128 Mr. R. F. Gwyther on 



and we find 



V = Av? + B(v 2 + w*), S = (A-B)tw, etc. 



To arrive at the required form for the body force, we 

 must put A + B=o, giving the form of Maxwell's result. 



It is easy to see that this stress can not be maintained 

 by displacements in an isotropic solid, nor, in fact, in any 

 medium of which we have any knowledge. 



4. Lastly, whatever be the character of a body of con- 

 tinuous matter under strain, the potential energy of the 

 strain at any point per unit volume must be a function V, 

 of a scalar character possessing permanency of form. Also, 

 if the strain is small, and the body acts for it as a 

 conservative system, V is a quadratic function of the 

 elements of the strain, and P is found from it by 



„ dV 

 P=— ,, etc. 



The coefficients in the expansion of V may now be 

 related to some directions fixed in the body, and will, 

 therefore, vary with the change of the axes. 



In the most general case, V will contain twenty-one 

 elastic coefficients, say 



2 V = K n e 2 + 2K l2 ef + 2K ls eg + 2K l4 ea + 2K 15 eb + 2K 16 ec 



+ K 22 / 2 + 2K 23 fg + 2K 24 /a + 2K 2B /6 + 2K 26 /c 



+ K 33 g 2 + 2K u ga + 2K s5 gb + 2K s6 gc 



+ K 44 a 2 + 2K 45 a6 + 2K 46 ac 



+ K BB & 2 +2K 66 be 



+ K 66 c 2 



The conditions that this may be an invariant scalar 

 function, considering the coefficients as varying, have now 

 to be expressed. Write Ai, for the differential operator as 

 far as it enters connected with the coefficients, and retain 



