ثلاث مسائل لو سألني عنها دون ان قتلته فانه يسألني عن المنصور ما اجبته - قال الفضل فبينما انا قاعد ان دخل ابو مع ام دلامة على انه دلامة على الرشيد باكيا وقد تعطا 2-9 يدخل على الرشيد وينعيها اليه وانها تذهب الى زبيدة تنعيه اليها فما رأة الرشيد باكيا قال له ما بالك تبكي 12. Translate into Arabic: As evening comes on, and it is time to rest, Abdel Hassan dismounts from his horse, and strikes his spear in the sand. This is the signal for all the party to stop. The camels kneel, the riders dismount, Yussuf and Ayesha are lifted out of their baskets, and the servants are busy putting up their tents, for in the desert there are no towns, no houses. The wandering Arabs live always in tents. These are made by putting up a few long poles, and throwing over them coverings of goat's hair or sheep skins. The coverings are fastened to the ground with pegs. There is a curtain for a door, and sometimes the tent is divided into two rooms by another curtain. The curtains and poles can easily be taken down, and folded so as to be packed on the backs of the patient camels. FELLOWSHIP EXAMINATION. Examiners. JOHN LEWIS MOORE, D. D., Vice-Provost. ANDREW SEARLE HART, LL. D. JOHN TOLEKEN, M. D. JOSEPH CARSON, D. D. THOMAS STACK, M. A. GEORGE LONGFIELD, D. D. JOHN H. JELLETT, M. A., Professor of Natural Philosophy. MICHAEL ROBERTS, M. A., Professor of Mathematics. JOSEPH A. GALBRAITH, M. A., Professor of Experimental Philosophy. JOHN K. INGRAM, LL. D., Regius Professor of Greek. Mathematics, and Mathematical Physics. PURE MATHEMATICS. DR. HART. 1. Determine the number of tangents to a curve of the fifth degree which are cut harmonically by the curve. 2. Show how to find the degree and class of the locus of the extremity of the polar subtangent of a curve of the mth degree. 3. Find the reciprocal polar, with regard to a cubic, of the Hessian determinant of the cubic. 4. If a curve of the fourth degree has four double points, show that it has also four double tangents, whose points of contact lie on a conic. 5. Show how to find the values of k, for which the equation ABC-kD 2E shall represent a cubic having a double point. 6. A cord of given length is attached to two fixed points at its extremities; find its form so that ds may be a minimum, p being the dis tance of the element ds from a fixed point. 7. Investigate the form of a curve of given length which passes through two fixed points and encloses the greatest possible area on the surface of a given ellipsoid. ap+ pa=a, Bp+ pß = b, yp + py = c, dp + pò = d. 9. Find the geometric meaning of a fourth proportional to three vectors not in the same plane. 10. Determine a, b, c as scalar functions of a, ß, y, so that Vaaßbyc = 0. PROFESSOR M. ROBERTS. 1. If the faces of a tetrahedron, self-conjugate with respect to touch the surface √BC yz + √AD x = 0, A (x − a)2 + B (y − ẞ)2 + C' (z − y)3 – D = 0 find the locus of the centre of the latter surface. 2. Prove that the quadrics represented by the following equations y2+ z2 + 2 (1 + sin 20) yz – 2 sin 2pxz + 2 sec 24xy + 2 cos 2px +(2+ sin 24) y = 0 x2 - z2+2 cos 2pxz - 2 sin 2px + sin 20y=0 have double contact; and if o be supposed to vary, find the locus of the chords of contact. 3. The centre of the ellipsoid A (x − a)2 + B (y − ẞ)2 + Cz2 = 1 describes a curve ẞ=4(a) in the plane xy; find the equation of the developable circumscribed to it along the curve common to two consecutive positions. 4. Prove that the differential equation of an umbilicar geodesic on the ellipsoid μ, v being the primary semiaxes of the confocal surfaces passing through any point of the geodesic, and w being the angle between the geodesic and the umbilicar section of the surface. 5. (u, k), (v, h) are two elliptic functions of the first kind, whose moduli are k and h, and which are connected by the relation ẞ (u, k) = (v, h), which expresses Jacobi's transformation for an odd number p; K denotes the complete value of (u, k), and k' is the complement of k; prove that 6. Hence prove that, employing the notation of the Fundamenta nova, how is the double sign to be used? and deduce Jacobi's formula 7. Prove Jacobi's theorem de additione argumenti Parametri for expressing by functions of a lower order the quantity - II (u, a) + II (u, b) – II (u, a + b) 8. Prove the following formulæ respecting the functions O, H, 9. If K, = denote, respectively, the invariants of the fourth and eighth degrees of the equation α0x5+5α1x+10α2x3 + 10α3x2 + 5α4x+A5=0 prove that the part of K2 - 32 which is independent of a5 is divisible by 6a0a4-15α1aз + 10α22. (a-B)2 (ay) (ô − e)3 + ( a − ẞ)2 (a − d)2 (y — €)2 +(a−ẞ)2 (a− e)2 (yo)2 + (a− y)2 (a–d)2 (ẞ— €)2 +(a− y)2 (a− ε)2 (ẞ − 8 )2 + ( a − d)2 (a — ε)2 (ẞ − y)2 is a factor of the resultant of the above equation, and of the equation (a0a4-4a1a3+3α22) x2 + (α0α5 − 3α1α4 + 2α2α3) x + α1α5 − 4a2α4+ 3α32 = 0 II. n quantities x1, x2, ....în are connected by the following n - 1 differential equations where fx is a rational and integral function of ≈ of the degree 2n - 1 ; let px = (x − x1) (X − X2) . .(x − xn), and let ø' (x) be derived of p (x); prove that ...... where is any root of the equation fx = o, and C is an arbitrary constant, is an integral of the above system of equations. 12. The theorem in the last question gives √(1−x)(1 + y + y2) − √(1 − y) (1 + x + x2) = C′(x − y) -X while Euler's equation furnishes as an integral of the same equation 1 √1-23-1-y3 = (x − y)√a-x-y deduce the relation between the constants C, a. I. Given - I particular integrals of a linear differential equation of the nth order between two variables, how is the nth particular integral found? If sin x, cos x are two particular integrals of the equation how is the complete integral determined? 2. If y1, y2 are the particular integrals of the equation prove that yiy2 = x3, and from this relation find the integral of the equation. 3. State the method of integrating the general partial differential equation of the first order between three variables Apply it to prove that the integral of the equation where U− (x + ay)2 - (x + ay) √1 −z2 + (a) - |