56 



therefore conclude, according to the last paragraph, that the planes 

 of lamination approximately coincide with those which were formerly 

 the planes of greatest tangential action. 



The author does not regard this mechanical action as the prohable 

 primary cause of the laminated structure, but rather as a secondary 

 cause, which may have had its influence in determining the positions 

 of the planes of lamination. He trusts that further evidence will be 

 collected on the subject. 



May 17, 1847. 



On the Symbolical Equation of Vibratory Motion of an Elastic 

 Medium, whether Crystallized or Uncrystallized. By the Rev. M. 

 O'Brien, late Fellow of Caius College, Professor of Natural Philo- 

 sophy and Astronomy in King's College, London. 



The object of the author in this paper is twofold : first, to show 

 that the equations of vibratory motion of a crystallized or uncrystal- 

 lized medium may be obtained in their most general form, and very 

 simply, without making any assumption as to the nature of the mo- 

 lecular 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 in representing the disarrangement (or state 

 of relative displacement) of the medium in the vicinity of the point 

 xyz by the equation 



8v= ^Sx+ ±$y+±Sz + l^L^+ ^fc% + &c.-&c. 

 dx dy dz 2 dx 9 - dxdy 



(where v=^a + ij/S-f-^, fyfc denoting, as usual, the displacements 

 at the point xyz, and a/3y being the direction units of the three 

 coordinate axes), and in finding the whole force brought into play at 

 the point xyz (in consequence of this disarrangement) by the symbo- 

 lical addition of the different forces brought into play by the several 

 terms of 8v, each 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 perpen- 

 dicular to AB ; let OM(=j) be the distance of any slice PP' from 

 any particular point O of AB, and suppose this slice to suffer a dis- 



