Definite Integrals, with Physical Applications. 391 



29. If we make n = l, then the law of internal tension is ex- 

 pressed by 4ttwi/3, and therefore the internal attraction is the con- 

 stant quantity 4nrm. Hence if a sphere AEBF (Fig. 4) be electrised 

 by the influence of a very distant body P, the force exercised by 

 P to separate the combined electricity in the interior of the sphere 

 may be represented by the constant —4nrm. The law of accu- 

 mulation is then A — Sm{2t—\) which vanishes when t=±; there 

 is therefore a transition line EF, which is a great circle having 

 its plane perpendicular to the direction of P's action ; this line 

 divides the sphere into two parts containing equal quantities of 

 the opposite electricities ; the maximum accumulation takes place 

 at A and B the poles of the transition circle. 



30. The expression for the accumulation A in the general 

 case of Art. 28, may be put under an elegant aud simple form, 

 which facilitates much the tracing of the actual distribution. 

 If we observe that 



i= 1.2%'Ldf and ( " +i )- < ° i.f:.»^ &c - we get 



_ m(2n + l)(-l)".d" «.(»-!) 



A ~ 1.2.3. ..ndt' \ r *•* + 1.2 '* ~ &C - ? 



;»(2h + 1)(-1)" el" , ,. 



= i.2.3...n -drW P utt,n * t=1 ~ L 



31. To determine the number of transition lines on the sur- 

 face of the sphere put 



j-^ . ~ (tt'y = P„, then A = m (2« + 1) ( - 1)".P„ ; 



and therefore the equation for determining the transition lines 

 is P„ = 0. 



