and the Laws of Plane Waves propagated through them. 135 



M. Cauchy afterwards gave the general equations peculiar to such a system, 

 and deduced from them Fresnel's wave-surface, as a first approximation to what 

 he considered as the more accurate laws of wave-propagation * 



In the year 1839, Mr. Green presented to the Cambridge Philosophical 

 Society a memoir, in which, by a modification of M. Cauchy's equations, he 

 obtained Fresnel's wave-surface as an exact deduction from the theory.f 

 This modification consists, as I have already stated, in restricting the system to 

 propagate normal and transverse vibrations. In M. Cauchy's or JVIr. Green's 

 theory, the vibration of the molecules is parallel to the plane of polarization. 

 In the same yeax Professor Mac Cullagh presented to this Academy a mecha- 

 nical theory of light, not founded on the hypothesis of attracting and repelling 

 molecules. The vibrations in this theory also are paraUel to the plane of pola- 

 rization, and the form of the wave-surface is that given by Fresnel. These 

 three theories of Hght, therefore, agree, so far as the laws of wave-propagation 

 are concerned ; and, excluding Fresnel's theory from the comparison (as the 

 vibrations perpendicular to the plane of polarization make it distmct from the 

 other two theories), there remain the mechanical theories of Mr. Green and Pro- 

 fessor Mac Cullagh, wHch are identical so far as the laws of wave-propagation 

 are concerned. The two theories are, however, really different in their ftmda- 

 mental assumptions ; and this remarkable agreement in the laws of wave-propa- 

 gation deduced from them admits of a simple explanation. I propose to account 

 for the agreement, and to suggest the direction in which we should look for a 

 true experimentum crucis between them. 



The function F used by Mr. Green, when reduced to its simplest form 

 will be ' 



and the simplest form of Professor MacCullagh's equations will be derived 

 from the function 



-2V=PX'+QY' + RZ\ (39^ 



It is evident from what I have stated in the first section, that (X, ju, v) will 



* Memoires de I'Institut. torn, x., 1830. 



t Transactions of Cambridge Philosophical Society, vol.vii. p. 121. • 



