38 Mi. 0. Heaviside. On the Transformation of [Jan. 18. 



The new surfaces (26), (28) may now be strained to their reci- 

 procals. Thus, take the first of (2b'), and put 



This makes 6*P -- = o. (30) 



~ 



Here the initial and final &*'s may be removed to the denominator, 

 and, since we also have 



(6p) = 6*p6*p = pfcp, (31) 



we bring the first of (26) to 



P =0. (32) 



1- '- 



[&- 1 ] (P&P) 



Now compare this with the second form of the same (26). They 

 are identical, except that 6 is now inverted. Consequently (32) re- 

 presents the index-surface corresponding to the wave-surface repre- 

 sented by the second of (26), and therefore by the first, since they 

 are the same. In a similar manner the strain (29) applied to the 

 second of (26) leads to the reciprocal of the first form. 



In like manner the simplex surface (28) is strained to its re- 

 ciprocal by 



Applied to the first form of (28), we get the second form with X in- 

 verted ; and, applied to the second form, we get the first, with X 

 inverted. These inversions of simplex wave-surfaces by homogeneous 

 strain are equivalent to Pliicker's theorem showing that the Fresnel 

 wave-surface is its own reciprocal with respect to a certain ellipsoid 

 (Tait, ' Quaternions,' 3rd Ed., p. 342). 



Transformation from Duplex Wave- to Index-surface by a Pure Strain. 



7. What is of greater interest here is the generalization of this pro- 

 perty for the duplex wave -surface itself. Take 



= c -> (c^- 1 )*. (34) 



Then we obtain 



c- 1 (cit'O'cc- 1 (c^- J )* = /i- 1 , (35) 



