206 The Rev. J. H. Jellett on the Equilibrium and Motion of an Elastic Solid. 



15. It is usual with writers upon the subject which has been here discussed, 

 to consider the problem of the transmission of undulations from one body to 

 another with which it is in mathematical contact. This problem, which, by an 

 extension of the phraseology of optics, has been denominated the problem of re- 

 fraction, has been investigated with special reference to a luminous ether, by 

 Professor Mac Cullagh, Cauchy, Green, and others ; and has been discussed 

 by Mr. Hadghton for the case of solid bodies in general. But all these inves- 

 tigations appear to me to be liable to an objection to which I am unable to con- 

 ceive any satisfactory answer. The nature of this objection, which has deterred 

 me from following in this particular the steps of the writers in question, I shall 

 now proceed to state. 



On referring to p. 189 it will be seen, that the /onra of the general equations 

 of motion, upon which the whole theory of undulation is based, depends upon 

 the fact, that the coefficients are constant quantities, a fact which is, as we have 

 seen, a result of the homogeneity of the medium; and the conclusions of p. 192 

 are evidently true, so long as the molecule under consideration is situated at a 

 finite distance from the bounding surface of the medium. The functions to be 

 intecfrated retaining the same form, and the limits of integration being the 

 same, it is evident that the definite integrals will have the same value for every 

 point. 



Let us now consider the case of two media in contact. For the sake of 

 simplicity, let the common surface of contact be an indefinite plane, which we 

 shall take for the plane of xy. Let a, a' be the radii of molecular activity for 

 the two media, and suppose that the molecule under consideration is situated 

 at a distance from the plane of xy less than the greater of these. If now two* 



• Instead of two spheres we may (as is easily seen) substitute a single sphere described with 

 a radius not less than the greater radius of molecular activity. This substitution does not, how- 

 ever, in any way affect the reasoning in the text. The single definite integral 



l\\A cos^ a cos a' dm 

 will still be replaced by 



\\\A COS?a cosa'dm + Jjji4i cos' a, cos a/dm; 



the first being extended through the upper segment of the sphere, and the second through the 

 lower segment. The value of the sum of these two quantities will evidently depend upon the dis- 

 tance ef the point from the surface of separation. 



