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tion a diminution of the magnetic moment of the needle. I think that this strange effect must be explained thus:-The direct extra current of the coil, without magnetic effect in its interior, momentarily increases the polarization of the pile, whence a current of very perceptible depolarization in consequence of the interruption; this current partially demagnetizes the needle. We have here, therefore, to do with a reflex effect, analogous to that of the condenser of the Ruhmkorff coil, although incomparably more feeble.
If the circuit contains two coils, P and Q, one very powerful, the other very feeble, the direct extra current of P and the reflex action of the pile succeed each other in Q, and produce a very singular effect. Interruption of the circuit augments greatly the magnetic moment of a needle magnetized in Q by a great number of passages; but the same operation diminishes the magnetic moment of a strongly magnetized needle placed in Q with its south pole to the left of the principal current. This effect is much more pronounced when P contains a core of soft iron. Example:
Needle magnetized in Q by passages. 1.16
Needle strongly magnetized
11.10 . 36.28
Thus one and the same complex instantaneous current magnetizes needles which were not magnetized, and partially demagnetizes needles which were strongly magnetized. The same thing can be easily reproduced by the alternation of a continuous direct current and a weaker inverse current.
We thus see that, in certain cases, magnetized and nonmagnetized steel needles permit us to analyze a complex instantaneous current, and recognize in it a change of sign.
[To be continued.]
XII. On the Mathematical Theory of Mr. Baillie Hamilton's
HE investigation which follows was completed more than a year ago (1873). The endeavour was then made to test the theory by experiment; but difficulties were encountered which, notwithstanding the assistance kindly afforded by Mr. Hamilton, were not got over. The subject has now been brought repeatedly before the public by Mr. Hamilton, who has read * Communicated by the Author.
papers before the Physical Society, the Ashmolean Society of Oxford, and the Musical Association; and in view of the interest generally expressed about it, I have thought that an account of the theory, as far as it goes, may be acceptable to the readers of the Philosophical Magazine. The two sets of observations given towards the close of these remarks were made by me at Mr. Hamilton's laboratory in November 1874.
It is not my purpose to describe the instrument here; it will be sufficient to point out that it produces continuous tones by means of combinations of strings and harmonium-reeds. There is a separate string to each note, and to some point on each string the extremity of a harmonium-reed is attached. The reed is then set in vibration by wind; and the problem is to determine the forms of vibration of the combination.
The method of the following investigation is substantially that employed at p. 139 of Donkin's 'Acoustics,' with the extensions necessary for the purposes of the problem.
The instrument may be regarded as a string loaded at the point of attachment to the reed, subject also at the same point to a force tending towards the position of rest, and varying directly as the displacement of the point from that position; this force represents the elasticity of the reed.
μ be the load at the point of attachment,
T the tension of the string,
p mass of unit of length of string,
a elastic force of reed per unit of displacement,
7 the length of the string,
b distance of point of attachment from one end,
Then the equation of motion of the point of attachment is
Assume (see Donkin's 'Acoustics,' pp. 119, 139)
y= sin m(l—b) sin mx(A cos amt + B sin amt)
from x=0 to x=b, and
y= sin mb sin m(l—x) (A cos amt + B sin amt)
from ab to x=1.
-a2m2 sin mb sin m (l—b) (A cos amt + B sin amt).
And taking the difference of the values of
given by the two
equations when x=b, we have
Hence, in order to satisfy (1), we must have
μa2m2 sin mb sin m(l—b) =Tm sin ml+a sin m(l—b) sin mb,
(ua2m2-a) sin mb sin m (l-b)=Tm sin ml.
This equation determines m, while A and B remain arbitrary unless the initial circumstances of the motion are given.
Put up, where X is the length of the string, the weight of which is equal to the load.
Put =a2v2, so that av= where is the periodic time of
the reed with the load vibrating alone. This combination will be referred to as "reed alone."
Then a=a22. pλ。, and equation (5) becomes
(m2 —v2)λ sin mb sin m(l—b) = m sin ml.
Now a is the velocity of transmission along the string. Let be the periodic time of the vibration actually sounded, λ the corresponding complete wave-length on the string (i. e. λ= twice the length of a single segment).
Similarly, if A be the wave-length on the string of the note of the reed alone,
This transformation is convenient; but it must be remembered that the notes denoted by λ and A are those which would be
given by single segments of lengths 2 respectively. Making the substitutions above indicated, the equation (6) becomes
which is the complete equation of the problem. The calculation of A in the general case presents great difficulty; and before a comparison 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 (λ). 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.
I. If n, rare integers and
bor 1-b, the left-hand side
of (7) vanishes; equating the right-hand side to zero, we must
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=λ, 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 λ be indefinitely diminished, we have ultimately the right-hand member of (7)=0, as before, and the string can sound any of its ordinary harmonics.
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; con
sequently it may be expected, from the symmetry, that minute exactness of the position is not of special importance. Putting, 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=1 (the reed the octave of the string), and
as a pair of values such as may easily occur, and con4TT 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
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.
Pitch, in equal. Pitch, referred to lowest note of combination.
Although it has not been possible to get a complete determination of the elements of any experiment, yet the following