1894.] Optical Wave-Surfaces by Homogeneous Strain. 39 



the first of which is obvious, whilst in the second we make use of 



fic~ l = (c/*" 1 )" 1 . (37) 



There are other ways in which this may be expressed, viz., 



all of which lead to /i0c0 = 1. (39) 



If this is self-conjugate, we see, by (17) and (35), that its use in 

 (20) brings us to 



_=0 = q SL_. (40 ) 



[c-'jfqcq) [ 



That is, the strain converts the first of (10) to the first of (40), and 

 the second of (10) to the second of (40). But the first of (40) is the 

 same as the second of (10) with /t and c inverted, and the second of 

 (40) is the same as the first of (10) with the same inversions. In 

 other words, the strain has converted the duplex wave-surface to its 

 corresponding index-surface. Observe that the crossing over from 

 first to second form is an essential part of the demonstration, which is 

 the reason I have employed two forms. 



In full, the strainer to be applied to r of the wave-surface to pro- 

 duce the vector s of the index-surface (or q in (40) ) is 



= 



[0] 



But to complete the demonstration it should be shown that this strain 

 is pure, because we have just assumed = 0' in equation (20) to 

 obtain (40). Now the purity of this strain is not obvious in the form 

 (41), nor in any of the similar forms in (38). But we may change 

 the expression for to such a form as will explicitly show its purity. 

 Thus, we have 



CflT 1 = C* . C*/*" 1 ^ . C~"*, 



identically, and this may be expanded to 



the right member reducing to the left by obvious cancellations. 

 Therefore 



by taking the square root. So, finally, 



= c- 1 (c/t- 1 )* = <r (c*/-y)<r*. (42) 



