322 Prof. D. N. Mallik on Elastic 



Hence 



A = 



_P fBX' BY' "bZ\ dx' dy' dz' 



4ir(X,+ 2j*) J "da:' + dy' + ~dz' ) r 



the integration extending over all space idaf dy' dz'. 

 Integrating by parts, 



where r 2 = (# — *')*+ (y_ y ')i + (r-s') 2 . 



Now, since B B Q 



A= , ,/ . [\ X'4 r -+ + 1 dx' dy' dz', 



if the surface integrals vanish on the implied condition for 

 the existence of body forces alone. 



If X', Y', Z' are constant over a sphere of radius a, and 



C 



then, since 



J 



9 



dx'di,'dz' = \. 



a 



V=§7r P ^r,i£R>a where R 2 = x 2 +y 2 + z\ 

 and is = %wp(3a*-'R 2 ), if R<a, 



P /V/ 3 ^« B 3a 2 -R 2 \ ,_ -o . 



For an ellipsoidal distribution (uniform), we shall have 



A= 3^+7 / T)( X '^ + + /J A ( 1_ a + X ""■•"/ 



• {(«' + *■) (// + X)(c 2 -fX)}'- 



and X = 0oru ( where -tt— + 7^ — + -> =1 ) 



V a' + yu, b' + fju r + fx 



according as the point is inside or outside. 



