Brard 





g„ ... m M 



(5) 



and we have 



i//i(M,t') = yi(M,t') + y;(M,t') . (6) 



The density y[(m, t ') is due to the sum of doublets rj(P, t ') on the part 

 S'(P,t') behind the arc P'm[,P which passes through m'; y[ depends upon t' 

 because, firstly, r^(P,t') depends also on the time, and secondly, because the 

 arc P'raJ.,P just mentioned above depends not only on m' , but also upon t ' when 

 m' is given. Consequently one has: 



y[(m',t') = r;[P(m',t'),t'], yl(m",t') = Tj [P(m", t ' ), t '] , 



where in the right members, P is a function of m' and of t'. 

 The potentials VjCM, t ') and y[(M, t ') satisfy the condition 



yj(m^,t') + yj(m^,t') = constant (7) 



(with respect to m.). Therefore, we have: 



7i(m', t ') 



7 



and 



7;(m',t') - 7';(m",t') = T^Cyp.t') (9) 



(m', m" infinitely close to p). 

 Moreover, we have 



^[y,(M,t') + y;(M,t')] = - ^ $„,(;.) (^) 



'l(m',t') 1 



= > -2yi(m.,t') + JJ A(m,mj) yjCm^.t') dS(mj) 



(8) 



= d^ *oi(M) (^) ,. x(M) = x(P)-0. (10) 



fj, ^ ' t 



observe that 



'Pj(m^,t') = -7^(m, t') + constant. (11) 



Let Pg be the point on s^ infinitely close to p. Putting 



836 



