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11, Inrestigation of the Equation to Fresnel's Ware Surface. By ARCHIBALD SMITH, Esq., Trinity College, Cambridge.

[Read May 18, 1835.]

“THE mathematical difficulties under which the beautiful and interesting theory of Fresnel has hitherto laboured are well known, and have been regarded as almost insuperable. He tells us in his Memoir (see the Memoirs of the Royal Academy of Sciences of Paris, tom. vii. p. 136.) that the calculations by which he assured himself of the truth of his construction for finding the surface of the wave were so tedious and embarrassing, that he was obliged to omit them altogether. A direct demonstration has since been supplied by M. Ampère (Anmales de Chimie et de Physique, tom. xxxix. p. 113.); but his solution is excessively complicated and difficult.” A geometrical demonstration , of considerable simplicity has been given by Mr M" Cullagh in a paper in the xv.1" Volume of the Transactions of the Royal Irish Academy, from which the preceding paragraph has been quoted.

The difficulties which were experienced in this problem arose from two causes of the same nature:–want of symmetry in the fundamental equations, and the use of the essentially unsymmetrical method of differential coefficients. By putting the fundamental equations of Fresnel under a symmetrical form, and by the use of the Method of Multipliers as it is employed in the Mécanique Analytique, the eliminations may be effected without difficulty.

To render what follows more intelligible, and to show in what it differs from the other methods, I shall give the fundamental equations

of Fresnel and Ampère, and state shortly the steps by which they are obtained.

In the Memoir of Fresnel referred to above it is shewn, that if a section of the “surface of elasticity”, whose equation is

(a' + y + x*) = a ++boy" + cox'......... (1) be made by the plane x = ma + n y............... (2),

the greatest and least radii vectores of the section will be the values of v, derived from the equation

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and that if a plane be taken, parallel to the former and whose distance from the origin is one of the values of v, this plane, whose equation is

x = ma + n y + v v^1 + m” + ......... (4) will be a tangent to the wave surface.

To deduce the equation to the surface we may solve (3) to find r, and substitute this value in (4), which will give the equation

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equation to the wave surface. This is the method which M. Ampère employed with success.

Instead of eliminating v at first, we may differentiate (4), considering v as a function of m and n determined by equation (3), we shall thus obtain the equations

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(x-ma' ny)* = w (1 + m + n’)........................ (2),

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- - - - - dr dr Between these six equations the five quantities m, n, c, dom' an’ to

be eliminated, and the resulting equation will be that of the wave surface. These are the equations given by Fresnel, but he was not successful in effecting the requisite eliminations.

The fundamental equations may be put under a symmetrical form by the introduction of an additional symbol.

If for m and n we substitute respectively – #. and — o and suppose l, m, n, connected by the equation /* + m” + n’ = 1, we shall, instead of (3) and (4), have the three equations lar + my + n x = w ............... (1), * + m” + n’ = 1 ............... (2),

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Differentiate these equations with regard to l, m, n, and r,
rd/4 ydm + xd n = dc ............ (4),
ldl + malm + n dn = 0 ............ (5),

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Multiply (5) by A, (6) by B, and subtract the sum from (4), making the coefficients of each differential equal to zero. We thus obtain

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This is the simplest form which the equation to the wave surface can aSSuline.

If we clear it of fractions, and replace r" by a "+y+x", we obtain the equation given by Fresnel, viz.:

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