>v 



Prof. J. J. Sylvester on Spherical Harmonics. 301 

 we know the bipotential is of the form H ( — ¥ , cos a j. 



For r, r' substitute respectively — . Then we may regard the 



case as that of an exterior and interior point of prise, and con- 

 sequently from the last case we have 



Hg,cos») = ^F(icos«). 



If we compare the two expressions Ff-^ 1 , cos a) and 



a 2 „/a 2 \ 



— 7 -b I — T? cos a ) 



rr \rr / 



respectively applicable to two internal and two external points 

 of prise, it will easily be seen that it leads to the following 

 theorem. Let there be two concentric spheres, and let any 

 two radii cut the first and second surfaces in the points P, Q 

 and P / , Qf respectively ; then the bipotential of the first sur- 

 face with respect to P', Q / as the points of prise, is to the bipo- 

 tential of the second surface with respect to P, Q as the points 

 of prise in the ratio of the squares of the radii of the two sur- 

 faces to each other. 



This is a theorem of precisely the same kind as Ivory's for 

 the comparison of the attractions (or, if we please, the potentials) 

 of two confocal ellipsoids in the particular case when they 

 become two concentric spheres, and may be verified by precisely 

 the same geometrical method. For we have only to divide the 

 two spherical surfaces into corresponding elements m, m f by 

 radii drawn in all directions to meet the two surfaces, and it 

 is evident that we shall have the distances inF f and m'Y equal, 

 as also mQf and m'Q. And, moreover, the ratio of any two 

 corresponding elements m, mf will be as the square of the radii, 

 which evidently establishes the theorem in question. It may 

 further be noticed that the relation between the bipotentials 

 in the three several cases considered, may be deduced from 

 the fact that each such radical as 



V 1 - 2Lv -2ky-2lz + lr + P + 1 1 



where /r + lc 2 + I 2 is greater than unity, may be put under the 

 form 



1 _ _ _ 1 



where lt n h n l, and h, h, I are the coordinates of two points tho 



