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We might substitute an ordinary galvanometer for the differential one, but balance the current by an independent battery, as we frequently do in the comparison of electroThe arrangement is that

motive force.

shown in fig. 3. A small change dp of p will show itself as a deflexion of the galvanometer. The sensitiveness is the same as before, with the exception that the E resistance 9 of the auxiliary battery is a disadvantage. Ifb is large, we now have

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Fig. 3.


If we can make a small, we come back to the original equation; and I have introduced this example merely to show that however much we may vary the form of the experiment, we always come back to the same limiting value.

The method most usually adopted for a comparison of resistances is the Wheatstone-bridge, which is generally assumed without further discussion to be the most delicate arrangement.

Fig. 4.

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It seems advisable, therefore, to discuss a little more fully the conditions under which the best effects can be obtained. If Р is the resistance to be measured, we see at once that if a change of resistance op produce a current ip/(p+g) in the galvanometerbranch, the resistance s must be small and q large compared to p and g. The arrangement is therefore very different from that we are usually told to adopt. To determine the effect of a small electromotive force dp in the branch p, let i1, i2, i3, i4, y be the currents in p, q, r, s, and g respectively; then, in the usual way, we find in terms of the current in p, (pr-qs)i

y= g(s + r) + s (q+r)'

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When the resistance p after being balanced is changed to Sp, the current through the galvanometer will be

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In order that dy should be as great as possible, s must be as small as possible. If it can be neglected compared with p, we

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If, further, 9 is large compared with p, and therefore r large compared with s, we obtain the fundamental equation,

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If all resistances are equal, which is often supposed to be the most favourable arrangement, the smallest detectable change of resistance dp is given by

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that is to say, we only obtain half the maximum sensibility. In the general case the best galvanometer-resistance is found to be

g= ?(p+s),


and by introducing this value into equation (4), we obtain

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This equation shows what we lose in sensitiveness by making s too large or q too small. If Carey Foster's method is adopted, p=q, and the sensitiveness is necessarily reduced in the ratio of 2:1.

The results here obtained are valid so long as we have sufficient battery-power at our command. If we are limited to a fixed and insufficient electromotive force, the conditions are altogether different, and in that case the problem has been fully discussed by Mr. O. Heaviside (Phil. Mag. 1873, xlv. p. 114).

M. Ch. Eug. Guye*, in a valuable paper, has discussed the best arrangement of a Wheatstone-bridge for bolometer measurements, but he only considered the special case of the equality between two and two of the four resistances. His results are in agreement with those given above.

In comparing the different methods of measuring resistance, we see that there is really no theoretical advantage of

* Archives des Sciences physiques et naturelles, January 1892.

the Wheatstone-bridge over any other balance method. We obtain for all of them, on the contrary, the same limiting value. When a differential galvanometer is used, the two opposing currents pass through the coils of the instrument. Whenever there is any danger of disturbances in the indications of the needle due to heating-effects within the galvanometer, the Wheatstone-bridge will have the advantage. I do not know whether there are any actual measurements as to the strongest currents we can send through a mirror-galvanometer; but it seems likely that when the resistances to be measured are such that weak currents only can be used, as with a platinum thermometer or bolometer, the differential arrangement will possess many advantages.

It appears that, in order to obtain the best effects from a Wheatstone-bridge, the resistance to be measured must adjoin one which is small and one which is great compared to itself; and these resistances must of course be constructed so that they are not overheated when the maximum current passes through the resistance which is to be measured. This we may not be able always to accomplish easily, and in practice the sensibility of the Wheatstone-bridge may have to be reduced still further. I think, therefore, that for measurements of resistance in which a high degree of accuracy is required, the differential galvanometer deserves more attention than it has so far received. If galvanometers can be constructed so that convection-currents due to heat-effects in the cell containing the suspended magnets can be avoided, it is very probable that the differential galvanometer will often prove to be the most suitable instrument.

With respect to the maximum current which we may be capable of sending through galvanometers of the same type, it is to be noted that the magnetic field at the centre of the coils is proportional to the square root of the electric work done in the instrument. When the heating-effect has reached the maximum allowable limit, the magnetic fields in different instruments having similar coils, but wound with different wires, will be the same.

The proposition embodied in equation (4) may give some useful hints in the construction of bolometers or platinum thermometers. Thus we may ask the question whether for certain purposes some other metal may with advantage be substituted for platinum.

The rise in temperature of a conductor St is connected with the electric work (W) by the equation


where s is the heat-capacity in electric units (water =449). For a certain allowable gain in temperature per unit time and a given mass, that material is the best which has the greatest specific heat. A metal of low atomic weight would therefore have the preference. If the volume of the substance be taken as fixed, we must make the product of density and specific heat as great as possible. This product is highest for iron and steel (90), then follows copper ('80), platinum (72), and gold (62). There would be an appreciable advantage in substituting a steel wire for a platinum one, if small differences of temperature are to be detected by a change of resistance, especially as the temperature-coefficient of electric conductivity is considerably higher than that of the other metals. There are, however, obvious disadvantages connected with the use of steel which in most cases would counterbalance its greater sensibility.

XVIII. Ronayne's Cubes.

By Professor H. HENNESSY, F.R.S.*

FEW years since I was presented by Mr. S. Yeates A with a pair of equal cubes, one of which could pass obliquely through the other. These objects had been for a long time in his father's possession, but there was no record as to where or by whom they were made. At first I paid little attention to the cubes, as I looked upon them merely as mathematical curiosities. Not long since, on perusing Gibson's History of Cork,' I came on a passage extracted from Smith's History of Cork,' published in 1750, in which a pair of interpenetrating cubes is distinctly referred to as the invention of Mr. Phillip Ronayne, of the Great Island near Cove (now Queenstown). The cubes were said to be constructed after Mr. Ronayne's design by Mr. Daniel Voster, who was at this time teacher of mathematics in Cork, and editor of the well-known book on Arithmetic, compiled during the early part of the last century by his father, Elias Vosterf. *Communicated by the Author.

+ Passage in Smith's History of Cork':

"Not far from the castle of Belvelly is Ronayne's Grove formerly called Hodnets Wood; [now Marino] a good house and handsome improvements of Philip Ronayne, Esquire. From the gardens one has a charming view of the river and shipping up to Cork, as also the town of Passage on the opposite shore. This gentleman has distinguished himself by several essays in the most sublime parts of mathematics; among others by a treatise on Algebra, which has passed through several editions, and is much read and esteemed by all the philomaths of the present time. He has invented a cube which is perforated in such a manner

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I was led by this passage in Smith's History' to more closely examine the cubes, together with the box in which they are contained. The box is mahogany, and it is fastened by hasps of an old-fashioned pattern; it is itself manifestly in an old style of work. One of the cubes is made of hard wood, the other consists of a brass shell or frame, in which two pieces of hard wood fill up the vacant space so as to constitute a complete cube. The brass work was not a casting, but the result of combining pieces of hammered or sheet-brass suitably cut out which were welded or soldered together, and the whole was manifestly finished by the use of file and hammer. The only mark on the brass is the number 1, placed on a part of the shell as a guide for inserting one of the supplementary wooden pieces. This figure is undoubtedly in the old style, and Mr. Yeates is distinctly of opinion that the brass work is old-fashioned, as he satisfied himself, by comparing it with old brass instruments, that it could be very well assigned a date in the early part of the eighteenth century. It was most probably the work of an amateur, or done under the direction of a mathematician for a special object; for if it had been made by an instrument-maker for the purpose of sale a number of similar brass cubical shells could more easily have been made by casting.

I have never seen any demonstration of the structure of the cubical shell, but it soon appeared manifest that it is based on the properties of a square. Lay off on the diagonal of one of the faces of the cube from its middle point two parts, each equal to half the side of the cube. Draw lines at the extremities of these perpendicular to the diagonal, and two isosceles rightangled triangles will be formed which constitute the bases of two equal triangular prisms, between which a cube equal to that from which the prisms were cut can slide. If the cube slides parallel to the faces of the original cube the prisms will be totally unconnected, and the whole problem turns upon their material connexion while allowing the sliding cube to pass between them.

The annexed figures show the cubes separately, and also

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that a second cube of the same dimensions may be passed through the same, the possibility of which he has demonstrated, both geometrically and algebraically, and which has been actually put in practice by the ingenious Mr. Daniel Voster of Cork, with whom I saw two such cubes." (Smith's History of Cork,' 1st edition, 1750, vol. i. p. 172.) This passage is quoted by Gibson, who says that Daniel Voster was probably father to Elias, the author of the Arithmetic. But this is not correct-Daniel was the son of Elias, as I find from the eighth edition published in 1766 that Daniel was editor of the book to which the name of Elias is prefixed as the author.

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