510 Mr. O'BRIEN, ON THE SYMBOLICAL EQUATION OF 



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 Mac CuUagh's equations *. If, on the contrary, we suppose the vibrations to be perpen- 

 dicular to the plane of polarization, we arrive at equations which agree exactly with Fresncl's 

 Theory in every particular f . 



If we introduce these six relations into the equations for crystallized media deduced from 

 M. Cauchy's hypothesis, that the molecular forces act along the lines joining the different particles 

 of the medium, it will be found that these equations are immediately reduced 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 

 M. Cauchy's hypothesis be true, a crystallized medium is incapable of propagating transverse 

 vibrations. 



Secondly, respecting the use of the Symbolical Method and Notation above alhided 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 ordinary equations of motion are included in the single symbolical equation. 



dH „ (d- d' d-] ,. J,, f d d d\ jd^ d,i rf(\ 



d¥-^{j?^dy'^^r'-^''-''^r^^'-^Ty^yd-J[r.-dy^^)- 



If we employ the notation A u . u, and assume the symbol 5B to represent the operation 



d d d 



d,v ay dz 



the equation of motion becomes 



dr 

 or, by using the notation Du . u also, it may be put in the form 



^= {JBAID - B(Dm.y]v. 



The symbol i3 written before any quantity U, which is a function of ayz, has a very 

 remarkable signification ; the direction unit of the symbol 59 C7 is that direction perpendicular to 

 which there is no variation of U at the point xyz, and the numerical magnitude of 19C7 is the 

 rate of variation of U when we pass from point to point in that direction. 



The symbols AlS-v and DlB.v have also remarkable significations. Al9.t' is a numerical 

 quantity, representing the degree of expansion, or, what is called the rarefiiction of the medium 

 at the point wyss. D'^.v represents, in magnitude, the degree of lateral disarrangement of the 

 medium at the point aiyx, and, in direction, the axis about which that displacement takes place. 



These two symbols may be found separately by the integration of an equation of the form 



dW „ fd'U d'U <PU 



_ Id' U d'U d" U\ 



\aJ' "^ ~df "^ ~d^^) ' 



dt^ Vrf.i" dy 



* Given in a Paper read to the Royal Irish Academy, December 9, 1839, p. 14. 



+ On this subject see a Paper by the late Mr. Greene in the seventh Volume of the Ctnnbridge Transactions, p. 121. 



