1894.] Optical Wave-Surfaces by Homogeneous Strain. 35 



_L . = o = r L. , (10) 



M (i>-'r) [ c ] (rc-'r) 



where, as before, two forms are given. Now, the final equivalence of 

 this transition from the index to wave-equation to mere inversion of 

 the two eolotropic operators is such a simple result that one would 

 think there should be a very simple way of exhibiting how the transi- 

 tion comes about. Nevertheless, I am not aware of any simple in- 

 vestigation, and, in fact, found the transition rather difficult, and by 

 no means obvious at first. I effected the transformation by taking 

 advantage of symmetrical relations between the forces and fluxes ; 

 in particular proving, first, that rE = = rH, or that the ray is 

 perpendicular to the electric and magnetic forces, comparing this 

 with the analogous property sD = = sB, and constructing a pro- 

 cess for leading from the former to the wave- equation analogous to 

 that leading from the latter to the index equation. It then goes 

 easily. However, we are not concerned with these details here. 



A caution is necessary regarding the interchangeability of /* and c. 

 They should be fully operative as linear operators. If one of them 

 be a constant initially, and therefore all through, we may not then 

 interchange them in the simplified equations which result. For 

 example, let /* be constant in (10). We have now 



- (11) 



pi* [c] (rc-'r) 



The first form is what we are naturally led to by initial assumption 

 of constancy of ft. Now observe that the interchange of ft and c in 

 the second form gives us the first form, after a little reduction, 

 remembering that [/t] is now /t 3 . But the same interchange in the 

 first form does not produce the second, because it is more general. 

 So we have gained a relative simplicity of form at the cost of 

 generality. The extra complication of the duplex wave-surface is 

 accompanied by general analytical extensions which make the 

 working operations more powerful. The equivalence of the two 

 forms in (11) may be established by the use of Hamilton's general 

 cubic equation of a linear operator, as done in Tait's work. Though 

 not difficult to carry out, the operations are rather recondite. On the 

 other hand, the much more general equivalence (10) is, as we saw for 

 the reason following (7) and (8), obviously true. This suggests that 

 some other transformations involving the general cubic may be 

 made plainer by generalizing it, employing a pair of linear operators. 



D 2 



