1894.] Optical Wave-Surfaces by Homogeneous Strain. 41 



-cE = Vv/^VvE, (48) 



and let r=JV, V=/~V, E =/->E', (49) 



so that (48) becomes 



-c/-'E' = Vy-V/i-'V/-^-'^. (50) 



Now employ Hamilton's formula 



Vmn = * V ***, (51) 



being here any self-conjugate operator. Take = / -1 , and we 

 transform (50) to 



-cf-'E' = V/'-'vV/Vv'E' x [/- 1 ] (52) 



= V/- V/- 1 (7/*- 1 /) Vv'E' x [/->] . (53) 



In this use Hamilton's formula again, with = f~ l , and we obtain 



= /Vv' (/A*- 1 /) Vv'E' x [/->] 2 . (54) 

 Or, more conveniently written, 



~ fi> = V '- 



So far, / is any pure strainer ; we can now make various special- 

 izations. For example, to get rid of p,~ l from the right side of (48), 

 and substitute c. Take 



/=c, then f ~ = ^ (56) 



which brings (55) to the form 



-^-'E' = Vv'cVv'E', (57) 



which should be compared with the other characteristic, that of H, 

 which is (8), or 



-^H = Vvc^VvH. (58) 



The above process is analogous to our transformation from the 

 duplex wave-surface to its reciprocal. As then, we have an inver- 

 sion of operators and also a crossing over from one form to another. 



Derivation of Index Equation from Characteristic. 



10. "We may also, in conclusion, exhibit how the index-surface 

 arises from the characteristic, when done in terms of V np to the last 



