212 THEORY OF NEWTONIAN FORCES. [PT. I. CH. V. 



This may be integrated by parts. 

 Call I 



J 



- 



Now ^(oo)- f(i)dt = 0, (since /( oo ) = 0), 



J 00 



4 <)-/-/($* 



./ <r 



f () = -/() 



Inserting these values 



or the variable of integration being indifferent, we may put u for 

 t in the first integral. 



Applying this to our integral, by putting C successively equal 

 to a 2 , 6 2 , c 2 , multiplying by a? 2 , 2/ 2 , 2 , and adding, 



Now the first three terms of the integrand are, by definition, 

 equal to 1, so that 



a 2 i/ 2 z z ) du 



(13) v 



This form was given by Dirichlet*. 



If the point x, y, z lies on the surface of the ellipsoid 



then a = and 



f " ( x 2 v 2 

 (14) F=7ra6cl jl -^-- y 



* Dirichlet, "Ueber eine neue Methode zur Bestimmung vielfacher Integrate." 

 Ablt. der Berliner Akad., 1839. Translated in Journ. de Liouville, t. iv., 1839. 



