58 



words, the coefficient C, which must be multiplied into c, in order 

 to give the force brought into play by the disarrangement cxy, is 

 equal to the coefficient of direct elasticity (A) minus the coefficient 

 of lateral elasticity (B). 



In the case of a crystallized medium, it may be shown that six 

 relations, corresponding to the relation C = A — B, are most probably 

 true, and are essential to Fresnel's theory of transverse vibrations ; 

 that is to say, the medium is capable of propagating waves of trans- 

 verse vibrations if these six conditions hold, but otherwise it is not. 



In employing the above considerations to determine the equations 

 of vibratory motion, the directions AB and C'C are always taken so 

 as to coincide with some two of the three coordinate axes ; and it is 

 this circumstance that makes the method peculiarly applicable to 

 crystallized media. Indeed, if it were necessary to take the lines 

 AB and C'C in any directions but those of the axes of symmetry, 

 the above considerations would not apply without considerable mo- 

 dification. 



The equations of vibratory motion obtained by this method for an 

 uncrystallized medium, are the well-known equations involving the 

 two constants A and B. The equations obtained for a crystallized 

 medium are perfectly free from any restriction of any kind, are appli- 

 cable to all kinds of substance, whether we suppose its structure to 

 be analogous to that of a solid fluid or gas, and hold for all kinds of 

 disarrangement, whether consisting of normal or transverse displace- 

 ments, or both. 



When we introduce the six relations between the constants above 

 alluded to, and moreover assume that the vibrations constituting a 

 polarized ray are in the plane of polarization, we arrive at Professor 

 MacCullagh's equations*. If, on the contrary, we suppose the vi- 

 brations to be perpendicular to the plane of polarization, we arrive 

 at equations which agree exactly with Fresnel's theory in every par- 

 ticular-]-. 



If we introduce these six relations into the equations for crystal- 

 lized media deduced from M. Cauchy's hypothesis, that the mole- 

 cular forces act along the lines joining the different particles of the 

 medium, it will be found that these equations are immediately re- 

 duced to the equations for an uncrystallized medium. From this it 

 follows that M. Cauchy's hypothesis cannot be applied to any but 

 uncrystallized media. In fact, it may be easily proved that if the 

 equations derived from this hypothesis be true, a crystallized medium 

 is incapable of propagating transverse vibrations. 



Secondly, respecting the use of the symbolical method and notation 

 above alluded to. 



The application of the symbolical method and notation to the subject 

 of vibratory motion is Very remarkable, and leads to equations of 

 great simplicity. In the case of an uncrystallized medium, the three 



* Given in a paper read to the Royal Irish Academy, Dec. 9, 1839, 

 page 14. 



•f On this subject see a paper by the late Mr. Greene in the seventh 

 volume of the Cambridge Transactions, p. 121. 



