ARITHMETIC (oral). A boat is rowed over a certain distance, when there is no tide, in 24 hours. With the tide, however, the same distance can be rowed over in 15 hours. Against the tide the boat can be made to go over 32 knots in 2 hours. Find the speed, accordingly, when it is rowed with the tide. ALGEBRA (oral). Solve the following simultaneous equations, (1+2k) x-(1+k)y=1−k, 3 (1+k) x-(3+k)y=3+k. GEOMETRY (oral). Draw a straight line meeting two straight lines not in the same plane and normal to a given plane. TRIGONOMETRY (written). 4. If the length of three bisectors of the three angles, A, B, C of a triangle ABC be respectively equal to p, q, r, prove that 5. Having given one angle, the perimeter, and the radius of the circumscribed circle of a triangle; solve the triangle. Application (3 hours). When the three sides of a triangle are known to be respectively, a=750.74 m., b=596.42 m., compute the three angles and the area. c=204.68 m., ANALYTIC GEOMETRY (3 hours). 1. Given a point (1, 1) and a straight line 3x+4y-6=0, the axes being rectangular. Form the equation of the curve of the second degree, having the point and the straight line for its corresponding focus and directrix and 5 for its eccentricity, and reduce it to the standard form. Such is the original. 2. Let N be the point of intersection of the normal at any point M on an ellipse and its major axis. Prove that the orthogonal projection of MN on the line passing through M and one of the foci is constant. 3. Prove that the four vertices of a parallelogram circumscribing an ellipse and its two foci are on the same equilateral hyperbola. 4. Given an ellipse and a circle concentric with each other, the radius of the circle being equal to the sum of half the major axis and half the minor axis of the ellipse. Prove that the locus of the point of intersection of the two normals to the ellipse at the points at which two tangents are drawn to the ellipse from any point on the circle is a circle. 5. Let N be the point of contact at which a tangent is drawn from the center M of a fixed circle to the circumscribed circle of a triangle self-conjugate with respect to the fixed circle. Prove that MN is constant. DIFFERENTIAL AND INTEGRAL CALCULUS (4 hours). = 1. If f'(x). ❤ (x)−ƒ (x). ø′ (x)=0 within the interval a≤ x ≤b, and ƒ (a) f(b)=0, then prove that (x) will become zero within the given interval at least once. Heref' (x) and ′ (x) are continuous within the given limits. 2. Let Y be the point at which the line passing through any point X on the diagonal AC of a parallelogram ABCD and the vertex B intersects the side AD or its extension. Find the minimum of the sum of the areas of the two triangles AX Y and BXC. 3. Take z as the function of two independent variables x and y; substitute x=r sin @ cos, y=r sin 9 sin ‚1 z=r cos 0 2 in√√1+(dz)2+(dz)2; taking 0 and as independent variables eliminate x, y, z. dx 4. B (l, m) represents Prove that f'z1(1-x)=-1 dr, 1 and m being positive. B(l,m)='+B(l+1,m). 5. Find the whole length of the space curve represented by the equations a and b being positive. ax=z (b+z), a2(x2+y2)= b2 22, 6. Take x=(u,v) and y=0(u,v), and change the variables of integration in Abel, N. H., 24, 32, 134. Abelian functions and integrals, 164, 182. Abraham, M., 184. Acta mathematica, 176. Adjunkter, 188–190. Adler, A., 26, 103. Agrégation, agrégé, 74, 75, 216, 255-266. Aldis, N. S., 146. d'Alembert's principle, 33, 256. Alexandroff, I., 103. Ämbetsexamen, 176-180, 222. Ampère, A. M., 134. André, P., 67. Anstellungsfähigkeit, 97. Apollonius of Perga, 25, 258. Appell, P., 70, 72, 146, 180, 183, 184. Archibald, R. C., 76. Archimedes, 10, 25, 150, 256. Association for the Improvement of Athénée royaux, 28f, 213, 228. Atlantic Monthly, 84. Australia, 3, 5-14, 212, 228, 230. Bayerische Zeitschrift für Realschulwesen, 99. Baynes, R. N., 184. Beier, A., 128. Beke, E., 137. Belgium, 3, 4, 28-38, 213, 214, 224-228. Beman, W. W., 10. Bergmann, F., 26. Bernoulli, J., 33, 179, 256. Berry, A., 60. Bessel's functions, 147, 183. Bettanzi, R., 143. Bézout, É., 133, 161. Bianchi, L., 32. Bibliography, Australia, 14; Astria, 26, Bioche, C., 76. Bismarck, 126. Austria, 3, 15-27, 77, 130, 134, 212, 213, Björling, C. F. E., 178, 179, 182. 218, 224, 225, 228. Babcock, K. C., 203. Blätter für die Gymnasialschulwesen, 99. Baccalauréat de l'enseignement second- Block, H. G., 186. aire, 67. Blutel, E., 254, 264. Baccalauréat ès sciences mathématiques, Bobynin, V. V., 160, 167. Brooks, R. C., 210. Brown, J. C., 4. Brown, J. F., 87, 95, 110, 129, 204, 207. Brown University, training of teach- Brunn, H., 107. Bruns, H., 109, 110. Bucherer, A. H., 98. Bulletin Administratif du Ministère de Bureau of Education (United States), bul- letins, 3, 4, 44, 60, 129, 200, 203, 207–210. Busse, 91. Busz, 91. Byerly, W. E., 146. Caesar, 131. Cacique, 72. Clapham High School, training of teach- Classes de mathématiques spéciales, 68-72, 74-76. Collège de France, 71, 73. College Entrance Examination Board, Collèges communaux, 28f, 66. Committee of Seventeen, report, 203, 204. 25. Concours, Belgium, 29; France, 71-75; Contemporary Review, 58. Coolidge, J. L., 143. Coore, G. B. M., 129. Coriolis, G. G., 256. Cambridge and Oxford local examina- Counts, G. S., 202, 209. tions, 6, 50, 231-236. Cambridge University, 51-55; entrance Candidatus magisterii, 41, 214. Candidatus philosophiæ, 41, 214. Cantor, G., 21, 87. Capen, S. P., 200, 202. Cardinaal, J., 157. Carslaw, H. S., 14. Cartan, E. J., 264. Castelnuovo, G., 219. Cauchy, A. L., 32, 62, 114, 134. Cauer, P., 91. Censors, 173, 174. Centralblatt für die gesamte Unterrichts- Verwaltung in Preussen, 83. Cesaro, G., 34. Ceva, G., 154. pan, 152; Netherlands, 155, 156, 220, Dodd, C. I., 137. Donder, T. de, 37. Drach, J., 182. Drude, P., 184. Dumur, J. P., 128, 211. Durège, H., 181. Eberhard, V., 87. École Normale Supérieure, Paris, 68f, 216. Eidgenössische Technische Hochschule, Encyclopaedia Britannica, 129, 163. England, 3, 45-60, 163, 215, 223-226, 228, Engström, F. A., 186. l'Enseignement Mathématique, 4, 26, 38, Eötrös, J., 135. Epstein, P., 117. Euclid, 9, 65, 169, 206. Euclid's elements, 6, 7, 41, 49, 50, 54, 172. "Euler's theorem" for polyhedra, 16, 154. Examination papers, Bavaria, Lehramts- Examinations for prospective teachers in |