Rodolfo Amedeo Lanciani.

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unequal, in order that outside the circular functions there may
be no terms containing the time as a fector or exponent and
which would therefore increase indefinitely ; and secondly, it is
necessary that the coefficients N may not be great in order
that the excentricity may not increase so as to render divergent
the series which have been assumed in the solution to be rapidly
convergent

The actual numerical solution by several eminent astrono-
mers, Lagrange, Pontecoulant and Leverrier, their results being
essentiaUv accordant in this respect, shows with a great degree
of probaoility that our solar system is a stable one, the law of
gravitation alone being considered; although to speak with
certainty on this point an analytical solution is to be desired.
But when it is required to compute for very remote epochs the
values of the elements of the orbits the co-efficients g which in
equations (1) are multiplied by the time must be carefdlly con-
sidered. These coefficients depend on the assumed masses of
the planets, and are generally determined by neglecting terms
of tne third order. The most complete investigation of this
subject is that given by Leverrier in tne Connaissance des Tems
for 1848 and 1844, and reproduced with some additions in the
memoirs of the Paris Ooservatory for 1856. In this work,
which is a masterpiece of astronomical calculation, Leverrier
shows that terms of the third order mav produce corrections of
the values of g amounting to three or tour tenths of a second.
Probable uncertainties in the assumed values of the masses of
the planets may give rise to errors of nearly two tenths of a
second. Hence, if we consider the forms of the general inte-
grals (I), we shall readily see that for very remote epochs, dis-
tant by millions of years, our calculations must be very untrust-
worthy ; since when the time is great the errors in the values of
g may completely change the character of the circular functions.



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372 F. E. SUmpaon on Farmer's Theorem.

For a satisfiictory solution of tliis problem, certainly one of
the most interesting in astronomy, our knowledge of the masses
of the planets must be greatly advanced In the case of the
interior planets it appears that we must wait patiently until the
theory of their own motions, or the motion of some one of the
periodical comets, shall fiimish the data for an exact determina-
tion of their masses. The masses of Mars and Jupiter will in
time be very accurately known fix>m the theories of some of
the minor planets. But in the case of Saturn, Uranus and
Neptune it appears to me that the instrumental means are
already at hana for making an accurate determination of their
masses, and a more complete investigation of the theories of
their satellites. When the novel and entertaining observations
with the spectroscope have received their natimJ abatement
and been assigned their proper place, it is to be hoped that
some of the powerful telescopes recentiy constructed may be
devoted to this class of observations, where a rich and an ample
field awaits the skillful observer. One could not wish a better
example than the beautiful work of Bessel on the satellites of
Jupiter.

August 2, 1870.



Art. XL. — Farmer's Theorem discussed; by Fbed. E.
Stimpson.

Toward the close of Prof Silliman's paper, " On the relation
between the intensity of light produced from the combustion
of illuminating gas and the volume of gas consumed " (this
Jour., xlix, 17), is the following: — "A comparison of the
foregoing results will show that the coincidences, with the
requirements of the theorem of Farmer are, within the limits
assigned, too numerous, and too closely accordant, to be
considered as otherwise than pointing clearly to its general
truth."

What I propose to examine now is, whether the data given
in the paper referred to, do warrant the conclusion reached.

According to the data given for the first experiment, two
lights were made equal, so that the disk of a Bunsen Photo-
meter stood midway between the flames, and the consumptions
were found to be 3*66 feet per hour.

" The screen was then moved upon the bar to a point just
four times as far from one flame as it was from the other, L a,
the bar being 100 inches, the screen stood at 80, i e., as 1 : 4.
The light from the distant burner was then increased, until the
disk again showed as an equality of illumination. On reading



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F, E, Stimpson on Farmer's Theorem,



878



the rate of the gas consumed by the two burners respectively,
one gave 8 '66 cubic feet and the other 7*82 cubic feet"

Now fix>m this it will be seen that the ratio of consumption
was as 1 to 2, but the ratio of the lights being as the squares of
the distances from the disk, becomes 200' : 80* or 1 : 16, and
the result on the paper should stand,

8-66*:7-82«=l:16andnot
8-66»:7-82«=l:4.

In other words, the lights were proportional to the fourth
powers of the consumptions and not to the squares.*

In experiments 2, 3 and 4, the ratio of the squares of con-
sumption is nearer to the ratio of the lights than the simple
ratio of the consumptions, but in experiment 2, the ratio of the
2*67 powers of the consumption almost exactly expresses the
ratio of the lights. In experiment 8 the ratio of the squares is
too small, and in experiment 4 it is too great; while in ex-
periment 6, the simple ratio is certainly much nearer than the
ratio of the squares.

The foUowmg table will, I think, suffice to show these various
relations :



Ko.
Xxp*t.



Bate of

ooniump-

tionper

bonr.



Intentltj

found by

expert-

ment.



Inten*
old Uw.



Intensltj
bjlaw ''

tbe
■qoAret.



.^ Jlnteniltyby
ofltbe f*«7 pow-
er of con-
snmptlon.



Dtiference between calcnlftted
and observed reeolU.
Law of a^
Old law. ■qoaree. power.



3-30
4-36
6136
6-556

8-72
4*884
6000
7-218

4-6
9-619

5-16
1006



1-
2-1
3-2
4-0

1-
2-
8-
4-

1-
4-

1-86
400



1-

1-32
1-55
1-68

1-

1-312
1-61
1-94

1-
2114

1-86
3-70



1-

1-73
2-42
2-83

1-

1-72
2-60
8-76

1-
4-46

1-86
5-72



1-

2-098
3-222
3-995




-0-78
-1-66
-2-32


-0-628
-1-39
-206


- 1-886


-0-30




-0-37
-0.78
-1-17


-0-28
-0-40
-0-24


+ 0-46


+ 1-72




-0002
+ 0-022
-0-006



In experiment 6, the reductions have been made by applying
the old rule to the correction for the candle in both cases ; the
correction by Farmer's Theorem bein^ applied only to the con-
sumption of the gas, and so applied gives the most concordant
results, the difference shown by the old rule however, (1-84
candles) is no greater than might have occurred between two
observations, even if the consumption in both cases had been

[* Since the above was written, I have been infonned that the error is not in the
proportion of (** 1 to 4 ") but in the statement of the position of the disk upon the
bar. It should read fwioe as far fhmi one flame as the other, L e., the screen stood
at 6S|, and thus the experiment supports the theorem.]



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874 F. E. Stimpson ati Farmer's Theorem.

eq^ual, i e., five feet per hour. But this experiment alone cer-
tainly does not warrant the conclusion reached, viz : — " That
this theorem applies with equal force to the weight of sperm
consumed by the standard candle as to the volume of the gas
burned in equal times," because the correction has not been
applied to the candle either here or in any other observation
given in Prof SiUiman's paper.

The confirmation obtained by experimenting upon Peytona,
Albert and WoUongong gas def)ends upon the assumption
that the true candle power of a rich gas can be obtained by
mixing it in definite proportions, with another gas whose illu-
minating power is known, and deducing fix>m the observed
candle power of the mixture, the candle power of the rich gas ;
until this assumption has been proved to be correct^ it is of
course useless in establishing Farmer's Theorem.

The next proof offered is drawn fix>m a tabular statement in
Sugg's Gas Manipulation. Prof Silliman says, " By this state-
ment the burner in question produced fix)m* five cubic feet of
gas exactly 15 (14?) candle power, but when reduced to 4*5
cubic feet consumption the candle when 'corrected to the
standard quality of gas by proportion ' was only 11-93 candles.
The values of the * correction ' referred to can only be conjec-
tured, but assuming that the observation made the uncorrected
rendering 11*32 candles (a very probable quantity), we find that
the law of the squares of consumption then makes the ratio as
follows: — 4'6'':il*82=5**14." Tne assumption here made is
not at all necessary because we can find the exact value of the
* correction' for the gas by reversing the proportion used^
thus : — 5 : 11*93— 4*5 : 10*73, and now applying Farmer's Theorem
the ratio becomes 4*5>: 10*73=5^: 13*24 only. The relation can

be exactiy expressed by the ratio 4*5* : 10-73=5* ; 13*997=14.
The last proof offered is drawn from Audoin and Berard's
experiments. But after several fruitless efforts to obtain the
results, as given under the head of " Intensities by law of the
squares of consumption," I am forced to conclude that these
figures are incorrect through some inaccuracy in applying that
law. The two tables referred to may be found in tne author's
original memoir on pages 439 and 441 of Annales de Physique
et de Chimie, voL Ixv, 1862. In these experiments they com-
pared the burning of two batwing burners at different rates of
consumption, with a " Bengel A^gand " whose rate of consump-
tion was nearljr constant The first three columns of the table
below give their experimental data.



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F. JE Stimpson on Farmer^ $ Theorem.
Tablb L Burner of the fifth aeries — dit J^ inch wide.



875



Contomp-
tion of the

under trial.


ConemnptloD

of ttie Bengel

Argand.


Comparative

Intenaltlee.

The Bengel

bnmer


Intenttles by
old formula

or direct ratio

of conaump-

tion.


Difference
between cal-
culated and
obaerved
results.


Intensities 1 Difference
Ifor law of between cal-
the squares culated and
of con- observed
sumption. results.


Liters per
hour.


Utersper
hour.








V




V




211


100


120


99-38


-0-62


82-31


- 17-69


189


96


110


97 64


-2-36


86-38


- 13-62


180


103


100


100-00


000


10000


0-0


166


104


90


104-86


+4-86


12217


+ 2217


142


104


80


102-39


+ 2-39


131-06


+ 31-06


124


101


70


99-66


-0-36


141-84


+ 41-84


102


101


60


103-82


+ 3-82


179-68


+ 79-68


88


102


60


99-30


-0-70


19719


+ 9719


68


100


40


102-79


+ 2-79


264-20


+ 164-20


67


102


30


90-73


-9-27


293-40


+ 193 40



Table IL Burner of


the same series— slit


^3 inch wide.


Consump-
under triaL


Consumption

of the Bengel

Argand.


Comparative
mtenslttes.


Intensities b7

or direct ratio

ofconsump-

tton.


Difference
between cal-
culated and

results.


the squares

of con-
sumption.


Difference
between cal-
culated and
observed
results.


"^JTurr


Liters per
hour.


264


106


200


100-4


+ 0-4


60-3


- 49-7


236


106


180


1001


+ 01


66-6


- 44-4


208


106


160


100-9


+ 0-9


63-6


- 36-6


182


106


140


100-9


+ 0-9


72-8


- 27-2


162


104


120


102-6


+ 2-6


87-7


- 12-8


130


104


100


100-0


00


100-0


00


112


104


80


92*8


- 7-2


107-8


+ 7-8


90


104


60


86-6


-13-4


1261


+ 26-1


76


104


60


86-6


-134


160-2


+ 60-2


66


104


40


78-8


-22-2


166-2


+ 66-2


43


104


20


60-6


-39-6


182-8


+ 82-8


28


104


10


46-4


-53-6


216-6


+ 116-6



Prom the first table it will be seen that the two burners
gave equal light (intensity 100), when the batwing consumed
180 liters per hour, and the standard 108 litera Correcting the
first line of this table for these consumptions, by the old rule
we have for the batwing 211 : 180= 120 : 1
and for the standard 108 : 100= l\V\

the total correction for both

will be therefore 211xl08:180xlOO=120x?:ZxZ';

the I cancels out and leaves

the proportion 211 X 108 : 180 X 100= 120 : V.

The second line of this table

becomes 189x108 :180x 96=110 :Z

and so on.

Now applying Farmer's Theorem, the proportions become
(211)» X(108)«:(180)« x(100)»=120 : V
(189)«X(108)«:(180)»X( 96)«=110:r.



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876 F. R Stimpson on Farmer's TTiearem.

The value of V and Z" from these proportions will be found
in the 4th and 6th columns of the above table.

It will be seen that in the second table the burners gave equal
light for consumptions of 130 and 104 ; these numbers nave been
used therefore in the corrections for that table.

The experiments show that the last term of the proportions
should be 100. The tables show that by the old formtda, this
term becomes about 100 for every experiment of the first table,
and for the first seven experiments of the second table, the
greatest difference being 7*2 ; and for the remaining five experi-
ments, the old formula gives the best approximate results except
for the last one, and here the old law gives a result which must
be muUiplied by 215 to make it correct, and Farmer's Theorem
gives a result tnat must be divided by 215 to make it correct

From a perusal of these various results, I am led to disagree
with Prof. Silliman, and say that * Farmer's Theorem ' is not
proven, and that the law of the squares does not in general give
any closer results than the old law of the direct ratio ; though
I entirely and heartily concur with him in the conclusion, thai
every photx>metric observe)* should recognize the importance of bring-
ing the consumption of gas and sperm to the agreed standard^ whoi
attempting to give the true candle power of any gas.

It is much to be desired that experimenters should turn their
attention to the matter of the relation of consumption of gas to
illuminating power, and I sincerely hope that Mr. Farmer will
not let the matter rest here, but will iniEtke and publish further
observations upon the same subject

Messrs. Aud!oin and Berard in their valuable experiments, to
determine the best burner for the city of Paris, proved that for
every consumption of gas there is a burner whicn is best suited
for that consumption. Now by the proper selection of a burner
for small consumptions, some simple relation may yet be found
between consumption and illuminating power.

In the above, reference has been made only to experiments
detailed in Prof SiUiman's communication. I have, however,
made a number of experiments myself upon the same subject,
besides coUatinc the results of some sixty or more independent
observations, which have been published during the last fifteen
years, and the results are curious, instructive and unexpected.
There are a few among them to which Farmer's Theorem mi^ht
be applied, quite a number to which the old law will apply ;
thougn many of them require a modification of the old law.

I hope in due time to prepare a paper giving some of these
resulta

April, 1870.



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JBw SiUiman — Note to Stimpson's Paper. 877



Abt. XLL — Note on Mr. StvmpeorCs Paper on Farmer's Theo-
rem; by B. Sii-liiMAN.

Mr. Stimpson's criticism on the first experiment of my paper
was induced by an obvious numerical error in the statement of
the data, which Mr. S. has himself corrected in a foot note on
p. 378. The experiment properly stated exactly sustains the
theorem.

I admit that in Experiment No. 2, the exponent comes nearer
to the third power than to the square, being 2 '689, 2*688 and
2*669 ; in the three cases average 2*661. But it also indicates
that the illuminating power of the standard 8*80 cubic feet was
very imperfect The experiments, however, appear to me to
demonstrate clearly the radical inaccuracy of the old rule for
photometric calculations, and that some ratio near to or greater
than the square gives often more trustworthy results than the
old rule.

In Experiment No. 8, the exponents are respectively 2*552,
2*297 and 2*092, coming in the last very near the square. It
shows that 4*88 cubic feet consumption gave the best results,
and also that 3*72 and 7*219 cubic feet consumption gave very
nearly the same degree of intensity of combustion.

In Experiment No. 4, the exponent 1*86 power for a con-
sumption of 9*519 consumption, Slows an imperfect combustion
of the fish-tail burner employed.

In Experiment No. 5, the experimental conditions were
wholly un&vorable to accuracy, owing to an inequality of
pressure unavoidable in the experimental method adopted, there
Deing one inch pressure on the 10*06 c. £ consumption, and
only half an inch on the 5*16 c. £ It is well known to all pho-
tometric observers how important a low pressure and an equal
pressure is to the results obtained. It was hardly fair to Mr.
Farmer to have quoted this trial, but I was desirous of exhib-
iting the entire range of observation, bad as well as good.

As in Experiment No. 6, the consumption of sperm was in
the two tests very nearly uniform ; the diflference would be very
trifling if the correction had been applied. Nor is it by any
means so certain that Farmers Theorem applies to the candle as
I supposed when the remarks quoted by Mr. Stimpson were
made, since, if a candle bums much over 120 grains it smokes
("tails off"), and then there is an end of all accuracy, and the
observation must be rejected ; since there is an imperfect com-
bustion giving an increase in consumption but not in intensity.

Mr. Stimpson rejects all the data given bjr me which are
founded upon the determination of the intensity of a rich gas
Am, Jour. Sol— Sbcond Sbkibs, Vol. L, No. 160.— Not., 1S70.
24



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878



B. SHUimun — Note to StimpstyrCs Paper.



by the * method of mixtures/ because, as he says, the accuracy
of this method has not yet been demonstrated by experiment
But this objection ceases to have force now that experiments,
made by Mr. Farmer and myself lately, prove that this method
is worthy of confidence. Inasmuch as the most important case
for the use of Fanner's Theorem is that of gas too rich to be
burned in the standard burner on a basis of 5 cubic feet per
hour, it is striking oS by all means the most valuable portion
of my contribution to photometrical methods if it could be
shown that the * methoa of mixtures ' was untrustworthy. I
am, therefore, glad of this occasion to reiterate my confidence
in this method, and to refer the reader curious in such matters
to a brief paper of mine upon this subject, which will be found
on page 8y9.

I care very little whether the results of experiment shall
show, when they are sufficiently accumulated, tbat the ratio of
consumption of gas is to the intensity produced as the squares
of consumption m a given case, or in some other ratio greater
than a simple ratio. I have desired chiefly to call al^ntion to
the general untrustworthiness of all photometric observations
which are made with volumes of gas much less than the normal
standard adopted, when these results are calculated on the
simple ratio of the consumptions. It is only by the accumula-
tion of carefuUy conducted experimental data that a law can be
established, ana these data are dow pretty rapidly accumulating.*

* In the proceedings of the American Association for Advancement of Science,
Salem meeting, I presented the matter referring to 8ugg^$ manipukttkm in a dif-
ferent form, giving the results of the obseryations with his Argand burner in a
tabular form as follows :







DlMerYfttloiii






ObeerrfttlonA




JS:


Cubic


corrected by


Differ-


Uncorrected


corrected bj


DIAfiT-


feet.


old rale
candles.


encee.


obeerratlons.


Fuiner*B
Theorem.


eacok


1.


5-


1400




1400


14-00




2.


49


13-78


0-22


13-604


14060


0-656


3.


48


13-74


0-26


13190


14-312


1132


4.


4-7


18*30


0-70


12-602


14-148


1-646


6.


4-6


13-04


0-96


11-996


14191


2*105


6.


4-6 •


11-93


207


10-738


13-256


2-617



The mean candle power of the 6th column is 13-99 candles; diflbrence 0*01 candlB.
The following will show the fractional power required to bring the uncorrected
observations (column 6) to 14*00 candles.



2.


4-91-88 :


13-604 :


6l-« :


14*00 oandlea.


3.


4-81-*?


13* 90 :


6l*7.


1400


4.


4*7l*»


12-602


6l»:


14-00


6.


4*61*88 :


11*996


: 6i*8e :


14-00 "


6.


4-6«-0 :


10*737


6^M8:


1400 "



This table shows that the 3rd and 6th tests have not been good ones, while ^
2d, 3d, 4th and 6th tests fall a littSe below the square or 2d power, and flie 6th
test £b considerably more.

New Haven, Julj, 1870.



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R SilMman on determining the Photometric power, etc. 879



Art. XLIL — On the Determination of the Photometric Power of
a rick gas by dilution, with a poor gas of known value: the
^^ method of mixtures;" by B. SiLLiMAN.

In a paper on * Fanner's Theorem/ * I have given several
examples of the method of determining the intensity or photo-
metric power of a rich gas by diluting it with several times its
own volume of a poorer gas of known intensity, and then calcu-
lating its value from the increment of intensity. Having dem-
onstrated in the paper before mentioned the worthlessness of all
determinations oi the intensity of gases of high illuminating
power made by burning them in volumes less than five cubic
leet, and then calculating their intensity by the rule of three up
to that volume, I have shown how much more exact results
were obtained when the results were calculated upon the theo-
rem of Mr. Farmer; this greater exactness being predicated
largely upon the confirmation drawn from parallel observations
UDon the same gases when measured by the method of mixtures.
Tne results thus obtained having, however, been questioned by
Mr. Stimpson,f on the ground that the method itself had not
been experimentally demonstrated, I have undertaken lately, in
connection with Mr. Farmer, to make some experiments calcu-
lated to test its accuracy.

The results which go to support the accuracy of the method
were obtained with the use of a new photometric apparatus,
constructed for the Manhattan Gas Co., under my direction, by
Sugg of London, and which was designed to embrace all the
best approved features which recent experience has indicated
in photometry. A discussion of these details would be out of
place in this connection. Before detailing our results, it will
DC proper to present the method of determination of intensity
for gas of hign illuminating power as practiced by Mr. Farmer
at the Manhattan Gas Works in New York, and which I have
called the method of mixtures.

To find the candle power of a gas having, for example, an
intensity sreater than 20 candles, mix the rich gas of untnown
power with a poorer gas of known power in such proportions
that the intensity of the mixture shall not be greater than 20
candles power, when consumed at the agreed rate of not over
five cubic feet per hour. Then to compute the candle power
(intensity) of the rich gas, —

* This Journal, II, xKx, 17 ; also Proceedings of American Association for Ad-
Tancement of Sdenee, Salem meeting, 1869, p. 149.
f See page 272, this velume.



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380 B. SiUiman on determini'ng the Photometric power of a

Let a=the percentage or volume of gas of low intensity.
^^ d=:the intensity in candles of the gas of low intensity.
^' o=the percentage or volume of rich gas used in the mixture.
^^ df=the mtensity in candles of the mixture as observed.
^^ a;=:the intensity in candles of the rich gas required.

Hence, ?*±^ = rf <^r*)?+rf=a,

And this expression is stated arithmetically in the following

Rule: — iSubtract the intensity of the poor gas Jrom the intensity
of the mixture; multiply the remainder by the volume of poor gas;
divide the product by the volume of rich gas; add to the quotient the
intensity of the mioced gas^ and the sum is the intensity of the rich
gas sought

Now when we reflect that in any given illuminating gas we
have always a certain volume of non-luminous combustible gas,
as the substratum to which is added, according to its source



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