Let ON (fig. 33) be the perpendicular from the origin on the tangent, say =p, P the point (x, y), The equation to a parabola, the focus being, the origin, will be and y2 = 4a (x+a), SP=x+2a, SA = a, and therefore SA. SP=a (x+2a)=p* or SY*. 8. In case of the contact of curves, when is contact said to be of the first, second, third orders? What order of contact has (1) tangent line, (2) a circle of curvature, with a plane curve. Obtain an expression for the radius of curvature of a plane curve at any point, either in terms of the rect angular coordinates, or in terms of the radius vector and perpendicular from the pole on the tangent at that point. Find the radius of curvature at that point of the ellipse where the curvature is least. The curvature is least at the extremity of the minor a" axis and at that point p=5, a and b being the semimajor and minor axes respectively. 9. Trace either of the following curves, and draw asymptotes: (1) Todhunter's Integral Calculus, Art. 14. dx x (a* dx 2 ; a2 + x2 1 -1 therefore (42) = 2a" (a + 2') + 2 tan¬¤. (a2 + x2)* x2) 2a3 (3) Todhunter's Integral Calculus, Art. 14. 11. Explain what is meant by a definite integral, and find 12. Find the differential coefficient of the area of a plane curve referred to polar coordinates. Find the whole area of the loops enclosed by the curve whose equation is r2 = a2 cos20. This curve (the Lemniscate) consists of two loops, and the two tangents to the curve through the origin make angles of 45° with the initial line. Area of half a loop of curve required 13. Explain why a plane is determined in space by a single equation, and why two equations are required generally for a straight line in space. If Ax+By+Cz=D be the equation to a plane, find the equations to a straight line perpendicular to the plane, and passing through a point whose coordinates are a, b, c, and show that the equations to the straight line may be expressed x-α y-b 2-c by = B C Find the equations to the straight line drawn from the origin perpendicular to the plane 2x + 3y + 4z = 5. = The line required is xyz. MIXED MATHEMATICS. WEDNESDAY, 27TH JUNE, 1877. 10 A.M. TO 1 P.M. 1. A square framework, formed of four uniform heavy rods hinged together, is suspended by one angle, which is connected with the opposite angle by a string of given length; find its tension. Let T (fig. 36) be the tension of the string connecting the two opposite angles; W the weight of each rod. Suppose the point C to be displaced to a slight extent x in a vertical direction, then the C of G of the framework will in consequence be displaced through a distance = x, and therefore, by the principle of virtual velocities, Thalf the weight of the framework = 2 W. 2. Enunciate the principle of virtual velocities, and apply it to find the condition of equilibrium of a smooth screw. Parkinson's Mechanics, Arts. 117, 126. 3. What is the relation between the tension of a uniform elastic string? relation been determined? Parkinson's Mechanics, Art. 43. length and the How has this A heavy straight bar is supported by two weightless elastic strings attached to its ends, their other extremities being fastened to a peg. If land r be the natural lengths, and l' and the stretched lengths of the strings, prove that in the position of equilibrium the strings being made of the same material. Let T be the tension of that string whose original length = 1, and let T, be the tension of that string whose original length = r. 2 then we have, by Hook's law, T‚=λ271, 1,=x"="; T but, from conditions of equilibrium of bar, we have 4. Prove that the radius of curvature of a catenary is equal to the portion of the normal cut off by the directrix. Let the tangent at any point P (fig. 37) of the |