228 PROCEEDINGS OP THE AMERICAN ACADEMY 



d(a^) _ 2 xp^' 

 clx a'^ ' 



+ -i^ 5^ «?.. + «'+ '^«> + ^^ 5P) («- + «'. + '?') = «' 



for all values of x, y, and z. 



=z — 2b 



,2 



d 



d(b 



r^iQa+Q. + Qc) 



Integrating, 



^« + c. + ^.=^- 



To determine this constant we must use the surface condition 



dV, , dV^ .„ 



-y^-f- -— = 4 90-. 

 dv^ ' a 1^2 



The X force which the shell exerts on a point in its interior is 



+2 Qpx8{a,b^c^Q^) = + 2 QpxaJ>,c,BQ^^ + 2 QpxQ^?i{aJ),c,), 



so that the sum of the X components on the two sides of the shell is 



7 ( dQa , dQb , dQc ) <j. o\ 

 2 8pxa,ViS^a = 2 Op^«^^ I ^(^ + ^(P) + J(^ I ^^"'>' 



Resolving the sums of the components along the normal, we have 

 2 8/=* {s^, (e. + e. + ej ^ + ^,3 ( <3„ + <?» + Qc) % 



where is the thickness of the shell and p is the density of the original 

 ellipsoids. 



Now g __ «S<^ 



