7. Four equal masses are attached at equal distances A, B, C, D at points on a light string, and so placed that ABC= ▲ BCD=120°, and the various parts of the string are straight; an impulse I is given to the mass at A in the direction BA, show that the impulsive tension in AB is I.

8. A heavy uniform beam, 8 feet long, weighs 20 lbs., and at one end it carries a load of 20 lbs. and at the other end a load of 10 lbs. The beam thus loaded is to be carried on a certain support. Find by a graphical construction the position of the support.

If the beam is to rest on two supports which are to divide the load in a given ratio, show how to find on your diagram any number of suitable pairs of points, and find the position of the supports when they are at a distance 4 feet apart, the pressures on them being in the ratio of 3 to 2.

9. A tripod consists of three equal rods, each of weight w, smoothly jointed at the upper ends. It is placed symmetrically on a rough horizontal table, for which the angle of friction is ø, and a weight Wis put on the top of the tripod. Show that the rods can not make an angle with the vertical greater than

10. Two equal uniform rods are fastened together so as to bisect one another at right angles. They rest in a plane at right angles to a rough wall, the one rod resting on the edge of the top of the wall, and an end of the other against the vertical side. Prove that the limiting inclination @ of the second rod to the vertical is given by the equation tan @ cos2λ=cos(λ+0) sin (λ—0),

where X is the angle of friction.

11. The figure ABCD represents a freely-jointed light plane framework, with forces acting in the plane at right angles to BCD. Show that it is in equilibrium, and prove, graphically or otherwise, that the stresses in AB, AC are equal and of opposite sign.

12. Two equal friction-wheels of radii R inches turn on axles B, C of radii r inches; the coefficient of friction between wheel and axle is tan o. A wheel of weight Wis attached to an axle of radius a inches which rests on the circumferences of the first two wheels as in the figure. Neglecting the weights of the wheels B and C, show that the sin 20 where sin sin : the centers

least couple which will rotate the wheel A is Wa

3

=

BO

ос

of the three wheels forming an equilateral triangle.

(ALTERNATIVE QUESTIONS IN PHYSICS.)

13. A diving bell has the form of a paraboloid of revolution cut off by a plane perpendicular to its axis at a distance h from the vertex, where h is the height of the water barometer. If the bell be lowered so that its vertex is at a depth 5h/2 below the surface of the water, find how high the water will rise in the bell.

14. A battery is connected to a galvanometer of 40 ohms resistance, and a certain current is observed. The galvanometer is now shunted with a resistance of 10 ohms, and the current in the galvanometer falls to one-half of the former value. Find the internal resistance of the battery.

15. The coefficients of cubical expansion of mercury and brass are .00018 and .00006 per degree centigrade. A mercury barometer with a brass scale reads correctly at 15° C., standing at 30 inches; what will be the error at 35° C.?

16. One cubic foot of air at a temperature of 500° C. absolute is expanded isothermally from a pressure of 120 lbs. per sq. in. to twice the initial volume; it is then expanded adiabatically to three times the initial volume. Find the pressure and temperature at the end of each stage and calculate, graphically or otherwise, the work done and the heat units taken in during each stage, taking the mechanical equivalent of one thermal unit as 1,400 ft. lbs., and the equation for adiabatic expansion of air as pvl. const.

APPENDIX C.

FRANCE.

CONCOURS FOR ADMISSION TO THE ÉCOLE NORMALE SUPÉRIEURE AND FOR THE BOURSES DE LICENCE IN 1913.

MATHEMATICS.1

GROUP I.

I.

(Time: 6 hours.)

Being given three axes of rectangular coordinates Ox, Oy, Oz, consider the surface (S) defined by the equation z=xy+x3 and the line (D) defined by the equations y=b, z=c, where b and c are two given constants, the second not being zero. In all that follows this line (D) remains fixed.

1. Show that the surface (S) is ruled and find its generators.

2. To each rectilinear generator (G) of the surface (S) establish a correspondence of the plane (P) drawn through the line (D) and parallel to the line symmetric to (G) with respect to the plane xOy. Determine the locus of the point of intersection of (G) and of (P), when the line (G) describes the surface (S).

Show that this locus is a curve (C) situated on a quadric (Q), and determine this quadric.

3. Form the equation of the fourth degree, giving the abscissas of the points of intersection of the curve (C) with a plane given by its equation ux+vy+wz+s=0. Calculate the elementary symmetric functions of the roots as a function of u, v, w, s. From this deduce the relation which the abscissas x1, X2, X3, X4, of four points of the curve (C) must satisfy in order that these four points should be in the same plane.

This relation will be useful in most of the questions which follow.

4. Deduce from the preceding relation the conditions which the abscissas X1, X2, X3, of three points of the curve (C) must satisfy in order that these three points shall be collinear.

Form the general equation of the third degree of which the roots are the abscissas of three collinear points of the curve (C). Show that the lines which cut (C) in three points generate one of the families of rectilinear generators of the quadric (Q).

5. Show that the necessary and sufficient condition that the osculating planes to the curve (C) in three given points cut on the curve (C) is that the three points are collinear.

6. Through any point M of the curve (C) there pass two planes enjoying the property of being tangent to the curve (C) at the point M and in another point (that is to say of being bitangent to the curve). Suppose M' and M" are the second points of contact of these two planes. Show that there exists a plane bitangent to the curve (C) in M' and M".

What conditions must be satisfied by the abscissas of the three points M, M', M", of the curve (C) in order that any two of them are points of contact of a plane bitangent to the curve (C)?

1 The solutions of the following problems are to be found in Nouvelles Annales de Mathématiques, tome 73, Oct.-Nov., 1914, pp. 467-482.

7. Form the general equation of the third degree whose roots are the abscissas of the points M, M', M", of the curve (C) subject to the preceding conditions. Express the coefficients of this equation by means of the abscissa e of the fourth point of intersection μ of the curve (C) with the plane (7) determined by the points M, M', M".

Calculate, in terms of the coefficients of the equation of the plane (7) and the coordinates of the point of concourse, A, of the tangents to the curve (C) at the points M, M', M". This point A is said to be the point associated with the point μ of the curve (C).

8. Show that there exists an infinity of quadrics, depending only on b and c, with respect to which the point A is the pole of the plane (π); determine these quadrics and show that one of them is the quadric (Q) already considered.

Determine the locus (T) of the point A, also the envelope of the plane (7), when the point μ describes the curve (C).

9. With any three collinear points μ1, M2, M3, on the curve (C) are associated the three vertices A1, A2, A3, of a triangle inscribed in the curve (F). Determine by supposing b=0, the envelope of the sides of this triangle when the line μ1 M2 M3 varies. Show that in the same hypothesis b=0, the circle circumscribed about the triangle A, A2 A3 passes through two fixed points.

II.

(Time: 4 hours.)

Given two rectangular axes, and the differential equation y—2xy′+y2y'3—0.

1. Show that this equation admits of an infinity of solutions, the curves C, of which the equation is of the form y2=f(x), f(x) denoting a polynomial in x. Write the gen

eral equation of the curves C; show that through every point of the plane there passes

either one or three curves C, and determine the region of the plane where the point ought to be found in order that the number of the curves which pass through it shall equal three; determine the locus of the points such that two of the curves C which pass through one of them are orthogonal.

2. Given the point A (x=0.5, y=0). Let P be that one of the curves C which passes through A and is concave toward the positive part of the axis Ox; let B be the point of the curve P which has for ordinate✔6. Suppose Q is that one of the curves C passing through B and concave toward the negative part of the x-axis; suppose finally that A' is the point where this curve cuts the axis Ox. Calculate the area bounded by the arcs of curves AB, BA', and the axis Ox.

3. A moving point, starting from A, traverses successively the arc AB of P, then the arc BA' of Q. Its tangential acceleration is constantly equal to its velocity, and its initial velocity is equal to 1; at the point B suppose that the velocity does not change in magnitude, but only in direction. Calculate to the nearest tenth the time taken for the point to traverse the arc ABA'.

4. At the point B, the acceleration of the moving point suffers a discontinuity. Calculate, by its projections on the two axes of coordinates, the geometric variation of the vector-acceleration at the point B.

Group II.
I.

Consider in a plane two rectangular coordinate axes Ox, Oy. A material point M, of mass equal to unity, is movable in a plane under the action of a force (F) of which the projections X and Y on the axes are X=x, Y=y-4x, x and y denoting the coordinates of the point M:

1. Form and integrate the differential equations of the motion of the point M. 2. Determine the motion of M in supposing that at the beginning of the time its coordinates are (a, 0) and that its velocity has -a and 2a for projections on the axes. Construct the trajectory (T) corresponding to this motion.

3. Calculate the time taken by the moving point in going from any point M of its trajectory (T) to the point M', where the tangent to the trajectory is parallel to the radius vector OM.

4. Prove that the hodograph of the motion is a homothetic curve of the trajectory (T) and calculate to the nearest tenth the ratio of homothety.

5. The trajectory (T) passes through the point 0. Evaluate, in terms of the abscissa of the point M, the area bounded by the arc of the curve OM and the chord OM, also the volume generated by this area turning about Oy.

NOTE. For interesting comment concerning the emphasis on analytical geometry in the above examination, compare E. Blutel's report, page 21 (Commission Internationale de l'Enseignement Mathématique. Sous-Commission Française, Rapports, vol. 2).