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provided by a pot of mercury well drowned in oil and condenser, only about one break in fifteen succeeded. On the other hand, of three bullets fired so as to cut the primary wire (no condenser) two succeeded; while for the failure of the third there was some explanation. The bullet without condenser was now distinctly superior to the best ordinary break with condenser.

The next step was the substitution of a rifle-bullet, fired from a service rifle. Here again the bullets were reduced to about one-half, and after cutting the wire were received in a long box packed with wet sawdust. At 60 mm., while the mercury-under-oil break with condenser gave only feeble brush-discharges, good sparks were nearly uniformly obtained from the bullet working without a condenser. At 70 mm. the bullet without condenser was about upon a level with the mercury-under-oil break with condenser at 60 mm. As regards the strength of the primary current, if there was any difference, the advantage was upon the side of the ordinary break with condenser, inasmuch as in the case of the bullet the leads were longer and included about 8 cm. of finer copper wire where the bullet passed.

In the next set of experiments upon the same Apps' coil excited by three Groves, the bullet was used each time, and the comparison was between the effect with and without the usual coil condenser. At 55 mm. the bullet without condenser gave each time a fair or a good spark, while with the condenser there was nothing more than a feeble brush scarcely visible in a good light.

The single pane of coated glass was next substituted for the usual condenser of the coil, with the idea that possibly this might be useful although the larger capacity was deleterious. But no distinct difference was detected when the bullet was fired with this or without any condenser.

In the last set of experiments now recorded the primary current was raised, six Grove cells being employed partly in parallel, and the wire was cut each time by a rifle-bullet. At 90 mm. no spark could be got when the coil condenser was in connexion; when it was disconnected, a spark, good or fair, was observed nearly every shot.

Altogether these experiments strongly support the view that the only use of a condenser, in conjunction with an ordinary break, is to quicken it by impeding the development of an arc, so that when a sufficient rapidity of break can be obtained by other means, the condenser is deleterious, operating in fact in the reverse direction, and prolonging the period of decay of the primary current. It is hoped that the establishment of this fact will inspire confidence in the theory,

and perhaps suggest improvements in the design of coils. The first requirement is evidently the existence of sufficient energy at break, and this implies a considerable mass of iron, well magnetized, and not forming a circuit too nearly closed. The full utilization of this energy is impeded by want of suddenness in the break, by eddy-currents in the iron, and (in respect of spark-length) by capacity in the secondary. It is to be presumed that in a well-designed coil these impediments should operate somewhat equally. It would be useless to subdivide the iron, or to reduce the secondary capacity, below certain limits, unless at the same time the break could be made more sudden. It would not be surprising if it were found that the tentative efforts of skilful instrument-makers have already led to a suitable compromise, at least in the case of coils of moderate size. The design of larger instruments may leave more to be accomplished.

LVIII. Applications of Elastic Solids to Metrology.
By C. CHREE, Sc.D., LL.D., F.R.S.

[Concluded from p. 558.]

Standards of Length.

§ 19. on a perfectly smooth and horizontal plane, we find

F a rectangular prism of horizontal length L be placed

from (36) for the elastic increment in the length at a height

h

2

+above the plane (h being the vertical dimension of the prism) the formula SL/L=―(1—2n) (p/E) +ng (p−p') (h/2E)—gz{np—(1—n)p'}/E. (66)

Here p represents, as before, the pressure in the liquid or gaseous medium surrounding the prism at the level of the C.G., i. e. at the height 1/2 above the plane. This result should apply to an ordinary standard yard, and to most commercial standards of length, if supported throughout the entire length on a perfectly smooth table. If, as in the standard yard, the scale is divided on the upper surface z=h/2, we have for its elastic stretching

8L/L= − (1 − 2n) (p—gp'}')

E

=-P/3k,

(67)*

[Oct. 1901.-A formula apparently equivalent to this is given without proof by Mr. Chaney, on pp. 86, 87 of the Procès Terbaux of the International Committee of Weights and Measures for 1899. It is given as applicable to the standard yard, without explicit reference to the method of support, and is illustrated numerically.]

where P is the pressure in the medium at the level of the divided surface, and k is the bulk-modulus.

When the prism is not of rectangular section, or when, being of rectangular section, it is supported on points or roller surfaces situated in a horizontal plane, we have to content ourselves with the mean change in the length of the longitudinal "fibres." For this we find from (1), H being the height of the C.G. of the section above the supports,

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SL/L=—(1—2n)(p/E)+ng(p−p') (H/E), (68) where p is the pressure in the surrounding medium at the height H.

In the case of the rectangular section, with the base uniformly supported, it follows from (66) that the mean change in the length is the change actually occurring at the level of the C.G.; but we are not entitled to assume that this is true generally. This is a question of some theoretical importance in view of the now common practice of dividing standards of length along the so-called "neutral" surface, i. e. the horizontal plane containing the C.G.

§ 20. To afford a more exact idea of the problems actually arising in metrology, I give in fig. 1 some representative

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forms of cross section in standards of length. The sides shown vertical in the figure are vertical in ordinary use ;

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M

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the numerals refer to the dimensions, and are in millimetres. A refers to a standard yard, B to the international prototype metres of the so-called X-section, C to a working standard belonging to the Bureau International, D and E to deflexionbars used in magnetometers. A and C are divided on their upper surfaces, B on the neutral plane, D on one of the vertical faces; E has holes on its upper surface into which a plug fits. B and C are copied from vol. vii. of the Bureau International's Travaux et Mémoires. Their shape is devised partly with the object of facilitating equalization of temperature throughout the bar. Most modern standards are supported not over the whole lower surface but on two symmetrical rollers, or on three points, one at one end of the bar, and two-in the same cross section-at the other end. This mode of support is intended to promote uniformity of temperature, the bar being surrounded by liquid, and to remove the uncertainty as to the distribution of surfacewhen a pressure bar rests on an ordinary table, and not on an ideal smooth plane *.

An exact solution of the elastic problem presented by a heavy bar supported on points or rollers has not yet been obtained even for a rectangular section, and the best thing to be done is probably to apply the ordinary approximate Bernoulli-Euler solution. From the researches of St. Venant, Pearson, and others we have grounds to believe that for bars like standard yards and metres, whose length is a large multiple of their greatest lateral dimension, the BernoulliEuler solution represents a high degree of accuracy except perhaps in the immediate vicinity of the supports. Actual observations on standards of the types B and C by Broch and Benoit (Trav. et Mém. 1. c.) seem to bear this out. The Bernoulli-Euler method of solution is so well known that it is unnecessary to describe it, and the results which I am now about to give are deducible from the solution without much difficulty. Some of them have, I find, been given by Airy and Broch, l. c., and possibly this remark applies to more than I am aware of; but I do not think they are generally known.

Applications of Bernoulli-Euler Method.

§ 21. In figs. 2 to 6, OB represents the half of a bar of uniform section w and total length 27 supported symmetrically at two points in a horizontal plane at a distance 2a

* See Airy (Phil. Trans. for 1857).

apart. From the symmetry the tangent at the middle point O is horizontal; this is taken as axis of x, the axis of y being drawn vertically downwards. In fig. 2 the support shown, A, is comparatively near the centre. In such a case O is the Fig. 2.

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highest point of the bar bent under its own weight, and the curvature is of one sign throughout. In figs. 3 to 6 the supports are at a greater distance from the centre, and the curvature changes sign between O and A; this is the normal condition in modern standards of length. The scale to which the ordinates are drawn is the same in all the figures 2 to 6, the bending being much exaggerated.

The notation employed is as follows:-E is Young's modulus, p the density, 21 the complete length of the bar, the cross section, w (= gpw) the weight per unit length, x2 the moment of inertia of the cross section about the perpendicular through the C.G. to the plane of bending. In the absence of gravity the bar would be strictly horizontal; y denotes the vertical displacement, when gravity Phil. Mog. S. 6. Vol. 2. No. 12. Dec. 1901.

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