[PENDULUM], where g represents the force of 'gravity : then the time in which the torsion wire vibrates once on its axis being made equal to the time in which a simple pendulum vibrates, we have (using the Kim 1 formula in ELASTICITY), V=3; therefore, as the momentum of inertia for a torsion wire suspending a body of a given form can be computed, and as 1 may be found from the observed time of a vibra tion, the value of a (the coefficient of the force of torsion) can be ascertained from this equation.
The torsion of slender wires was first employed by Coulomb for the purpose of determining the intensities of forces in nature and the laws of their action in circumstances+ which render direct methods in applicable : his experiments were performed with an instrument which be invented, and which lie designated a torsion balance. Under ELECTROMETER is given a description of the instrument and of the method of employing it in finding the laws of electric attractions and repulsions ; and it will, therefore, be sufficient in this place to explain its application in determining those of magnetic action. For this purpose Coulomb adapted to the suspending wire, which was of copper, a small stirrup, as w, also of copper, in which could be placed a meg netiaed needle of steel, Before this was done, however, a copper needle, equal in weight to the magnetised needle which was to be used in the experiment, was placed in the stirrup, and the plate D at the top of the glass case was turned round till one extremity of the copper needle, which turned with the plate," was brought to the zero of the gradua tions on the horizontal circle b a c in the case, the suspending wire being in an untwisted state ; the whole case was afterwards turned round till the needle, still pointing to zero, was in the direction of the magnetic meridian, which had been previously determined. The copper needle was then taken away, and the magnetised needle put in the stirrup ; and as soon as It was at rest iu the magnetic meridian, the suspending wire was twisted by turning the stem E, to which it is attached at the upper extremity of the case, till the index there had passed over some given number of degrees, which in one experiment was 360'. The suspended needle was thus made to deviate from its previous position 104 degrees, in which state the horizontal force of terrestrial magnetism was in equilibrio with the force of torsion ; and the angle of torsion was then equal to 3494° (=860*-104°). On turning the index at E through two revolutions, the needle was observed to rest between the opposing forces, at 211° from its original place, when consequently the angle of torsion was 698i* (= 720' Obtaining in like manner several other angles of torsion with the correeponding"deviations of the magnetic needle, and comparing them together, Coulomb found that the forces of torsion are constantly proportional to the sines of the deviations of the needle.
In order to discover the law of magnetic action with respect to the distances between the attracting or repelling bodies, Coulomb placed a magnetised needle in the stirrup of the balance, and after twisting the wire by turning the micrometer stem at F. on its axis through a certain number of degrees, he observed where the needle rested between the opposing forces of torsion and the horizontal component, of terrestrial magnetism : assuming then that the deviations of the needle were proportional to the forces of torsion, he found that, hi order to make the needle deviate one degree, it was necessary to employ a force of torsion expressed by 35*. The wire being then untwisted, and the magnetised needle placed in the magnetic meridian, Coulomb intro duced in the glass case, in a vertical position, and also in the plane of the magnetic meridian, a niagnetised needle of the same dimensions as the other, so that if the two needles could have approached each other they would have been in contact at about an inch from the extremity of each; but the poles of the same denomination in the two needles being presented to each other, a repulsion took place, and the sus pended needle came to a state of rest between the opposing forces of torsion and of magnetic repulsion. When the micrometer at 13 was allowed to remain in its actual position, the suspended needle was repelled 24 degrees, and consequently it was prevented from returning to the zero point by a force of torsion expressed by the sum of 24 degrees and of the horizontal force of terrestrial attraction (=24 x 35°, or 8401; thus the whole force of magnetic repulsion was expressed by 864°. In a second experiment, the wire being twisted by making the stem at x perform three revolutions (= 1080°) in a direction contrary to that of the 24' before mentioned, the needle rested at 17* from zero : the force of magnetic repulsion was then expressed by the sum of 1097 degrees and the value of terrestrial attraction (=17 x 35°, or 595°), that is, in all, 1692 degrees. On comparing together several experiments of the same nature, and also several similar experiments in which the poles of a contrary denomination were presented to each other, Coloumb found, neglecting small differences which may be supposed to have arisen from the extent and configuration of the needles, that the forces of magnetic repulsion and attraction vary inversely as the squares of the distances.
The "bifilar magnetometer" which was invented by M. Gauss, is a species of torsion balance : it is described briefly in the article TERRES