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Either (3) or (1) gives for x=b
=-a’m2 sin mb sin m(1-6)(A cos amt + B sin amt).
dt2 And taking the difference of the values of
given by the two
dx equations when x=b, we have
- msin ml(A cos amt + B sin amt).
dx Hence, in order to satisfy (1), we must have ma?msin mb sin ml—b)=Tm sin ml + sin m(l—b) sin mb,
(ua?m? — a) sin mb sin m(1–5)=Tm sin ml. (5) This equation determines m, while A and B remain arbitrar unless the initial circumstances of the motion are given.
Put f=pło, where , is the length of the string, the weight of which is equal to the load.
27 Put =a?y?, so that ay= where t is the periodic time of р M
t the reed with the load vibrating alone. This combination will be referred to as reed alone.” Then a=a?y? .pro, and equation (5) becomes
(m? — v2), sin mb sin m(2—6) =m sin ml. (6) Now a is the velocity of transmission along the string. Let I be the periodic time of the vibration actually sounded, a the corresponding complete wave-length on the string (i. e. d= twice the length of a single segment). Then, from the form of (3), (4),
λ" Similarly, if A be the wave-length on the string of the note of the reed alone,
A This transformation is convenient; but it must be remembered that the notes denoted by 1 and A are those which would be
λ Λ given by single segments of lengths
respectively. Making the substitutions above indicated, the equation (6) becomes
which is the complete equation of the problem. The calculation of x in the general case presents great difficulty ; and before a conparison of theory with experiment could be effected, it would be necessary to measure the length, weight, and tension of the string, the distance of the point of attachment from one end, the note of the reed alone (A), and the load (90). The last element cannot be directly ascertained by any means with which I am acquainted.
The following are particular cases in which equation (7) is satisfied.
ηλ 1. If n, r are integers and =b or =1-6, the left-hand side
2 of (7) vanishes; equating the right-hand side to zero, we must ra
a Z have also =l, or
2 λ Now
is the length of a segment of the string which would give the note actually sounded. Hence in this case the note sounded is a harmonic of the string alone. And the case arises when the point of attachment is a node.
It is obviously true that the string can vibrate in one of its ordinary harmonics if the point of attachment remains at rest; for in this case neither the load nor the elastic force of the reed comes into play. But the case is not a solution of the problem.
II. Again, if A=X, or the note sounded be that of the reed alone, the right-hand side of (7) is to be equated to zero, and the note sounded is a harmonic of the string alone, as before. The case is that in which the reed and string would, if independent, vibrate simultaneously. For suppose the attachment severed; the reed will go on speaking its own note and the string its harmonic; and as these are the same note, the motion will go on as if the attachment continued to subsist. This is obviously a possible solution of the problem.
III. if nö be indefinitely diminished, we have ultimately the right-hand member of (7)=0, as before, and the string can sound any of its ordinary barmonics.
This is the case in which the effect of the peculiar arrangement of the reed is insufficient to modify sensibly the normal properties of the string. It is possible that this case may be realized by the employment of a thick and heavy metallic wire.
IV. On the assumption of certain relations between the elements, the formula (7) reduces in complexity. The simplest assumption that can be made is, that the point of attachment is at the middle of the string. According to experiment no discontinuity in the nature of the results arises at this position; consequently it may be expected, from the symmetry, that minute exactness of the position is not of special importance. Putting,
(8) which gives an infinite number of values of when A, o are assigned. I now assume A=l (the reed the octave of the string), and
I o= 47
as a pair of values such as may easily occur, and convenient 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
1 The numbers placed under the head in the Table which
ã 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.
1 The second column contains the values of the ratios reduced
λ 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.
Pitch, in equal. Pitch, referred to lowest note of combination.
-5868 1:441 2:357 3.295 4.25
15-556 25.362 29.876 34.282
Flat major tenth.
Although it has not been possible to get a complete determination of the elements of any experiment, yet the following 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.
Pitch of overtone, segments. observed.
1 3 5 7
1 3 5
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 middlc 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
a I 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.)
Number Ratio of middle segof
ment to any other
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 in harmonious 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*.
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.
* Comniunicated by the Author.