Impulsive Motion of Electrified Systems. 137 



AY lien a>b, we put a 2 — h q = c-, and then we have 



T 1 , a + c 



J,= -loo- . 



1 c ' «— c 



The case when a<b is considered in § 26. 



If in J 2 we put b <2 (l — n 2 ) = a <2 — l <2 } and change the variable 

 from x to ?/, where 



P -n-cr 1 



we find, after some reductions, 



J,- " * 



2 ~ vT 2 J H- 



where ™ . , n*(a*-P)(P-n*a 9 ) 



R2= ^+- ( i- n y? • 



Thus, when Z 2 is positive, or a 2 >(l — ?r)/> 2 , that is, when 

 the ellipsoid is prolate or less oblate than the Heaviside 

 ellipsoid, 



3-=l 



j 2= i[i og(2 , + R)] = ilo g £±|, 



.r=-l 



The case when I 2 is negative is considered in § 26. 



The third integral J 3 can now be easily found, for on 

 differentiating the integral Jo with regard to n, we find that 

 ndJ 2 /dn = J s -J 2 . Thus 



J 3 = — (nJ 2 ), 

 da 



Since dl/dn=nb 2 jl, we find 



T l 2 —n 2 b 2 . a + 1 2an 2 b 2 



(o w b fl -r l\a 2 -l 2 ) 

 Collecting these results, and using /LtKt* 2 = l, we have 



The electrostatic energy of the ellipsoid at rest is U G , where 



U = r^- log — • 

 § 25. AYith the notation of § 24, the momentum carried 



