Paraweter Values of Potential Theory. 177 



III. — The (jeneralised potential. 



§9. — Fundaweutal formula. — Tlif siiiipk' forms taken bv the in- 

 tegrals of §6 depend upon tlio fonnidu (2i)), whieh is true only for 

 the ordinary jjotential. 1 now propose to find the value of the 

 integral 



n,{tp)dt 



when the potential is -generalised. corres{)ondin^ to the equation 

 (2). In Green's formula 



put U = l, and \=y[qj)), q being a variable point and y a fixed 

 point. If in (42) the integration is extended over a closed surface 

 and }} is outside the surface we find, since r/iqp) satisfies (2) 



(43) fh{tp)dt= - }^/<j{qp)dg 



where dq is the element of volume at q. The integration in the 

 second member being extended throughout the volume enclosed by 

 the surface, the integral represents the potential at p due to a 

 uniform distribution of mass of unit density throughout that 

 volume. We shall denote this potential by X(/>). 



If, however, p is inside the closed surface we must surround j^ hy 

 a small sphere Q, of radius ?•, the surface integration of (42) now- 

 including the surface of this sphere, and the volume integration 

 extending only throughout the volume between the sphere and the 

 original surface. At the small sphere the positive direction of the 

 normal is that of /• increasing, so that (42) becomes 



/h{tp)dt + ky{gp)d^=-J^,g(sp)ds=:l/2^Je-'^'{ l+^.jds 

 il il 



and when the radius of the sphere becomes vanishingly small the 

 second member is equal to 2. Hence when /; is within the closed 

 surface 



(44) fh{tp)dt = -2-ky'g{q2y)dq='2-k^X{p) 



the volume integral of the second member being convergentl since- 

 the subject of integration becomes infinite at p — q only as \ir. 



To find the value of fh{ts)dt where s is a point on the boundary 

 we observe that fh{tp)dt is a double stratum potential of unit 

 moment over the boundary. Hence its value at a point on the 

 surface is the mean of its values at points infinitesimally close to- 

 this, one just inside and the other just outside. So that 



(45) fh{ts)dt= 1 -hrfg{gs)dq 



= l-A;^X(s) 



1 Cf. Leathern. " Volume and surface intejrrals used in Physics," p. H (Canibrid<fe Tract, 19U5)- 



