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the problem of determining the function which (f, F being given functions, and the limits a, ß of the integration being also given) satisfies the equation

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He observes, that, unlike the methods employed in his former me-
moir, and the solutions there employed, which are quite rigorous,
the methods of the present memoir depend upon developments into
series, the strictness of which has been contested by some mathema-
ticians; but that passing over these difficulties, he has solved the
famous problem, the solution of which has been vainly sought after
for the last two hundred years, because on the above-mentioned
equation depends the integration of the generally linear equation of
any order whatever of two variables, and consequently the whole
Integral Calculus. The solution first obtained by the author, and
which he afterwards exhibits under a variety of different forms, is as
follows:
Theorem I.--The equation being given,

F(x)
f(x, 0) Ø(x+0) do = F(x),

F.(2) where f(x, 0) is a given function of x and 0; F,(x) is a given function of x such that the equation F(x)=cannot hold good for any finite value of x; F.(x) a given function of x containing all the factors which render F(x) infinite, and the function F(x) being absolutely arbitrary; and a and ß being given constants (independent therefore of x and 6), the expression for Øx which satisfies the preceding equation is F.(a)

F.(a)

*P(a) $x = fila,)F,'(a) e

embay) + &c., +

fi(a,)F' (az) where f(x) is determined by

fr(x) = (c—a,) e*P(ar)

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solved relatively to m,, and a,, Qz, az, &c. are the roots of

F(x) = 0.

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and the solution of a linear equation is at once made to depend upon this as follows : viz. given for the determination of the function or the equation

do (x)
f(x, 0) $(x) + f(x, 1) + &c. =

F(x),

dx Assume

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a, ß being constants, and 4(0) a function of 0 to be determined. It is always permitted to assume this equation. By this means, writing for shortness

(d, 0)=f(d, 0)+f(a, 1) 0 + &c., the equation becomes

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which is of the desired form.

A solution which occurred to the author after the memoir was drawn up, is as follows : viz. given, as before, the equation

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then (WV - 1 being determined by the equation

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1
ex BW –1) F(W V

w – 1)
x =

dw 2π

f(WV-1,wV ) gives the solution of the problem.

The above-mentioned formulæ are selected out of a great number of very general results contained in the memoir.

III. Letter from Dr. W. BIRD HERAPATH to Professor STOKES,

“On the Detection of Strychnia by the formation of lodostrychnia.” Communicated by Professor Stokes, Sec. R.S. Received June 12, 1856.

Bristol, June 7, 1856. MY DEAR SIR,– Will you do me the favour to announce to the Royal Society, that I have been engaged during some time past in the application of my discovery of the optieal properties of iodostrychnia to the detection of this alkaloid in medico-legal inquiries? I find it is perfectly possible to recognize the 10,000th part of a grain of strychnia in pure solutions by this method, even when experimenting on very minute quantities. In one experiment I took To oth of a grain only, and having produced ten crystals of nearly equal size, of course each one, possessing distinct and decided optical properties, could not represent more than the tobooth part of a grain ; in fact, it really represents much less, inasmuch as one portion of the strychnia is converted by substitution into a soluble hydriodate, and of course remains dissolved in the liquid.

I had hoped to have been able to complete this matter during this summer, but I now find it impossible to do so in time for this session of the Royal Society. I trust to be able to do so before Christmas, however. Will you oblige me by getting this notice inserted in the ‘Proceedings,' as a new test for strychnia at this juncture possesses considerable interest, the colour-tests having been so dubiously spoken of recently by toxicologists ?

In order to operate in this experiment, it is merely necessary to use diluted spirit of wine, about in the proportions of one part of spirit to three of water, as the solvent medium, and to employ the smallest possible quantity of the tincture of iodine as the reagent, and after applying heat for a short time, to set in repose. On spontaneous evaporation or cooling, the optical crystals deposit themselves, and may be recognized by the polarizing microscope, according to the description given of this substance in a former notice to the Society in June last.

You may remember that this proposition was also contained in my paper on iodo-strychnia, which was withdrawn from the Royal Society by me in June last in consequence of a necessity for revision and the completion of experiments requisite to settle the formula of that peculiar substance, and the introduction of an abstract of the literature concerning it.

I remain, &c.,

W. Bird HERAPATH.

IV. “Dynamical Illustrations of the Magnetic and the Helicoi

dal Rotatory Effects of Transparent Bodies on Polarized Light.” By Professor W. Thomson, F.R.S. Received May 10, 1856.

The elastic reaction of a homogeneously strained solid has a character essentially devoid of all helicoidal and of all dipolar asymmetry. Hence the rotation of the plane of polarization of light passing through bodies which either intrinsically possess the heliçoidal property (syrup, oil of turpentine, quartz crystals, &c.), or have the magnetic property induced in them, must be due to elastic reactions dependent on the heterogeneousness of the strain through the space of a wave, or to some heterogeneousness of the luminous motions* dependent on a heterogeneousness of parts of the matter of lineal dimensions not infinitely small in comparison with the wave length. An infinitely homogeneous solid could not possess either of those properties if the stress at any point of it was influenced only by parts of the body touching it; but if the stress at one point is directly influenced by the strain in parts at distances from it finite in comparison with the wave length, the helicoidal property might exist, and the rotation of the plane of polarization, such as is observed in many liquids and in quartz crystals, could be explained as a direct dynamical consequence of the statical elastic reaction called into play by such a strain as exists in a wave of polarized light. It may, however, be considered more probable that the matter of transparent bodies is really heterogeneous from one part to another of lineal dimensions not infinitely small in comparison with a wave length, than that it is infinitely homogeneous and has the property of exerting finite direct "molecular" force at distances comparable with the wave length: and it is certain that any spiral heterogeneousness of a vibrating medium must, if either right-handed or left-handed spirals predominate, cause a finite rotation of the plane of polarization of all waves of which lengths are not infinitely great multiples of the steps of the structural spirals. Thus a liquid filled homogeneously with spiral fibres, or a solid with spiral passages through it of steps not less than the forty-millionth of an inch, or a crystal with a righthanded or a left-handed geometrical arrangement of parts of some such lineal dimensions as the forty-millionth of an inch, might be certainly expected to cause either a right-handed or a left-handed rotation of ordinary light (the wave length being wth of an inch for homogeneous yellow).

* As would be were there different sets of vibrating particles, or were Rankine's important hypothesis true, that the vibrations of luminiferous particles are directly affected by pressure of a surrounding medium in virtue of its inertia.

But the magnetic influence on light discovered by Faraday depends on the direction of motion of moving particles. For instance, in a medium possessing it, particles in a straight line parallel to the lines of magnetic force, displaced to a helix round this line as axis, and then projected tangentially with such velocities as to describe circles, will have different velocities according as their motions are round in one direction (the same as the nominal direction of the galvanic current in the magnetizing coil), or in the contrary direction. But the elastic reaction of the medium must be the same for the same displacements, whatever be the velocities and directions of the particles; that is to say, the forces which are balanced by centrifugal force of the circular motions are equal, while the luminiferous motions are unequal. The absolute circular motions being therefore either

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