594 
Proceedings of the Royal Society 
4. On some Definite Integrals. By Professor Tait. 
The integrals referred to occur in connection with applications of 
the Method of Electric Images. A curious special case was given 
to the Society in July 1875, hut was not inserted in the “Pro- 
ceedings.” It depended on the fact that the image of a sphere, 
whose surface density is inversely as the cube of the distance from 
a point, is another sphere with a similar law of distribution. But 
any desired number of integrals, whose values can be at once 
assigned, may be obtained by various applications of the following 
process. Take any centrobaric distribution of electricity, and calcu- 
late directly the density induced by it at any point of an uninsulated 
sphere. This must be inversely as the cube of the distance from 
the centre of gravity of the given distribution. 
Take, as a simple example, a uniformly charged sphere of non- 
conducting matter with unit charge, radius a, at a distance p from the 
centre of an uninsulated sphere of radius r. Suppose r >a+p , so 
that the inducing sphere is wholly internal. We see by the method 
above that the density of the induced charge at points defined by 
radii making an angle a with the line of centres is represented by 
1 
either of the following expressions, multiplied by 
87r 2 r 
rr 
(r 2 -a 2 -p 2 -\- 2 ap cos 0) sin 6d6d(p 
2tt (r 2 —p 2 ) 
[r 2 + a 2 +p 2 - 2 ap cos 0 - 2 rp cos a + 2 ar (cos a cos 0 - sin a sin 0 cos <f>) ]f (r 2 + _ 2 rp 
cos 
i.e ., the double integral is independent of a. 
Again, if the unit charge on the small sphere be distributed 
inversely as the cube of the distance from the centre of the large 
one, we have obviously for the induced density on the large sphere 
the expression — 
87r 2 r. 
yr, 
(a 2 - p 2 ) (r 2 - a 2 -p 2 + 2 ap cos 0) sin Oddd<p 
0 0 
[a 2 + p 2 - 2ap cos 0]i[r 2 + a 2 +p 2 - 2ap cos 0 - 2 rp cos a + 2 ar (cos a cos 0 - sin a sin 0 cos 
But if the small sphere include the centre of the large one, f.e., if 
a>p, the induced density is uniform; so that the value of the 
integral is 
5j7T 
ar 
