Electric Theory of Light and of a Quasi-labile ^thei/'. 239 



It is the object of this paper to compare this new theory 

 with the electric theory of light. In the limiting cases (that 

 is, when we regard the velocity of the missing wave in the 

 elastic theory as zero, and in the electric theory as infinite) 

 we shall find a remarkable correspondence between the two 

 theories ; the motions of monochromatic light within isotropic 

 or seolotropic media of any degree of transparency or opacity, 

 and at the boundary between two such media, being repre- 

 sented by equations absolutely identical, except that the 

 symbols which denote displacement in one theory denote force 

 in the other, and vice versa*. In order to exhibit this corre- 

 spondence completely and clearly, it is necessary that the 

 fundamental principles of the two theories should be treated 

 with the same generality, and, so far as possible, by the same 

 method. The immediate consequences of the new theory will 

 therefore be deduced with the same generality and essentially 

 by the same method which has been used with reference to 

 the electric theory in a former volume of the American 

 Journal of Science (vol. xxv. p. 107). 



The elastic properties of the sether, according to the new 

 theory, in its lirniting case, may be very simply expressed by 

 means of a vector operator, for which we shall use Maxwell's 

 designation. The curl of a vector is defined to be another 

 vector so deri\ed from the first that if ^l, v, w be the rectan- 

 gular components of the first, and u', v', to' those of its curl, 



du .^. 



dy" ' ^^^ 



where x, y, z are rectangular coordinates. With this under- 

 standing, if the displacement of the sether is represented by 

 the vector ^, the force exerted upon any element by the sur- 

 rounding aether will be 



— B curl curl Cc dx dy dz, . . . . (2) 



where B is a scalar (the so- called rigidity of the sether) having 

 the same constant value throughout all space, whether pon- 

 derable matter is present or not. 



Where there is no ponderable matter, this force must be 

 equated to the reaction of the inertia of the a3ther. This 

 gives, with omission of the common factor dx dy dz, 



A(^=-Bcurlcurl(g, (3) 



where A denotes the density of the sether. 



* In giving us a new interpretation of the equations of the electric 

 theory, the author of the new theory has iu fact enriched the mathematical 

 theory of physics with something which may be compared to the cele- 

 brated principle of duality in geometry. 



, dio dv 



1 du dw 



f dv 



''-d^~dz' 



dz dx ' 



dx 



