Results of the Contraction of a Solid Globe. 9 



I. 



To find the Depth of the Level of no strain. 



The volume of the shell at the depth x is 4<7rx 2 dx, and it 

 will contract in the interval dt through 



E4,irx 2 dxf dt. 

 at 



The whole contraction of the sphere interior to this shell 

 will therefore be nr ,/„ 



E4tt1 (r-xfj t dtdx. 



So that the volume of the sphere interior to this shell will 

 become, observing that -^- is negative, 



1^-^)3+ | E 47r C(r- X y^ t dtdx ; 

 and, neglecting E 2 , its radius will be 



The circumference of the interior sphere will therefore be 

 diminished in the interval dt by 



j^]y-*y%dtdx. 



It is evident that, if the diminution of this circumference is 

 equal to the horizontal contraction of the shell next above it, 

 that shell will neither be compressed nor extended. But the 

 horizontal contraction of that shell will be 



2ire(r — x) j- dt dx. 



Hence the condition that the shell at x is situated at the level 

 of no strain will be, since E = 3e, 



Se C r , .„ dv , , ^dv 



-, ^ I (r— xY^-dx—e(r— x)-j -. 



(r—xyj x v J dt v J dt 



It will be observed that the position of this level of no strain 

 does not depend on the coefficient of contraction, which will 

 divide out. 

 According as 



3e f r , v , dv j > , N dv 



*=i?)( r — ?■**'%<—•'>■*' 



so will the shell be compressed, not strained, or extended. 



