] 48 The Aether as an Elastic Solid. 



remained to supply the boundary-conditions at an interface, 

 which are required in the discussion of reflexion, and the 

 relations between the elastic constants of the solid, which are 

 required in the optics of crystals. Cauchy seems to have con- 

 sidered the question from the purely analytical point of view. 

 Given certain differential equations, what supplementary con- 

 ditions must be adjoined to them in order to produce a given 

 analytical result ? The problem when stated in this form 

 admits of more than one solution ; and hence it is not surprising 

 that within the space of ten years the great French mathe- 

 matician produced two distinct theories of crystal-optics and 

 three distinct theories of reflexion,* almost all yielding correct 

 or nearly correct final formulae, and yet mostly irreconcilable 

 with each other, and involving incorrect boundary-conditions 

 and improbable relations between elastic constants. 



Cauchy's theories, then, resemble Fresnel's in postulating 

 types of elastic solid which do not exist, and for whose 

 assumed properties no dynamical justification is offered. The 

 same objection applies, though in a less degree, to the original 

 form of a theory of reflexion and refraction which was, 

 discovered about this timef almost simultaneously by James 

 MacCullagh (6. 1809, d. 1847), of Trinity College, Dublin, 

 and Franz Neumann (b. 1798, d. 1895), of Konigsberg. To 

 these authors is due the merit of having extended the laws 

 of reflexion to crystalline media; but the principles of the 

 theory were originally derived in connexion with the simpler 

 ease of isotropic media, to which our attention will for the 

 present be confined. 



* One yet remains to be mentioned. 



f The outlines of the theory were published by MacCullagh in Brit. Assoc. Rep. 

 1835 ; and his results were given in Phil. Mag. x (Jan., 1837), and in Proc. 

 Royal Irish Acad. xviii. (Jan., 1837). Neumann's memoir was presented to the 

 Berlin Academy towards the end of 1835, and published in 1837 in Abh. Berl. 

 Ak. aus dem Jahre 1835, Math. Klasse, p. 1. So far as publication is concerned, 

 the priority would seem to belong to MacCullagh; but there are reasons for 

 believing that the priority of discovery really rests with Neumann, who had 

 arrived at his equations a year before they were communicated to the Berlin 

 Academy. 



