10. Define an electric image, and find the surface density on an uninsulated spherical conductor (radius a) under the influence of a quantity e of electricity at an external point at a distance ƒ from the centre of the sphere.

3

When the sphere is insulated and the whole charge on the sphere is e

α

find the position of the line of no electrifica

tion on the surface of the sphere, and the quantities of electricity on each side of this line.

(Maxwell, Electricity, § 157).

α

a2

When uninsulated the charge induced on the sphere is -e, and the surface density at any point is ea where r is the distance from the influencing point.

Hence if the sphere be insulated and have a charge

Απαγ

we must superpose on the preceding system a charge uniformly distributed with surface density e

elf

At the line of no electrification on the surface of the sphere r=f; hence if A be the influencing point (fig. 79), B the

image, the centre of the sphere, and EE' the line of no electrification; then AE= AC, and therefore BE= EC. The quantity of electricity on EDE'

and

r2 = a2 – 2aƒ cos 0 +ƒ3, af sin Ode=rdr. Therefore the quantity of electricity on EDE'

= 4e (1-2), (-) dr

= de (1+) (+),

xi. What is meant by the specific inductive capacity of a dielectric ?

Investigate the electrostatic capacity, per unit of length, of a submarine cable, the diameter of the core being d, and the external diameter of the insulating sheath D.

Assuming the leakage to bear, at all points, the same ratio to the charge, form the equation for the transmission of electric potential along the cable: and shew from it that, ceteris paribus, the time necessary for a definite electrical operation is as the square of the cable's length. If the leakage be considerable, how must the battery-power depend on the length of the cable in order that slow signals may be of a given intensity?

(Maxwell, Electricity, SS52, 126; Stokes and Thomson, Proceedings of the Royal Society, VII.)

Let c be the electrostatic capacity per unit of length, so that cul is the quantity of electricity required to charge a length of the cable up to potential v.

Then c=

1 K

capacity of the dielectric.

where K is the specific inductive

Let k denote the galvanic resistance of the cable, and let y denote the strength at the time t of the current at a point P of the cable at a distance x from one end.

Let h denote the ratio of the leakage to the charge per unit of time.

The potential at the outside of the cable may be taken at each instant as zero; hence at the time t the quantity of electricity on a length dx of the cable at P will be cvdx.

dy

The quantity that leaves the element de in the time dt for the adjacent parts of the cable will be dtdx, and the leakage in the same time will be hcvdxdt.

the differential equation for the transmission of electric potential along the cable.

This is the differential equation for the propagation of the temperature v in a bar, of which c is the specific heat per unit

1

of volume, and h the coefficients of interior and exterior

conductibility of the bar per unit of length.

If we assume v=e", the differential equation becomes

The consideration of the dimensions of this equation shews that two cables will be similar, provided the squares of the lengths x, measured to similarly situated points, and therefore the squares of the whole lengths 7, vary as the times divided by ck; or the time of an electrical operation is proportional to ckl.

Taking into consideration the leakage, the potential diminishes as eht, and the time varies as the square of the length of the cable; hence the battery power must vary as ext

THURSDAY, Jan. 21, 1875. 9 to 12.

MR. COCKSHOTT. Roman numbers.

MR. GREENHILL. Arabic numbers.

1. EXPLAIN the general principle of reciprocal polars. Shew that the reciprocal of a circle with respect to a point is a conic section, and determine the nature and magnitude of this conic.

The diagonals of a quadrilateral inscribed in a circle intersect at right angles in a fixed point. Prove that the sides of the quadrilateral touch a fixed conic.

The angular points of a rectangle circumscribing a conic lie on the director circle.

Reciprocating with respect to a focus proves that if the diagonals of a quadrilateral inscribed in a circle intersect at right angles in a fixed point, the sides of the quadrilateral will touch a conic, of which the fixed point and the centre of the circle are foci.

ii. Prove by changing the order of integration, or otherwise, that