1350 |
1344 |
1342 |
LMO |
Lilt |
1337 |
LSM |
0 0 1 |
1 1 1 |
1 |
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75 |
1333 |
1332 |
1328 |
1326 |
1325 |
1321 |
1319 |
1318 |
001 |
1 1 1 |
1 2 |
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76 |
• |
1114 |
111] |
1309 |
1307 |
l :;•.:, |
L804 |
LJ02 |
1300 |
0 0 1 |
1 1 |
1 |
|
77 |
1299 |
12 '.'7 |
i •_".' i |
H |
1189 |
L287 |
1285 |
1284 |
000 |
1 1 |
1 |
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78 |
- |
1171 |
1177 |
1276 |
U74 |
LI 71 |
1171 |
1269 |
I2t;r |
000 |
1 1 |
1 |
|
79 |
r.-r.i |
L261 |
1259 |
1258 |
i •.•:.t; |
12:.:. |
1253 |
12:. 2 |
000 |
1 1 |
1 |
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80 |
LM7 |
1141 |
L 144 |
124] |
L1S9 |
L1M |
i-.'.-u; |
ii n u |
1 1 |
1 |
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81 |
1229 |
L117 |
1221 |
1224 |
1222 |
1221 |
u it n |
1 1 |
| |
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82 |
1117 |
1215 |
1214 |
ini |
111] |
1208 |
1206 |
u ii n |
1 1 |
1 |
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83 |
1205 |
1200 |
1199 |
1 1 I"'. |
1 1 •.•:. |
1193 |
1192 |
it u it |
1 ! |
1 |
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84 |
1190 |
L18f |
1186 |
L188 |
1181 |
1181 |
1179 |
1178 |
n n n |
1 1 |
1 |
||
85 |
1 1 7:. |
1114 |
1 1 7'.' |
1171 |
L1M |
L167 |
1166 |
1164 |
|
1 1 |
1 1 |
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86 |
il'-.i |
1160 |
! 1 N |
II. -.7 |
1 1:.:. |
1 1 :.-.' |
L1M |
1 1 |
1 1 |
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87 |
11 in |
1147 |
1145 |
Illl |
1 1 12 |
11 i" |
1139 |
li::» |
1 1 |
1 1 |
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88 |
1136 |
1134 |
1133 |
Hi".. |
L117 |
L1M |
1125 |
000 |
1 1 |
1 |
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89 |
1122 |
1121 |
1120 |
1119 |
1117 |
ill*. |
in:. |
111 1 |
1 II 2 |
<» i» n |
1 1 |
1 |
|
90 |
mi |
111" |
1109 |
1107 |
LlOi |
L1M |
ll"| |
11"! |
• |
1 1 |
1 |
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91 |
LOM |
LON |
1005 |
L0»] |
1089 |
IMSX |
M n ii |
'i 1 |
1 |
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9-J |
1085 |
1088 |
L07I |
I".', |
1 |
1 |
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93 |
1076 |
1"7I |
1071 |
1072 |
L071 |
1 1 |
ii n n |
1 |
1 |
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94 |
: ",i |
LOU |
LOOO |
1055 |
n ii (i |
1 |
1 |
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<»5 |
LOM |
1052 |
1050 |
: MM |
1044 |
L041 |
1 |
1 |
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96 |
L04J |
1040 |
LOU |
1 |
1 1 |
||||||||
97 |
LOU |
ion |
LOM |
1 |
1021 |
n n n |
1 |
1 1 |
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!».H |
1018 |
!••! ; |
I'M | |
I"!'-1 |
1"11 |
1 |
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99 |
1010 |
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1007 |
1 1 |
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