FresneFs Theory of Double Refraction. 27 



A cos a = K cos a, 

 H cos a = 0, 

 C cos y = K cos y. 



Hence if K does not again = C, we have y = — , and . • . 



« 5= 0, H = 0, and K for this axis = A ; if, however, K again 

 = C, which cannot happen, as appears from considering the 

 first of these equations, unless A = C y, and therefore a re- 

 mains indeterminate, H vanishes, and K = C = A. If, then, 

 A and C are unequal, this second axis is X ** the first ; but 

 if A = C, any axis in a certain plane passing through the first 

 possesses the required property ; and in either case the value 

 of K for this axis is A ; similarly with respect to B and C. 

 Giving then the name of axes of elasticity to a set of three rec- 

 tangular axes, through a molecule possessing the property un- 

 der consideration, it follows (since the same inferences which 

 have resulted from supposing the axis of z determined as 

 above would have been deduced from the same supposition 

 with respect to the axes of a: and ?/,) that there is at least one 

 set of such axes through every molecule of a medium. Take 

 these then as axes of coordinates ; then if A, B, C are unequal 

 there can be but one set ; if any two of A, B, C, e.g. A, B, are 

 equal, there are an infinite number of sets, having one axis, 

 that of 3, common to each ; if A, B, C are all equal, any set 

 of rectangular axes through m are axes of elasticity. 



Retaining the axes of elasticity as coordinate axes, we have 

 F = G = H = ; and the initial forces put in play on m by 

 separate displacements w, v, w respectively || axes of ar, j/, z 

 will be A M, B t;, C wj respectively, and will act in the direction 

 of the displacements; and.*, the whole initial force put in 

 play on w by a displacement p, making *:s a, /3, y with the 

 axes, will be the resultant of the three, p A cos a, p B cos /3, 

 p C cos y, and will . • . 



= p K = p VK^ cos^ a + B^ cos2/3 + C2 cos2 y, 



in a direction making fe s X, /a, y with the axes, such that 



K cos A = A cos a, K cos jtt r= B cos /3, K cos v = C cos y, 



and therefore not generally coincident with the direction of 

 displacement. 



It follows also from a comparison of some of the chief pro- 

 perties of the principal axes of rotation, that 



1. The resolved part in the direction of an axis, inclined at 



E5^ 



