Rodolfo Amedeo Lanciani.

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mass — ^not delicate differential resulta By Joule, Maxwell,
and Olausius we know that the average velocity of the mole-
cules of oxygen or nitrogen or common air, at ordinary atmos-
pheric temperature and pressure, is about 50,000 centimeters
per second, and the average time from collision to collision a
five-thousand-millionth of a second. Hence the average length
of path of each molecule between collisions is about tttsVitt
of a centimeter. Now, having left the idea of hard globes,
according to which the dimensions of a molecule and the
distinction between collision and no collision are perfectly
sharp, something of apparent circumlocution must take the
place of these simple terma

First, it is to be remarked that two molecules in collision
will exercise a mutual repulsion in virtue of which the distance
between their centers, after being diminished to a minimum,
will begin to increase as the molecules leave one another. This
minimum distance would be equal to the sum of the radii, if
the molecules were infinitely ham elastic spheres; but in reality
we must suppose it to be very different in different collisions.
Considering only the case of equal molecules, we might, then,
define the radius of a molecule as half the average shortest
distance reached in a vast number of collisions. The defini-
tion I adopt for the present is not precisely this, but is chosen
so as to make as simple as possible the statement I have to



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44 W. Thompson on the size of Atoms.

make of a combination of the results of Olau^us and MaxwelL
Having defined the radius of a gaseous molecule, I call the
double of the radius the diameter ; and the volume of a globe
of the same radius or diameter I call the volume of the
molecule.

The experiments of Cagniard de la Tour, Faraday, Regnault,
and Andrews, on the condensation of gases do not allow us to
believe that any of the ordinary ^es could be made forty
thousand times denser than at ordinary atmospheric pressure
and temperature, without reducing the whole volume to some-
thing less than the sum of the volume of the gaseous molecules,
as now defined. Hence, according to the grand theorem of
Clausius quoted above, the average length of path fi-om col-
lision to collision cannot be more than five thousand* times the
diameter of the gaseous molecule ; and the number of mole-
cules in unit of volume cannot exceed 25,000,000 divided by
the volume of a globe whose radius is that average length of
patL Taking now the preceding estimate, tttVtt ^^ * centi-
meter, for the average length of path fi'om collison to collision^
we conclude that the diameter oi the gaseous molecule cannot
be less than -^zr^Tiizzzz ^^ a centimeter; nor the number of
molecules in a cubic centimeter of the gas (at ordinary density)
greater than 6 X 10* ' (or six thousand million million million).

The densities of known licjuids and solids are from five
hundred to sixteen thousand tmies that of atmospheric air at
ordinary pressure and temperature ; and, therefore, the numbw
of molecules in a cubic centimeter may be from 8 X 10 '^ * to
10* * (that is, from three million million million million to a
hundred million million million million). From this (if we
assume for a moment a cubic arrangement of molecules), the
distance from center to nearest center in solids and liquids may
be estimated at from y^^^ J^^^y to ^T^TTirvTnr of a centimet^?.

The four lines of argument which I have now indicated,
lead aU to substantially the same estimate of the dimensions of
molecular structura Jointly thev establish with what we can-
not but regard as a very high degree of probability the con-
clusion that, in any ordinary Bquid, transparent solia, or seem-
ingly opaque solid, the mean oistance between the centers of
contiguous molecules is less than the hundred-millionth, and
greater than the two thousand-millionth of a centimeter.

To form some conception of the degree of coarse-grainedness
indicated by this conclusion, imagine a rain drop, or a globe of
glass as large as a pea, to be magnified up to the size of the
earth, each constituent molecule being mamiified in the same
proportion. The mamified structure would be coarser grained
than a heap of small snot, but probably less coarse grained than
a heap of cricket-balls.



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W. GHhha — Miscellaneotis Optical Notice 46



Abt. YIL — Miscellaneous Optical Notices ; by WOLCOTT GiBBS,
M.D., Rumford Professor in Harvaxd University.

§1.

On the measurement of wave lengths hy means of indices of re-

fraciion.*

In a brief notice f communicated to the British Association
for the Advancement of Science at its meeting in 1849, Prof
Stokes has given a method for measuring wave lengths, which
depends ufjon the feet that, in substances of medium refractive
power, the increment of the index of refraction in passing from
one point of the spectrum to another is nearly proportional to
the mcrement of^ the square of the reciprocal oi the wave
lengtL The author showed that even when the intervals were
taken much longer than necessarv, the error in the wave length
was usually only in the eighth place of decimala At the (uite
of the puWication of this notice the subject of wave lengths
possessed but little interest The recent development of the
spectral analysis of light has given a new impulse to this branch
of optics, and has rendered necessary the construction of a
nomxiEd map of the entire solar spectrum. This has been most
successfrJlv accomplished bv Angstr6m,:t but an attentive study
of his work, as well as of the elaborate researches of Van der
Willigen§ and Ditscheiner,| will show that new measurements
will be far from superfluoua The imperfections even of the
best ruled glasses are so great that it may be reasonably doubted
whether the wave lengths of verv fine lines can be satisfec-
torily measured directly. Metho<ls of determining such wave
lengths, depending upon the comparison of the refraction and dif-
fraction spectra, have been given by myself^ and by Thalen.**
As it seems at least desirable to multiply such methods, I will
here give first a discussion of the method of Stokes in its one-
inal form, and afterward a simplification of that method whicn
will also have its usea

If Cauchy's formula for dispersion, n=^a + p + ji >

* Read before the National Academy of Sdenoes, April 12th, 1870.
f Beport of tiie British Assooiatioa for the Advaiicement of Sdeooe for 1849.
Notices and abstracts, p. 10.

iBeoherohee sor le Spectre Solaire. Berlin, 1869.
Arohiipes da Mus^ Teyler, toI. i, pw 1.
Sitzongsberiohte der k. k. Akad. der Wliwensohaften Bd. 1, 1864.
This Journal, zlvii, Mardi, 1869.
** M^olre sur la determination des longneors d* onde des raies m^taOiques, 1 868.



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46 W. GHhhs — Jfiscdlaneotis Optical Notices.

be reduced to its two first terms, and if we then eliminate the
constants a and b firom three equations of the form

we shall obtain the three following equations, involving only
wave lengths and indices of refraction:

1» = Kz^j) _ (1)

K-»i)j| + («3-«i)x2

*;= ^^^^^ r (2)

{«,-n,)j5 + (ng-na)j5

*8 1

i? = ^^^^^^ r (3)

Of these ecjnations (1) and (8) serve for extrapolation and (2)
for interpolation. To test the degree of accuracy attainable in
determining wave lengths by these formulas, f have selected
the measurements made by van der Willigen.* The indices of
refiraction determined by the Dutch physicist are in feet the only
indices which are at once sufficiently exact and sufficiently nu-
merous. In addition they have the great advantage of having
been made with reference to lines in the solar spectrum the wave
lengths of which had been measured by the same observer.
There can therefore be no question of identity. As a first ex-
ample of the method, I give a determination oi the wave length
of C, taking B as one of the lines exterior to C, and taking in
succession 7 other exterior lines more refi»ngible than C, to
combine with B. Formula (2) was therefore employed, and
with the following data and results : —

B 1-61079 687-48

C 1*61252 656-56

D 1-61486 628-11 656-70 +0-14

11 1-61537 613-96 656-71 4-^-15

13 1-61560 610-52 656*56 0-00

14 1-61728 589-56 656*71 +015

16 1-61978 561-80 656*76 +0*20

17 1-62064 553-19 656*79 +0*23
19 1-62148 545-83 656-87 +0-31



Mean of the errors +0-17

In this table the first column gives the designation or num-
ber of the lines, the second its index of refirSetion, as deter-
mined by a Steinheil prism of 60°, the third the corresponding

* Archives da Mua^ Teyler, yd. i, p. 70.



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W. Oiths — Misce^neoua Optical Notices. 47

wave lencth, according to Yan der WiUigen, and the fourth
the wave length as found bjr formula (2) by combming B with
each line after C in succession.

The mean of the seven values of the wave le ngt h of C thus
found is 656"70, which is in excess of Van der Willigen's own
determination of the value of C by 014. From this it appears
that the method may be applied with a tolerable degree of ap-
proximation, even in the case of a flint glass prism of high dis-
persive power, and for indices of refraction which refer to lines
at considerable angular distances from each other. The increase
in the computed values of C, as the intervals between B and
the second line of comparison are increased, will however clear-
ly appear from the table. The following results were obtained
with the indices of a second Steinheil pnsm. No. 2, of 46® 52'
26''-8, also of flint glasa



B


1-60621


B and 8a


666-21


—0-86


C


1-60694


Band 11


666-33


—0-23


8a


1-60872


Band 13


666-81


—0-26


11


1-60973


B and 16


666-38 .


-0-18


13


1-60998


B and 17


666-47


—0.09


16


1-61408








11


1-61495









The mean of which is 656'28, the error being —0-28. To deter-
mine to what extent the method applies, when flint glass prisms
are usgd, and the indices are selected from the more refrangible
portion of the spectrum, the following data were assumed : —

F 1-62917 486-39 F and G 438*88 4-0.30

F and 39 438-76 +0-18

F and 38 438*82 -f-0*24

G and 36 438*76 +0*18

35 and 38 438-75 -t-0-17

36 and 39 438*67 -f-0.09

In this table, line 87 is taken as the middle line in applying
formula (2), and the results obtained by combining the other
lines in pairs are given in columns 4, 5 and 6. It ydll be seen
that, as m the case of the less refrangible portion of the spec-
trum, the results obtained are with ttus prism always too high.
For the purpose of comparison, I have computed the same wave
length from the indices of refraction of the second prism. The
data and results are as follows : —



36


1-63244


467-00


37


1-63828


438-68


38


1-63931


434*28


39


1-63966


432-74


G


1-64006


431-12



F 1-62332 486*39 F and G 489*07
85 1-62657 467*00 F and 89 438*89
37 1-63221 438-68 F and 38 438-92



1-63824 434*28 G and 35 439*00 +0*42



39 1-63358 432*74 36 and 38 438*89
G 1-63400 431*12 35 and 39 438*84



.0*49
0*31
0-34



•0*31
•0*26



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48 W. GHbba—Miscdlcmeaus Optical Hbtices;

In the case of the first prism the mean of the errors is +0'21,
while for the second the mean of the errors is +0*85. From
this it appears that in the more refrangible portion of the roec-
trum the errors are considerably greater thim in the less renun-

S'ble portion, even for equal differences of wave length, and
rther, that the advantage in precision is with the prism hav-
ing the higher dispersive power. As the probable errors of the
measurements of the incuces of refraction are not riven, it is
impossible to determine to what extent the errors in tne commu-
ted wave lengths are due solely to want of precision in the in-
dices. It is also to be remarked that, while with the second
prism the errors in the less refrangible portion of the spectrum
are affected with the sign — , in the more refrangible portion they
are largely positive. The dose agreement in the value of the
wave length of 87, as found by v an der Willigen, with the
values as found by Ditscheiner and Angstrom — 488*27 and
488*28 — ^proves that the source of error is not an erroneous de-
termination of this quantity. It seems therefore certain that the
nearly constant errors noted above are due in part to the &ct
that the indices of refraction are determined only to five places
of decimals, and in part to the high dispersive powers of the
prisms employed, wmch would render it necessary to employ more
than two terms in Cauchy's formula to obtain a closer approxi-
mation- As the formulas for interpolation would in this way be
rendered extremely complicated, it is better, in the case of any
series of observations embracing a particular part of the'scale^
simply to determine the mean of the errors, and to apply this
mean with its proper sign to the computed values of tne par-
ticular wave length to Be determined by the measurement of
indices of reframon. If we apply such a correction in the
cases of the four series of data ana results given above, we find
for the corrected values of the wave-lengths the following nu-
merical results : —



(1)


(2)


(8)


W


666-58


656-49


438-67


488-72


666-54


656-61


438*56


488-64


666-88


666-69


438-61


488-57


656-64


666-66


488-66


438-66


666-69


666-76


488-64


488-64


656-62




488-44


438-49



The true values being respectively 666'56 and 488-5ft These
results are, I think, sufficient to show that a valuable control
for the accuracy of measurements of wave lengths may be ob-
tained even when prisms of high dispersive power are employ-
ed, provided that the intervals taken are not too laiga It
seems at least probable that a greater degree of precision is at-
tainable in measuring indices in the case of substances of high



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[iinea


X


Indices.


25b


518-63


1-62459


26


617-61


1-62472


21^


61707


1-62479



W. Oibbs — Miscellaneous Optical Notices. 49

than in those of low dispersive power, partly because the angu-
lar deviations to be measured are larger, and partly because the
spectral lines are less crowded together.

The following example will serve to illustrate the advantage
of taking shorter intervals: —

X Indices. X

1-61882

517.61 1-61895 517-56
1-61901

The data are here also taken from Van der Willigen*s meas-
ures with the same prisms.

When the angular distances between three spectral lines are
not too great, the angular deviation of the lines may, as I find,
be substituted for the indices of refraction in formulas (1), (2)
and (3). The differences between the angular deviations are of
course to be converted into seconds. The following results will
show the degree of accuracy attainable by this method, the data
being taken from Ditscheiner's* measurements of the indices of
a flint prism by Steinheil of refracting angle 60° 4' 59''.

Sirchhoff's scale. X Angular deyiationB. X Indices. X

B 593 687-06 47'' 40' 56" 1-61858

C 694 655-95 47° 51' 19" 655-97 1-61587 655.82

.. 877 618-67 48° 8' 8" 1-61824

From this it appears that the error in the determination of
the wave-length of the middle line C is only ■fO-02 when the
angular deviations are employed, but amounts to —0-18 when
the indices of refraction are taken as tiie elements of the calcu-
lation. Yet the interval between B and 877 is very large.

The following data are taken from anotiier part of the scale,
the measurements being made witii the same prism : —

Kirchhoff *8 scale. X Angular deviations. X Indices. X

b 1648-8 51713 49^ 4' 16" 1-62775

1655-6 516-58 49** 4' 44" 516-56 1*62782 516-61

1698-8 51408 49° 6' 47" 1-62817

Hence the error in the determination of the wave length of
1655*6 is, when the angular deviations are taken, only —0-02
and when the indices are taken +0-03. It must be borne in
mind that in all the above mentioned examples the angles are
those of minimum deviation. As the numoers upon Earch-
hoff 's scale also represent angular though not minimum devia-
tions, it seemed worth while to determine how fiir for a short
interval, these could be employed. Taking the tiiree scale num-
bers of the last example, the error in the wave-length of

* Bestimmung der WeUenlangen der nraunhofersQhen Limeii des Sonnenqwc-
trnms, p. 43.
Am. Joxtb. Soi.~8BOoin> Sbbibs, Vol. L, No. IIS.—Jult, 1870,
4



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50 W. Otbba — Miscellaneous Optical Notices.

1655*6 was found to be — 0*88, and when the scale numbers
were taken as the sines or tangents of corresponding angles,
+0-09.

The following data are taken from the more refrangible part
of the spectrum, the measurements being also those of Ditsch-
einer, and made with the same prism : —

2822-8 483.84 60° 34' 67" 1-64287

G 2864-7 430-88 60° 37' 52" 430-68 1-64384 430-83
2969-7 429-90 60° 38' 47" 1-64362

In this case the error in the wave-length of the middle line,
2854*7, is —0*20, as determined from the angular deviations,
and — 0*05, as determined from the indices. It must be borne
in mind that, in this part of the spectrum, the determination
both of wave lengths and of indices of refraction is difficult on
account of the feeble intensity of the light

Since only the differences between the angular deviations of
the spectral lines are employed in the formulas above given, it
follows that in determining wave lengths by the method in
question, it is not necessary to employ a spectrometer with a
divided circle and appliances for the measurement of large
anglea A common spectroscope will be sufficient, if the ob-
serving telescope be provided with a filar micrometer by means
of which the an^ar distances of any given line fix)m two
other lines of which the wave lengths are known may be meas-
ured. The researches of Angstrom leave nothing to be de-
sired as regards the wave-lengths of standard lines, and the
method given may prove a convenient means of determining
with aU requisite precision the wave lengths of metallic lines.



On liquids of high dispersive power, — Of the liquids which have
hitherto been proposed for the construction of prisms, bisulphid
of carbon imquestionably presents the greatest advantages. It
is cheap, colorless, and unites a moderately high mean ren-active
to a very high dispersive power. By tacit consent a prism of
60° filled with this liquid has come to be adopted as a sort of
standard. The disadvantages of the bisulphid are equally well
known, and I have spent no little time and labor in tne en-
deavor to find a liq^uia with a still higher dispersive power, less
volatile, less sensitive optically to changes of temperature, and
less offensive in odor. In these efforts I have not been alto-
gether successful, no one liquid examined possessing all the
qualities desired. Many organic liquids witn high mspersive
powers are difficult to prepare in a state of purity and speedily
oecome discolored by absorption of oxygen from the air. Such
are oil of cassia, the colorless oil obtainable from balsam of



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W, Oibbs — MisceUaneous Optical Notices. 51

Peru and others. The thallic alcohol of Lamy* is fiw too
costly. The solution of silico-tungstate of sodium,f of meta-
tungstate of sodium % a^^d of soluble tungstic acid § as obtained
by dialysis, all promised good results from their extraordinary
densities, but aU proved difficult to prepare in a state of purity
and extremely easy of decomposition.

A solution of phosphorus in bisulphid of carbon has, accord-
ing to Messrs. Dale and Gladstone,]] a dispersion of 0*225,1" or
nearly one and a half times as great as bisulphid of carbon
alone, but becomes turbid on exposure to sunliffht from the for-
mation of amorphous phosphorus. It occurred to me, that, by
dissolving sulphur with the phosphorus, the formation of amor-
phous phosphorus might be prevented, and experiment proved
that this was the case. The solution, as thus obtained, has a
pale yellow color, but is perfectly clear and undergoes no change
by the action of light even when long continued. I have been
in the habit of preparing it by dissolvmg one part of dry flowers
of sulphur and two parts of phosphorus, in four or five parts of
bisulphid of carbon, and filtering the liquid through a well
dried ribbed paper filter, which is easiljr done. The refractive
and dispersive power of the solution will of course vary with
the quantity of phosphorus and sulphur dissolved. By a gentle
heat the whole, or nearly the whole, of the bisulphid of car-
bon may be driven off, a liquid compound of sulphur and phos-
phorus remaining which Eas so high a mean refractive power
that it cannot be employed with prisms having a refractive ancle
of more than 45°-50°. The same end may, however, also oe
attained bv continually adding phosphorus to a saturated solu-
tion of sulphur in bisulphid of caroon, in which phosphorus
appears to oe soluble witnout limit

With a strong and probably saturated solution of sulphur in
CS, the angle between Li and D was 0° 50' 10". When phos-
phorus was added the angle was 2° 25' 80", the refracting angle
of the prism being 60°. in this last case the angle between Isa ,
and !Na, was 2' 20". The spectrum was perfectly clear,
the definition of the dark lines leaving nothing to be desired.
In consequence, however, of the yellow color of the liquid there
is always a marked absorption of the violet end of the spectrum.

In working with the above described solution I have em-
ployed hollow glass prisms with refractinjg plates cemented on
with a mixture of glue and molassea These were found to

* Ann. de Ohimie et de Physique, 4th series, vol. iii, p. 373.
• ' Ano. de Ohimie et de Physique, 4th series, vol. iii, p. 5.

: iScheibler in Journal fiir prakt Chimie, Izxziii, p 273.
[ Graham in Journal of the ChemicaJ Society, vol ii, p. 318
L. and E. PhiL Mag., yoL xviil, p. 30.

* \ The number 0'226 is the differeDce between the indices for the extreme red
and yiolet rays.



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62 W. Oibbs-^-MisceUaneous Optical Notices,

be perfectly tiffht and to last for months without change.
The great disadvantf^e in the use of a solution of sulphur
and phosphorus consists in the danger of breaking the prisms ;
the hquid taking fire spontaneously when it has been a few
seconds in contact with any porous material like wood or
paper. On the other hand, however, the large quantity of sul-
phur present prevents the fire from spreadmg, a drop placed
upon ajMece of wood leaving after combustion only a charred
spot When not in use the prisms should be kept in an iron
pot with a tight cover. In tms manner I have employed and
preserved two during a lontj and hot summer. The viscid, or
rather oily, nature of the solution serves to prevent, to a great
extent, the formation of ascending and descending currents &om
slight changes of temperature, and when the prisms are well
shaken before use the definition remains perfect for a lon^ time.
In my spectroscope the prisms rest upon a plate of glass mstead
of upon one of metal.

§8.

On an advantageoics form of apparatus for the study of the ab-
sorption of light in colored liquids.
In his examination of the spectra of colored fluids, Mr. Glad-
stone employed a hollow wedge of glass, the two ref5racting sur-
feces of which made with each other an acute angla The
wedge was filled with the liquid to be studied and so placed
that the refracting edge of the analyzing prism was at right
angles to the line of intersection of the two fiwjes of the wedge.
In this manner a beam of light was obtained which represented
difierent thicknesses of the absorbing liquid, and the resulting
spectrum became a complete absorption diagranu In using
this apparatus I found the angular deviation produced by the
wedge a source of considerable inconvenience. In addition it
is easy to see that the wedge itself produces a certain amount of
chromatic dispersion. To remedy these defects and at the same
time retain the advantages of the method, I have devised what
may be termed a double wedge. Two hollow wedges, of glass
or metal, are placed together in such a manner that
the first and last surfaces of the bounding plates of
j^lass are parallel The two wedges are separated
by a single plate of glass with parallel surfaces.
The base of each wedge, or acute-angled prism, is
bored for the insertion of a cork or stopper. The
construction of the apparatus wiU be readily un-



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