32 ELECTRICITY [ELECTKOSTATICAL THEORY. He finds a solution, bh (i l *~l)e 1-t di It is then easy to find F(r), and write down the general expressions for the potential. Poisson goes on to show that the density at the point of contact of the spheres is zero. He finds, for the mean density on the two spheres 1 and b respectively, r* - bh f t 1- -l i+ 6 y LI *. 1+6 this being, in fact, the value of /(O), A / e and B = 6(1 + 6) 1+6 _ , - dt . 1 - t He shows that the calculation of the ratio of A to B may be reduced to the calculation of the first of these integrals only. For the difference 4ir6 2 B - 4irA between the charges on 1 and b he finds the elegant expression 1 + 6 T+l from which it follows that the whole charge is always greater on the sphere of greater radius. He then calculates the value of for various values of b, and its limit for 6 = 0, and next the ratio of the densities at the two points diametrically opposite the point of contact, and finds for the mean density on each of two equal spheres in contact A = A log 2. He also calculates for this last case the ratio of the greatest to the mean density. In the case of two unequal spheres, the ratio of the greatest density on the smaller to the mean density on the larger.is found for various values of b. He then passes on to investigate the densities for various values of /i. Plana All these results are compared with the measurements and of Coulomb, and found in satisfactory accordance with them. In his first memoir, Poisson considers the case where the distance between the spheres is great compared with the radii ; and in a subsequent memoir he considers the case of two spheres at any distance. Plana (Sur Id distribution de I electricite a la surface dea deux Spheres, Turin, 1845) extended the calculations of Poisson, using much the same methods. He also calcu lated approximately the mean densities in the case of several spheres in contact, and arrived at results which agreed satisfactorily with the experiments of Coulomb. For a table of his results, see the end of the first volume of Biess s Reibungselectricitdt. An account of the work of Roche, who also followed in the footsteps of Poisson, will be found in Mascart, t. i. p. 290 sqq. Synthe- The researches of Green led him to a very valuable tical me- synthetical method, by means of which we can construct thod of an i n fi n ite number of cases where we can find the electri cal distribution. Suppose that we take any distribution whatever of electricity, for which we know the potential at any point, and consequently the level surfaces. Take any level surface, or parts of level surfaces, inclosing the whole of the electricity, and suppose these level surfaces to become actual conducting sheets of metal. Suppose the electrical distribution inside to be rigid, and connect the sheets of metal with the earth, so as to reduce them to potential zero. The sheets will become charged in such a way that the whole potential at every point in them and external to them is zero. Let now U be the potential at any external point due to inside distribution, and V that due to the charge on the sheets, then we have everywhere on or outside the sheets, U + V = 0, or V = - U. Now U is constant at every point of each sheet ; hence V is so also. Hence the distribution to which V is due is an equilibrium distribution per se. Removing now our internal distribution, and changing the sign of that on the sheets, we have a distribution of electricity in equilibrium on a set of conductors of known form, the potential of which at any external point is V = U, where U is known. Also the potential V is clearly constant inside every conductor. Hence, applying the characteristic surface equation, we get for the density at any point of any of our conductors the expression ~fa ~dv We might make this a little more general, and state our result thus : If we distribute on a level surface or sur faces of any electrical system, completely inclosing that system, electricity with surface density at every point <r j -T- , this distribution ivill of itself be in equili brium, and the potential at any external point will be kU. We have given a physical demonstration of this import ant theorem. The mathematical reader will easily see the application to this case of the general reasoning about the solution of V 2 V = 0, of which we have already given examples. For a simple but interesting case of this general theorem, see Thomson and Tait s Natural Philo sophy, vol. i. 508. To Sir William Thomson we owe the elegant and Methodi powerful methods of "Electric Images" and " Electric " f Sir ^ Inversion." By means of these he arrived, by the use of SOI) ni simple geometrical reasoning, at results which before had Electric required the higher analysis. We shall endeavour to images, illustrate these by two simple examples. We do not follow the methods of the author (for which, see his papers), but take advantage of what we have already laid down. Let A be any point outside a sphere (fig. 12) of radius a, and centre C. Let AC=/, and take B in CA such that CB CA = a 2 , = -; then it is easily proved that, if P be any point on the sphere, BP_a AP~/ Hence if E be any quantity of electricity, we have Fig. 12. _7 E 0. E AP BP Therefore, if we place a quantity E of electricity at A, and a quan tity - -%E at B, the sphere will be a level surface of these two, that, namely, for which the potential is zero. Another level surface of the system is evidently an infinitely small sphere surrounding A. Hence it follows, from the theorem of Green which we have jus-^t discussed, that a distribution of electricity on the sphere, the T> density of which is given by g -r- (where R is the resultant force due to E and - ^ E at any point of the sphere), together with a quantity E at A, gives a system in equilibrium, the potential due to which at any point outside the sphere is the same as that of E at A, and _ - E at B. It appears, therefore, that the action of the electricity induced on the uninsulated sphere by the electrified point A is equivalent at all external points to the action of - ^E at B. The electrified point B is called by Sir Wil liam Thomson the electrical image of A in the sphere. It is obvious that the whole charge on the sphere is - ?E, and we can very easily find the density at any point. In fact, resolving along CP, which we know to be the direction ng e, th of resultant force, the forces due to A and B, we get
E