400 Mr Sea vie, On the Coefficient of Mutual Induction, etc. 
§ 4. To find the potential energy of Q in the field of S, 
I follow a method suggested by § 131 b in Maxwells chapter on 
Spherical Harmonics. We have simply to multiply the mass of 
each particle of Q by the potential at that particle due to S. 
That is, if U denote the potential due to S, we must find the sum 
2m U. 
Now, whatever be the form of the system S, we know that at 
any point for which r < s, where s is the least distance of any 
particle of S from the origin, the potential can be expanded in the 
“absolutely” convergent series of spherical harmonics 
U =Y 0 + rYj + r 8 Y 2 + . . . (r < s) (5), 
when Y 0 , Y x ... are functions of the two angular coordinates 6, (f) 
employed to fix the direction of the radius vector r. 
Hence when the angular coordinates of the axis OT are 6, <£, 
the potential due to $ at a point on OT at a distance h from 0, is 
U = Y 0 + hY 1 + h?Y 2 + ..., 
so that 
W = 2m U = 2m ( F 0 4- h 7, + h? F 2 + . . . ) 
= F 0 2m + YlZmh + Y 2 2m& 2 + 
Hence by (4) 
W = g 0 Y 0 + g x Y x +'g. 2 Y 2 + (6). 
By what has been proved in § 3, this series expresses the mutual 
potential energy of the systems S and T. 
We see that any term of the series, as g n Y n , is obtained by 
multiplying g n by the value of Y n corresponding to the direction 
of the axis of T. 
§ 5. It is easily shewn that, under the condition s > t, imposed 
by § 3, the series (6) found for W is absolutely convergent. For 
since .9 > t, we can take a length q such that s > q >t, and then 
each of the series 
Y 0 -\-qY 1 -V q 2 Y 2 + ..., 
gt/q + ffi/tf+g*/q‘ + -~, 
is absolutely convergent. Hence, using j | to denote the numerical 
magnitude of a quantity, 
| Yn+i/Y n I <V?. \9n+llgn\<q, 
so that | F„ +1 £r m+1 /(F„</„)l ^ 
The last result shews that the series for W is absolutely con- 
vergent if the least distance from 0 of any particle of S exceed 
the greatest distance from 0 of any particle of T. 
