and the Laws of Plane Waves propagated through them. 1 1 3 



composed of two terms, coefficients of the original function 0^ ; these comjjound 

 coefficients will, in the equations of motion, be equivalent to simple coefficients, 

 so that the total number of distinct constants in the equations of wave-motion 

 will be reduced to thirty-six ; this reduction cannot, however, be introduced 

 into the conditions at the limits, because each coefficient will at the limits be 

 multiplied by a different function of (,r, ?/, z). Hence may be deduced an im- 

 portant theorem, which I shall prove in a general manner. 



" Two bodies, different in their molecular structure, may have the same 

 laws of wave-propagation, but cannot have the same laws of reflexion and 

 refraction." 



The laws of wave-propagation (5) depend upon the differential coefficients 



of f -T— ,&c. j, taken with respect to {j-, y, z) ; but the laws of reflexion and 



refraction (6) depend on the quantities f-p,&c. J themselves. Each of these 

 quantities is of the form 



dx dy dz dx dy dz dx dy dz 



If, therefore, for example, in the first of the equations (5), there be in (— - 



\da, 



a term of the form [B'-p], there will be in ^- a term of the form (a' ^^^ 



dy) da. \ dxj 



then, differentiating the first with respect to .r, and the second with respect to 



y, we shall obtain, as part of the equations of motion, (B -f .4')t-^, while 



dxdy 



tlie corresponding part of the conditions at the limits will be [[^ -^^ li,dydz 



■^^A' -j-l^dxdz. If, now, we suppose another body such that [^I'-j^J and 



[B-jA are terms in (-^— ] and (-r-j. the part resulting from these quan- 



titles in the equations of motion will be (A' + B\ -^V , which is identical with 



dxdy 



the former value, while the corresponding portion of the conditions at the 



VOL. XXII. Q 



