Ellipsoidal Shells and of Solid Ellipsoids. 407 



the lateral surface also contributes nothing, the outer cap 

 gives F'ds' if F' be the field-intensity there. The matter 

 within the space is ads, and therefore by Green's (or Gauss's) 

 theorem of the surface integral of normal force over a closed 

 surface Y f ds' = — ±irads. But if matter were distributed 

 over d$', and similarly over the rest of the surface of the 

 confocal, so that the field remained unaltered, the surface 

 integral oyer the short tube just described would still be 

 F7/s', and the matter on ds' would now be ads. This 

 distribution is unique for the given level surface, and the 

 given field external to it. Since the surface S' is ellipsoidal, 

 and the potential is constant within it, the distribution upon 

 it effected as described, by carrying the matter out from the 

 initial homoeoid, is also homceoidal ; otherwise the distribution 

 over the surface producing uniformity of potential would not 

 be, as it can be proved to be, unique. 



It is easy to verify this latter point as to the nature of the 

 distribution. The matter on c/s / is now ads, and for a we 

 may write ftp where /3 is a constant. Hence the new 

 surface-density a r — ftpds/ds' . But if p' be the length of the 

 perpendicular from the centre for <^s', and a , b', c' = \/a? + du, 

 ^b 2 -\-du, \/c 2 -r-du, we have p'ds f =pds . afb'c'labc, so that 

 a' = /32j'abc/a'b'c l , thus a' varies as p\ that is, the distribution 

 is homoeoidal. 



We can now imagine a further step of potential taken 

 from the surface S' to a succeeding confocal and so on, until 

 we have carried the whole distribution of matter to a surface, 

 every point of which is at an infinite distance from the 

 original surface. There the surface-density will be zero, 

 and the potential at infinity, which has not been altered by 

 the transference, will be zero. It is thus seen that the equi- 

 potential surfaces of the external field of the original 

 homoeoid are ellipsoids confocal with the original homoeoid, 

 a well-known result which has been otherwise established 

 above. 



The matter in the transference passes from confocal to 

 confocal, so that the matter on an element of one is carried 

 to the corresponding element on the next, and so on. Thus 

 at any stage of the transference when the distribution is on 

 a given confocal the matter which was originally on the 

 element ds of the original homceoid is situated on the element 

 of the confocal which corresponds to ds. The transference 

 is along the hyperbolas which are the orthogonal trajectories 

 of the confocals, or, as they are often called, the lines of 

 force of the field. 



29. Another way of dealing with this problem of equivalent 



