40 Mr. 0. Heaviside. On tlie Transformation of [Jan. 18, 



This is of the form 0i0s0i, where t is pure. Its conjugate is there- 

 fore 0j0'j0i. This reduces to itself if 0j is pure. But 0j is pure, 

 because it is also of the form 6\0^\, where O l and 2 are both pure. 

 So our single strain depending on is pure. 



Substitution of three successive Pure Strains for one. Two ways. 



8. This is dry mathematics. But it is at once endowed with in- 

 terest if we consider the meaning of the expression of the strain as 

 equivalent to the three successive strains 0i, 02, and t . First, the 

 strain 



a - -A r - c ~* r (43) 



^'--' 



converts the duplex wave-surface to a simplex surface. This was 

 done before, equation (28). Next, the strain 



p. q - (44) 



CRT 



converts the simplex surface q to another simplex surface whose 

 vector is p, and which is the index-surface corresponding to the 

 wave-surface q. This strain (44) is, in fact, the same as (33), and 

 the result is 



P - -i - = = p-P_, (45) 



i- 



where X = c~*/ic~*. Finally, the strain 



converts the simplex surface p to a duplex surface 8, which is the 

 reciprocal of the original duplex wave-surface, the result being (40). 



The interchangeahility of ft. and c shows that we may also strain 

 from r to 8 by a second set of three successive pure strains, thus, 



= /( H(, t c-y)y-*. (47) 



This is the same as first straining the surface r to the simplex surface 

 (26) ; then inverting the latter, which brings us to the simplex sur- 

 face (32) ; and finally straining the last to the duplex surface 8. 



Transformation of Characteristic Equation by Strain. 



9. In connexion with the above transformations, it may be worth 

 while to show how they work out when applied to the characteristic 

 equation itself of E or H. Thus, take the form (7), or 



