Rodolfo Amedeo Lanciani.

The American journal of science and arts online

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The following Table 5, in which the numbers are arranged
according to the temperatures, is obtained by reducing the
values contained in Table 4 in the same manner as those of
Table 2 of the exampla

The coefficients a and b are derived by means of least squares
from the values contained in the third and fourth columns of
Table 5, as in the example. Designating by «p the modulus of
elasticity at 0°, the modulus for the temperature t is found to be,

lor iron, « = e^ (1 — 0-000447t - 0-00000012r«)

for copper, « = «^j (1 — 0-000520r— 0-00000028t«)
for brass, « =e^,(l - 0-000428t — O'OOOOOl Set*)

The numbers of the next to the last column of Table 5 were
calculated with these values. The probable error of each value
of the decrease of the coefficient of elasticity for 1° for a given
temperature calculated from two groups of observations amounts
accordingly to d= 0*000014.

Ax. JouB. Soi.— Sbcond Sbbibs, Vol. L, No. 150.— Nov., 1870.

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862 Eohlrausch and Loomis— Influence of Temperature

Table 5.

Decrease of tbe ooeff. I

of elMtle. for an Incr.
of temp, of 1^ M nine.

Obeerred. ' Calcul.






0000467 0000454

+ 0-000018





.+ 27





- 07





- 42





~ 18





- 04





+ 31





- 05






+ 0-000028





- 10





- 02





- 26





- 16





+ 24





- 01






+ 0-000005





- 10





- 09





+ 37





- 30





+ 02





+ 06





- 06





+ 07

If the other definition of the modulus of elasticity is em-
ployed (as was done by Wertheim) referring to the section, and
not, as above, to the mass of the unity of the length of the
wire, the coefficients of tiie temperature will be slightlv altered
in accordance witii the observation on page 857. If the coeffi-
cients of dilatation for V are assumed to be for iron =0-000012 ;
for copper = 00000176 ; and for brass = 0*000019, the coeffi-
cients of T become,

for iron, 0"000483,

for copper, 0-000672,

for brass, 0000486.

The coefficients of the quadratic terms remain sensibly un-

It is evident from the results of these observations tiiat the
mean variation of the elasticity for tiie three metals investigated
differs but littie from that of tiie temperature. For a change
of temperature from 0"* to 100** the diminution is for

Iron, 4*6 and 6*0 per cent.

Copper, 6-6 ." 6-0 «
Brass, 66 " 6-2 "

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on the Elasticih/ of certain Metah 86$

where the numbers of the second column relate to the second
definition of the modulus of elasticity.

If the variation in elasticity is compared with the influence
exercised by the temperature on other properties of bodies it
will be remarked that it is much greater than the cubical dila-
tation as well as the variations of refraction. It is of about
the same order as the variation of the permanent magnetism
caused by the temperature, as well as that of specific heat The
increase of the galvanic resistance on the other hand is much

It results ftirther from the sign of the quadratic term that
the variation of elasticity for all three metals is the most rapid
in the higher temperaturea While, however, the increase for
iron is almost imperceptible, and is also very small for copper,
it is quite considerable for brass. The decrease for 1° is for
At o*». At 60^ At 100^

Iron, 0*0447 00459 0*0472 per cent

Copper, 0-0620 0-0648 0-0676 "
Brass, 00428 0*0664 0-0699 "

It will be observed that the differences in the variation of
the coefficient of elasticity for the three metals investigated,
are in the order of the height of their melting pointa

The results of these investigations show no trace at all of
the remarkable phenomenon of a maximum, alluded to at the
beghming of this article, which would seem to be indicated for
iron by the investigations of Wertheim. If, therefore, different
varieties of iron do not manifest a totally different behavior, or
if the modulus of the longitudinal elasticity does not undergo
a very different change m consequence of temperature from
that of the torsional elasticity, this anomaly must be accounted
for by the imperfect method of observation employed by Wer-
theim. This supposition is confirmed fiirther oy the observa-
tions of Kupffer (see below), as well as by a very simple experi-
ment I^ namely, two tuning forks are in vibration, ana one
of them is heated, the number of vibrations changes in the
manner demanded by the assumption of a decrease of elasticity
for increasing temperatures.

It is to be remarked here that Wertheim's calculation of the
heat generated by condensation during vibration, loses thus all
its value.*

Observations of Kupffer, 'f — These investigations appear to
be much less general^ known than they deserve, for they
contain much varied and valuable material for practical use.
Kupffer's observations are in general on bars vibrating trans-

* Pogg. Aim., Bd. 11, a 32.

t Mem. d6 TAoad. de St. Peterab. 1856, 6 ser. T. vi, p. 400.

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864 Kohlrausch and Loomis — Influmce of Temperature

versally at the temperatures of about —10% +15° and +80**.
Unfortunately, however, very long intervals (at least a yearj
occur between the observations, and it would seem that he dia
not always employ the same bars, so that his inv^tigations
permit but a hmited conclusion in r^ard to the law of the
variations of elasticity as modified by temperature. Finally, the
dilatation caused by temperature was disregarded in the nu-
merical values riven by JKupflfer, a source of serious errors in
view of the metnod of observation which was employed.

A few of the results derived by Kupffer were obtained h^
means of torsional vibrations. Among these are the coem-
cients of temperature for the elasticity of an iron wire (piano
cord), of a copper wire and of a brass wire. After having
apphed to these values the correction reauired for the dilatation
of the vibrating disc, the decrease of the elasticity for 1° ex-
pressed in parts of the total elasticity is found to be

for iron,


("menu'^S. 446)

" copper,


S. 464

« brass,


( S. 467)

These values agree tolerably well with those found by us, with
the exception of the copper wire, in regard to which it is pos-
sible that the divergence may be caused by the circumstance
that we employed a chemically pure metaL

5. The aosolute moduli of elasticity of the iron, copper and brass
wires calcukUed from torsional vibrations and from the velocity of

To determine from our investigations the absolute moduli of
torsional elasticity, it is necessary to know the dimensions and
mass of the wires, as well as the moment of inertia of the vi-
brating weight *

The latter is to be calculated from the mass of the perforatea
lead cylinder =1818 grs. and the interior radius =0-34 c. m. and
the exterior radius =5*07 cm. It is found to be

^(5^*+ 0^*)= 284^0 grs. D c. m.

To this must be added the moment of inertia = 40 grs. nam.
of the connecting pieces and the mirror, which was calculated
from their size and form. The total moment of inertia there-
fore is K=28510 grs. n c m.
Further was determined

Iron. Oopper. Brass.

The length of the wire, fc=20'80 20-76 20-80 c. m.

The mass of one c. m. length, m=0-00d01 0*00655 0-00403 grs.
The density, ^^=7-82 d-00 8-41

The time of vibration at 20% «=l7-35 12-23 20-55 sec.
The radius of the wire, r=00111 0-0162 0-0123 c m.

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on the Elasticity of certain Metak. 866

The modulus of torsion T is derived from these data by
means of the formula, p. 857,

T=!^ -J—
g mr^t^ '

It is found to be

for iron =444 kilometers.

" copper=217 "

" brass =190 "

If it is desired to follow the definition customaiT in practice,
and to refer the modulus to the section instead ol to the mass
of the unity of length, these numbers must be multiplied by
the corresponding densities ; whereby they become

for iron, 3470 knog rammet

' qaadratmlUimeter.

" copper, 1960 **

" brass, 1600 "

The velocity of sound was measured in the same species of
wires by stretching them in a Weber's monochord, and oy vary-
ing the length, comparing the longitudinal tone with tuning
forks. The normal tone of the set of tuning forks was deter-
mined by comparison with two normal tuning forks of Appunn.

The modulus of dilatation of the wire (i a the length of a
wire of the same species possessing the weight necessary to .
double the original length), is obtained from the velocily of

sound c by means of the formula A=— , where g denotes the

acceleration due to gravity. The value A' ordinarily employed
in practice, and made use of by Wertheim, (L a, the weight
which hung on a wire whose section is unity would double the
length), is obtained as above by multiplication with the density.
Thus it was found

Velocity of sound. Modulus of elastidty.

A. A'.

Iron, 6060 meters. 2580 kilometers. 20310 ^ ^m .

Copper, 3640 " 1360 « 12140 "

Brass, 3380 " 1170 " 9810 "

Wertheim gives A' for iron wire =19445, copper =12636,
brass =9000 g^

According to these observations, the moduli of torsion bear
to the modmi of dilatation, for all three wires very nearly the

same ratio. The ratio 7=- is

for iron, 6*86

" copper, 6-23

« brass, 6-16

Mean, 6*07

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866 J, J. Woodward on (he Oxy-cdlcium Light

The deviations of these values fix)m the mean are perhaps dne
to errors of observation, especially in the determination of the
densities, since in order to be sure of investigating the same
material as in the original experiments, but small masses could
be employed. It is possible too that the deviation of the mean
from tne whole number 6, is to be souffht in these errors.

Without entering on a more extended investigation of this
point, and without discussion as to whether the customary
theory of elasticity is applicable to such thin wires, or whether
the lack of isotropy, or the relations of superficies, forbid this
application, it may be remarked that aocoroing to the theory


where /u denotes the ratio of the transversal contraction to the
longitudinal dilatation. By a comparison of the torsional and
longitudinal vibrations, there would result accordingly for our
three wires very nearly the same value for m. If it is assumed
that 4(lH-|w)=6 there results f*=i. This is the extreme value
permitted by the theory, which would correspond to a volume
xmchanged by dilatation.

It may suffice to have indicated the fitct It would lead us
too far m)m the chief object of our investigation to develope
the subject more in detail

Art. XXXYHL — On the Oxy- Calcium Light as applied to Photo-
Micrography ; by Lieut Col J. J. Woodward, Assistant Sur-

feon, U. S. Army. Report to the Surgeon General of the
r. S. Army, dated June 4, 1870.

Since the preparation of my report of January 4, 1870, on the
use of the Magnesium and Electric lights in Photo-micrography,
I have made some experiments with the Oxy-calcium, or Harems
light, as a source of illumination for the same purpose, and have
succeeded in obtaining excellent pictures with powers as hi^h
as a thousand diameters. This result appears to me of consid-
erable importance, both because of the comparative cheapness
of this lignt, and because the apparatus for its production is so
common as to be practically within the reacn of every mi-
croscopist In adcution to these advantages the oxy-calcium
lijght possesses the quality of steadiness to a greater d^ree than
either the electric or the magesium lamp and requires much less
trouble and skiU to managa

For the purposes of nay experiments, I made the hydrogen
as I consumea it, in a Hare's self-r^ulating ffenerator, by the
action of dilute sulphuric acid on scraps of orainary sheet-zinc.

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as applied to Photo-micrography. 867

The oxygen was sometimes made in the usual way from
chlorate ot potassa, sometimes purchased compressed in iron
cylinders ; in either case it was transferred to a large sheet-iron
gasometer for use. The gases were burned under a pressure
equal to a column of water fourteen inches higL I used for
lainp one of the first-class magic-lanterns, manufactured by
J. W. Queen k Co. (No. 924, Chestnut street, Philadelphia,
Pennsylvania,) in which the disc of lime is revolved bv clock-
work before the burning jet of gas and a fresh surmce con-
stantly presented to the nama I simpler removed from the
lantern the lens intended to magnify the image on the slides,
when the apparatus is in ordinary use, and allowed the cone of
liffht proceeaing from the large condenser of the instrument to
mil upon the achromatic condenser of the microscope, in the
same manner as described and figured for the ma^esium lamp
in my report of January 4th, a reference to which will render
any description of the arrangement of the microscope and of
the sensitive plate unnecessary in this place.

I employed the ammonio-sulphate cell, as I do in taking
Photo-micrographs with other sources of liffht, but found I
could dispense with the ground glass which is necessary in
photographing so many objects, if sunlight or the electric lamp
IS employed ; a large portion of the lime disc being luminous,
the resulting mixed divergent pencil, like that obtained from
the magnesium lamp, does not produce the interference phe-
nomena which result when tissues and many other objects are
illuminated by powerful parallel raya This circumstance, how-
ever, renders the calcium light inferior to the sun and the elec-
tric lamp, in the resolution of the Nobert's plate and certain
lined test objects.

I did not find the time of exposure differed materially fex)m
what I had given in making photographs of the same objects
with the magnesium lamp, and th^ pictures produced were not
inferior to these in quality. This arose from the fact that the
greater steadiness oi the ciftlcium liffht permitted the use of con-
densers which concentrated the light to a greater degree than I
had found advantageous with the magnesium lamp, and not
from equality in the actinic power of the two sources of illumi-
nation. I have recently made some experiments with the view
of obtaining positive information with regard to the comparative
actinic energy of the electric, magnesium and calcium lamps
which I enmloy. For this purpose all condensers being re-
moved the divergent pencil proceeding from each lamp in turn
was permitted to fell, for the space oi five seconds, on an ex-
posed circtdar portion of a sensitive plate thirty feet distant

The whole operation was completed in less than a minute,
when the plate being developed m the ordinary way three cir-

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868 J. J. Woodward on the Oxy-cab^xwrn Light

cular spots appeared as the results of the exposures. The spot
produced by the electric light was intensely black, that by the
magnesium of a rich middle-tint, while the circle impressed by
the calcium light was extremely pale. Want of time prevented
me from continuing these experiments and obtaining as I de-
sired numerical values for the relative actinic powers of these
sources of illumination under definite conditions ; this I have
however regretted the less, as the actual energy of the naked
flames is not really the measure of their avaUaDility in photo-
micrography ; here the question of steadiness, involving, as it
does, the possibility of great concentration, plays a most impor-
tant part and materially modifies the result

So far as I know, tne Calcium light has never before been
successfully employed as the source of illumination for making
Photo-micrographs in this country. My friend Dr. R L. Mad-
dox, however, writes me that it has been experimented with in
England by Dra Abercrombie and Wilson. He thinks they
used powers as high as an eighth with pleasing resulta Th^
information has directed my attention to the essay of Dr. Wil-
son in the Popular Science Review for 1867, volume vi, page
64, in which that gentleman gives in detail the process em-
ployed by himself and Dr. Abercrombia He experimented
witn an oil lamp and with the Oxy-calcium and magnesium
lights : " I can scarcely think it would be used now 3iat the
more active light of magnesium is within the reach of every
ona" And of the magnesium: " The light fjdls only in stead-
iness, and if some means could be devised for burning the metal
uniformly and at a fixed point nothing would be left to desire."
Dr. Beale (How to work with the Microscope, 4th edition, p.
248) tells us that some of the pictures of these gentlemen were
remarkably good, " they possess a peculiar delicacy in the half
tones and the shadows, with much roundness of the objects, but
the definition, as might be expected, does not quite equal in
some of the finest markinffs, prints obtained from sun n^ativea"
A perusal of Dr. Wilson^ paper will show that my process dif-
fers from his in the use of the following precautions : the inter-
polation of an ammonio-sulphate cell to exclude the non-actinic
rays, the use of lenses specially made for photography for all .
powers from the Jth down, the use of mucn larger condensers
to concentrate the light and so to shorten the exposure, and in
the case of the magnesium light, in the use of^ a clock-work
lamp to increase the steadiness of the illumination. Each of
these points are in my judgment essential to obtain the best

I learn from the same letter of Dr. Maddox that he had him-
self made experiments with the magnesium lamp some time be-
fore those of Abercrombie and Wilson. He used powers as

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as applied to Photo-TmcrogTaphy. 869

Idgh as a fifth, and appears to have obtained better results than
I supposed any one had done prior to the publication of my re-
port He gives me the follo¥ring account of his experience :
" The first picture I took with the magnesium light was done in
a very rude way. An inch and a quarter of wire was held in a
small spirit flame and advanced by hand as burnt The objec-
tive was Beck's fds, the object the sycamore-leaf insect, and
about Jths of an inch of wire remained after use. I sent a print,
with a sun print of the same to the British Journal of Photogra-
phy, and m the number for July 1, 1864, you will find some
remarks by myself and the editors. Now to try and meet any
error that might arise fi'om wliat we may term want of correc-
tion, I used the |ds with the correcting lens, which is excellent
for sunlight ; the picture was soft, full of half tone, but wanted,
as in other pictures I had seen by artificial light, the decision of
definition in the outlines,*' " After this I us^ the ith ¥rith the
little apparatus sketched in Beale's book (page 276) and which
I venture to think, embraces all that is required for its use, pro-
vided the condenser has its focus at the burning point, and that
the reflector has the sama" With the }th, fibers of cotton and
muscular fibrillas of boiled shrimp, ¥rith several other objects
were tfJcen, but I did not use any mgher power, nor indeed pay
much attention to tlie subject as I gave the preference to the
sunlighted prints and negatives" i give these extracts with
great pleasure as showing the experience in this direction of one
of the most distinguished laborers in the field of Photo-micro-
graphy, and regret that I was not acquainted with them at the
time of publishing my first report The method of Dr. Maddox,^
however, differed from mine m the same essential points as that
of Abercrombie and Wilson, and the peculiar ntness of the
magnesium light for photographing the animal tissues and those
objects generally, which require the use of ground glass when
sunlight is employed, would appear to have escaped the observa-
tion of these accomplished gentlemen, and to have remained un-
noted until the publication of my report

In conclusion I append to this paper two illustrative photo-
graphs. The first, which represents tne 6th square of the Hol-
lers type-plate of the diatomaceas, taken with Wales's IJ inch
objective, arranged to give thirty-five diameters, will serve for
comparison with the photograpliJB of the same object with the
same leAs taken by sunlight and by the electric and magnesium
lamps, which were published with my former report The sec-
ond represents the Navicular Lyra, taken with the Powell and
Lealand's immersion j\ arranged to magnify 1000 diameters.

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870 A. Hall an the Secular Pertarhations of the Planets.

Abt. XXXIX. — On tlie Secular Perturbations of (he Planets;
by Asaph Hall.

In view of the recent consideration in geological speculations
of the secular inequalities of the excentricity of the earth's
orbit, it may be worth while to state briefly the method of
treating secular perturbations, and pur present knowledge of the

Denoting by e, ^r, 9, and the excentricity, the longitude of
the perihelion, the inclination of the orbit and the longitude of
its ascending node, it has been found easier in the discussion of
the problem to substitute for these Quantities four others which
are simple functions of them ; and tnus are assimied

h=ie sin n, p-=itSLng <p sin d.

Izzze cos n, ^=ztang 9 cos B.

Neglecting terms of the third and higher orders of the excen-
tricities and inclinations in that part of the development of the
perturbative function from which the secular perturbations
ari e, linear diflferential equations of the first order are estab-
lished containing the first diflferential co-efficients of A, i, j? and
q with respect to the time ; and it is by the integration of these
equations that the secular perturbations are obtained. By this
process also the equations are separated into two distinct classes
admitting of separate treatment ; the one class containing the
diflferential co-emcients of h and I and the solution furnishing
the values of the excentricities and the longitudes of the peri-
helia, and the other containing those of p and q and the solu-
tion giving the values of the inclinations and the longitudes of
the ascendinff nodea In this way the discussion of the prob-
lem is much simplified. The numerical co-efl5cients in the
diflferential equations depend on the semi-major axes of the
orbits, and on the masses of the planets. Considering the eight
principal planets of our solar system, there will be of course
eight values of A, Z, p, and j, and these are usually distinguished
by the addition of accents to the symbola

In order to eflfect the integration, Lagrange, whose method is
stiU followed, assumed

A= N sin {a ^-f j?), I =N cos {gt+^y •

A'=N' sin {g t+(t) l'=W cos (^ t-\^), Ac

Diflferentiating these equations and substituting the value of

/7A /7A' /27

JtT) "^ff jTt &C., in the diflferential equations and then eliminating
cCt at at

the ratios of the coeflScients N, N', N", Ac, a numerical equa-
tion is obtained for the determination of ^. This equation will

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A. Hall on the Secular Perturbations of the Planets. 871

rise to the degree denoted by the number of planets considered,
and in our solar system will be of the eiffhth degree. If we
denote by gr,, a,, <7, - — g^ the roots of this equation the
general integral will be

A=Ni sin (^i<+ft)-f N, sin (^,^-(-/?,)-f ....+Ng sin (^rf+ft),
l=N, co8(^i«-H?i)+N, cos (^,«+fe)+.— +Ngcos078«+ft), ,,.
A'=N\sin (^le+ftHNVin (^^+/?,)+.. -fN'eBin (ff^t+^s), ^^^
r==N\cos(^,e+ft)+N'|Cos(^,<+ft)+...- + N'8Cos(^8«+ft),

The arbitrary quantities Ni, N'l /?i, /?», &a, are determined

by the initial values of h and L The solution for p and q is
quite similar to that for h and I

The conditions necessary for the stability of the system are,
first, that the eight roots of the equation for g shall be real and

Online LibraryRodolfo Amedeo LancianiThe American journal of science and arts → online text (page 95 of 109)