fulcrum supposing W to vary in arithmetic progression. 7. A body G is kept at rest by three forces proportional to AG, BG, CG; G is centre of gravity of the triangle formed by joining A, B, C. A B 8. If with centre of gravity of any number of bodies as centre, and with any radius a circle be described, the sum of the products of each body, and the square of its distance from any assumed point in circumference is constant. 9. Prove that in perfect elasticity Aa2+Bb2= Ap2+Bq2, where a and b are the velocities of A and B before impact, and p and q after. Compare also elasticity and compression when Aa+Bb"= Ap"+Bq". 10. In a single moveable pulley, where the strings passing under the moveable pulley are not parallel, compare P and W; first, when the strings are equally; secondly, when they are unequally inclined to the horizon. 11. An imperfectly elastic ball falls perpendicularly from a height (a); Required whole space by ball after 5 rebounds, and the greatest height after last rebound. 12. Assuming the time of oscillation, to equal the time of describing semi-circle, &c., investigate the actual value of the time of oscillation, and thence, compare it with time down axis. 13. In inclined planes, P: W:: W's velocity : P's. 14. Determine the expressions for range and greatest height, upon a plane passing through point of projection; and compare greatest height of all parabolas with a given velocity to farthest 16. If a body is kept at rest by three forces, and lines be drawn at any equal angles to the directions in which they act, forming a triangle, the sides of the triangle represent the quantities of 17. If 3 forces are represented by 3 sides forming the solid angle of a parallelopiped, the resulting force is the diagonal of the parallelopiped. If the 3 forces are equal and act in planes perpendicular to each other, compare the compound force with them. 18. A is vertex of If triangular pyramid, G 19. A uniform beam AB is moveable about fixt point A, and supported by given weight P over fixt pulley C; AC is equal to AB and parallel to horizon. Required position in which AB rests. 20. Make a body oscillate in a given cycloid. radius; 21. VP MN perpendicular the diameter. Cycloidal area MVN hexa gon inscribed in the circle. M N V 22. Compare times of describing vertical diameter and any other; required also that diameter, the time through which 2 time down vertical diameter. 23. If the number of oscillations performed in same time by two pendulums (whose lengths are Land) be as m: m+n. Compare force of gravity at the two heights. 24. If one pendulum is at a distance of (n) radii from the earth's centre, at what point below the surface, must another of equal length be placed to vibrate in same time? About half-past nine the Hall is again empty, and the Examinees are again refreshing, some their memories, others their bodies. For my own part, feeling now pretty easy as to the result of the contest, I spent the entire evening with a few of my most esteemed friends in the utmost conviviality; retiring, however, at about eleven, to recruit for the next morning. That epoch having arrived, the Sophs work the Papers following. TRINITY COLLEGE. NEWTON AND CONICS. 1. EXPLAIN by short examples, the method of exhaustions, of indivisibles, and of prime and ultimate ratios. 2. Prove that if a radius vector be drawn bisecting any arc, it must ultimately bisect the chord. 3. If a straight line EDA make with the curve CBA a given angle at the point A, and the ordinates CE, BD be drawn; the triangles ACE, ABD, are ultimately in the duplicate ratio of the sides. 4. Let AB be the subtense of the arc, AD the tangent, BD the subtense of the angle of contact perpendicular to the tangent, as in the 11th lemma: then let a scries of curves be drawn in which DB AD4, AD, ADo, &c., the angle of contact in each succeeding case will be infinitely less than in the preceding. 5. If the areas described by the radius vector are not proportional to the times, the revolving |