128 On the Doctrine of Extraneous Force. 



These formulas are not directly convenient for finding /, m, n 

 from X, /ju, v given (the ordinary formulas for doing so need 

 not be written here) ; but they give \ } ^, v explicitly in terms 

 of /, m, n supposed known ; that is to say, they solve the 

 problem of finding the wave-front of the simple laminar wave 

 whose direction of vibration is (I, m, n). The velocity is 

 given by 



'-»£+?+£) <** 



It is interesting to notice that this depends solely on the 

 direction of the line of vibration ; and that (except in special 

 cases, of partial or complete isotropy) there is just one wave- 

 front for any given line of vibration. These are precisely in 

 every detail the conditions of Fresnel's Kinematics of Double 

 Refraction. 



22. Going back to (35) and (3G), let us see if we can fit 

 them to double refraction with line of vibration in the plane 

 of polarization. This would require (36) to be the ordinary 

 ray, and therefore requires the fulfilment of (38), as did the 

 other supposition ; but instead of (37) we now have [in order 

 to make (36) constant] 



A = B = C (59), 



and therefore each, in virtue of (14), zero ; and 

 a=/3 = 7 =l ; 



so that we are driven to complete isotropy. Hence our present 

 form (§ 7) of the stress theory of double refraction cannot be 

 fitted to give line of vibration in the plane of polarization. 

 We have seen (§ 21) that it does give line of vibration per- 

 pendicular to the plane of polarization icith exactly FresnePi 

 form of wave-surface, when fitted for the purpose, by the 

 simple assumption that the potential energy of the strained 

 solid is expressed by (52) with k constant! It is important 

 to remark that k is the rigidity-modulus of the unstrained 

 isotropic solid. 



23. From (58) we see that the velocities of the waves cor- 

 responding to the three cases, 1 = 1, m=l, n = l, respectively 

 are </ (kju\ \/(k/(Z), V(k/y). Hence the velocity oi any wave 

 whose vibrations are parallel to any one of the three principal 

 elongations, multiplied by this elongation, is equal to the 

 velocity of a wave in the unstrained isotropic solid. 



