Writing a T for λ in the coefficient of e, since e is small,

Therefore the rate of rotation of the plane of polariza

If r+s is even, sin (r+s) 1⁄2π is zero, and there is no rotatory polarization.

If r is even and s odd, then the rotation of the plane of polarization changes sign with a the velocity of light, representing the effect of turpentine or quartz.

If r is odd and s even, then the rotation of the plane of polarization is unaltered by reversing the ray, representing the state of things in the magnetic field.

8. Form the equation for the conduction of heat in a bar, on the supposition that the temperature is the same throughout a transverse section, and that the rate of loss by surface radiation and convection is, at each point, directly as the excess of temperature over that of the surrounding medium. Point out the dimensions of the various quantities introduced. Integrate the equation completely in the two following cases, where the bar is very long, and is supposed to be heated at one end

(a) periodically, supposing the conductivity, specific heat, density, &c. unaltered with temperature, and the temperature of each transverse section a periodic function of the time,

(b) steadily, supposing the conductivity inversely as the absolute temperature, density, &c. unaltered with temperature, the surrounding medium at absolute zero, and the flow of heat steady.

Let c be the thermal capacity per unit of volume; k, h the coefficients of interior and exterior conductivity per unit of area; the temperature at the distance x from the origin; A the sectional area; and 7 the perimeter of the sectional

area.

The quantity of heat which enters the element dx of the bar in the time dt from the adjacent parts of the bar is dv

d

(Aku) ddt, and the loss of heat from the surface in

dx dx

the same time is hlvdxdt.

The increase in the quantity of heat in the element dx d in the time dt is (Acv) dt dx.

dt

Therefore, equating the gain and loss of heat, we obtain the equation

If [Z] be the unit of length, [T] the unit [] the unit of temperature, and [H] the unit

the dimensions of k will be

[H]

and of c will be

[Le]

of h will be

of time, of heat, [H] [L' TO] '

[H] [LT]' (Maxwell, Theory of Heat, Ch. XVIII.).

(a) If A, c, h, k, l be constant, the differential equation may be written

where

dv

d'v

dt dx2

– Hv,

We must assume as the general integral of the equation

4

where Tis the period and V is the mean temperature. Substituting in the equation we must have

These equations determine Pr and In

and A and B are

determined from the given circumstances of heating.

(b) The differential equation reduces to the form

do

Let v = then is the area of the curve representing

the temperatures, and the differential equation becomes

d'v dp*

=

1

Hence can be found in terms of x and then v=

аф

dx

xii. When a substance is melting at the absolute temperature t under pressure p, if I be the latent heat of fusion, u' the volumes of unit of mass of the substance corresponding to its liquid and solid states, prove that

and

where J is the dynamical equivalent of heat.

Show how J. Thomson's discovery of the dependence of the temperature of fusion of ice on pressure is connected with this relation.

(Briot, Théorie mécanique de la chaleur, §§ 127-129).

for the perturbation in radius vector of a planet.

Integrate this equation so as to obtain a first approximation to δη.

Explain why this and the similar equations for longitude and latitude cannot be employed with advantage in the calculation of secular variations or of long inequalities.. (Cheyne, Planetary Theory, § 112).

ii. Assuming the equation

dę dn

+ +

dz) dz=

dx dy dx dy dz = [[(15+mn + n5) ds,

where l, m, n are the direction-cosines of the outward-drawn normal to the element ds of the surface of a closed space S, throughout which and over whose surface the integrals are taken, prove Green's Theorem; and show how to adapt it to the case in which one of the potentials is many-valued, and S is multiply-connected.

Hence show that the whole exhaustion of potential energy of any number of gravitating particles, originally scattered at infinite distances from each other, is