16 Prof. Maxwell on the Theory of Molecular Vortices 

 Let jjl be the coefficient of cubic elasticity, so that if 





(80) 



dy d. 

 Let m be the coefficient of rigidity, so that 



^-fe = ^(J-|),&c (81) 



dy, 



Then we have the following equations of elasticity in an isotropic 

 medium, 



with similar equations in y and z, and also 



*=t®+D> ;■••:• <*> 



In the case of the sphere, let us assume the radius = a, and 



E=«4 7?=^, r=/(^ + y 2 )+^ 2 + ^. . (84) 

 Then 



p*x= : 2(p—im)(e+ff)z+mez=p &!f , 



Pzz = 2{ps-im){e+y)z + 2mgz, 



^=2^ + 2 /)^ > . . . (85) 



771 



Pzx=-j( e + 2 f) Z > 



P*ff = 0. 



The equation of internal equilibrium with respect to z is 



ip-+ip»+-BP-=°> • • • • (86) 



which is satisfied in this case if 



m(e + 2f+2ff) + 2bi-$m){e+ff)=0. . (87) 

 The tangential stress on the surface of the sphere, whose 

 radius is a at an angular distance 6 from the axis in plane xz f 



T = {p m —p„) sin 6 cos 6 +p xx (cos 2 6 - sin 2 6) . . (88) 



= 2m(e+/-#)«sin0cos 2 0-^(e + 2/)sin0. . (89) 



In order that T may be proportional to sin 6, the first term must 

 vanish, and therefore 



<? = e+f, (90) 



ma 



T=- T (e + 2/)sin0. 



(91) 



