192 On the Stresses in a Heavy Spherical Shell. 



Substituting in the general equations above, we obtain 



a?(*r>=x+Y, (5) 



J^ 2U )=S-f^ s - Y )- • • • i> 



To make the conditions more precise, multiply equations 

 (1), (2), (3), and (4) by r, and then integrate all the 

 equations from r=a to r=b along the edges of the cone for 

 which cos = x. 



From (3) and (4) we obtain the equations for the equilibrium 

 of that portion of the shell which lies within this cone, with 

 the conditions 





\ Xdr 



= and 



f 6 

 Ydr 



J a 



= 



• • 



. . 



00 



From (5) we 



obtain 



r' 



! p=|"(s 



l + Y)dr, 











remembering 

 and from (6) 



thatP = 



both w. 



lien r = a < 



md 



when 



r = b 





r*U= ~ {'Xdr- j2- 9 {\x~Y)dr. 



dxj a l — ^Ja 



Since Q and U are both to vanish when a? = a, it is also 

 necessary that X shall have the form 



. u 1 v dx, 



J a 



and that 



x C r 

 uvdx + —- — * 1 Ydr 

 1— #M« 



j: 



shall have the same analytical form. The formal solution is 

 thus completed. 



The expressions for Q and R involve essentially gravitational 

 terms which are not dependent upon boundary conditions. 

 If it is assumed that the stresses are purely elastic, as is 

 assumed by all writers on that section of Elasticity usually 

 referred to as "under bodily forces/ - ' these essential 

 gravitational terms are excluded by the form of the consequent 

 solution, which gives the stresses in zonal harmonics with 

 coefficients of a definite kind when arranged in powers of r. 



As the conclusion that the stresses between the parts of a 

 heavy body are not purely elastic stresses, is contrary to the 

 theory usually adopted, it appears desirable to mention that 



