REPORT ON PHYSICAL OPTICS, 387 



M. Ampere. The results alluded to are contained in two short 

 papers read to the French Academy in the year 1828, and since 

 embodied into one, and published in ihe Annales de Chimie*. 

 Fi'esnel had arrived at the equation which belongs to all the 

 tangent planes of the wave- surface, and had shown in what man- 

 ner the equation of the surface itself might be thence deduced 

 by differentiation and elimination. This direct process, how- 

 ever, he seemed to think would involve complicated and embar- 

 rassing calculations. The method which he substituted for it 

 consisted in verifying the equation, to which he was led by rea- 

 sonings not altogether rigorous, and proving (by calculations 

 which he found too tedious to transcribe), that it satisfied the 

 conditions already assigned. M. Ampei'e has supplied the direct 

 demonstration, and deduced the equation of the wave-surface in 

 the manner originally pointed out by Fresnel. From this equa- 

 tion he has derived also the beautiful geometrical construction 

 given by Fresnel, and which the latter had obtained indirectly. 



A very concise demonsti*ation of the same theorem, and of the 

 other principal points of Fresnel 's theory, was given not long 

 after by Mr, M'Cullaght- This writer has shown that both the 

 magnitude and direction of the resultant elastic force, called into 

 action by any displacement, may be represented by means of an 

 ellipsoid whose semiaxes are the three principal refractive in- 

 dices of the medium ; and from this ellipsoid, by the aid of a 

 few geometrical lemmas, he has deduced in a clear and simple 

 manner the leading results arrived at by Fresnel. The axes 

 of this ellipsoid coincide in direction with, and are inversely 

 proportional to, the axes of Fresnel's generating ellipsoid; and 

 Mr. M'Cullagh has demonstrated the truth of Fresnel's construc- 

 tion for the wave- su.rf ace, by means of a simple geometrical 

 relation between its tangent planes and tlie sections of the two 

 ellipsoids. 



In the third supplement to his *' Essay on the Theory of 

 Systems of RaysX," Professor Hamilton has pi*esented that 

 portion of Fresnel's theory, which relates to the fundamental 

 problem of the determination of the velocity and polarization of 

 a plane wave, in a very elegant analytical form ; and from the 



• " M6moire sur la Determination de la Surface courbe des Ondes lumineuses," 

 &c., torn, xxxix. 



t " On the Double Refraction of Light in a crystallized medium according to 

 the principles of Fresnel," TransactioHs of the Royal Irish. Academy, vol. xvi. 

 A further development of the principles of this memoir has been recently given 

 by the author in the 17th vol. of the same Transactions, under the title " Geo- 

 metrical Propositions applied to the Wave-theory of Light." • 



I Transactions of the Royal Irish Academy, vol. xvii. 

 2c2 



