VIBRATORY MOTION OF AN ELASTIC MEDIUM. 



519 



18. 



We may easily shew that the disarrangement y -^y^ SxSy brings no force into play 



upon O ; for it is perpendicular to the plane of the paper, and its nature 



is as follows. Draw two physical lines QQ' and PP through O equally 



inclined to JlJ^'; then in consequence of the disarrangement the lines OP 



and OP will become bent upwards (i. e. upwards considering the plane 



of the paper to be horizontal), and the lines OQ and OQ' will be bent - 



downwards; also the curvature of POP will be exactly the same as 



that of QOQ', only opposite in directions. Hence the two forces 



brought into play on O by the curvature of the two physical lines PP' 



and QQ' will be equal and opposite ; and the same may be said of every 



other pair of physical lines drawn through equally inclined to JCJ^'. It is therefore manifest 



that no force will be brought into play on O by this disarrangement. 



19. Thus it appears that the force brought into play by tiie disarrangement, 



d-v 

 dwdy 



SxSy , 



ill be 



dxdy 



{a^ + fir, + y^)S^vSy, 



dxdy dxdy 



Hence the force brought into play by the disarrangement, 



d-v 



^v = 



dxcy +— — — dydx + 



dxdy ^ dyd 

 will be expressed by a symbol of the form 



dzdx 



IzLv 



^' (C„7 + C,'^/3) + '^' 



dydx 



dz dee 



iC,^a + C/^7) + 



d' 



dxdy 



(U")- 



20. Hence, collecting these three results, the general equation of vibratory motion will be 



+ U" (8). 



d-v 



21. We have seen that, in the case of an uncrystallizcd medium, the constant C (i.e. the 

 constant to which the different C's in U" become equal when the medium becomes uncrystallizcd) 

 is equal io J - B ; in other words, C is the difference between the coefficients of direct and lateral 

 elasticity ; and it is easy to explain how this is on simple mechanical principles, which appear to 

 apply to a crystallized medium as well as to an uncrystallizcd, and which therefore will furnish us 

 with certain probable relations between the coefficients involved in equation (S). These relations, 

 as we shall presently shew, have a very important [jhysical signification. 



22. Referring to the figure in p. Jl, we may explain tiie jiliysical meaning of the relation, 

 C == A - B as follows : — 



The disarrangement represented in this figure consists of an increasing expansion of the 

 medium as we go along the line YY', caused, not by direct displacements (;. e. displacements 

 parallel to VY), but by /a<er«/ displacements (i.e. displacements perpendicular to 1"]'). C'onse- 

 quently the fierce brought into play u|)on by this increase of ex])ansion will be modified by the 

 lateral elasticity of the medium, which tends to restore the piiysital lines PP', QQ', &c. to their 

 equilibrium |)ositioiis .S'.S'', '/'T', &c. In fact the uneijual expansion cau^e(l by the disarrangement 



Vol.. VIII. I'AKT IV. 3 X 



