PROFESSOR HORACE LAMB ON THE PROPAGATION OF 



PART I. 



Two- DIMENSIONAL PROBLEMS. 

 3. The equations of motion of an isotropic elastic solid in two dimensions (x, y] are 



where u, v are the component displacements, p is the density, X, /x, are the elastic 

 constants of LAME, and 



These equations are satisfied by 



provided 



8</> 0\li 8</> 3// /.-> * 



u = -- + J , v = 5- a ....... (25),* 



3x fy cy ex 



-ML^vty 3V ^/fv 2 ^ ....... (26). 



In the case of simple-harmonic motion, the time-factor being d?*, the latter 

 equations take the forms 



(V* + A*) $ = 0, (V 2 + F)^, = ...... (27), 



where 



70 jJ fJ 22 7 ^ r r 2 JtS/ 



~ r+ 2 l j.~ p< p. ~ p ' 



the symbols , 6 denoting (as generally in this paper) the views-slownesses,^ i.e., the 



reciprocals of the wave-velocities, corresponding to the irrotational and equivoluminal 

 types of disturbance respectively. 



The formulas (25) now give, for the component stresses, 



-=A-(-2 = K~(f> 2 55- -p 2 -~ 

 /A p. ox By* cxdy 



&* = ^ + ^ = 2 - u ^ - ^ - 2 "- .... (29). 



fi ox d?/ dx dy dx~ 



ii 11 ty a* 2 axa y - 



* GREEN, 'Camb. Trans.,' vol. 6 (1838); ' Math. Papers,' p. 261. 



t The introduction of special symbols for wave-slownesses rather than for wave-velocities is prompted by 

 analytical considerations. The term " wave-slowness " is accredited in Optics by Sir W. R. HAMILTON. 



