240 THEORY OF NEWTONIAN FORCES. [PT. I. CH. V. 



its own charge. But if the distribution should be cut up into 

 small parts, new surface densities would appear on each part. 

 To prevent this the distribution must be supposed split up into 

 infinitely thin shreds along the lines of polarization on separating 

 these from each other no new surface densities would be formed, 

 so that the energy as calculated would be the work obtained 

 by letting these shreds be bodily removed to infinite distances 

 from each other. Similarly polarized shreds side by side of 

 course repel each other, so that this energy is positive. If we 

 should further break up each shred into infinitely short lengths, 

 and separate these from each other, we should have to do positive 

 work to pull them apart, and if we should remove all the parts 

 to infinite distances from each other, it has been shown by Lord 

 Kelvin* that we should have to do exactly as much work as was 

 before obtained by separating the shreds. Consequently the 

 energy must be defined by the first operation alone. 



127. Development of Potential of Polarized Body in 

 Spherical Harmonics. We have seen in 123 (7) that 

 the potential due to a doublet placed at the origin is a spherical 

 harmonic. We may develop the potential due to any polarized 

 distribution in a series of spherical harmonics. If we call r and r' 

 the distances from the origin of the attracted point x, y, z, and the 

 point of integration a, b, c, so that 



7-2 _ tf + y2 + ^ /2 _ a 2 + fc3 _|_ c2j 



we have for the distance between the two points, by 100 (22), 

 if r < r, 



where //-, the cosine of the angle between r, r, is 

 (ax + by + cz)/rr'. 



Inserting this value and those of P , P lt P 2 , 100, we 

 have 



1 _ 1 asc + by 4- cz 1 3 (ax + by + czf r 2 / 2 

 = + ~ ~~ ~ + ~ ~~ + 



* Thomson. "On the Mechanical Values of Distributions of Matter, and of 

 Magnets." Papers on Electrostatics and Magnetism, p. 437. 



