220 Methods for the Solution of Special Problems [CH. vm 



Expansions in Legendres Coefficients. 



256. THEOREM. The value of any function of 6, which is finite and 

 single-valued from 6 = to 6 = TT, and which has only a finite number of 

 discontinuities and of maxima and minima within this range, can be 

 expressed, for every value of 6 within this range for which the function is 

 continuous, as a series of Legendres Coefficients. 



This is simply a particular case of the theorem of 240. It is therefore 

 unnecessary to give a separate proof of the theorem. 



The expansion is easily found. Assume it to be 



/(/A) = a + a 1 ^+a a 5+... + a,5+ (170), 



then on multiplying by P^ (/A) dp, and integrating from /* = 1 to /u = + l, 

 we obtain 



I +1 P n (p)f(ti dp = Y a s ( +1 P, 



J -1 4 = J -1 



every integral vanishing, except that for which s = n. Thus 



a n = = I P n (p)f(p)dfj, (171), 



* j -i 

 giving the coefficients in the expansion. 



If /(/A) has a discontinuity when /x = yt/, , the value assumed by the 

 series (168) on putting /* = /u, is, as in 240, equal to 



M/i(/A>)+/ 2 G"o)} (172), 



where yi (yu, ), f^(^) are the values of /(AI) on the two sides of the discon- 

 tinuity. 



HARMONIC POTENTIALS. 



257. We are now in a position to apply the results obtained to problems 

 of electrostatics. 



Consider first a sphere having a surface density of electricity S n . The 

 potential at any internal point P is 



~ 



S n dS 



PQ J J Va 2 2ar cos 6 + r 2 



= 1 1 (l + - 7? (cos 0)+,% (cos 6) + ..'. J dS 



JJ a \ a a~ / 



= 4?ra 2 - ( ri (S n ) cos0=l) by the theorems of 237 and 255, 



CL 



a"-~ 

 this expression being evaluated at P. 



(173), 



