Theory of a Contractile JEitlter to Optical Problems. 523 



exactly the expression assumed by MacCullagh in his theory 

 of reflexion, and which Stokes has shown (B. A. Report, 1862, 

 p. 278) to be impossible as an expression for the energy of a 

 strained solid, for it leads to the equations T Xi ,= — T yx , &c, 

 instead of T XJ/ = T yx , where T xy means the stress parallel to 

 y on a plane normal to x ; how, then, can it represent the 

 energy of the strained medium ? 



The explanation of this point is simple. The second ex- 

 pression for W only gives the energy of the whole solid under 

 certain surface-conditions. Each element of the integral is 

 not an expression for the energy of the corresponding element 

 of the solid ; to find this we have to take into account the 

 surface-integrals introduced by the transformation. These 

 surface-integrals it is true vanish when the whole medium is 

 considered; but in calculating the stresses on each element 

 they are of importance, and when they are taken into account 

 the true values are found for T xy &c. We cannot get these 

 values from the transformed expression directly, for that is 

 only true under certain conditions. 



A second point is the following : — The integral 



$(&*■ ■>•* 



is transformed into 



flJS^ + -)* , * lb 



+ certain surface-integrals. 



These surface-integrals vanish if u, v, w are all zero at the 

 surface. They also vanish ivhenever u, v, w are functions of 

 the same function of x, y, z and t. Thus, as I pointed out in 

 a paper on the Reflexion and Refraction of Light (Proc. 

 Camb. Phil. Soc. vol. iii. 1880), if W denote the true ex- 

 pression for the work W', the transformed expression W = 

 W ' + M, where M is a quantity which may be negative, but 

 which vanishes if u, v, w are functions of the same function 

 of the variables. 



There remains the third point. Let us suppose that, in 

 transforming, as is done by Sir William Thomson, the inte- 

 gral for W we pass across a surface at a finite distance from 

 the origin, in crossing which the rigidity changes from B to 

 B'. Unless either there is no motion over this surface 

 which is impossible, or certain relations hold in addition to 

 the ordinary surface-conditions among the stresses, implying 

 of course the existence of surface-tractions &c. other than 

 those which arise from the strains, the surface-integral oc- 

 curring in the transformation does not vanish, and the surface 



