XXXVIII. On the Symbolical Equation of Vihrutory Motion of an Elastic Medium, 

 whether Crystallized or UncrystalUzed. By the Rev. M. O'Brien, late 

 Fellow of Caitis College, Professor of Natural Philosophy and Astronomy 

 in King's College, London. 



HRead March 15, 1847.] 



Preliminary Observations. 



The object of the following Paper is twofold ; Jirst, to shew that the equations of vibratory 

 motion of a crystallized or uncrystallized medium may be obtained in their most general form, 

 and very simply, without making any assumption as to the nature of the molecular forces ; and, 

 secondly, to exemplify the use of the symbolical method and notation explained in two Papers read 

 before the Society during the present academical year. 



First, with regard to the Method of obtaining the Equations of Vibratory Motion. 



This method consists, first, in representing the disarrangement (or state of relative displacement) 

 of the medium in the vicinity of any point xyz by the equation 



„ dv dv dv . 



dv = --- dx + -~ dy + —- COS + 

 ax dy dz 



•^ dn? 



drv 

 dwdy 



Sx^y + &c.. 



(where v =^a + ti(i + ^y, ^ t] ^ denoting, as usual, the displacements at the point wyx, and 

 a (i y the direction units* of the three co-ordinate axes), and, secondly, in finding the whole force 

 brought into play at the point wyz (in consequence of this disarrangement) by the symbolical 

 addition of the different forces brought into play by the several terms of Sv, eacfi considered 

 separately. It is easy to see that these different forces may be found with great facility, without 

 assuming anything respecting the constitution of the medium more than this, that it possesses direct 

 and lateral elasticity. By direct elasticity we mean that elasticity in virtue of which direct or 

 normal vibrations take place, and by lateral that in virtue of which lateral or transverse vibrations 

 take place. 



The forces due to the several terms of ^v are obtained by means of the following simple 

 considerations : — 



Let AB be any line in a perfectly uniform medium, and conceive the medium to be divided 

 into elementary slices by planes perpendicular to AB ; 

 let OM (= «:) be the distance of any slice PP' from 

 any particular point O of AB, and suppose this slice to 

 suffer a displacement equal to \ca)' (c being a constant) 

 in the direction AB, and the other slices to be similarly 

 displaced. Then it is evident that the medium suffers 

 by these displacements a uniformly increasing expansion 

 in the direction OB, and a uniformly increasing con- 

 densation in the direction OA, the rate of increase both of 

 the expansion and condensation being c. Now in all known substances, whether solid, fluid, or 



' i, e. Three lines, each a unit of length, drawn parallel to the three axes. 



