32 Mr. O. IIca,viKi<le. On the Transformation of [Jan. IS. 



Conversely, therefore, if we start with the duplex wave-surface 

 corresponding to homologous permittivity and inductivity, and homo- 

 geneously strain it, the strainer being proportional to c*, we convert 

 it to a simplex wave-surface whose one eolotropy is homologous with 

 the former two. 



Remembering that the equation of the duplex wave-surface is 

 symmetrical with respect to the two eolotropies, so that they may be 

 interchanged without altering the surface, it struck me on reading 

 Dr. Larmor's remarks that a similar reduction to a simplex wave- 

 surface could be effected by a strainer proportional to /**. This was 

 vrritinl on examination, and some more general transformations 

 presented themselves. The results are briefly these : 



Any duplex wave-surface (irrespective of homology of eolotropies), 

 when subjected to homogeneous strain (not necessarily pure), usually 

 remains a duplex wave-surface. That is, the transformed surface is 

 of the same type, though with different inductivity and permittivity 

 operators. 



But in special cases it becomes a simplex wave-surface. In one 

 way the strainer is c*/[c*], where the square brackets indicate the 

 determinant of the enclosed operator. In another the strainer is 

 /**/[/**] These indicate the strain operator to be applied to the vector 

 of the old surface to produce that of the new one. 



Now, these simplex wave-surfaces may be strained anew to their 

 reciprocals with respect to the unit sphere, or the corresponding 

 index-surfaces, which are surfaces of the same type. So we have at 

 least four ways of straining any duplex wave-surface to a simplex one. 



Furthermore, any duplex wave-surface may be homogeneously 

 strained to its reciprocal, the corresponding index-surface, of the 

 same duplex type. The strain is pure, but is complicated, as it in- 

 volves both c and /*. The strainer is c~ l (cfiT 1 )*, divided by the 

 determinant of the same. This transformation is practically the 

 generalization for the duplex wave-surface of Plucker's theorem re- 

 lating to the Fresnel surface, for that also involves straining the wave- 

 surface to its reciprocal. 



Instead of the single strain above mentioned, we may employ three 

 successive pure strains. Thus, first strain the duplex wave-surface to 

 a simplex surface. Secondly, strain the latter to its reciprocal. 

 Thirdly, strain the last to the reciprocal of the original duplex wave- 

 surface. There are at least two sets of three successive strains which 

 effect the desired transformation. The investigation follows. 



Forms of the Index- and Wave-surface Equations, and the Properties of 

 Inversion and Intercluingedbility of Operators. 



4. Let the electric and magnetic forces be E and H, and the 



