GLOBE AND HEMISPHERES. 417 in the cases where the repulsion is supposed inversely as the square of the distance, which I could not have done otherwise, but it is evident that the phenomena will be nearly of the same kind if the planes are not infinitely extended. For if the distance ag be small in respect of the length and breadth of the plane Aa, a particle placed at a will be repelled by the plane Aa with very nearly the same force as if the plane was infinitely extended. It is plain that these 6 last cases agree very exactly with the laws of electricity laid down in the 3rd and 4th hypotheses [Thoughts... Art. 202]. If the lines Bb and Dd touch one another so that [Here the MS. ends. ED.] NOTE 19, ART. 234. Cavendish's Experiment on the Charge of a Globe between two This experiment has recently been repeated at the Cavendish Laboratory in a somewhat different manner. The hemispheres were fixed on an insulating stand, so as to form a spherical shell concentric with the globe, which stood inside the shell upon a short piece of a wide ebonite tube. By this arrangement, since during the whole experiment the potentials of the globe and sphere remained sensibly equal, the insulating support of the globe was never exposed to the action of any sensible electromotive force, and therefore had no tendency to become charged. If the other end of the insulator supporting the globe had been connected to earth, then, when the potential of the globe was high, electricity would have crept from it along the insulator, and would have crept back again when, in the second part of the experiment, the potential of the globe was sensibly zero. In fact this was the chief source of disturbance in Cavendish's experiment. See Art. 512. Instead of removing the hemispheres before testing the potential of the globe, they were left in their position, but discharged to earth. The effect on the electrometer of a given charge of the globe was less than if the hemispheres had been removed, but this disadvantage was more than compensated by the perfect security from all external electric disturbances afforded by the conducting shell. The short wire which formed the communication between the shell and the globe was fastened to a small metal disk hinged to the shell, and acting as a lid to a small hole in it, so that when the lid and its wire were lifted up by means of a silk string, the electrode of the electrometer could be made to dip into the hole in the shell and rest on the globe within. The electrometer was Thomson's Quadrant Electrometer. The case of the electrometer, and one of the electrodes, were permanently connected to earth, and the testing electrode was also kept connected to earth, except when used to test the potential of the globe. To estimate the original charge of the shell, a small brass ball was placed on an insulating stand at a distance of about 60 cm. from the centre of the shell. The operations were conducted as follows: The lid was closed, so that the shell communicated with the globe by the short wire. A Leyden jar was charged from a machine in another room, the shell was charged from the jar, and the jar was taken out of the room again. The small brass ball was then connected to earth for an instant, so as to give it a negative charge by induction, and was then left insulated. The lid was then lifted up by means of the silk string, so as to take away the communication between the shell and the globe. The shell was then discharged and kept connected to earth. The testing electrode of the electrometer was then disconnected from earth, and made to pass through the hole in the shell so as to touch the globe within without touching the shell. Not the slightest deflexion of the electrometer could be observed. To test the sensitiveness of the apparatus, the shell was disconnected from earth and connected to the electrometer. The small brass ball was then discharged to earth. This produced a large positive deflexion of the electrometer. Now in the first part of the experiment, when the brass ball was connected to earth, it became charged negatively, the charge being about of the original positive charge of the shell. When the shell was afterwards connected to earth the small ball induced on it a positive charge equal to about one-ninth of its own negative charge. When at the end of the experiment the small ball was discharged to earth, this charge remained on the shell, being about of its original charge. Let us suppose that this produces a deflexion D of the electrometer, and let d be the largest deflexion which could escape observation in the first part of the experiment. GLOBE AND HEMISPHERES. 419 Then we know that the potential of the globe at the end of the first part of the experiment cannot differ from zero by more than where V is the potential of the shell when first charged. But it appears from the mathematical theory that if the law of repulsion had been as (+), the potential of the globe when tested would have been by equation (25), p. 421, 0.1478 x qV. 1 d Hence q cannot differ from zero by more than ± 72 D Now, even in a rough experiment, D was certainly more than 300d. In fact no sensible value of d was ever observed. therefore conclude that q, the excess of the true index above 2, must either be zero, or must differ from zero by less than We may Theory of the Experiment. Let the repulsion between two charges e and e' at a distance r be where (r) denotes any function of the distance which vanishes at an infinite distance. The potential at a distance r from a charge e is V =e · © [TM* $ (r) dr. Let us write this in the form (2) where and f(r) is a function of r equal to fr[["$(r) dr] dr. (3) (4) We have in the first place to find the potential at a given point B due to a uniform spherical shell. Let A be the centre of the shell, a its radius, a its whole charge, and o its surface-density, then α = Απασ. (5) Take A for the centre of spherical co-ordinates and AB for axis, and let AB = b. Let P be a point on the sphere whose spherical co-ordinates are and, and let BP=r, then and the potential due to the whole shell is therefore (8) a sin e do do. (9) 0 0 the upper limit r, being always a + b, and the lower limit r, being a b when ab, and b-a when a<b. We have next to determine the potentials of two concentric spherical shells, the radius of the outer shell being a and its charge a, and that of the inner shell being b and its charge B. THEORY OF THE FIRST EXPERIMENT. 421 Calling the potential of the outer shell A, and that of the inner B, we find by what precedes, In the first part of the experiment the shells communicate by the short wire and are both raised to the same potential, say V. = Putting A B = V and solving equations (16), (17), we find for the charge of the inner shell In the original experiment of Cavendish the hemispheres forming the outer shell were removed altogether from the globe and discharged. The potential of the inner shell or globe would then be In the form of the experiment as repeated at the Cavendish Laboratory, the outer shell was left in its place, but was connected to earth, so that A = 0. In this case we find for the potential of the inner shell when tested by the electrometer Let us now assume with Cavendish, that the law of force is some inverse power of the distance, not differing much from the inverse square, that is to say, let If we suppose q to be a small numerical quantity, we may expand f(r) by the exponential theorem in the form and if we neglect terms involving 9, equations (19) and (20) become |