209 



-mil 



d^ if?- y^; — 



where ^ represents the area of the sphere and N its normal 

 ^rawn towards a». 



Now we must integrate witli respect to / from a high negative 

 value ^1 to a high positive one ^2 '). For the arbitrary function 7^^ occur- 

 ring in X we take a function which is zero for all values of the 

 argument except for those very near zero (there we must pass to the 

 limit) in this way however that for zero the integral of jF over that 

 small domain has just the value J. By interchanging the passage to 

 the limit and the integration ') and by contraction of the sphere the 

 identity becomes 



or after a partial integration 



{Q)t=-L 



x^^P,(t=o) = — j J dco -—^ 4- J do) 





A translation of the point ^ = then gives the result we want. 



1) If we want to be accurate the extension > must also be delimited at the 



r 

 outside. For the largest value of r which occurs ti +— must still be negative. 



Only afterwards we pass to the limit of an infinite extension. 



2) This interchange which is not further justified will be known to be charac- 

 teristic of Kirchhoff's method. Here we shall simply borrow it from Kirchhoff. 

 If we want to execute the integration rigorously, we shall have to avail our- 

 selves of a method given by J. Hadamard: Acta Math. 31 (1908) p. 333; 

 especially § 22. Gomp. for further literature J. Hadamard, Journ. de Pliys. 1906. 



