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We can construct in the first place out of the Schering potential 

 the potential of two double points, in P^ and P,, the positive 

 poles of which are both directed toAvards Q^ (so that in opposite 

 points unequal poles correspond). Let us determine a point S of the 

 hypersphere by the distance PS = r and /^ QPS = (p (where for P 

 and Q the index 1 or 2 must be taken according to S lying with 

 Pi or with Pj on the same side of P), then this potential (a) becomes 



cos w 

 sin^ r 

 where the sign + ( — ) must be taken for the half hyperspheres 

 between P^ (Pj and B. 



This field has no other divergency but that of the double points 

 P, and P,. 



If we now reverse the sign of the potential in the half hyper- 

 sphere on the side of P,, we obtain the potential (^): 



cos<p 

 sin' r 



The divergency of this consists in the first place of two double 

 points, one directed in P^ towards Q^ and 0)ie directed in P, towards 

 Q, (so that now in two opposite points equal poles correspond) ; 

 and then of a magnetic scale (indeed a potential discontinuity) in 

 sphere B varying in intensity according to cos y. 



II. By the side of this we wish to find a potential, the divergency 

 of which consists of only such a magnetic scale in sphere B with 

 an intensity proportional to cos (f. Now a field of a magnetic scale 

 in B with an intensity varying according to an other zonal sphe- 

 rical harmonic, is easy to find. Let us namely take in each "meridian 

 sphere" PQH as potential of a point S the angle PHS=: ^ .t — ^ QHS 

 (P and Q to be provided with indices in the way indicated above 

 according to the place of S) = tan~^ \cos (p tan r\, then we have such 

 a potential : in the hypersphere it is a zonal spherical harmonic about 

 PQ as axis ; on the sphere B it has its only divergency in the 

 shape of a magnetic scale, the intensity of which varies according to 

 a zonal spherical harmonic with pole Q. 



Let us now take in turns all the points of the sphere B as pole 

 Q of such a potential function, and let us integrate all those poten- 

 tials over the solid angle about P each potential being multiplied by 

 cos Q'Q, then according to a wellknown theorem on spherical har- 

 monics the integral is a zonal harmonic of form cos(pf{r), wiiere 



ƒ (ƒ•) = i cos<p . tan-^ [cos (f tan r\ chxt , {dm representing the element 



