is increasing, and, & being positive, tani is a minimum; the increase takes place while

aty changes from π+ to π-ε,

arriving at the mean position 8, where i is a minimum.

The motion of the nodes is exhibited geometrically in fig. (93).

8. Draw the course of a small pencil of parallel rays, passing at such an angle through a biaxal crystal cut with parallel faces, that external cylindrical refraction takes place.

How may the constants a, b, c, corresponding to the axes of elasticity be obtained experimentally?

If the two faces of a prism, formed of a biaxal crystal, be perpendicular to each other, and one contain the two axes of elasticity a, c, and the other b, c; and if μ, μ¿ be two refractive indices for the ordinary ray when the planes of refraction are perpendicular to the axes of a and b respectively; shew that D, the minimum deviation of the extraordinary ray, is given by the equation

Let OA, OB (fig. 94) be the projections of the faces containing (a, c) and (b, c) respectively,

QR, RS, ST, directions of normals to the extraordinary wave front at incidence, 1st and 2nd refraction,

TDq = D, the deviation,

u = velocity of wave in air,

v = velocity of extraordinary wave in the crystal;

and by (7), 2u* = a + b2 - a cos 24-b cos2p;

.. a2 + b2 - 2u a cos 24+ b2 cos20,

0 = a2 sin 24-b" sin 24;

·. (a2 + b2 — 2u2)* = a* + b* + 2a2b2 cos 2 (Þ+¥) ;

.. 2a2b2 {1 — cos2 (4 + $)} = 4 (a2 + b2) u2 — 4u*,

.. by (2),

EXAMINATION PAPERS FOR THE

MATHEMATICAL TRIPOS 1854.

TUESDAY, Jan. 3. 9...12.

1. THE complements of the parallelograms, which are about the diameter of any parallelogram, are equal to one another.

If K be the common angular point of these parallelograms, and BD the other diameter, the difference of the parallelograms is equal to twice the triangle BKD.

2. Divide a given straight line into two parts so that the rectangle contained by the whole line and one of the parts shall be equal to the square of the other part.

Produce a given straight line to a point such that the rectangle contained by the whole line thus produced and the part produced shall be equal to the square of the given straight line.

3. The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles.

If the opposite sides of the quadrilateral be produced to meet in P, Q, and about the triangles so formed without the quadrilateral circles be described meeting again in R; P, R, Q will be in one straight line.

4. Describe an isosceles triangle having each of the angles at the base double of the third angle.

Upon a given straight line, as base, describe an isosceles triangle having the third angle treble of each of the angles at the base.

5. If four straight lines be proportionals, the rectangle contained by the extremes is equal to the rectangle contained by the means.

EA, EA' are diameters of two circles touching each other externally at E; a chord AB of the former circle when produced touches the latter at C', while a chord A'B' of the latter touches the former at C: prove that the rectangle contained by AB, A'B' is four times as great as that contained by BC′, B'C.

6. If a solid angle be contained by three plane angles, any two of them are greater than the third.

Within the area of a given triangle is described a triangle, the sides of which are parallel to those of the given one. Prove that the sum of the angles subtended by the sides of the interior triangle at any point not in

the plane of the triangles is less than the sum of the angles subtended at the same point by the sides of the exterior triangle.

7. Prove that the rectangle contained by the latus rectum of a parabola and the abscissa of any point in the curve is equal to the square on the ordinate drawn to the axis.

If N be the foot of the ordinate, SY the perpendicular from the focus on the tangent, and T the point where the tangent meets the axis produced, NY is equal to TY.

8. Define the tangent to an ellipse, and shew that it makes equal angles with the focal distances of the point of contact.

If NP be the ordinate of P, Y, and Z, the points where the tangent at P meets the perpendiculars from the foci, NY: NZ :: PY : PZ.

9. The tangent at a point P of an ellipse cuts CA, CB produced in T, t respectively, and PN, Pn are the respective perpendiculars from P upon CA, CB; prove that CT. CN= AC2, and that Ct. Cn = BC2.

Shew that the subnormal is a third proportional to CT and BC.

10. The rectangle contained by the abscissæ of the major axis of an hyperbola is to the square on the ordinate as the square on the major axis is to the square on the minor axis.

If A, M be the extremities of the major axis of an ellipse, PP' a double ordinate, and AP, P'M be produced to meet in Q; Q will lie in an hyperbola having the same axes as the ellipse.

11. Parallelograms, whose sides touch an hyperbola and its conjugate, and are parallel to conjugate diameters, have the same area.

If CP, CD be conjugate semi-diameters, and through C a straight line be drawn parallel to either focal distance of P, the perpendicular let fall from D on this straight line will be equal to half the minor axis.

12. If two spheres exterior to each other be inscribed in a right cone touching it in two circles on the same side of the vertex, and a plane be drawn touching the spheres and cutting the cone; shew that the section is an ellipse, that the points of contact of the spheres with the plane are the foci, and that the planes of the two circles contain the directrices.

TUESDAY, Jan. 3. 1...4.

1. DIVIDE

48 7,31 by 1085,70 174,

1.0714285; and find what decimal of a guinea is equivalent to 2835 of a pound sterling.

2. The capital of a firm consists of £713. 3s., £964. 17s., £2391. 3s., subscribed by three partners; divide £2231. among them in proportion to their several capitals.