Mr. O'BRIEN, ON THE SYMBOLICAL EQUATION, ETC. 509 



gaseous, a disarrangement of this kind would bring into play on the slice O a force along the 

 line AB proportional to the rate of increase c, i.e. a force Ac, A being a constant depending upon 

 what we may call the direct elasticity of the substance. 



Again, suppose that the slice PP receives a displacement i cw' in the direction OC perpen- 

 dicular to AB, and the other slices similar displacements. Then the line AB will become curved 

 into a parabola A'OB', and all the lines of the medium parallel to AB will be similarly curved, the 



radius of curvature being equal to -, and perpendicular to AB. Now in all known substances* a 



c 



disarrangement of this kind would bring into play upon the slice O a force in the direction OC 



proportional to the curvature c, i. e. a force Be, B being a constant depending upon what we may 



call the lateral elasticity of the substance. 



Lastly, suppose that MP = y, and that the point P of the medium receives a displacement cxy 

 parallel to AB, and the other points similar displacements. Then the slice PP' will, in consequence 

 of this kind of displacement, turn through an angle tan"' (esc) into the dotted position, and the 

 other slices will suffer similar rotations, those on the other side of O, such as QQ', turning the 

 opposite way. Now it is easy to see that a disarrangement of this kind produces a uniformly 

 ncreasing expansion in the direction OC, and a uniformly increasing condensation in the direction 

 OC', the rate of increase both of the expansion and condensation being c. But the expansion and 

 condensation here described are quite different from those previously noticed, since they are pro- 

 duced, not by displacements parallel to C'C, but by lateral displacements, i.e. perpendicular to C'C. 

 On this account all that we can assert without further investigation is, that the force brought into play 

 upon an element at O by this disarrangement acts along the line C'C, and is proportional to c, i. e. 

 equal to Co, where C is a constant evidently depending in some way both upon the direct and 

 lateral elasticity of the medium. 



There is, however, a very simple way of finding the precise value of the force brought into 

 play by a disarrangement of this kind ; for, if we turn the axes of ,v and y in the plane of the paper 

 through an angle of 45", it will be found, that this disarrangement is nothing but a combination of 

 the two kinds of disarrangement previously noticed, and from this it immediately follows, in the 

 case of an uncrystallized medium, that the force brought into play at O is (A - B)c ; in other 

 words, the coefficient C, which must be multiplied into c in order to give the force brought into 

 play by the disarrangement cwy, is equal to the coefficient of direct elasticity (A) minus the 

 coefficient of lateral elasticity (B). 



In the case of a crystallized medium it may be shewn that six relations, corresponding to 

 the relation C = A - B, are most probably true, and are essential to Fresnel's Theory of Transverse 

 Vibrations; that is to say, the medium is capable of propagating waves of transverse vibrations, if 

 these six conditions hold, but otherwise it is not. 



In employing the above considerations to determine the equations of vibratory motion, the 

 directions AB and C'C are always taken so as to coincide with some two of the three co-ordinate 

 axes, and it is this circumstance that makes the method peculiarly applicable to crystallized 

 media. Indeed, if it were necessary to take the lines AB and CC in any directions but those of 

 the axes of symmetry, the above considerations would not apply without considerable modification. 



The equations of vibratory motion obtained by this method for an uncrystallized medium are 

 the well-known equations involving the two constants A and B. The equations obtained for a 

 crystallized medium are perfectly free from any restriction of any kind, are .ipplicahle to all kinds 

 of substance, whether we suppose its structure to be analogous to that of a solid, fluid, or gas, and 

 hold for all kinds of disarrangement, whether con.sisting of normal, or transverse displacements, 

 or both. 



• Fluidii and KMes pomtess lateral cltiKticiiy ns well as solidH, only in a foniiianitivi-lv tVtblt di-grce. 



