4902. (C. W. Merrifield, F.R.S.)—Can a sphere be touched by more

than twelve other equal spheres ?


5067. (S. Tebay, B.A.) - Let xy + x2 + ••••+ n 1, where

X1 > X2>.

... > Xn; find the mean value of rm.


5090. (C. Leudesdorf, M.A.)-Evaluate (1) the equation
(ax2 + by2 + c+2fy + 2gx + 2hxy) (ax^2 + by2 + c + 2fy' + 2gx' + 2hx'y')

[(ax' + hy +9) x + (hx' +by' +f)y + (gx' + fy' + c)]?,
when li' + my' = 0, and x' and y' become infinite ; and (2) give
the geometrical interpretation......

5101. (A. Martin, M.A.)—An auger-hole is made through the centre

of a sphere; show that the average of the volume removed is,
in parts of the volume of the sphere, 1 - T.


5111. (Professor Wolstenholme, M.A.) — 1. If a, B be two angles

such that [1+2 (cos a)8] [1 + 2 (cos B)] = 3............ (A),

(1–8 coss all_ (1 + 8 cosó Bji

sin3 a cos a

sin3 B cos ß

2. A circle and a rectangular hyperbola each passes through

the centre of the other, and a, B are the two acute angles of in-

tersection of the curves at their two real common points ; prove

that a, B will satisfy the equation (A), and that the squares of

their latera recta are in the ratio

(1 + 8 cos5 a) : 8 sin3 a cos a..................(B).

3. If a circle and a parabola be such that the circle passes

through the focus of the parabola, and its centre lies on the

directrix, prove that their angles of intersection satisfy the

equation (Ā), and their latera recta are in the ratio (B).

4. If a rectangular hyperbola and a parabola be such that

the centre of the hyperbola is the focus of the parabola, and the

directrix of the parabola touches the hyperbola ; then, if their

acute angles of intersection be a- - 2a, – 2B, prove that a, B

will satisfy the equation (A), and that the squares on the latera

recta are in the ratio (B).


5146. (S. Roberts, M.A.)—Given a pencil of rays and a system of con-

centric circles; prove (1) that if one set of intersections range

on a straight line, the other intersections lie on a circular cubic,

having a double point at the origin of the pencil and the double

focus at the common centre of the circles; and (2) determine

therefrom, with reference to a system of parabolas having the

same focus and axis, the locus of the points the normals at which

intersect in a fixed point.


6173. (H. T. Gerrans, B.A.)-Find the sums of the infinite series
20 26 212

x 24

x3 205 X7 x9

+ &c.,


2 5 8 11 14

3 3.5 5. 7 7.9
6192. (H. T. Gerrans, B.A.)-AB is a fixed diameter of a circle, OA

a chord, ON an ordinate of the diameter, AP a line drawn so

that ZOAP ZOAN, and AP=AN; find the locus of P... 30
5212. (Professor Wolstenholnie, M.A.)-A circle is drawn touching

both branches of a fixed hyperbola in P, P', and meeting the
asymptotes in L, L', M, M': prove that (1) LL' =MM'=major





+ &c.

5299. (L. H. Rosenthal.)-Solve the simultaneous equations,

23— ax2 + (6 2y) x + ay - ( = = 0.


xay -axy - (y^ by + d) = 0.......... .(2). 70

5304. (Professor Clifford, F.R.S.)-Prove that the negative pedal of

an ellipse, in regard to the centre, has six cusps and four nodes;

find their positions, and the length of the arc external to the

ellipse between two real cusps; and account fully for the

apparent reduction of the curve to a circle and two parabolas

respectively, in special cases......


5315. (Colonel A. R. Clarke, C.B., F.R.S.)-A straight line inter-

sects a cube; show that the chance that the intercepted segment


is less than the side of the cube is



5320. (J. J. Walker, M.A.)—If normals to the ellipse

b*ir* + aạya a«bo = 0
be drawn from any point on the curve

(a x2 + bạya (4)3 + 54a-b°c4x*y* = 0,

prove that they form an harmonic pencil.


5331. (Professor Wolstenholme, M.A.)—Prove that (1) the evolute

of the first negative focal pedal of the parabola y'=c (ic – æ)
(where c = 4a the parameter) is the curve 27 (y2-8cx — (2)

8cx (8x + 9c)”; (2) the equation of the pedal itself is
27aya = (3a — «) (+ 6a)?; (3) the normal of the pedal exceeds
the ordinate by a fixed length; (4) the arc measured from the
vertex to any point is equal to the intercept of the normal on
the axis of y; and (5), if a heavy uniform chain be tied tightly
round a curve, such that the pressure per unit is equal to the
weight of a unit of length of the chain, this curve must be the
first negative focal pedal of a parabola.

5339. (Hugh McColl, B.A.) — In the quadratic equation

30° +20+ y = 0, the coefficient x is taken at random between
0 and 3, the coefficient y between - 1 and 4, and the coefficient
z between – 3 and 3; show that the chance that the following





events will simultaneously happen-namely, that x will be the

greatest (algebraically) of the three coefficients, and that the

roots of the equation will be real-is

to log 2 + 4) = .196495...


5347. (R. Tucker, M.A.)-Prove that


- B ),

where B is the symbol of the first Eulerian integral.

5367. (Elizabeth Blackwood.)—If X, Y, Z be random points taken

respectively in a sphere, in a great circle of the sphere, in a
radius of the sphere; show that the respective chances of X,
Y, Z being farthest from the centre are as 3, 2, 1.

5380. (W. Gallatly, B.A.)—If a circle A touches internally another

circle B at P, and a tangent to A at the point Q intersect B in
R1, R2, prove that _R,PQ = _R,PQ.

5395. (C. Smith, M.A.)—If P, Q are points on two confocal conics,

such that the tangents at these points are at right angles, show
that the line PQ envelops a third confocal.


5428. (Professor Elliott, M.A.)—Prove (1) that the highest point on

the wheel of a carriage rolling on a horizontal plane moves

twice as fast as each of two points in the rim whose distance

from the ground is half the radius of the wheel; and (2) find

the rate at which the carriage is travelling when the dirt

thrown from the rim of the wheel to the greatest height attains

a given level, explaining the two roots of the resulting



5436. (Dr. Booth, F.R.S.)-Express E (sec A) and (cosec A) in

terms of the radii of circles connected with the given triangle



5437. (Christine Ladd.)—If Iy, I2, I3 be the points of contact of the

inscribed circle with the sides of a triangle ABC; O1, 02, 03

the centres of the escribed circles; ri, re the radii of the circles

inscribed in the triangles 111,13, 0,0,03; and a, b, g the dis-
tances 0,03, 0:01, 0,02; prove that

a + b + c

a + B +7

5438. (The Editor.)—If A, B, C, D are four points taken at random

on the perimeter of a regular n-gon, find the respective proba-

bilities that AB, CD will intersect (1) inside, (2) on, (3) outside

the perimeter.


5441. (D. Edwardes.)-Prove that

sin 2a sin (B-y)cos (6 -- a) + sin2B sin(y - a) cos(0-B)

+ sin 2g, sin (a- -B) cos (0-5)

=tan (a + B +.7-6).
sin 2a sin (B—y) sin (e--a) + sin sin (y-a) sin (0-B)

+ sin 29 sin (1-B) sin (0-7).


5445. (C. H. Pillai.) -A point D is taken in the side BC produced of

an equilateral triangle ABC, and CE is drawn cutting AD in E,
and making the angle ACE ADC: prove that
(1) AE+EC BE; (2) AE . EC = BE BC?;
(3) AE : EC = BC: CD.






Page 5456. (Professor Minchin, M.A.)—If a body P moves in a plane orbit,

so that the direction of its resultant acceleration is always a
tangent to a given curve, prove that, if Q is the point of contact
of this tangent, p the perpendicular from Q on the tangent at P,
w the angle subtended at P by the radius of curvature at Q, 0
the angle which PQ makes with a fixed line, and h a constant,

[blocks in formation]







6458. (Professor Wolstenholme, M.A.) – Find the locus of the inter

section of perpendicular tangents to a cardioid, and trace the

resulting curve. 5462. (Professor England, M.A.)—A variable circle passes through two given points, and through one of these pass two given

find the envelope of the chord that joins the other points
where the circle intersects these lines.
5466. (The Editor.)—If two random points be taken one in each of

(1) the arcs, (2) the areas, of two semicircles that together make
up a complete circle ; find, in each case, (a) the average dis-
tance between the points, and (b) the probability that this

distance is less than the radius.
5467. (Christine Ladd.) – If three conics touch each other and

have a common focus, prove that the common tangent of any
two will cut the directrix of the third in three points which lie

on one straight line.
5470. (C. W. Merrifield, F.R.S.)—Prove that broken stone, for roads,

cannot weigh less per yard than half the weight of a solid yard
of the same material, assuming that none of the broken faces
are concave, and that it is shaken down so that there shall be

no built-up hollows........
5472. (Cecil Sharpe.)—Show how to draw a straight line terminated

by the circumferences of two of three given circles, and bisected

by the third circumference. 5474. (Rev. A. F. Torry, M.A.)-Find what normal divides an

ellipse most unequally.
5475. (C. Taylor, M.A.)—Prove the following construction for

tangents to a conic :- Take a point T at a distance TN from
the directrix, and divide ST in t so that St : ST = AX : TN,
where A is the vertex, and X the foot of the directrix. About
S draw a circle wi radius SA, and from t draw tangents to
the circle cutting the tangent at A in V, V'. Then TV, TV'

will touch the conic.
6478. (C. B. S. Cavallin.)—Three straight lines are drawn at random

across a triangle ; show that the probability that cach line cuts
unequal pairs of sides is 16 (a + b + c)-4A2, where A is the area

of the triangle and a, b, c its sides.
6479. (A. W. Panton, M.A.) - If circles be drawn on a pair of

opposite sides of a square; prove that (1) the polars of any point
on either diagonal with respect to the two circles meet on the
other diagonal, and (2) the four tangents from the point form
an harmonic pencil ..









+ ...






Page 6480. (Nilkantha Sarkar, B.A.)-C is the centre of an ellipse, CB its

semi-minor axis ; also a circle is drawn concentric with the
ellipse, and touching its two directrices, and meeting CB pro-
duced in A; determine (1) the eccentricity of the ellipse in order
that CA may be bisected in B, and (2) whether the ellipse
2x2 + 3y2 = ca answers the condition.

41 5484. (A. Martin, M.A.)-Find three positive integral numbers the

product of any two of which, diminished by the sum of the same
two, shall be a square.

90 5487. (Byomakesa Chakravarti.)-If SMPR be a semicircle on SR, and the chords SP, RM cut each other in N; prove that SR2 SN. SP + RN. RM.

25 5492. (M. HERMITE.) - Soit

x? F(x) = 1+

+1(n + 1)(n + 2)(n + 1) (n + 2)(n + 3)
on demande de démontrer qu'on a
F (2) F(-x) 1

n (n + 1)2(n + 2) 'n (n + 1) (n +2,2 (n + 3) (n +4)

76 5496. (Professor Crofton, F.R.S.)—Prove that the mean value of the

reciprocal of the distance of any two points within a circle of
radius r is


5497. (Professor Wolstenholme.)- A heavy uniform chain rests on a

smooth arc of a curve in the form of the evolute of a catenary, a
length of chain equal to the diameter of curvature at the vertex
of the catenary hanging vertically below the cusp ; prove (1)
that the resolved vertical pressure on the curve per unit is
equal to the weight of a unit-length of the chain ; (2) that the
resolved vertical tension is constant (the chain being, of course,
fixed at its highest point]; and (3) that the former property
is true for a uniform chain held tightly in contact with the
curve whose intrinsic equation is s = a sin q (1 + cos2 ) -5,
where p is measured upwards from the horizontal tangent, and
the directrix (or straight line from which the tension is mea-
sured) is at a depth a s2 below the vertex.......

104 5498. (Professor Minchin, M.A.) — If E is the complete elliptic

function of the second kind, with modulus k, and if k' is the
complementary modulus, prove that, if n assume all values from
1 to 00 ,
E = $(mk2) {1+3 (2n +1)

.3.5 ... 2n-1


2.4.6 2n 5499. (Professor Townsend, F.R.S.)—A solid circular cylinder of

uniform density and infinite length, being supposed to attract,
according to the law of the inverse sixth power of the distance,
a material particle projected, with the velocity from infinity
under its action, from any point external to its mass, in any
direction perpendicular to its axis ; show that the particle will
describe freely, under its action, a circular arc orthogonal to the
surface of the cylinder.

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........ 91

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