156] POTENTIAL OF ELLIPSOID. 417 



Now 



<p (oo) = / f(f) dt = 0, [since f(oo) = 0], 



CO 



QO 



9>W= ff(t}dt, 



?>'() = - /"() 

 Inserting' these values in 29), 



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

 in the first integral. 



Applying this to our integral 27), by putting C successively 

 equal to a 2 , IP, c 2 , multiplying by # 2 , i/ 2 , # 2 , and adding, 



31) V= 



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

 equal to 1, so that 



32) V 



This form was given by Dirichlet. 1 ) 



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



. 



then = and 



33) V=itabcf {l_-j-_--j rr-l-: ~^~ = 



^y I a z +% o*-}-u c -fu} y^^. u j^^. u j^^_ u ^ 



o 

 We find for the derivatives of F, 



oo 



fly /* d u 



^ = CX J (a 2 + W )V(^R)(& 2 + ^)(c 2 -H*) 



a 



-, dai* x* y* z* \ 1 



xabc^l 1 asjQ pV^ ^2jr^| / g 2 8 



1) Dirichlet, "Uber eine neue Methode zur Bestimmung vielfacher Integrate." 

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



WEBSTER, Dynamics. 27 



