376 Cambridge Philosophical Society. 



constants, but involving also functions of x, the only result which 

 appeared to the author worthy of notice is the solution of 



D%+6Dm+c2m-w(w+ 1)3;^= X ; 



from a particular case of which, the general solution of Laplace's 

 equation, 



may be found in the simple form 



with a similar function using — V^ — l for v — 1. 



CAMBRIDGE PHILOSOPHICAL SOCIETY. 

 [Continued from p. 311.] 



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 uncrj^stal- 

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

 simply, without maldng 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 jioint 

 X1/Z by the equation 



i. dv X' , dv r. , dv ^ I d-v ^ ,, , d"v j i> , n -o 



ov= --dx+ — 01/ + — dz-\ ?x^+ dxdy + &c. — &c. 



dx dy dz 2 dx'^ dxdy 



(where i' = ^a + ij/3 + ^y, ^15^ denoting, as usual, the displacements 

 at the point xyz, and ajSy being the direction units of the three 

 coordinate axes), and in finding the ivhole 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 Bv, 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 



