On the Vibrations of a Crystalline Medium. 149 



in the crystalline medium *. I shall adopt the latter hypo- 

 thesis, and shall consider only the attraction or repulsion 

 exerted on the molecule which is close to (pa, qb, re) by 

 the six | nearest molecules of: the crystal, namely those 

 close to ((p±l)a, qb, re), (pa, (q±l)b, re), {pa, qb, (r±l)c). 



§ 2. The case of the crystalline medium will be understood 

 more clearly if we first concentrate attention on a few simple 

 problems. 



Suppose we have a light uniform elastic string OZ, 

 fastened at its ends under tension P, of stretched length 

 (>i + l)c and of unstretched length (n-\-l)d; and that to 

 this string are attached at equal intervals n particles, 

 each of mass M. Suppose, now, that the particles receive 

 small displacements from their equilibrium positions, and 

 that we wish to determine their vibrations about these 

 positions. 



Taking the axis of z along the string and any two 

 perpendicular lines through as axes of x and y, suppose 

 that the coordinates of the rth particle are (x r , y r , rc-{-z r ), 

 where x r , y r , z r are small. Then, if T is the kinetic energy 

 of the system of particles, and U the work done in stretching 

 the string from the equilibrium position, we readily find 

 that 



T = iM r xVv 2 + 2/, 2 + ^ 2 j, (i.) 



r = l 



while approximately 



U = - \ P/2C } \ X, 2 + (X 2 - X,) 2 + • • • + (X n - flfc-l) 2 + Xn 2 } 



-\V/2e\ [y^A- (y 2 -y 1 ) 2 + . . . + .(#, -y n -i) 2 +y^\ 



-{-p/2(e-d)\{z 1 2+(z 2 -ztf +■ . . . +(**-s*-i)* + ***\ • 



. . . (iij 

 Tlie Lagrange equations 



give the motion of the particles. It is obvious that the 

 longitudinal vibrations given by the quantities z r , and 

 the two lateral vibrations given by x r and y r are inde- 

 pendent of one another ; and that the equations giving 

 the quantities z r are similar to those giving x r or y r . 



* See a paper *by Lord Kelvin in the Philosophical Magazine, 

 July 1902, p. 139. 



t A smaller number than six, if p is 1 or /, &c. 



