1894.] Optical Wave-Surfaces by Homogeneous Strain. 37 



These three, (17) to (19), reduce (16) to 



q- -V- - = = q- -3 (20) 



[XJ (qX-'q) 



where the second form is got from the first by interchanging X and 6, 

 which is permissible on account of the interchangeability of p. and c. 

 Comparing (20) with (10), we see that there is identity of form. 

 Consequently (20) represents a duplex wave-surface whose operators 

 are 6 and X, provided they are self-conjugate. They are, for, by 

 the elementary formula 



(abc}' = c'b'a', (21) 



it follows that 0c0' = (0c0')', (22) 



and similarly for the other one. 



In case the strain is a pure rotation, we may take the form of 

 (following Gibbs) as 



= I.i+J.j + K.k, (23) 



where i, j, k is one, and I, J, K another set of coperpendicular unit 

 vectors. For, obviously, this makes 



0r = I.ir+J.jr+K.kr = Ix+Jy + Kz. (24) 



Special Cases of Reduction to a Simplex Wave-surface. 



6. Now take some special forms of 0. We see, by inspection of 

 (17), that we can reduce either of 6 or X to a constant. Thus, first, 



= p-*, X = 1, 6 = /t-*c/i-*. (25) 



Then (20) reduces to 



q SL_ = o = q- -^ , (26) 



showing that the original duplex wave-surface is reduced to a simplex 

 one involving eolotropy 6, given by (25). 

 Similarly, a second way is 



= c -*, 6 = 1, X = <rVe-*, (27) 



which reduces (20) to the simplex wave-surface 



HL 



[X] (qv-iq) 

 involving the eolotropy X. 



x-4 



(28) 



