520 Mr. O'BRIEN, ON THE SYMBOLICAL EQUATION OF 



is resisted, and, to a certain extent, balanced (so to speak) by the lateral elasticity, and therefore 

 the unequal expansion has not its full effect in producing force upon O, but a certain part is spent 

 upon the lateral elasticity. 



If there was no lateral elasticity the force on O would be the same as if the displacements were 

 direct (i.e. parallel to Y'Y), for then the unequal expansion would produce its full effect ; in other 

 words the force brought into play on O would be 



dxdy 

 observing that the rate of increase of the expansion of the medium as we go along Wis 



dccdy 



To find the force actually brought into play upon 0, allowing for the lateral elasticity, we 



must diminish this force by a certain quantity depending upon the lateral elasticity, which quantity 



d'P 

 must of course be proportional to — \- . It is clear therefore that the force actually brought into 



dx'dy 



play upon O is expressed by a symbol of the form 



dxdy 

 P being a certain constant depending upon the lateral elasticity. Art. 6 shows that P = B. 



This evidently explains the physical meaning of the relation, C = A — B, for this I'elation 



indicates that the force brought into play by the disarrangement a - — —SxSy is, not the force 



d'f 



A — &, which is the force due to the full effect of the unequal expansion, but the force 



dxdy 



d-p 

 (A — B) - — ^ /3, which is equal to the former force diminished by a quantity depending on the 



UiL G/'tJ 



lateral elasticity, and proportional to the rate of increase of the expansion. 



23. From this explanation of the meaning of the relation C = A — B, it is very probable, 

 I think, that a similar relation holds wlien the medium is crystallized ; for it does not seem essential 

 to this explanation that the medium shall be perfectly uniform in all directions; all that seems 

 really necessary is, that the medium shall be symmetrically arranged with reference to the two axes 

 XX' and W. We must take care, however, in applying this explanation to a crystallized medium, 

 to give A and B their proper values, namely A-i and B^ ; for by A is to be understood the coefficient 

 of direct elasticity in the direction OY, that is A,, and by B the coefficient of lateral elasticity 

 brought into action by the unequal rotation of physical lines parallel to OY about the axis of z, 

 that is S, (for B, is the coefficient of lateral elasticity for the curvature of such lines about the axis 

 of 2^). Hence the relation, C = A — B, transferred to a crystallized medium, is Cj = A.^ - B^, and 

 therefore, writing down this relation for the six CTs in the expression U", we have the following six 

 relations, viz. : — 



•(9). 



24. We shall now shew, that, if these six relations hold, the forces brought into play by any 

 system of transverse viljrations constituting a single wave, are always perpendicular to the direction 

 of propagation of the wave ; but if these relations do not hold, the forces will not be perpendicular 

 to that direction. In other words, we shall shew that these six relations are essential tn FresneCs 

 Theory of Transverse Vibrations. 



