192 The Rev. J. H. Jellett on the Equilibrium and Motion of an Elastic Solid. 

 and eliminating I, m, n between tlie equations (0), we find 



(*-n,)(s- *o) (5-*-3)-i>3>^=(s-no-^in3 {s-^,)-n.,<^,{s-^^) 



- n. <i>3 >i^, - Hs Oi >i/^, = 0. (P) 



The value of s being determined by this equation, those of cos I, cos in, cos n 



are found at once from (0), combined with the condition 



cos- 1 + cos'- m + COS" n = 1. 



Hence, the equation (P) being of the third degree, it appears that for each di- 

 rection of wave plane there are in general three directions of molecular dis- 

 placement ; of these directions one is necessarily real, while the remaining two 

 may be either both real or both imaginary. The vibration will not, however, 

 be necessarily real, because its direction is so, as it is further necessary that the 

 corresponding velocity of wave propagation should be real. Hence, as 



S = — ev', 



it is plain that at least one value of s must be negative. We infer, therefore, 

 generally, that no body will be capable of transmitting a plane wave propa- 

 gated by parallel rectiUnear vibrations, unless equation (P) have at least one 

 real negative root. 



The surface of wave slowness, being the locus of a point upon the wave 

 normal whose distance from the origin is inversely as the velocity of wave pro- 

 pagation, will be found by putting 



J_ 

 r'- 



in the general equation (P). It is evidently a surface of the sixth order. In 

 fact, if we put 



P =: A^,^,x^ + A^,,,y^ + A^.^.z''- + 2^1^,„,^z -f 2A,.,,,a;z + 2A^i,^,a:y, 



and denote by P, Q', R', P", Q", R" the expressions derived from these by 

 replacing a' by /3' and y successively, we shall have, as the equation of the 

 surface, 



