structed through simply joining and intersection, and in particular without the further use of a right angle?

DESCRIPTIVE GEOMETRY.

In a vertical plane of projection a semi-ellipse is given, of which the shadow is a semicircle. It is subjected to a rotation in space about a vertical axis. Construct the normal curve and the meridian of the surface of rotation so constructed, i. e., the section of any plane (1) perpendicular to the axis of rotation or (2) through the axis. (Suppose the proportions given.) Give a free drawing in india ink of the two curves sought, also a short description of the corresponding construction.

ANALYTIC MECHANICS.

1. On three mutually perpendicular weightless sticks, rigidly fastened together at a point 0, "mass-points" P, Q, R are placed at distances a, b, c from 0.

The system is turned about an axis through O which at a given instant, when the angular velocity is w, makes angles a, B, y with the three sticks.

γ

How great is the vis viva of the turning motion of the system at this instant? Through what single force at O and what pair of forces (what axis-moment) may the three centrifugal forces be replaced?

Discussion of the results for the case a=B=y.

2. The velocities of three noncollinear points of a rigid body moving with freedom of the first order, are given for a certain instant as vectors, i. e., in size and direction. Show how to find, either by drawing or by calculation, for the same instant: (a) the velocity of a fourth point of the body, also considered as a vector, and (b) the central axis or rotation axis, i. e., the locus of the points of the body which, for the instant, of all of its points has the least velocity.

TRIGONOMETRY AND MATHEmatical GeogRAPHY.

1. On two points A and B of equal height and 25 meters from each other rests a smooth thin band of steel which sags 50 centimeters in the middle. How many millimeters (to the nearest tenth of a millimeter) is the band longer than 25 meters? The calculation is to be carried through, and the exactitude of the result proved, without employing any tables. (The calculation is to be carried through in numbers.) 2. On a simple rod standard a vertical reference-plane F is determined by a strong white endless thread passing through two rings 0, and 02, and stretched with the help of a hanging stone 03. The azimuth of the two directions of the thread differs by exactly 180°. The azimuth is not exactly known, but it corresponds about (according to the compass within 5°) to the prime vertical.

At the point of observation (in middle Germany), determining the northern latitude on an evening toward the end of November, 1895, the following times of transit through the plane of the thread are observed in siderial time:

y Lyra (about westward) S1=22h 29m 59s,

y Andromeda (about eastward) S2=22h 35m 45o.

The apparent AR [right ascension] and ♪ [declination] of the two stars on the evening of observation were:

Lyra a1=18h 55m 2s, d1=+32° 32′.8,

y Andromeda a2=1h 57m 32, 82=+41° 50′.1.

What is the northern latitude of the place of observation? What was the azimuth of the plane of the thread? What kind of stars are to be chosen for such determination of the northern latitude by transits through one and the same vertical first eastward and then westward? Calculation of in numbers is desired.

(The candidate is allowed a logarithm table.)

THEORETICAL PHYSICS.

1. Discussion of the physical foundations of the Newtonian definition of the quantities "force" and "mass."

2. Derivation of the formula of an adiabatic curve.

3. Foundations of a theory of plane diffraction grating for parallel light rays.

4. How and with what exceptions is Helmholtz's law of induction derived on the basis of the law concerning the conservation of energy?

II. Mathematical Questions of the Chemistry-biology Division.

ALGEBRA AND LOWER ANALYSIS.

1. A merchant buys wares for a certain sum, has in addition 5 per cent expenses, and sells them again før 504 marks, and thereby gains a twentieth part of the purchasing price. What did this amount to?

2. A left a fortune of 100,000 marks. From this his children must receive 6,810 marks annually for 10 years from the first payment within a year after his death. The capital after 10 years is to be devoted to school purposes. How large will this amount be?

3. Express the fraction

as the sum of two positive fractions with denominators

13 and 23. What are possible solutions?

4. In the equation ra— 11x3+px2+qx−60=0 the root 2+i is given. Find the other roots and the coefficients p and q. (Check by means of Horner's method).

1.

Lim (ex-esinx) x=0(x−sinx)

DIFFERENTIAL AND INTEGRAL CALCULUS.

=?

2. Express y=log (x+√1+x2) as a power series in x and discuss the convergence of the series.

3. Given the hyperbola-2-1-0.

Through the origin and with different points of the hyperbola as centers, circles are described. Find the equation of the envelope of these circles.

ELEMENTARY GEOMETRY.

1. Describe a circle which touches a given line in a given point P and cuts a given circle K at the ends of a diameter. (Analysis, construction, proof.)

2. Given a point and two lines L and L', which cut one another in A. On L find a point X such that the perpendicular X Y from X on L' is a mean proportion between A Y and PX. (Construction and proof.)

3. Given a right circular cone of which an axial section is an equilateral triangle. Produce the surface beyond the circular base, such that the whole surface of the added conical shell is to the surface of the whole cone as 5 is to 6 (including construction of the calculated result).

4. B and C are the middle points of two spheres of radii r, and r2 (10 and 14 cm.). To an observer at A the spheres appear under angles of sight S, and S2 (3° 37′ 20′′ and 4° 16′ 30′′). Angle BAC is 71° 4' 40". How great is the distance between B and C, and how large are the angles ABC and ACB?

ANALYTIC GEOMETRY.

1. The curve with equation

x3-2x2y+2y3-y2x-4y2+2xy-x+2y=0

breaks up into three lines, of which one is x-2y. What are the equations of the other two? What are the coordinates of the vertices of the corresponding triangle; find its area.

x2 y2

2. In the hyperbola 2-2-1-0 a diameter with slope u is given. What is understood by a diameter conjugate to this diameter and how is the slope of this diameter derived? How large are the semi-diameters corresponding to these two directions? 3. Discuss and sketch the curve with equation

x3+y2x-2x2+y2=0.

4. Given a circle of radius r and a line which is at a distance a from the middle point of the circle. A second circle touches the line in its intersection with the line drawn through the middle point of the first circle perpendicular to the line; the common tangents of the circles touch the second circle in points, the locus of which is required. (The given line is to be taken as x-axis, and the foot of the perpendicular as origin of coordinates.)

DESCRIPTIVE GEOMETRY.

An ellipse is projected horizontally as a circle of 10 cm. diameter, vertically as a line (30° toward the right with the base line). The horizontal trace of a given plane makes an angle of 60°, the vertical trace of 45°, both toward the right with the base line. Construct the shadows of the ellipse on these planes. (Both projections of rays of light coming from the left make angles of 45° with the base line.)

The two projections of the shadows are to be constructed independently of one another; at any given point construct the tangent and find the nature of the curve in order that its conjugate diameters can be determined.

EXPERIMENTAL PHYSICS.

1. A locomotive weighing 20,000 kilograms moves on a track 1 meters wide and its center of gravity is 13 meters above the rails; what is the greatest velocity that the locomotive may attain in order that, on a curve of 80 meters radius, it shall not leave the rails? What is the maximum velocity, if the outer rail be so raised that the plane of the rails is inclined to the horizontal plane with an angle of 5°?

2. For determining the temperature of a smelting furnace, a platinum sphere of 100 grams is put in it and then thrown into a mixing calorimeter which contains 800 grams of water at 10° C. What is the temperature of the furnace if the brass calorimeter tub weighed 250 grams and the final temperature reached 14.8°? (Specific heat of brass, 0.0926; of platinum, 0.0326.)

3. Two biconvex lenses with focal lengths ƒ1=4cm. and ƒ1⁄2=6cm. are arranged from left to right, such that the distance of their optical middle point amounts to 5=1 cm.; the thickness of the lenses may be neglected. To the left of the first lens is a luminous substance AB 1 cm. high. Construct the picture of the object which is thrown through the pair of lenses and also determine the distance b1 of the picture from the second lens. How great is the common focal distance ƒ of the system of lenses counted from the second lens; and what advantage is there in such a combination of lenses over a simple lens with the same focal distance?

APPENDIX F.

JAPAN.

The following mathematical papers were set in Tokyo for the twenty-fifth examination for teachers' licenses, in 1911.

PRELIMINARY EXAMINATION QUESTIONS.

ARITHMETIC (3 hours).

1. Find three fractions, A, B, and C equal to 3, 1, and fr, respectively, such that A's denominator is equal to B's numerator, and B's denominator to C's numerator. Find the simplest forms of such three fractions.

2. A certain company, dividing its capital in the ratio of 3:57, carried on its business in three divisions. At a semiannual settlement it was found that the first division had made 2,600 yen, and the second had earned 8 per cent a year on its capital, but that the third had suffered a loss of 5 per cent a year of its capital. However, the net result was found to be a gain of 6 per cent a year on the total capital. What was the amount of the capital?

3. A steamer, bound for a certain port, had its engine damaged when one-fifth of its voyage had been completed. As it had to reduce its speed by 10 knots for the rest of its course, the average speed was found to be less than the first by 4 knots. What was the initial speed?

4. By evaporating 600 grams of water containing 3 per cent of salt, one containing 5 per cent of salt was to be obtained. It was found, however, that 70 per cent of the water had already evaporated. How much water containing 3 per cent of salt must be added in order to obtain the solution of required strength?

5. Of a cylindrical vessel holding one shō, the height and diameter of which are equal, find the height to the hundredth place.

1

x+y-az-b=0,

3. Cut a triangle and a rectangle, having equal bases on a straight line, by another straight line [parallel to it] 1 so that the sum of the areas cut out between the parallel lines shall be equal to the area of the triangle. Find the distance between the parallel lines.

4. In how many different ways can 10 balls be arranged in a straight line, provided that 2 special balls must in all cases be placed so as to occupy alternate positions? 5. Let a1, a2, Az, be an arithmetic progression, and b1, b2, bз, ...... be a geometric having all its terms positive. Prove that a, is not greater than b,, if a, b, and a b2.

These words do not occur in the original.

'It is not clear what is meant by "alternate positions."

GEOMETRY (3 hours).

1. Let two circles touch internally at A. From any point P in the circumference of the external circle draw a tangent P M to the internal one, and prove that P A:P M is constant.

2. Given a vertical angle, the radius of the inscribed circle, and the area, construct the triangle.

3. The vertex A of the rectangle ABCD is a fixed point, and B and D are on the circumference of a fixed circle. Find the locus of the point C.

4. Find the limit of the position of a point, the ratio of whose distances from two fixed points is less than a given ratio.

5. Of a quadrilateral whose four vertices are not all in one plane, three are fixed and one moves along a straight line. Find the locus of the intersection of the lines joining the middle points of its opposite sides.

FINAL EXAMINATION QUESTIONS.

ARITHMETIC, ALGEBRA, AND GEOMETRY (written).

Part I (3 hours).

1. The sum of a certain irreducible fraction and its reciprocal is equal to

3. If a, b, p, and q be real, prove that the following equation has real roots,

4. Solve the following inequality, x−b>√a(a−2x), where a and b are positive, and√represents the positive square root.

5. Prove that the following three equalities are consistent with one another,

1. If rectangles ABDE, ACFG be externally constructed on the two sides AB and AC of the right angle A of a right-angled triangle ABC, prove that the straight lines BF and CD intersect with each other on the perpendicular from A to the hypotenuse BC.

2. Draw a circle with its center on a straight line passing through the center of a given circle, intersecting this circle at right angles and passing through a given point. 3. Of a triangle ABC, the vertex A is a fixed point on an edge of a trihedral angle and the other two vertices B and C move respectively along two other edges. Find the locus of the center of gravity of the triangle.