272 FUESNEL ON DOUBLE REFRACTION. 



and 



^^hn±hm^ 



c -\- gm -\- jn 



From equation (2.) we get 



Ill — , J 



gn — h 



substituting this value of in in equation (1.), and getting rid 

 of the denominators, we have 



g.i-f.n''+ {b-c)n+fY-^fn{gn-h) [-/.?22 + (6-c) w +/] + 

 c{-a){gn-h) l-fn'' + {b-c)n+f] -hn{gn-hf-g{gn-hf = 0. 



This equation in {n), which under this form appears of the 

 fourth degree, falls to the thiixl on effecting the multiplications, 

 because then the two terms containing (w"*) mutually destroy 

 each other ; hence we are sure that it has at least one real root. 

 There is therefore always one real value of [n), and consequently 

 one real value of {m). Consequently there is always at least one 

 straight line which satisfies the condition that a small displace- 

 ment of a material point along this straight line gives rise to a 

 repulsive force — the general resultant of the molecular actions — 

 the direction of which coincides with that of the displacement. 



To those directions which possess this property we give the 

 name of Axes of Elasticity. 



Proceeding from this result, it is easy to prove that there are 

 still two other axes of elasticity perpendicular to one another and 

 to the former. In fact, take this last-mentioned one for axis of 

 X, the components parallel to y and z, produced by a displace- 

 ment in the direction of the axis x, will be nothing ; so that we 

 shall have ^ = 0, A = 0; and the equations (1.) and (2.) become 



m{c ^ a + fn) = 0, 

 and 



n^ - (j~^) n-l=0. 



The former equation gives m = ; and the second gives for (w) 

 two values which are always real, the last term (— 1) being a 

 negative quantity. Hence we see that besides the axis of x there 

 are two other axes of elasticity ; they are perpendicular to the 

 axis of X, since for both one and the other m = 0, that is to say 

 their projections on the plane x z coincide with the axis of z ; 



