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angle average axes axis becomes centre chance chord circle circumscribed column common condition conic conjugate containing coordinates corresponding cos² cubic curve determine diagonals diameter distance divided double drawn ellipse envelope equal equation figure fixed point foci focus four given given conic gives Hence hyperbola inscribed intersection inverse joining latter length locus meet middle point normal obtained once origin pair parabola parallel passes perpendicular plane polar pole positive Professor Proposed prove putting quadric question radius reduces reference respectively resultant right angle roots sets sides sin² Solution squares straight line substituting suppose surface tangent tetrahedron theorem touch triangle values vertex vertical whence WOLSTENHOLME write
Page 101 - ... normal to the tangent plane at the bearing. (This condition implies that, of the normals to the tangent planes at the bearings, no two coincide ; no three are in one plane, and either meet in a point or are parallel ; no four are in one plane, or meet in a point, or are parallel, or, more generally, belong to the same system of generators of an hyperboloid of one sheet. The conditions for five normals and for six are more complicated...
Page 31 - A circle is drawn so that its radical axis with respect to the focus S of a parabola is a tangent to the parabola ; if a tangent to the circle cut the parabola in A, B, and if SC, bisecting the angle ASB, cut AB in C, the locus of C is a straight line. Solution by the REV. J. WOLSTENHOLME, MA Through the foci S, S...
Page 93 - ... the locus of the middle points of a system of chords of a conic which subtend a right angle at a fixed point is another conic. Solution by the Rev. J. WOLSTJJNIIOLMK, MA Taking the fixed point for origin, let equation of circle be (x— c)" + j/2 = a2, and let tx + my = 1 be a chord.
Page 26 - ... it has one pair of opposite angles equal. 3. If in a quadrilateral ABCD, AB be equal to AD and BC to DC, the diagonal AC bisects each of the angles BAD, BCD. 4. If in a quadrilateral ABCD, AB be equal to AD and BC to DC, the diagonal BD is bisected at right angles by the diagonal AC. 5. Prove that the triangle, whose vertices are the middle points of the sides of an equilateral triangle, is equilateral. 6. Prove that the triangle, formed by joining the middle points of the sides of an isosceles...
Page 54 - Find the average square of the distance between the centres of the inscribed and circumscribed circles of a triangle inscribed in a given circle.
Page 19 - If a chord of a circle is divided into two segments by a point in the chord or in the chord produced, the rectangle contained by these segments is equal to the difference of the squares on the radius and on the line joining the given point with the centre of the circle.
Page 46 - PIO. 2. Note.— Each of the eight sets of lines is connected with a Pascal's line by the relation that, if two triangles be such that the lines joining corresponding vertices meet in A point, the three intersections of corresponding sides will be in the same straight line. In Fig. 2 the two triangles are GKM, LNH. It will be found that the line ^=? is the Pascal ABEDCF, and that *Lt mnmn is the Pascal DBEACF.
Page 39 - X(px + c0 = 0, or, which is ultimately the same thing, of ex' + c0 = 0 ; the latter of fx = 0. The number of imaginary roots of the infinite group is therefore v or v — 1 according as v is even or odd ; ie, is the same as the number of changes of sign in the upper series. Hence the number of imaginary roots belonging to fx=0 cannot be less than the number of sign-changes in the lower series; but this last number will not be affected if we multiply each quantity, ««,«!, <*3, ..... °y the common...
Page 101 - ... the same system of generators of an hyperboloid of one sheet. (Salmon's Geometry of Three Dimensions, p. 179.) Hence, the centre being the pole of the plane at infinity, we see that if through each of the given poles we draw lines parallel to the intersection of the other two planes, the hyperboloid of which these are generators passes through the intersection of the three planes, and the locus of centre is the generator of the same system at that point. 2597. (Proposed by WHH HUDSON. MA)—...