Parameter Values of Potential Theory. 173 



is always sinj^ular. We shall assume that the surfaces are all 

 exterior to one another, so that X. = — 1 is not a characteristic 

 number. The functions P(^s) and Q{ts) assume simple values at the 

 pole Ao=l. For the functions i/'i(«), ^i{s), • . • , >/'/«(*) are such that 

 i/',.(,s-) is equaP to + 1 over the rth surface and zero over all the 

 other surfaces; while 4)r{t) is a distribution of electricity over 

 the surfaces giving constant values over each of the surfaces 

 and rli)-oughout each of the vi inner regions. This distribution 

 <f)r(/) has a total charge +1 over the rth surface, and zero over 

 each of tIic otlieis. It tliere-fore represents the electric distribution 

 over the ;/( surfaces regarded as conductors, due to unit charge on 

 the rth sui-face. Hence, if we use an index to denote the particular 

 value of the pole A..,. 



F+^{ts) = (f>y{t) rr^l, 2, .... Ill 



according as .»>• is on the 1st. 2nd, wth surface. Fui'ther, the func- 

 tion if/r(p), being ecpial to the potential of a double stratum of unit 

 moment over the rth sui'face. is given bv 



(30) if,,{/>)=jli{tp)</t = '2, 1, or 



according as /> is within the rth surface, on its boundary, or out- 

 side that surface. The jjotential <I>r(^) due to the distribution 

 <f>r(^) is the conductor jxitential refei'red to. We shall denote it by 

 rr(^). So tliat 



(31) i F + Uj.p) = 2cl>,{f), ^At), or 



) Q + \>/p) = 2Vr{(j), VrW), or 

 according as p is witiiin the rth surface, on its boundary, or in the 

 outer region. This of course is a particular case of (13) and (14). 

 We may prove several interesting properties of the functions in- 

 volved in (5), (7) and (8), making use of the relation (29). If in 

 the first of (7) we replace p by a boundary point «, multiply by dt 

 and integrate over the boundary we find 



/P(fs)dt = XjP{Os)dO 

 Hence 



(32) /P(ts)dt=0 Ao±l. 



By the .same process we deduce from (5) thaf^ 



(33) {l~\)/H(ts)dt=\ 



Substituting from (6) and putting Ay = 1 we have 

 (33') (1 -\)/ H+\ts)dt + jP + \ts)dt=l 



1. Plemelj. Loc. cit., Kap. 16. 



* In (32) X may be replaced by a i)oiiit p. The same may be done in (33) and (34') provided the 

 second member be changed to 2 for p in the inner reyion, and to for p in the outer region. Cf. § 10. 



