the generating point from the centre of the moving circle being c. 1849. In a three-cusped hypocycloid whose cusps are A, B, C, a chord APQ is drawn through A: prove that the tangents at P, Q will divide BC harmonically, and their point of intersection will lie on a conic passing through B, C ; also the tangents to this conic at B, C pass through the centre of the hypocycloid. 1850. A tangent to a cardioid meets the curve again in P, Q: prove that the tangents at P, Q divide the double tangent harmonically, and the locus of their common point is a conic passing through the points of contact of the double tangent and having triple contact with the cardioid (two of the contacts impossible). [The equation X ̄1+ Y ̃3 + Z ̄1 = 0 will represent a cardioid when X=x+iy, Y=x-iy, Z= a; and a three-cusped hypocycloid when X=x+y√3, Y=x-y√3, Z=9a-2x.] 1851. Chords of a Cartesian are drawn through the triple focus: prove that the locus of their middle points is (23 — bc) (μ3 — ca) (r2 — ab) + a2b3c2 sin' 0 = 0, a, b, c being the distances of the single foci from the triple focus which is the origin. 1852. Two points describe the same circle of radius a with velocities which are to each other as m : n (m, n being integers prime to each other and n>m); the envelope of the joining line is an epicycloid whose vertices lie on the given circle and the radius of whose fixed circle is n-m a When n + m m is put for m, the points must describe the circle in opposite senses, and the envelope is a hypocycloid. Hence may be deduced that the class number is m + n. 1853. An epicycloid is generated by circle of radius ma rolling upon one of radius (n − m) a, m, n being integers prime to each other, and in the moving circle is described a regular m-gon one of whose corners is the describing point; all the other corners will move in the same epicycloid, and the whole epicycloid will be completely generated by these m points in one revolution about the fixed circle. The same epicycloid may also be generated by the corners of a regular n-gon inscribed in a circle of radius na rolling on the same fixed circle with internal contact. 1854. In an epicycloid (or hypocycloid) whose order is 2p and class P+q, tangents are drawn to the curve from any point 0 on the circle through the vertices: their points of contact will be corners of two regular polygons of p and q sides respectively inscribed in the two moving circles by which the curve can be generated which touch the circle through the vertices in 0. 1855. The locus of the common point of two tangents to an epicycloid inclined at a constant angle is an epitrochoid, for which the radius of the fixed and moving circles are respectively and the distance of the generating point from the centre of the moving circle is where a, b are the radii of the fixed and moving circles for the epicycloid, and a is the angle through which one tangent would turn in passing into the position of the other, always in contact with the curve. 1856. The pedal of a parabola with respect to any point 0 on the axis is a nodal circular cubic which is its own inverse with respect to the vertex A, the constant of inversion being the square on O4. If 00' be a straight line bisected in A, PQ a chord passing through O', OY, OZ perpendiculars on the tangents at P, Q, then A, Y, Z will be collinear and AY.AZ AOo. = [If OAb and 4a be the latus rectum, the distances of the two single foci from O, the double focus, are given by the equation x2 + 4ax = 4ab, and the vector equation is, for the loop, r, being the distance from the internal focus. The difference of the arcs 8, 8, from the node O to corresponding points Y, Z on the loop and sinuous branch is determined by the equation ds, ds, sin 0 - = do do cos 0 (a + b) (a + b + 3a - b cos3 0).] INTEGRAL CALCULUS. 1857. The area common to two ellipses which have the same centre and equal axes inclined at an angle a is 1858. Perpendiculars are let fall upon the tangents to an ellipse from a point within it at a distance c from the centre: prove that the area of the curve traced out by the feet of these perpendiculars is 1861. Prove that the arc of the curve y√– b2 ( 1 = COS between x 0, x = 2b, is equal to the perimeter of an ellipse of axes 2a, 2b: and determine the ratio of a b in order that the area included between the curve and the axis of x may be equal to the area of the ellipse. [ab=2: √3.] 1862. Find the whole length of the arc enveloped by the directrix of an ellipse rolling along a straight line during a complete revolution; and prove that the curve will have two cusps if the excentricity of the ellipse exceed √5-1 2 being the angle through which the directrix turns.] 1863. A sphere is described touching a given plane at a given point, and a segment of given curve surface is cut off by a plane parallel to the former prove that the locus of the circular boundary of this segment is a sphere. 1864. Two catenaries touch each other at the vertex, and the linear dimensions of the outer are twice those of the inner; two common ordinates MPQ, mpq are drawn from the directrix of the outer: prove that the volume generated by the revolution of the arc Pp about the directrix is equal to 2 × area MQqm. 3 1865. The area of the curve r = a (cos 0 + 3 sin 0)2 ÷ (cos 0 + 2 sin 6)3 included between the maximum and minimum radii is to the triangle formed by the radii and chord in the ratio 781 : 720 nearly. 1866. Prove the results stated below, A denoting in each case the whole area, a and 7 the co-ordinates of the centre of inertia of the area on the positive side of the axis of y; (1) the curve (a2 + x2) y2 - 4a3y + x* = 8a", (3) the curve y3 (a3 + x2) − 4a3y + (x2 − 2a3)2 = 0, π 9-4/3 a 33 (21√3-16); (4) the curve y' (a2 + x2) − 2max3y+x* = 0, (m>1), 2 4a (m2 - 1) + 3 {/m2-1-m log (m + 3п consists of three loops, the area of one of which is equal to the sum of the areas of the other two. 1868. For a loop of the curve x'y2 — 4a3y + (3a2 — x2)2 = 0, A = a2 (2π- 3√3), x=4a {/3-log (2+√/3)} a 9/3-4π 1869. The area of a loop of the curve y=32π-3/3 y' (4a2x2)-4a3y + (a2 - x2)' = 0 is a (3-2 log 2). [This curve breaks up into two hyperbolas.] the internal area included by the four branches is 2√3a2. y 1872. Trace the curve whose equation is x = 2a sin ; x and prove that each loop has the same area Ta and is bisected by the straight line joining the origin to the point where the tangent is parallel to the axis 1873. The area of the loop of the curve a2- y = n3x3′′ 2n-2 n is indefinitely increased, is 2 a'; and the area between the curve a2¬3 y2 = nx2" and the asymptote, when n is indefinitely increased, a-x 1874. The areas of (1) the loop of the curve yTM (a + x) = x2′′ (α − x), (2) the part between the curve and the asymptote differ by 1875. Prove that the whole arc of the curve Sa3y' = x2 (a2 - 2x2) is Ta; and for the part included in the positive quadrant, the centre of gravity of the area is 5a (3/2); the centre of gravity of the volume generated by Ба revolution about the axis of x is $ (854/2, 0); ; and that of the area of |