Sidebilder
PDF
ePub

No.

Page 5502. (W. S. B. Woolhouse, F.R.A.S.)—(1) Two chords are drawn in

a circle; all that is known, appertaining to them, being that
they intersect within the circle; determine the respective
probabilities that a third random chord shall intersect neither,
only one, or both of them within the circle. (2) Two chords
are drawn in a circle ; all that is known, appertaining to them,
being that they do not intersect within the circle; determine
the respective probabilities before stated.

58 5504. (Dr Booth, F.R.S.)—If a be the area of a plane triangle, prove that 4A?

abco, where a, b, c are the sides of the original triangle, and o the semiperimeter of its orthocentric triangle.

34 5506. (The Editor.)--If A, B, C, D, E, F are six points taken at ran

dom in the perimeter of a regular n-gon, find the probability
that the three intersections of (AB, CD), (CD, EF), (EF, AB)
will all lie inside the perimeter.

45 5515. (E. P. Culverwell, M.A.) If be one of the four normal

distances of a point P from an ellipse, and p the parallel central
perpendicular on a tangent line; prove that, if E{(pr)-'}
vanishes, then P lies on the director circle of the ellipse; and state
the corresponding theorem for an ellipsoid.

58 5519. (S. Roberts, M.A.)—If S and T are the fundamental invariants,

and H is the Hessian, of the cubic curve U=0, prove that the
twelve lines on which the inflexions lie in threes are represented
by S2U4 + TU3H-18 SU2H:-27H' =0.

44 5520. (William R. Roberts, M.A.)-Show that the six inflexional

tangents to a unicursal quartic all touch the same conic. 24 5527. (W. S. B. Woolhouse, F.R.A.S.)—Project a given triangle orthogonally into an equilateral triangle.

83 5534. (Christine Ladd.)- If four conics, S, A, B, C, have one focus

and one tangent, D, cominon to all, and if a tangent common to
S and A intersects D on a directrix of A, a tangent common to
S and B on a directrix of B, and a tangent common to S and C
on a directrix of C, prove that the common tangents of A, B, C
will meet in three points in a straight line.

43 5540. (Professor Lloyd Tanner, M.A.)—If M, N are two numbers of

n digits each, and the numbers formed by prefixing M to N and
N to M are as a to b; find M, N, and indicate the conditions
required to ensure (1) at least one solution, and (2) only one
solution.

94 5542. (Professor Minchin, M.A.)—A solid triangular prism is placed,

with its axis horizontal, on a rough inclined plane, the incli-
nation of which is gradually increased; determine the nature
of the initial motion of the prism......

87 5544. (J.J. Walker, M.A.)— Writing HERMITE's series [Quest. 5492] 1

x2

23
n (n + 1) n (n + 1) (n +2) n(n + 1)(n + 2)(n+3)
1

203
if f (x)

(n + 1). 1 (n +2) 1.2 (n+3) 1.2.3 F(x)

109 f (x)

[ocr errors]

F(x)

+

+

+ n

t...,

+

+

n

prove that

[ocr errors]

Pagu

No.
5548. (Elizabeth Blackwood.)-P, Q, R are three random points

within a circle, and their respective distances from the centre
are p, q, r; show that the chance that the roots of the

equation pxa-qx+r = 0 are real is tá + 1 log, 2.
5549. (A. W. Panton, M.A.)--A plane cuts a spheroid of revolu-

tion, and makes an angle a with the axis; prove that the
eccentricity of the section is equal to e cos a, where e is the
eccentricity of a section through the axis. .........

75 5553. (R. A. Roberts, B.A.)--Prove that the points of contact of

parallel tangents to a Cartesian oval lie on a conic which passes
through four fixed points......

86 5557. (W. H. H. Hudson, M.A.)–Find the shape of a uniform wire

such that the moment of inertia of any portion of it bounded
by two radii vectores about an axis through the pole perpen-

dicular to its plane, may vary as the angle between them....... 102 5558. (C. Leudesdorf, M.A.)-Given a number of equal-sized spheres

in the closest possible contact ; find how many each would
touch.

85 5563. (Professor Sylvester, F.R.S.) – If at two points in a cubic curve,

lying in a straight line with a point of inflexion, tangents be
drawn to meet the curve again, prove that their intersections
with it will also be in a straight line with the same point of in-
flexion.

.52, 93 5564. (Professor Benjamin Peirce, F.R.S.)— Find the probabilities

at a game of a given number of points, which is played in such
a way that there is only one person who is the actual player,
and when the player is successful he counts a point, but when he
is unsuccessful he loses all the points he has made and adds one
to his opponent's score.

72 5569. (Professor Tanner, M.A.)—Solve the functional equation (xy) = xq (y) + yo (2)..........

41 5571. (Christine Ladd.)-Find the locus of a point with regard to

which the reciprocal of a fixed triangle has a constant area,

provided the radius of the auxiliary circle remains constant. 5572. (Elizabeth Blackwood.)— The centre of a given circle is 0,

and P and Q are two random points within the circle ; find

the chance that the triangle POQ is less than a given area. 106 5573. (Professor Monck, M.A.)—The three edges and the diagonal

of a rectangular parallelepiped are integer numbers ; show how
to obtain a series of parallelepipeds possessing the same
quality.

74 5575. (The Editor.)—If of four triangles A,B,C, A,B,C2, A3B3C3,

A,B4C4, the first is in perspective with the second, the second
with third, the third with fourth, and the fourth with first, in
such a way that the vertices of the same letters are correspond-
ing; and if the four centres of perspective lie in a straight
line; prove that the four perspective lines meet in a point...... 53

65 No.

Page 5580. (S. Roberts, M.A.)-If the sides of a variable triangle pass

through three fixed points in a straight line, while one vertex
moves on another straight line, and a second vertex describes
a given curve, prove that the locus of the third vertex is a

homographic transformation of the given curve. .................. 96 6586. (R. E. Riley, B.A.)--If a series of circles be described concentric

with an ellipse of eccentricity e, prove that the chords of contact
of tangents drawn to the ellipse from points in these circles
envelop a series of concentric similar ellipses of eccentricity
e (2-2)*.....

46 5588. (T. Mitcheson, B.A., L.C.P.)—If on each side of a triangle ABC

triangles similar thereto are drawn, so that the angles adjacent
to A are each equal to C, those adjacent to B each equal to A,
and those adjacent to Ceach equal to B : prove that three circles
drawn round the outer triangles, and the three lines that join
their vertical angles with the opposite angles of ABC, all inter-
sect at a point O, such that the angles OBC, OCA, OAB are

equal to each other; and that cot OBC = cot A + cot B+cot C. 98 5590. (S. A. Renshaw.)—If a quadrilateral be inscribed in a conic,

show that a point may be found on each of its sides such that,
the four points being joined, a quadrilateral inscribed in the
former will be formed whose opposite sides produced will meet
on the directrix.

103 5594. (Professor Townsend, F.R.S.)—Prove the following pairs of

reciprocal properties of a system of two conics :

(a) When two conics are such that two of their four common points subtend harmonically, the angle determined by the tangents at either of the remaining two, they subtend harmonically that determined by those at the other also.

(6) When two conics are such that two of their four common tangents divide harmonically the segment determined by the points of contact of either of the remaining two, they divide harmonically that determined by those of the other also.

(c) The associated conic, envelope of the system of lines divided harmonically by the two original conics, breaks up, in the former case, into the point-pair determined by the eight tangents to them at their four common points.

(d) The associated conic, locus of the system of points subtended harmonically by the two original conics, breaks up, in the latter case, into the line-pair determined by their eight points of contact with their four common tangents.....

88 5599. (The Editor.)—Calling the straight line that passes through

the feet of the perpendiculars on the sides of a triangle from a
point (P) on its circumscribed circle the Simson line of the
point P, (from the name of the geometer who seems to have
been the first to mention it,) and putting li, la, lz for the
segments of the Simson line that are included within the angles
A, B, C of the triangle, respectively, and a? + b2 + 2 = 20$;
prove that (22-27)* = (62 -1,2) + (02-13

(1),
(od-a)/2 + (r– 62) 7,2 + (09-c%) 4 = (28) ............(2),
(p?– a2) a2. APP+(03–62) 62. BP + (oʻ—c) c. CP? = a_b%ce ...(3). 80

[ocr errors]

log (

(11m)

No.

Page 5600. (Christine Ladd.)–Required the envelop of the Simson line.... 80 5602. (Professor Monck, M.A.)—Two chords of given length are

drawn at random within a given circlo; find the chance that they will intersect within the circle.

85 5605. (J. C. Malet, M.A.) If a quadric V intersects another

quadric U in the planes L and M, and passes through the pole
of L with respect to U : prove that it will also pass through
the pole of M with respect to U.

103 5606. (R. E. Riley, B.A.)-A particle slides down a rough parabola

whose axis is vertical, starting from an extremity of the latus

rectum. If it stops at the vertex, prove that i = 7-1 loge 4... 108 5607. (J. Royds, A.C.P.)—Find x from the equation (a-x)+(a-t = b.

87 5609. (A. W. Panton, M.A.)-The four common tangents to two

circles being supposed such that the two opposite airs, external
and internal, are at right angles to each other; show that
their eight points of contact with the circles lie on two straight
lines, every point on each of which subtends the circles in an
harmonic system of tangents.

101 5610. (J. Hammond, M.A.)—Prove that

dx

2

72 u =

79 1-22

16
5611. (Professor Wolstenholme, M.A.)—Having given that
(yaz2 +1) + (y2 + 2%) _ ("x2 + 1) + (z2 + x2) e (x+y + 1) + (x2 + y)
y%

XY
prove that k = 2-1, and that
y2 + 2x + xy = (yz)-1+(zr) -1 + (xy)-1.

100 5613. (R. A. Roberts, M.A.)—Prove that the locus of the centroid

of a triangle inscribed in a conic and circumscribed to a
parabola is a straight line.........

99 5621. (D. Edwardes.)—If a circle be drawn through the centre of the

inscribed circle and the centres of any two escribed circles of a
triangle, prove that its radius is double that of the circum-
scribed circle of the triangle.

73 5623. (C. K. Pillai.)- ABCD is a parallelogram ; a point E is

taken in the diagonal BD, and a point F in CE; also FG is
drawn parallel to DC meeting the diagonal in G, and GH is
drawn from G parallel to BF, and meeting AB in H; prove
that AH : AB DG : DE...

109 5625. (Professor Cayley, F.R.S.)-The equation

{q2 (x+y + 2)2– yz-zx-xy}' = 4 (29+1) xyz (x+y+z) represents a trinodal quartic curve having the lines x=

= 0, y=0, x=0, x + y +z = 0 for its four bitangents; it is required to transform to the coordinates X, Y, Z, where X=0, Y=1, Z=0 represent the sides of the triangle formed by the three nodes.

96 5654. (T. Mitcheson, B.A., L.C.P.)—If within a triangle a point O be taken such that LACO = BAO = CBO = 0, prove that cot 0 = cot A +cot B + cot C........

98

[ocr errors]

..........

MATHEMATICS

FROM

THE EDUCATIONAL TIMES.

WITH ADDITIONAL PAPERS AND SOLUTIONS.

[ocr errors]

NOTES ON RANDOM CHORDS. By the EDITOR. Since the publication of Miss BLACKWOOD's Solution of Question 5461, with Mr. WOOLHOUSE's notes thereon (Reprint, Vol. XXVIII., pp. 108–110), we have received several communications on the subject of random chords, from which we now select for publication the following remarks :

1. Colonel CLARKE's views on the subject are expressed as follows:

A question very similar to 5461 was proposed by Mr. WOOLHOUSE in the Lady's and Gentleman's Diary for 1856, in the following terms :-“In a dark room two persons each of them draw a chord at random across a circular slate; what is the chance that they will intersect ?” The solutions adopted at the time gave į as the answer; therein agreeing with Miss BLACKWOOD's. But there is another solution which gives į as the chance; and these are the solutions of these problems :

(A) “Two random points P1, P, on the circumference of a circle are joined by a straight line, and two other random points Q1, Q2 are joined by a straight line ; what is the chance that these lines intersect within the circle ?” The answer here is, without question, $. Or, expressed otherwise, thus :-(A) “Through each of two random points P, Q on the circumference of a circle, a chord is drawn in any direction at random, what is the chance that these chords intersect within the circle ?"

(B) “Two lines are drawn at random across a circle, what is the chance that they intersect within the circle ?” The answer here is, without ques

In (A) the distribution of lines or chords is that of lines drawn through random points on the circumference of a circle. But such lines are not random lines ; since for a given direction they lie closer together towards the circumference, and are further apart at the centre; they are not random lines, or they could not have this particular law of density. In Question 5461, the expression is “all possible chords being supposed equally probable.” Here we must suppose an even distribution of chords. In problem (A) the chords are distributed evenly as to direction, but not as to distance from the centre; and we might evidently have even distribution as to direction combined with any arbitrary law of distribution as

tion, ž

[blocks in formation]
« ForrigeFortsett »