( 



490 RICE 



ART. K 



Now we rotate the axes of reference to OU, OM', ON'. Let the 

 strain-functions for these axes now be cxi, a2, as, on, as, ae and 

 the local coordinates X', yJ , v' . Of course ai is not in general 

 equal to ai, nor a^ to a^., etc.; for ai is the ratio of elongation 

 parallel to OU , while ai is that parallel to OX', etc.; and 

 aii/{ocia2)^ is the shear of OL' and OM' while a6/(aia2)^ is the 

 shear of OX' and OY', etc. But the equation 



aiX'2 + «2m" + oizv'^ + 2a4/x'''' + 2a5/X' + 2a6X'M' = ^' 



represents just the same elongation-ellipsoid as before, situated 

 in the same way in the body. Let the function which expresses 

 the strain energy in terms of ai, 02, ... a& be 0(ai, a^, ... aa). 

 Exactly the same function of ax, ai, ... a a must also be equal 

 to the strain energy. This must be so on account of the isoiropy. 

 In the illustration above, assume the sphere to be strained 

 homogeneously for simplicity, and refer to any axes of reference. 

 Keeping the forces as it were "in situ," we rotate the sphere and 

 axes. The energy is unchanged. But the mathematical con- 

 s "derations leading us to a certain function of ai, 02, ... Oe which 

 is equal in value to the energy will lead us in the second case to 

 just the same function of ai, ai, ... ae; for the general oper- 

 ations are unchanged by a change of axes and just the same re- 

 lations exist between the stress-constituents and the strain-co- 

 efficients for any one set of axes as for another. Once more that 

 is the essence of isotropy. 



We are thus naturally led at once to the purely mathematical 

 question of trying to solve the following problem : 



"An ellipsoid referred to OX', OY', OZ' has the equation 



air^ + ai-n" + azt" + 2a,v'^' + 2af,^'^' + 2a,^'rj' = k\ 



When referred to another set of axes OL', OM', ON' its equation 

 is 



q:iX'2 + aofx'^ 4- aa/' + 2a4M'/ + 2ayX' + 2a6XV' = k\ 



What function of ai, 02, as, ai, a^, as is equal in value to the same 

 function of ai, a2, as, ai, as, aa?" 



That problem we have implicitly solved in the note on 



