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§ 43. When there is no load, and an algebraic type of vibration is assumed, the application of the Lagrangian equations is even simpler. Taking, for example, the case of a shaft supported at both ends, we have for the displacement (cf. (b) case 2)

whence

y=nx (l3 — 21.x2 +a3), .

(17)

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(win) (31/630)σpl + (24/5) ŋEIP=0.

Assuming cos kt, and so /n-k2, we have (cf. (6), § 12)

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§ 44. Under certain circumstances an equation of type (1), § 1, may be shown to be true for vibration frequencies. The ordinary differential equation for a frictionless simple harmonic motion is

Md2x/dt2+ Fc = 0, .

(21)

where M is a quantity of the nature of a mass, and F a force of restitution, such as is exerted by a spring. The frequency k/2π of the corresponding vibration is given by

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Suppose, now, that the force of restitution remains the same whether we apply one or a series of loads, M1, M2, &c. When the loads are put on one at a time, the corresponding frequency equations are

1/2 = M1/F,

1/k2=M/F; . . . (23)

when put on all together we have for the frequency equation

1/k2 = (M1 + M2+. . . )/F

=1/k12+1/k2+...

(24)

This is analogous of course to Dunkerley's hypothesis, but it is far from amounting to a proof. Even if we assumed that what is true of transverse vibration frequencies is true of whirling velocities, we should have to prove that the addition of pulleys at different parts of a shaft is equivalent to varying the load without affecting the forces of restitution.

§ 45. The following investigation would seem to show that the result is not in general strictly true, though it may be, and not improbably often is, a close approximation to the truth.

Suppose that a massless shaft of length 1, supported at its ends A and B, carries a mass M, at C (AC=a), and a second mass M2 at D (CD=c, BD=b), the effect of the moment of inertia being negligible in either case.

Measuring from A, and a' from B, we may assume the following types of displacement-derived by considering the bending of the shaft under the weight of the two loads :from A to C, y=n[M1(b+c)x{ l2 — (b+c)2 — x2}

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Taking as usual for the angular velocity, and applying Lagrange's equations, we find after algebraic manipulation 1/ (k2+w2) = M1{a2(b+c)2/3EIl} + M2{b2 (a+c)2/3EI} —R, (26) where

R=M1M2(M1+ M»)a2i2c2(4ab+4ac+4bc+3c2)

÷12EI/{M13a2(b+c)2 + M22b2(a+c)2

+M,M,ab (2ab+2ac+2bc+c2)} . . (27)

For the critical angular velocity answering to whirling we put k=0, and find

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w12=3EI÷M1a2(b+c)2, w2=3EIl-M2b2(a+c)2. Referring to (9) or (10) §13, we see that w, and we are the critical velocities for the shaft when loaded with the mass M, and when loaded with the mass M2.

In order that (28) should agree with Dunkerley's hypothesis R should vanish. It is, however, obvious that R is positive for all values of M/M1, and for all values of a, b,

and c. It vanishes, it is true, when e vanishes, but the two loads then coincide in position. As R is positive, the application of Dunkerley's hypothesis gives a larger value for 1/w, and so a smaller value for w2, than does (28).

This does not entirely disprove Dunkerley's hypothesis, because we are not entitled to assume that (25) accords absolutely with the true type of displacement, and we know from Rayleigh's general theorem that, unless this is the case, the value given by (26) for 2 must be somewhat in excess, and consequently the value (28) for 1/2 somewhat too low. We may however expect, in accordance with Rayleigh's general reasoning, that (26) is a very close approach to the truth; and whilst R is usually much smaller than 1/w + 1/w, it is by no means negligible, unless one of the loads be much less than the other, or one of the three lengths, a, b, and be small.

Considering, however, the various sources of uncertainty, it must be allowed that in the present instance Dunkerley's hypothesis gives at least a fair first approximation. Taking, for example, the fairly representative case presented when the two loads are equal, and a, b, c all equal, we find

1/w2 = (15/16)(1/w? +1/w?).

§ 46. In carrying out investigations in cases where there are two, three, or more loads, the physical significance of the processes is more easily seen by adopting a generalized notation. In the above case, for instance, it will be found that the displacements at the points where the loads occur are really of the types

n'(M1y11+ M2Y12) and n'(My12+ M2Y22),

where Y11 and 12 are the displacements at the point where M1 occurs, due respectively to unit loads at this point and at the point where M2 occurs. (By a well-known general theorem Y12 12 and y2 are equal.) The kinetic and the potential energies vary respectively as

M1(M1y11+ M2/12)2 + M2(M1712+ M222)2 and

(May11+2M1 M2Y 12 + M3⁄4Y22), and the function which appears in the expression for 1/w2 really varies as

M1711+ M2/22

(Y11/22 — Y†2) M, M2 (M1 + M2) ÷ { My11+2M1M2Y12+ My22}. The sign of R in (26) and (28) really turns on the sign of (Y114 22-yi2) •

LXI. Energy of Secondary Röntgen Radiation. By CHARLES G. BARKLA, M.Sc. (Vict.), B.A. (Cantab.), King's College, Cambridge; Oliver Lodge Fellow, University of Liverpool*.

IN

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Na paper on Secondary Radiation from Gases subject to X-rays"† experiments were described which led to the following conclusions:

All gases subject to X-rays are a source of secondary radiation, the nature of which is similar to that of the primary radiation. The absorbability of the secondary radiation is (within the limits of possible error-about 10 per cent. of the absorption coefficient for aluminium) the same as that of the primary radiation producing it.

For a given primary radiation the intensity of secondary radiation from different gases at the same pressure and temperature is proportional to the density of the gas from which it proceeds.

The opinion was expressed that the secondary radiation is due to a kind of scattering of the primary by the corpuscles constituting the molecules of the gas.

Results similar to those which led Sagnac‡ to conclude that the secondary radiation from air was more absorbable than the primary radiation producing it had been obtained, but the evidence was then considered insufficient to lead to a definite conclusion as to the difference in character of the two radiations. A direct method of comparison did not indicate the slightest difference in the absorption of the primary and secondary radiations by similar plates of aluminium.

As the experiments of Townsend§ and Sagnac]] on secondary radiation from metals showed that this radiation was more absorbable than the primary radiation producing it-the change in penetrating power, however, depending on the metal-and as the results referred to led to the probability of a transformation of the radiation by air, further and more careful experiments were made on the subject.

The following method was employed :

:

A beam of X-rays passed through rectangular apertures in two parallel lead screens A and B (see figure). Two screens, C and D, were placed in planes perpendicular to the others,

* Communicated by the Physical Society: read March 25, 1904.
+ C. G. Barkla, Phil. Mag. [6] v. p. 685 (1903).

Comptes Rendus, cxxvi. pp. 521-523 (1898).
Proc. Camb. Phil. Soc. x. p. 217 (1899).

|| Comptes Rendus, cxxv. p. 942 (1897).

in positions shown in the figure, the screen C being just outside the primary beam and D parallel to C at a distance of 15 centimetres.

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In C and D were two square apertures, R and S, 5 cm. sq., placed so that the lines joining corresponding points of the two were perpendicular to their planes.

Behind the aperture S was placed an electroscope M which was carefully shielded from radiations proceeding from all directions except through the two apertures R and S. This radiation entered by a thin paper and aluminium window. Lead plates E and F protected it from secondary and tertiary radiation from metals. The rate of fall of the gold-leaf in electroscope M was then only affected by secondary radiation from air in the primary beam opposite aperture R.

A second similar electroscope N was placed behind thick leaden screens Z in the primary beam, and received a narrow pencil of primary radiation through a small hole in this screen. Absorbing plates could be placed immediately in front of both electroscopes. The rays then entered the primary electroscope N by passing normally through the absorbing plate. The secondary rays of greatest obliquity entering the electroscope M, in their passage through the

* For description see previous paper.

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