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carry out my e of Plates and

ed (in a memor ansactions of

eden (viz. putti gbeen gratef cret committee

the resistances of the different branches of the bridge arrange-
ment-under the limiting supposition, however, that the line
used for duplex working was perfect in insulation, or, more gene-
rally, that the real conduction-resistance of the line could be neg-
lected against the resistance of the resultant fault*.

It now remains, therefore, to investigate if the simple relations given are generally true; or if not, what they become

paper, that Phi case the line has an appreciable leakage. In fact as a dearty

the case of practical importance; since all overland aes, essers
ally long ones, even if constructed on the best known prizepes,
will always have a very considerable leakage; i. e. the rentance
of the resultant fault (1) will generally be by no means very arge
in proportion to the real conduction-resistance L of the ne
In order to obtain the best general solation of the printem,
must conduct the investigation with great catan taat s
e must be careful not to introduce beforehand any reason 6-
een the different variables, however couvement, that s 105
essarily a consequence of the paramount ennaran a le in-
d for duplex telegraphy, i. e. regularity of aquats.
hus it will be seen that the present general estigation
be conducted somewhat differently from the special me
in the First Part.

confessedly wit Spectrum (obser

vellously minute t from the surfaces tract from large v aces of hydrocarb ; for I adopted a chemical purity ned with prejudice rming those conclus riginal investigation overed by eyes tran a gentleman occup will either support hdraw them altoge

must, however, be understood from the beginning har
er the best relations may be which should enst between
ferent resistances of the bridge method when used in m
et line, these relations must revert to the mean me
efore if we put i=x; and this fact affords a certain meck
correctness of the new relations to be found.

al solution of the first problem for the Brudge Metrot
nexed diagram (p. 110, represents the general tase:

Theory of Duplex

d from vol. xlviii. P. therefore I shall refer in the present paper.

is investigation xlviii. p. 138) the be ver as Swan worked, Then shed on spectrosc rum in flames not conta at I discoved, the spectr azzi Smyth's communic of regret that my duties appointed within a month um of carbon, have quite s. That regret is much t k would have been done far Morren, Lielegg, Troost liged to add that neverth estigation in other d is the only notice 1. A. Societ

eral mathematical question when she saved fr
graphy has been stated as follows:-

HITY OF SIGNALS-D and S are two functions which
lly equal to zero when no variation in the system iezara
any given variation in the system must be 19 mal
nd approximate rapidly towards zero as die voration
becomes smaller and smaller.

ese two functions D and S were expressed, my fir


EN 1 A

E'N' μ' m2TM"

on of the terms "resultant fault."

inction," "real insulation," "measured nslation...
**real conduction. *
frequent occurrence in this paper, see my Teating in-
M. Section I.

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sequently it may be expected, from the symmetry, that minute exactness of the position is not of special importance. Putting, ι

then, b=l―b=2, (7) reduces to


which gives an infinite number of values of λ when A, λ are assigned.

I now assume A=7 (the reed the octave of the string), and ι as a pair of values such as may easily occur, and con

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venient for calculation, for the sake of seeing the general nature of the results to be expected.

The equation (8) can then be put in the form

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tan π

5868 1.441




referred to


The numbers placed under the head, in the Table which follows, are approximate values of the first five roots of the above equation. Proceeding further, we should find a root lying between every consecutive pair of integers.

The second column contains the values of the ratios reduced







to equal-temperament semitones; it gives the pitch of the note sounded with reference to the octave of the string.

The third column gives the pitch of the note sounded with reference to the lowest note of the combination, both in equaltemperament semitones and by description.


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Pitch, in equal. Pitch, referred to lowest note of combination.



perament semitones.






Flat major tenth.

Sharp minor sixteenth.

Flat two octaves and tritone.

Sharp two octaves and minor seventh.

Although it has not been possible to get a complete determination of the elements of any experiment, yet the following

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observed successions of overtones may not be devoid of interest. There are four notes of the arrangement which I shall call I., and three notes of II. The pitch in semitones is appended for comparison. In both cases the point of attachment was nearly, though not exactly, in the middle of the string.

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Comparing these numbers with the overtones indicated in column 3 of the calculated Table, it will be seen that they follow, as far as they go, the general course indicated by theory in the hypothetical case assumed; and it may be inferred that this case furnishes a rough representation of the circumstances of the two experiments examined.

The above results are the only experimental ones which I know of.


The calculation of the length of the middle segment in the hypothetical case follows easily from the numbers in column 1 of the calculated Table. The fundamental, of course, has for its segment the whole string. In the other cases, expressing


λ 4.714.



λ 7 in terms of, which is the length of each segment except the 2' middle one, we get the middle segment at once, since we know the actual number of segments. (It is hardly necessary to remark that the numbers of segments in the successive overtones are the odd integers, by the symmetry.)


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Note observed.


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Pitch of overtone, in semitones.








Ratio of middle segment to any other segment.





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So that, as the pitch of the note sounded rises, the reed diminishes more and more the segment to which it is attached, as compared with the others. Of course this remark is confined for the present to cases resembling the hypothetical case.

The note employed may be either the fundamental or any one of the overtones. As these are in general all inharmonious to each other, only one can be used at a time. But it is probable that, in particular cases, some two or more may become harmonious; and they would then be capable of combining in a true periodic motion.

XIII. Carbon and Hydrocarbon in the Modern Spectroscope. By
W. MARSHALL WATTS, D.Sc., Physical-Science Master in the
Giggleswick Grammar School*.

N the January Number of the Philosophical Magazine appears a paper with the above title by Professor Piazzi Smyth, which calls for a few words from me by way of explanation.


1. Professor Smyth inquires "why, since for cometary work the reference-spectrum should be of feeble intensity, I do not examine it in that shape, viz. as given by the blue base of the flame of a small alcohol lamp, or the all but vanishing globule of flame when a common gas-light is on the point of going out from inanition?" The answer is simple, that with a spectroscope of six prisms the loss of light is so great that in the spectrum of a blowpipe-flame there would not be more than one line (5165.5) bright enough to be measured, and it was my object to employ as large a dispersive power as possible in order to secure as great accuracy in the determinations as I could.

The same reason explains why, "although the spectrum consists notably and notoriously of five bands, viz. the orange, citron, green, blue, and violet," I only give measurements for three of the bands: the orange and violet bands were not bright enough to be measured accurately.

2. An equally simple explanation solves the "strange problem" why the lines 5165.5 and 5585-5 are the best-determined.

5165.5 happens to fall close to the magnesium-line b, whose wave-length we know with great accuracy from the labours of Ångström.

5585.5 happens to be exactly coincident with an iron line in the solar spectrum. The first band of the citron group, although brighter than the second, does not fall near to any marked line in the solar spectrum which could be used as a reference-line; and its determination is therefore not quite so exact.

* Communicated by the Author.

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3. Chemical Parentage of the Spectrum under discussion.—I freely admit the force of Professor Piazzi Smyth's remarks on the difficulty of volatilizing carbon; but that does not appear to me to affect the experimental evidence for my assertion that "this spectrum is the spectrum of carbon, and not of a hydrocarbon or any other compound of carbon." That evidence is very simple; this spectrum can be obtained alike from compounds of carbon with hydrogen, with nitrogen, with oxygen, with sulphur, and with chlorine.

Whether or not the spectrum is produced by the vapour of carbon is another question; but if this spectrum is, as Professor Piazzi Smyth asserts, that of a hydrocarbon, will Professor Piazzi Smyth explain how it is possible to obtain it from cyanogen, a compound of carbon and nitrogen, when no hydrogen is pre sent? I have just repeated the experiment with cyanogen for perhaps the fiftieth time. Dry mercuric cyanide was heated in a test-tube, and the gas evolved was dried by passing through a tube containing phosphoric anhydride; it then passed through a tube provided with platinum wires, the end of which dipped below warm and dry mercury. On passing the discharge from an induction-coil between the platinum wires a spark was obtained which gave the spectrum in question brilliantly, the gas being decomposed and carbon being deposited.

Professor Piazzi Smyth says that in May 1871, in a paper sent to the Royal Astronomical Society, he "gave such extracts from the authorities on either side as showed that the spectroscopists declaring for pure carbon, in opposition to those pronouncing for carbohydrogen, were blundering little less than the perpetual-motion men of last century." Permit me to quote from a paper communicated by myself to the Journal of Science for January 1871:

"At first sight it would appear that carbon is an element unlikely to yield a discontinuous spectrum, inasmuch as it is not known in the gaseous condition; and that if we obtain discontinuous spectra from carbon compounds, they must be due to some compound of carbon. Thus the bright blue lines observed by Swan (1856) in the spectrum of the Bunsen-flame might be supposed to be more probably due to carbonic oxide or carbonic acid than to carbon itself. But we find that these same lines occur not only in the spectrum of the flame, but also in the spectra obtained by passing the electric spark either through carbonic oxide, or olefiant gas, or cyanogen, and the lines thus found to be common to compounds of carbon with different elements must of course be due to carbon itself. Whether they are really produced by carbon in the gaseous state is a question which cannot yet be certainly decided. If the carbon is in the solid

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