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'. OXFORD MUSEUM. /
LIBRARY AND READING-i^OM.
TlHIS Book belongs to the " Studeut's
Library." \
It may not be reniqvecl from the
Reading '^iboin withoat^jwrmiaeion
of the Librarian.
\
C
/?7i-
f ^
MATHEMATICAL PROBLEMS.
MATHEMATICAL PKOBLEMS
OS
THE FIRST AND SECOND DIVISIONS OF THE
SCHEDULE OF SUBJECTS
rem TiiM
CAMBRIDGE MATHEMATICAL TRIPOS EXAMINATION.
DEVISED AND ARRANGED
I»Y
JOSEPH WOLSTENHOLME, M.A.
LATE FEIJX)W AND TUTOR OF CnRT8T*8 COLLKOK ;
BOMSTISTK rELLOW OF BT JOHN*S COLLKOB;
rBOrESSOB or mathematics in THI BOTAL IXDLAN BNOIKEZBIKa OOLLEOK.
'* Tricks to shew the stretch of hnman hrain,
Mere curious pleasure, or ingenious pain.**
Pope, Euay on Man,
SECOND EDIT ION, GREATLY ENLAROED,
MACMILLAN AND CO.
1878
[The Riyht of Travffaflon is rttcrrfd.]
-*. •'
/
PBXRTXD BT 0. J. CLAT, V.A.
AT THX UKITBB8ITT PBBBB.
PREFACE TO THE FIRST EDITION.
This '' Book of Mathematical Problems" consists, mainly, of ques-
tions either proposed by myself at various University and College
E x a mina tions during the past fourteen years, or communicated to
my friends for that purpose. It contains also a certain number,
(between three and four hundred), which, as I have been in the
habit of devoting considerable time to the manu£Eu;ture of pro-
blems, have accumulated on my hands in that period* In each
subject I have followed the order of the Text-books in general use
in the University of Cambridge; and I have endeavoured also,
to some extent, to arrange the questions in order of difficulty.
I had not sufficient boldness to seek to impose on any of my
friends the task of verifying my results, and have had therefore to
trust to my own resources. I have however done my best, by
solving anew eveiy question from the proof sheets, to ensure that
few serious errors shall be discovered. I shall be. much obliged
to any one who will give me information as to those which still
remain.
I have, in some cases, where I thought I had anything ser-
viceable to communicate, prefixed to certain classes of problems
fragmentaiy notes on the mathematical subjects to which they
relate. These are few in number, and I hope will be found not
altogether superfluoua
This collection will be found to be unusually copious in
problems in the earlier subjects, by which I designed to make it
useful to mathematical students, not only in the Universities, but
in the higher classes of public schools.
VI PUEFACE.
I have to express luy best thanks to Mr R. Morton, Fellow of
Christ's College, for his great kindness in reading over the proof
sheets of this work, and correcting such errors as were thereby
discoverable.
NOTICE TO THE SECOND EDITION.
The present edition has been enlarged by the addition of such
other problems from my accumulated store as seemed to myself
worthy of preservation. About one hundred of these, and pro-
bably the most interesting, have appeared in the mathematical
columns of the " Eiducational Times," and many of the others have
already been used for Examination purposes. The " Fragmentary
Notes" have been increased, and I hope improved. Answers are
given in the great majority of cases and sometimes hints for the
solution. I have taken much pains to avoid mistakes, and
although, from the nature of the case, I dare not venture to
expect the errors to be few in number, I hope they will not often
be found of much importance. The greater number of the proof
sheets have been read over by my colleague, Professor Minchin,
and many improvements are due to his suggestion. I am deeply
grateful for his kind and efficient help.
I shall be thankful for information of misprints or other
mistakes which are not in the list of Errata.
B. L E. Ck)LLKOB, Nov. 8, 1878.
CONTENTS.
PBODLBMB
1-117.
118—123.
L
124 (1-74).
n.
125—142.
ITT.
14a— 163.
IV.
164—199.
V.
200—223.
VI.
224 233.
VIL
234—261.
vm.
262—261.
IX.
262—281.
X.
282—296.
XI.
297—307.
xu.
808-317.
XIII.
318—347.
XIV.
348—361.
XV.
362-376.
XVI.
876—401.
xvu.
PAOB
GEOMETBY (Eudid) 1
ALGEBRA.
Highest Common Divisor 12
Equatiomi 18
Theory of Quadratic Equations 17
Theory of DiviKors 21
Identities and Equalities 22
Inequalities 81
Proportion, Variation, Scales of Notation ... 85
Progressions 86
Permutations and Combinations 88
Binomial Theorem 89
Exponential and Logarithmic Series .... 42
Summation of Series 46
Becnrring Series 50
Conyergent Fractions 58
Poristio Systems of Equations 61
Properties of Numbers 64
Probabilities 65
PLANE TRIOONOMETBY.
402—428.
I.
429—484.
n.
486—510.
III.
611—624.
IV.
626—584.
V.
585—698.
VI.
599—629.
VU.
630—635.
VIII
Equations 70
Identities and Equalities 75
Poristio Systems of Equations 85
Inequalities 91
Properties of Triangles 94
Heights and Distances. Polygons .... 102
Expansions of Trigonometrical Functions. Inverse
Functions 104
Series 109
Vlll
CONTENTS.
PBOBLBMS
636— 712. I.
713— 843. n.
844—882. UL
CONIO SECTIONS, GEOMETBICAL.
PAGE
Parabola 115
Central Conies 122
Beotangolar Hyperbola 134
883— 913.
914— 976.
976—1088.
1089—1121.
1122—1152.
1163—1235.
1236—1287.
1288—1400.
1401—1455.
1466—1540.
1641—1571.
CONIC SECTIONS, ANALYTICAL.
I. Straight Line, Linear Transformation, Circle . 138
n. Parabola referred to its axis .... 146
III. Ellipse referred to its axes 163
IV. Hyperbola, referred to its axes or asymptotes . 178
y. Polar Co-ordinates 183
YI. General Equation of the Second Degree . . 188
yn. Envelopes (of the second class) . . . 204
Yin. Areal Co-ordinates 216
IX. Anharmonic Batio. Homographio Pencils and
Banges. Involution . . . . . 236
X. Beciprocal Polars and Projections . . . 244
XI. Invariant Belations between Conies. Covariants 258
1572—1641.
THEOBY OP EQUATIONS
266
1642—1819.
BIFFEBENTIAL CALCULUS 281
1820-1866.
HIGHEB PLANE CUBYES
310
1867—1993.
INTEGBAL CALCULUS 319
SOLE) GEOMETBY.
1994—2030. L Straight Line and Plane 347
2031 — ^2071. U. Linear Transformations. General Equation 364
2072—2134. m. Conicoids referred to their axes 361
2136—2156. lY. Tetrahedral Co-ordinates 371
2157—2173. Y. Focal Curves : Beciprocal Polars 374
2174—2179. YL General Functional and Differential Equations 377
2180—2187. YH. Envelopes 378
2188—2207. Yin. Curvature 379
CONTENTS.
IX
1»B0BLBM8
3206—2243.
I.
S244— 2257.
n.
2258—2279.
m.
3280—2295.
IV.
229&— 2304.
V.
8805—2829.
VL
STATICS.
Compoeition and Besolation of Forces
Centre of Inertia .
Eqailibrioin of Smooth Bodies
Friction
Elastic Strings
Catenaries, Attractions, Ac. .
PAOB
888
890
898
896
897
DYNAMICS, ELEMENTABY.
2880—2358. I. Rectilinear Motion : Imptilses
2357—2879. H. Parabolic Motion under OraTity .
2880—2391. m. Motion on a smooth Cnrve nnder GraTity
402
406
409
3892—2486.
NEWTON. (Sections I. n. HI.) .
412
3437—3477. I.
247S— 3512. n.
3518—3559. m.
3560—3578. IV.
3574—3596. V.
3597—3618.
I.
3614—3683.
n.
3688—3657.
III.
3658—3693.
IV.
3693—3764.
3765—3784.
3786—3815.
DYNAMICS OF A PABTICLE.
Rectilinear Motion, Kinematics 418
Central Forces 434
Constrained Motion on Cozres or Sorfaoei: Partidea
joined by Strings 439
Motion of Strings on Cnnres or Sorfaoes 487
Resisting Media. Hodogn^hs 440
DYNAMICS OF A RIGID BODY.
Moments of Inertia, Principal Axes .... 444
Motion about a fixed Axis 446
Motion in two Dimensions 449
Bfiscellaneous 454
HYDROSTATICS 468
GEOlfETRICAL OPTICS 474
SPHERICAL TRIGONOMETRY AND ASTRONOMY 477
EBBATA.
Page 73, question 415 is wrong.
107, line 6 from the bottom, for '*aU yalnea of $" read ** Yaloes of
between - ^ and ^ ."
108, line 10, for +336n+162, read -336fi+161.
109, line 6, dele the eecond ''that."
211, qneetion 1267, insert ** Prove that."
273, line 9, for ac+o^, z+o, read x+&|, x+b^.
289, qnestion 1686, for tt*-** read «*•"*; and for x*"**, read
327, line 6 from the bottom, for in (m - 1), read m (n - 1).
„ line 3 from the bottom, for + ... +dx., read ix^,
480, question 2619, for » - 0, read » + 0.
444, line 3 from the bottom, for 23 read 32.
446, line 2 from the bottom, for M read S,
GEOMETRY.
1. A point IB taken within a polygon ABC...KL; prove that
OA9 OBf,,,OL are together greater than half the perimeter of the
polygon.
2. Two triangles are on the same base and between the same
paraUels; through the point of intersection of their sides is drawn a
straight line paxullel to the base and terminated by the sides which do
not intersect: prove that the segments of this straight line are equal
3. The sides AB, AC of a triangle are bisected in 2), B^ and CA
BB intersect in F: prove that the triangle BFC is equal in area to the
quadrangle ADFB.
4. AB, CD are two parallel straight lines, B the middle point of
CZ>, and F, G the respective points of intersection of AC, BB, and of
AB, BDx pxx)ve that FG is parallel to AB.
5. Through the angular points of a triangle are drawn three parallel
straight lines terminated by the opposite sides*: prove that the triangle
formed by joining the ends of these lines will be double of the original
triangle.
6. Ko^ 5, c be the middle points of the sides of a triangle ABC,
and if through ^, J?, C be drawn three parallels to meet 6c, ea, a& re-
spectively in A\ B, C, the sides of the triangle A'BC will pass through
A^ B, C respectively, and the triangle ABC will be double of the
triangle A'BC.
7. In a right-angled triangle the straight line joining the right
angle to the centre of the square on the hypotenuse will bisect the right
angle.
8. Through the vertex of an equilateral triangle is drawn a straight
line terminated by the two straight lines drawn through the ends of the
base at right angles to the base, and on this straight line as base is
described another equilateral triangle: prove that the vertex will lie
either on the base of the former or on a fixed straight line parallel to
that base.
^ All Biraight lines are supposed to bo prodaeed if necessary.
/
W.P.I ^ 1
2 GEOMETRY.
9. Through the angle (7 of a parallelogram ABGD is drawn a
Btraight line meeting the two sides AB, AD in P, Q : prove that the
rectangle under BP^ DQ is of constant area.
10. In any quadrangle the squares on the sides together exceed the
squares on the diagonals by the square on twice the line joining the
middle points of the diagonals.
11. If a straight line.be divided in extreme and mean ratio and
produced so that the part produced is equal to the smaller of the seg-
ments, the rectangle contained by the whole line thus produced,, and the
part produced together with the square on the given line will be equal
to four times the square on the larger segment.
12. Two equal circles touch at A, a circle of double the radius is
drawn having internal contact with one of them at B and cutting the
other in two points : prove that the straight line AB will pass through
one of the points of section.
13. Two straight lines inclined at a given angle are drawn touchihg
respectively two given concentric circles : their point of intersection will
lie on one of two fixed circles concentric with the given circles.
14. A chord CD is drawn at right angles to a fixed diameter AB of
a given circle, and DP is any other chord meeting AB in Q : prove that
the angle PCQ is bisected by either CA or CB,
15. AB \a the diameter of a circle, P a point on the circle, PM
perpendicular on AB ; on AM^ MB as diameters are described two circles
meeting AP^ BP m Q^ R respectively : prove that QR will touch both
circles.
16. Given two straight lines in position and a point equidistant
from them, prove that any circle through the given point and the point
of intersection of the two given lines wall intercept on the lines segments
whose sum or whose difference will be equal to a given length.
17. A triangle circumscribes a circle and from each point of contact
is drawn a perpendicular to the straight line joining the other two : prove
that the straight lines joining the feet of these perpendiculars will be
parallel to the sides of the original triangle.
18. From a fixed point of a given circle are drawn two chords
OP, OQ equally inclined to a fixed chord : prove that PQ will be fixed in
direction.
19. Through the ends of a fixed chord of a given circle are drawn
two other chords parallel to each other: prove that the straight line
joining the other ends of these chords will touch a fixed circle.
20. Two circles with centres -4, B cut each other at right angles
and their common chord meets AB in C ; DE is a chord of the first
circle passing through B : prove that il, 2), J^, C lie on a circle.
21. Four fixed points lie on a circle, and two other circles are
drawn touching each other, one passing through two fixed points of the
four and the other through the other two : prove that their point of
contact lies on a fixed circle.
QEOMKTBY. 9
22. A circle A passes through the centre of a circle B : prove that
their common tangents will touch A in points lying on a tangent to B;
and converselj.
23. On the same side of a straight line AB are described two seg-
ments of circles, AP, AQ are chords of the two segments including an
angle equal to that between the tangents to the two circles at A : prove
that P, Q, B are in one straight lina
24. The centre A of a circle lies on another circle which cuts the
former in. B^ C ; AD is a chord of the latter circle meeting BC in E and
from jD are drawn DF, DO to touch the former circle : prove that Gf E^ F
lie on one straight line.
25. If the opposite sides of a quadrangle inscribed in a circle be
produced to meet in P, Q^ and if about two of the triangles so formed
circles be described meeting again m R: P^ R^Q will be in one straight
line.
26. Two circles intersect in A and through A any two straight lines
BAC^ BAG' are drawn terminated by the circles : prove that the chords
Bff^ GG* of the two circles are inclined at a constant angle.
27. K two circles touch at A and PQ be any chord of one circle
touching the other, the tswaa or the difference of the chords APj AQ will
bear to the chord PQ a constant ratio.
28. Four points A, B, G, P are taken on a circle and chords
PA\ PB, PG* drawn parallel respectively to BGy GA, AB : prove that
the angles APA\ BPB'y GPG' have common internal and external
bisectors.
29. Two circles are drawn such that their two common points and
the centre of either are comers of an equilateral triangle, P is one
oonmion point and PQ^ PQ tangents at P terminated each by the other
circle : prove that QQ will be a common tangent.
30. On a fixed diameter AB of a given circle is taken a fixed point
C from which perpendiculars are let fiJl on the sti*aight lines joining J,
B to any point of the circle : prove that the straight line joining the feet
of these perpendiculars will pass through a fixed point.
[If jD be this fixed point and the centre, the rectangle under OG^
OD will be half the sum of the squares on 0(7, OA.^
31. Four points are taken on a circle and the three pairs of straight
lines which can be drawn through the four points intersect respectively
mEf Fy G\ prove that the three pairs of straight lines which bisect the
an^es at E^ F^ G respectively will be in the same directions.
32. Through one point of intersection of two circles is drawn a straight
line at right angles to their common chord and terminated by the circles,
and through the other point is drawn a straight line equally inclined to
the straight lines joining that point to the extremities of the former
straiglit Une : prove that the tangents to the two circles at the points on
tlufl latter straight line will intersect in a point on the common chord.
1—2
4 GEOMETRY.
33. Two circles cut each other at A and a straight line B AC is
drawn terminated by the circles ; with B, C as centre are described two
circles each cutting at right angles one of the former circles : prove that
these two circles and the circle of which BO is a diameter will haye a
common chord.
34. Circles are described on the sides of a triangle as diameters
and each meets the perpendicular from the opposite angular point on
its diameter in two points : prove that these six points lie on a circle.
35. The tangents from a point to a circle are bisected by a straight
line which meets a chord PQ of the circle in E : prove that the angles
BOP, OQB are equal.
36. A straight line PQ of given length is intercepted between two
straight lines OP, OQ given in position ; fiirough P, Q are drawn straight
lines in given directions intersecting in a point R, and the angles POQ,
PRQ are equal and on the same side of PQ (or supplementary and on
opposite sides) : prove that R lies on a fixed circle.
37. From the point of intersection of the diagonals of a quadrangle
inscribed in a circle perpendiculars are let &11 on the sides : prove that
the sum of two opposite angles formed by the straight lines joining the
feet of these perpendiculars is double of one of the angles between the
two diagonals.
38. K OP, OQ be tangents to a circle, PR any chord through P,
then will QR bisect the chord drawn through paiullel to PR,
39. Two chords AB, AC of & circle are drawn and the perpendicu-
lar from the centre on ^^ meets AC in D: prove that the straight line
joining D to the pole of BC will be parallel to AB,
40. A circle is drawn subtending given angles at two given points :
prove that its centre lies on a fixed circle with respect to which the two
given points are reciprocal ; and conversely that if a circle be drawn with
its centre on a given circle and subtending a given angle at a Hxed point
it will also subtend a fixed angle at the reciprocal point.
41. Prove the following construction for finding a point P in the
base BC of a triangle ABC such that the ratio of the square on AP to
the rectangle under the segments BP, PC may be equal to a given
ratio -.—Take the centre of the circle ABC and divide -40 in 0' so that
the ratio of AC ix> O'O may be equal to the given ratio, the circle whose
centre is O and radius O'A will meet BC in two points each satisfying the
required condition. K P, Q be the two points AP, AQ will be equally
inclined to the bisector of the angle A and will coincide with this
bisector when the given ratio has its least possible value, which is when
CO is to 40 in the duplicate ratio of BC to the sum of the other two
sides. Also the construction holds if 0' lie in OA produced, AP, AQ
being then equally inclined to the external bisector cf A and coinciding
with it when the given ratio has its least possible value outside the
triangle, which is when 00' haa to OA the duplicate ratio of BC to the
difference of the other two sides.
GEOMETRY. 6
42. If a circle touch each of two other drclee the straight line passing
through the points of contact will cut off similar segments fixim the two
circles.
43. Two circles have internal contact at A, a straight line touches
one circle at F and cuts the other in Q, Q'l prove that QF^ FQ[ sttbtend
equal angles at A.
K the contact be external, FA bisects the external angle between
QA, Q'A.
44. A straight line touches one of two fixed oiroles which do not
intersect in F and cuts the other m Q, Q'l prove that there are two fixed
points at either of which FQ^ FQ subtend angles .equal or supple-
mentary.
45. Any straight line is drawn through one comer AiA% parallelo-
gram to meet the diagonal and the two sides which do not pass through
A hn F^ Q^ R\ prove that AF will be a mean proportional between
FQ, FR.
46. In a triangle ABC are given the centres of the escribed circles
opposite By C and Uie length of the side BG : prove that (1) ^ lies on a
fixed straight line; (2) AB^ AG are fixed in direction; (3) the cirole
ABC is given in magnitude ; and (4) the centre of the circle ABC lies
on a fixed equal circle.
47. Any three points are taken on a given circle and from the
middle point of the arc intercepted between two of the points per-
pendiculars are let fall on the straight lines joining them to the third
point : prove that the sum of the squares on the distances of the feet
of these perpendiculars from the centre is double the square on the
radius.
48. At two fixed points A^ B are drawn AC^ BD at right angles to
AB and on the same side of it and of such magnitude that the rectangle
ACy BD is equal to the square on AB : prove that the circles whose
diameters are AC^ BD will touch each other and that their point of
oontact will lie on a fixed circle.
49. ABC is an isosceles triangle right angled at C and the parallelo-
gram ABCD is completed ; with centre D and radius DC a circle is
described : prove that if P be any point on this circle the squares on
FA, FC will be together equal to the square on FB.
50. A circle is described about a triangle ABC and the tangents to
the circle at B^ G meet in ^'; through A' is drawn a straight line meet-
ing ACy AB in the points B, C: prove that BB, CC will intersect on
the circle.
51. If jD be the middle point of the side BC of a triangle ABC and
the tangents at B, C to the circumscribed circle meet in A\ the angles
BAA\ DAG will be equal.
52. The side BC of a triangle ABC is bisected in Z>, and on DA is
taken a point F such that the rectangle DF^ DA is equal to the rectangle
BD, DC : prove that the angles BFC, BAG are together equal to two
light angles.
6 GEOMETRY.
53. If lihe circle inscribed in a circle ABC touch BC in 2), the
circles inscribed in the triangles ABD, BAG will touch each other.
Also a similar property holds for the escribed circles.
64. Given the base and the vertical angle of a triangle : prove that
the centres of the four circles which touch ^e sides of the triangle will
lie on two fixed circles passing through the extremities di the base.
55. A circle is drawn through B^ C and the centime of perpendicu-
lars of a triangle ABG ; D is the middle point of BG and AD is produced
to meet the circle in E : prove that AE is bisected in D,
56. The straight lines joining the centres of the four circles which
touch the sides of a triangle are bisected by the circumscribed circle ;
also the middle point of the line joining any two of the centres and that
of the line joining the other two are extremities of a diameter of the cir-
cumscribed circle.
57. With three given points not lying in one straight line as centres
describe three circles which shall have three common tangents.
58. From the angular points of a triangle straight lines are drawn
perpendicular to the opposite sides and terminated by the circiimscribed
circle : prove that the parts of these lines intercepted between their point
of concourse and the circle are bisected by the corresponding sides respec-
tively.
59. The radii from the centre of the circumscribed circle of a tri-
angle to the angular points are respectively perpendicular to the straight
lines joining the feet of the perpendiculars.
60. Three circles are described each passing through the centre of
perpendiculars of a given triangle and through two of the angular points :
prove that their centres are the angular points of a triangle equal in all
respects to the given triangle and similarly situated : and that the rela-
tion between the two triangles is recipro<»i.
61. K the centres of two of the circles which touch the sides of a
triangle be joined, and also the centres of the other two, the squares on
the joining lines are together equal to the square on a diameter of the
circumscribed circle.
62. The centre of perpendiculars of a triangle is joined to the middle
point of a side and the joining line produced to meet the circumscribed
circle : prove that it will meet it in the same point as the diameter
through the angular point opposite to the bisected side.
63. From any point of a given circle two chords are drawn touch-
ing another given circle whose centre is on the circumference of the
former : prove that the straight line joining the ends of these chords is
fixed in direction.
64. ABC is a triangle and the centre of its circumscribed circle ;
A!BC' another triangle whose sides are parallel to OA^ OB, OC ; and
through A\ E, C are drawn straight lines respectively parallel to the
corresponding sides of the former triangle : prove that ^ey will meet in
% point which is the centre of one of the circles touching the sides of the
riangle A'BC\
QEOMETRT. 7
65. A triangle is drawn having its sides parallel to the straight
lines joining the angular points of a given triangle to the middle points
of the opposHe sides : prove that the relation between the two triangles
is reciprocal.
66. Two triangles are so related that straight lines drawn through
the angular points of one parallel respectively to the sides of the other
meet in a point : prove that straight lines drawn through the correspond-
ing angular points of the second parallel to the sides of the first will also
meet in a point ; and that each triangle will be divided into three tri-
angles which are each to each in the same ratia
67. The diameter AB of a circle is produced to C so that BC s AB^
the tangent at A and a parallel to it through C are drawn and any point
F being taken on the latter the two tangents from P are drawn forming
a triangle with the tangent at A : prove that this triangle will have a
fixed oentroid.
68. A common tangent ^^ is drawn to two circles, CD is their
common chord and tangents are drawn from A to any other circle
through Cf D : prove that the chord of contact will pass through B.
69. Four straight lines in a plane form four finite triangles : prove
that the centres of the four circumscribed circles lie on a circle which
also passes through the conmion point of the four circumscribed circles.
70. A triangle ABC is inscribed in a circle and AA\ BB^^ CC are
chords of the circle bisecting the angles of the triangle (or one internal
and two external angles) and meeting in E : prove that RC\ GA\ A'Bf
respectively bisect EA^ EB^ EG at right an^^es: also the circles KBQ\
EBC will touch each other at Ey EA being l£e common tangent.
71. Two of the sides of a triangle are given in position and the
area is given; through the middle point of the third side is drawn a
straight line in a given direction and terminated by the two sides :
prove that the rectangle under the segments of this straight line is
constant.
72. In the hexagon ABGA*BC* the three sides AB^, CA\ BC are
parallel, as are also the three BA\ CF, AC : prove that AA\ BE, CC
¥riU meet in a point.
73. Two parallelograms A BCD, A' BCD' have a common angle B :
prove that AG\ A'C, DD* will meet in a {xAnt ; or, if the parallelograms
be equal, will be parallel.
74. On two straight lines not in the same plane are taken points
Af Bf C; A', By C respectively : prove that the three straight lines
each of which bisects two corresponding segments on the two straight
lines will meet in a point.
75. Four planes can be drawn each of which cuts six edges of a
given cube in the comers of a regular hexagon, and. the other six pro-
duced in the comers of another regular hexagon, whose area is three
times that of the first, and whose sides are respectively perpendicular to
the central radii drawn to the comers of the first.
8 GEOMETBY.
76. Given the circumscribed circle and the centre of perpendiculars
of a triangle, prove that the feet of the perpendiculars lie on a fixed
circle, and the straight lines joining the feet of the perpendiculars touch
another fixed circle.
77. Given the circumscribed circle of a triangle and one of the
circles which touch the sides, prove that the centres of the other three
circles which touch the sides will lie on a fixed circle.
78. 11 Of K he the centres of the circumscribed and inscribed
circles of a triangle, L the centre of perpendiculars, and OK be pro-
duced to i?^ so that Off ia bisected in K, then will ffL = E ^2r, where
B, r are the radii of the two circles.
79. In any triangle ABC, 0, O* are the centres of the inscribed
circle and of the escribed circle opposite A ; 00' meets BG in 2), any
straight line through D meets AB, AG respectively in 6, c, Ob, O'c in-
tersect in F, O'b, Oc ia Q: prove that F, ^, Q lie in one straight line
perpendicular to 00\
80. The centre of the drcuniscribed circle of a triangle and . the
centre of perpendiculars are joined : prove that the joining line is
divided into segments in the ratio of 1 : 2 by each of the straight lines
joining an angular point to the middle point of the opposite side. .
81. The side BG of a triangle ABC is bisected in i>, a straight line
parallel to BC meeting AB, AG produced m F, F respectively is divided
in Q, so that FQ, BD, QF are in continued proportion, and through Q
is drawn a straight line RQR' tei-minated hj AB, AC and bisected in Q :
prove that the triangles ABC, ARR are equal.
82. On AB, AG two sides of a triangle are taken two points D, F;
AB, AG are produced to F, G bo that BF is equal to AD and GG to
A£; BG, GF, FG are joined, the two former meeting in H\ prove that
the triangle FUG is equal to the two triangles BffC, ABE together.
83. K two sides of a triangle be given in position, and their sum
be also given, and if the third side be divided in a given ratio, the point
of division will lie on one of two fixed straight lines.
84. Two circles intersect m A, B, FQ is a straight line through A
terminated by the two circles : prove that BF has to BQ a constant
ratio.
85. Through the centre of perpendiculars of a triangle is drawn a
straight line at right angles to the plane of the triangle : prove that any
tetrahedron of which the triangle is one face and whose opposite vertex
lies on this straight line will be such that each edge is perpendicular to
the direction of tiie opposite edge.
86. A, B, C, J) are four points not in one plane, and AB, AC
respectively lie in planes perjiendicular to CD, BD : prove that AD lies
in a plane perpendicular to BG; and that the middle points of these six
dges lie on one sphere which also passes through the feet of the shortest
istances between the c^posite edges.
GEOMETRY. 9
87. In a certain tetrahedron each edge is perpendicuLur to the direc-
tion of the opposite edge : prove that the straight line joining the centre
of the circumscribed sphere to the middle point of anj edge inlL be equal
and parallel to the straight line joining the centre of perpendiculars of
the tetrahedron to the middle point of the opposite edge.
88. Each edge of a tetrahedron is equal to the opposite edge : prore
that the straight line joining the middle points of two opposite edges is
at right angles to both : also in such a tetrahedron the centres of the in-
scribed and circumscribed spheres and the centres of gravity of the
volume and of the surface of the tetrahedron coincide.
89. If from any point be let fall perpendiculars Oa, Ob, Oc, Od
on the fieuses of a tetrahedron ABCD, the perpendiculars from A, B^C^ D
on the corresponding faces of the tetrahedron abed will meet in a point
(/, and the relation between and C is reciprocal
90. The greatest possible number of tetrahedrons which can be
constructed having their six edges of lengths equal to six given straight
lines all unequal is thirty; and when they are all possible the one of
greatest volume is that in which the three shortest edges meet in a
pointy and to them are opposite the other three in opposite order of
magnitude.
91. Two tetrahedrons A BCD, abed are so related that straight lines
drawn from a^ b^ c, d perpendicular to the corresponding faces of A BCD
meet in a point : prove that straight lines drawn from A^ B, C, D per-
pendicular to the coiTe8|)onding faces of abed will meet in a point o, and
that vol. OBCD : vol. ABCD :: vol obcd : vol. abed.
92. A solid angle is contained by three plane angles : prove that
any straight line through the vertex makes with the edges angles whose
sum is greater than half the sum of the containing angles, uid extend
the proposition to any number of containing angles.
93. Two circles are drawn, one lying altogether within the other;
0, (/ are the two points which are reciprocals with respect to either
circle, and FQ is a chord of the outer circle touching the inner : prove
that if FP*, QQ' be chords of the outer circle passing through or ff,
F'Qf will also touch the inner circle.
94. The circles described on the diagonals of a complete quadri-
lateral as diameters cut orthogonally the circle circumscribing the
triangle formed by the diagonals.
95. Four points are taken on the circumference of a circle, and
through them are drawn three pairs of straight lines, each intersecting
in a point: prove that the straight line joining any one of these {>oints to
the centre will be perpendicular to the straight line joining the other twa
96. A sphere is described touching three given spheres : prove that
the plane passing through the points of contact contains one of four fixed
stnJ^t lines.
97. Four straight lines are given in position : prove that an infinite
number of systems of three circles can be found such that the points of
10 GEOMETRY.
intersection of the four straight lines shall be the centres of similarity
of the circles taken two and two.
98. In two fixed circles are drawn two parallel chords FP^
PQj PQ' are joined meeting the circles again m E, S; P, S*y respec-
tively : prove that the points of intersection of QQ^^ RR" and of PP', SIS'
lie on a fixed straight line, the radical axis of the two circles.
99. The six radical axes of the four circles taken two and two
which touch the sides of a triangle are the straight ^ lines bisecting
intemallj and extemallj the angles of a triangle formed by joining the
middle points of the sides of the former triangle.
100. If two circles have four common tangents the circles de-
scribed on these tangents as diameters will have a common radical axis.
101. Four points are taken on a circle and from the middle point
of the chord joining any two a straight line is drawn perpendicular to
the chord joining the other two : prove that the six lines so drawn will
meet in a point, which is also common to the four nine points' circles of
the triangles each having three of the points for its angular points.
102. Given in position two sides of a triangle including an angle
equal to that of an equilateral triangle ; prove that the centre of tiie
nine points' circle of the triangle lies on a fixed straight line.
103. Given in position two sides of a triangle and given the sum of
those sideSy prove tluit the centre of the nine points' ciix;le lies on a fixed
straight line.
104. The perpendiculars let fall from the centres of the escribed
circles of a triangle on the corresponding sides meet in a point.
105. The straight lines bisecting each a pair of opposite edges of a
tetrahedron A BCD meet in and Uirough A, B, G, JD respectively are
drawn planes at right angles to OA, OB, 00, OD : prove that the faces
of the tetrahedron bounded by these planes will be to one another as
OAiOBiOC \ OD.
106. A straight line meets the produced sides of a triangle ABC in
A\ B, G respectively : prove that the triangles ABB, ACC\ ACC\
A'BB will be proportionala
107. A point is taken within a triangle ABC, and through
A, B, C are drawn straight lines parallel to those bisecting the angles
BOC, CO A, AOB : prove that these lines will meet in a point.
108. Straight lines AA\ BB\ CC are drawn through a point to
meet the opposite sides of a triangle ABC : prove that the straight lines
drawn from A, B, (7 to bisect B'C\ C'A\ A'Bf will meet in one point ;
and that straight lines drawn from A, B, C parallel to FC\ C'A\ A'B'
will meet the respectively opposite sides in three points lying on one
straight line.
109. If two circles lie entirely without each other and any straight
line meet them in P, -P ; QjQ' respectively, there are two points such
that the straight lines bisecting the angles POF, QOQ' shall be always
ght angles to each other.
GEOMETRY. 11
110. Given two circles which do not intersect, a tangent to one at
any point P meets the polar of P with respect to the other in P* \
prove that the circle whose diameter is PP' will pass through two
fixed pointa
111. A point has the same polar with respect to each of two
circles : prove that anj common tangent will subtend a right angle at
that point.
112. Given two points J, B,a. straight line PAQ is drawn througfi
A so that the angle PBQ is equal to a given angle and that BP has to
BQ a given ratio : prove that P, Q will lie on two fixed circles which
pass through A and B.
113. If be a fixed point, P any point on a fixed circle and the
rectangle be constructed of which OP is a side and the tangent at P a
diagonal, the angular point opposite will lie on the polar of 0,
114. If 0A\ OB', OCy be perpendiculars from a point on the
sides of a triangle ABC, then will
J^ . ^(7 . CJ' + ^C . C'J . ^'^ = 2 A ^'J^C X diameter of the
circle ABC,
115. From a fixed point are let fall perpendiculars on two con-
jugate rays of a pencil in involution : prove that the straight line join-
ing the feet of these perpendiculars passes through a fixed point.
116. If be a fixed point, P, P* conjugate points of a range in
involution and PQ, FQ be drawn at right angles to OP, OF ; Q will lie
on a fixed straight line.
117. In any complete quadrilateral the common radical axis of the
three circles whose diameters are the three diagonals will pass through
the centres of perpendiculars of the four triangles formed by the four
straight linea
ALGEBRA.
I. Highest Common Divisor.
118. Beduce to their lowest terms the fractions
(1)
(3)
(6)
(7)
(9)
(11)
(1)
(2)
(3)
(4)
Ila;*-f24g'-fl25
a;*+24a;+55 '
^x^-k- 5g- 2
27«*-45a;*-16'
23:^-1 la:*- 9
4a;*+lla;*+81'
g*- 209a; 4- 56
56a;*-209a^+l'
16g'-a;'-fl6a^-i-32
32a^+16a^-a;*+16'
l+a:*
\^/ 1 OK^
65a:*+ 24iC»+ 1
125aJ*+24aj + l*
9x'-hlla^-2
81a?*+llaj+4*
a;^4-lla^-54
a:*+lla; + 12'
8a;"^ 377g'4- 21
21a:'-377a;*+8'
ai*+2a^+3aJ*-2a:'4-l
(6)
(8)
^^^^ 6a»+a;^+17a:*-7ai'-2'
(a + Aa;)« + (A + 6a:)«
a + 2Aa; + 6a;* a(A + &c)*-2A(a + Aa5)(A + 6aj) + 6(a + Aa;)*'
119. Simplify the expressions
a;(l-y»)(l-O-l-y(l-O(l-g«)-l-g(l-a:^(l-y0-4a;yg
o(6-fc-a)*+6(c4-a-6)*4-c(a4-6-c)*4-(64-c-o)(c4-a-5)(a4-6'-c)
a*(6+c-a)+6*(c+a-6)+c'(a+6-c)-(6+c-a)(c+a-6)(a+6-c) '
a»(5»-c^ 4- 5'(c'~ g') -f c'(a'-5')
a* (6 - c) + 6'(c ^a)-¥c'{a-'b) *
1 1
(a-6)(a-c)(a-rf) (6-c)(6-(i)(6-a)
1 1
+ (c-rf)(c-a)(c-6)"*'(ci-a)(ci-6)(ci-c)'
ALOEBBA« 13
(5) / i\T — w j\ + 7t — T7i — j\ii \ + *^o mmilar terms,
^ ' {a — b){a — c){a — d) {b — e)(o — d){b-a) '
Hi\ .. (q + ^Xa-K) , ,. (^ + e)(b + a) , (e + a)(e+b)
^ ' (a-b){a-e) {b-e)(b-a)'^ (c-a){c-6)'
m «» <fL!: *> ('* + *'> 4. ;.» (^ •»•<)(> + «) . . (« +«) (iji.«
^ ' (o-6)(a-c) (6-c)(6-a) (c-o)(c-*)'
/«\ «♦ (i±.>)(«±f) . ft« (ft + e)(6 + a ) (e + o)(c + J)
^ ' (a-6)(«-«) (6-c)(6-a) " (e-o)(c-'6)'
^ ' (a+6 + c)(-o + 6+c)(a-6 + c)(a + 6-e) *
120. Prove that
( a5-erf)(a'- 6' 4-c'-<04-(a4;~ftd f) (a« 4- 5' - c* -■ cfQ
_ (6 + c)(a + rf)
121. Prove that
(ft-cXl-fft'Xl-fO-f (c--o)(l-i-c«)(l +0*) + (a-6)(l H-a^Xl +6^
a(6 -c)(l +6')(1 + c*) +6(c-a)(l +?)(! +a') + c(a-.6)(l W)(l +6^
_ 1 "bc — ea — ab
^ a + 6 + c-aic *
122. Prove that
{(g +b)(a 4- c) 4- 2a (6 4- c)}' - (a - 6)«(ct ■ c)«
a
_ {(5-hc)(5-*-a)4-26(c4-a)}«-(6-c)'(6-a)'
6
^ {(c 4. g) (c 4- 5) -f 2g(a 4- 5)}' - (e > a)'(c ~ by
" c
= 8(6 + c)(c + g)(a4-6).
123. Prove that
{(6-c)« + (c-g)«4-(g-6)«}{g«(6-c)« + 6«(c-g)« + c'(g-6)«}
= 3(6 -c)«(c-g»-6)« 4- {g(6- c)« + 6(c -g)«4. c(g- 6)T.
II. Equations,
124. Solve the equations
(1) (ar+l)(a;+ 2)(a;+ 3) = («- 3)(a; + 4)(« + 5),
(2) («+l)(a? + 2)(a: + 3) = («-l)(aj-2)(«-3) + 8(4aj-l)(«+l),
14
ALGEBRA.
(3) (« + «)(« + a + 6) = (« + 5) (a? + 3a),
w
(5)
(«)
(7)
(8)
(9)
5
a;+l a; + 5 a; + 3 » + ?'
20 10 15 6
a+lO a; + 20 a; + 6 a;+15'
05+ 6a a? — 3a x + 2a x + a'
X h
a5 + 6-a 05 + 6 — c
a 6
=1,
(10
(11
(12
(13
(U
(16
(16
(17
(18
(19
(20
(21
(22
(23
a—c 6+c
1 + = r+ ,
x+b—e x+a—c x+b x+a
( a -by {b-cy ^ (e-jy
{a-b)x + a-P (b-e)x + P-y {e-<l)x + y-S
, (^-«)' -0.
((£ — a) 85 + 8 — a
4(a;-a)« = 9(«-6)(a-6),
2 (a? - 2a)« = (3aj - 26) (3a- 6),
aj (aj-5) (a:- 9) = (aj-6)(a:'- 27),
(« + 7) (x^-4)= («+ 1) («• + 14aj+ 22),
aj" + 3 a^-aj+l ^a:*-2«+l
+ ,r- =2
a;-l
a-2
a;-3
a"-a; + l a"-3a: + l „ 1
+ = — =2aj-
aj-l
a;-3
4a:-8'
05 + 1 a; + 2 g lla; + 18
111 1
a 6 05 a+6 + x
- , aj — a ,
+ 6'? = «•,
,aj— 6
a-ft 6-a
(a;-9)(a;-7)(aj-5)(a;-l) = (a;-2)(a;-4)(a;-6)(a;-10),
(a + a;)* + (6 + a;)* = (a-6)*,
(a + a;)* + (6+a;)* = (a-6)*,
(ft-^)' (c-«)' («">)• _g
(6-c)«-(a:-a)«"^(c-a)«-(a;-6)« (a-6)«-(a;-c)" '
{a;(a + 6-a;)}* + {a(6+«-a)}* + {6(a + «-6)}* = 0,
ALGEBRA. 15
(24) (5±?) (?±.^) + (a?-«)(a?-^ ) ^ (g4-c)(g4-c0 ^ ^-c) («- rf)
^ ^ (a;-a)(a5-6) (a5 + a)(aB + 6) (a5-c)(aB-d) (a5+c)(a5 + rf)'
(34;
(36)
8
(25, ^ ^ —
^ ' x-i-2 05 + 4 OB + l « + 3 a: + 5'
(26) (a^- 18a; -27)* = (« + !)(« + 9)*,
(27) (a:«-27)* = (a;-5)(a:-9)-,
(29) (a! + 3)'(a:'-9x + 9)(2a:'-6a!+9) + (a:' + 3x-9)» = 0,
(30) x{x+i) + \(^ + 4y^l0,
(31) ai'+l+(x + l)* = 2(a!' + a! + l)»,
/32X _^ L.+ 9 _Ji_ + _L-o
^ ' lOx-60 a!-6 a;-7 «-8 as-9" '
(33) 13«'=10^^-^\-12 il^,,
^ ' «*-4« + 5 a:*-6a: + 6'
4 4 11
aj-1 «-2 a;-3 a:-4 30'
^^^^ «'-.7a; + 3"«' + 7a; + 2"^'
2 3 a?
«*+2aj-2 «*^2aj + 3 2*
/^7\ ^5 22a5+15
^'^^^ a:* + 3a; + 3"a;'+a:+i = ^*'
/^ft\ y^ + ^Q 2a;+4 _ .
^'^^^ «--4a;+5"a:»-2x+2^^'
/Qo\ ''*"* 72a; -32 CSa;* -.
^^^^ (?TT"a;'-4a; + 8'^"y""'
(40) 3(a;' + 2) + -^ + 7-^, = -ii-?^
^ ^ ^ ' «-l (as-l) ar + x+1'
(41) (2a;»- 7a;*4. 9«-6)« = 4 (a;*-a;+ 1) (a:* - 3a; + 3)*,
(42) («-2)(« + l)«(a;* + 2a; + 4)(a;*-a;+l)'+15aJ* + 8 = 0,
(43) af + l+(a;+l)» = 2(a;*4.a; + l)*,
(44) iB'' + l + (a;+l)»' = 2(a;« + a;+l)' + 15a;'(a;* + a;+l)*,
(45) 16a; («+ l)(a;+ 2)(«4. 3) = 9,
(46) a^ + 2«»-llaf + 4a; + 4=0,
IG ALGEBRA.
.,^. 40 20 8 12 ,
^^ a:« + 2a;-48 a" + 9a; + 8 "*■«?- + 10a: a;' + 5x-60
,. . 1 2 ^6 8 -^^
^^ «(«-!) "*'(a;-l)(«-3) (a;-l)(a; + 2) ■^a'-4'*" '
(51) («• + !)•= 4 (2a;-. 1),
(^2) ??-_! ?_
^ ^ 3 "ar' + Sx-T a;" + x-3'
/Ko\ Q«j 4x+69 9x+23
<^^> ®'^ = a;' + 2x + 3"a;' + x+l'
(54) a;'=6x + 6,
(55) 4x* = 6x + 3,
(56) x* + 6a;"=36,
(57) a;y(x + y) = 12x+3y, xy(4x + y-a^)= 12 (x + y-3);
(58) xJT^'^yJT^=xy-jr^J\^=l',
(59) x' + 6a:3^ + 2y" = 90, y(a;* + xy + y) = 21;
(60) «(y + «-x) = a", y(« + x-y) = 5*, «(x + y-«) = c*;
(61) a^ + 2ay + 2y' = l4-J;+2.
(62) a;* + 2y« = a, y" + 2«a; = 5, «' + 2xy=c;
(63) M* + r* + 2xy = a, a;* + y* + 2ut7 = 5, iia;+vy = c, vx + uy = d;
(64) ay« = a(y*+«^ = 6(«' + a;7=c(a;* + 3^;
(65) cy+5«=---, a« + ca; = j~p, 6a; + ay = ---;
(66) «+ = y + =« + = Jxyz:
^' y«aa; xy
(67) x + y + « = 7nay», y« + aa + a?y = n,
(l+a;«)(l+y^(l + «') = (l-n)';
(68) 10y* + 13««-6y« =242,
5«» + 10a;*-2aa; =98,
13a;* + 5y" -16a:y = 2;
proTing that an infinite number of solutions exist ;
ALGEBRA, 17
(69) X* + 2xjt^ + ^xjn^ = a, ,
x^ + 2a5^j + ^^6 =" ^t»
a:,* + 2a:^«, + 2a5^«i = «,,
35/ + 2a;^a5, + 2x^x^ = a^,
(70) a?/ + X* + 2jj^. + 2a?^, = a, , a;,ar. + xjt^ + a?,^!^ -- ft, ,
»«* ■*• V ■*• 2«,«, + 2a;^a?. = a„ a:^, + a;^ai, + a;^a?^ --- b^,
a;,* + a;,' + 2«^», + 2aj^a;j = a„ aj^a?, + aj^«, + »^,i«« - ^3 ;
(71) aB"+y* + «" + 6a>y« = -4 op a,
ay* -^ysf-^zaf =s 5 or 5,
as* +yaj' 4-«y* =-lopc;
(72) x^ + 3a;,a;/ + Sr, (a?/ + »/) + Caj^a?^^ = 1 7 or « , ,
x^ + 3a?^/ + Zx^ {x* + «,*) + 6a;^^, = 13 or a,,
ajj^H- 3«^j* + Sajj (a>/ +aj,*) + Ox^^x^= 15 or a,,
a?/ + 3a;^aj,* + 3a;,(aj,* + a5,*) + 6aj^»,aj,= 19 ora^;
and ahew how to solve STstemB of equations like (69) and (71) with any
odd number of unknown quantities, and systems like (70) and (72) with
any even number.
[From (69) may be obtained
where ci> is any fifth root of unity, and a like method applies to all such
systems.]
(73) «*-3ay-y»=:rt, ay(aj+y) = 6;
[{x - o>y)' = a - 3«6, where «• + cd + 1 = ;]
(74) a^-6«»y'-4a:y» = a, y*-6«"y»-4B'y = 5;
[{x - ort/y - a + «ft.]
III.
125. In the equation
a h
= 0,
x — tnd x—mc x-i-nib x-fma
prove that, ifa + & + c4-(2==0, the only finite value of a; will bo
m{ac-\-bd)
a + 6
w. p.
18 ALGEBRA.
126. In the equation
^1 ^« 0>d ^A ^
05 + 6, a; + 6, a + 6, sb + 6^
prove that, if
ai + a, + a3 + a^ = 0, and aj>^-^aj!>^-^a^^'¥aj>^-0j
the only finite value of x will be
127. The equation
18 equivalent to
oa* + 5^ + cjs" = a^ + 2a^.
128. Find limits to the real valUes of x atnd y which' can satisfy
the equation
«■ + 120^ + 43^ + 4a: + By + 20 = 0.
[a cannot lie between — 2 and 1, nor y between - 1 and ^.]
129. If the roots of the equation
oaf + 2Aa5 + 6=0
be possible and different, the roots of the equation
(a + 6)(aa" + 2Aa; + 6) = 2(a6-.A')(a:"+ I)
will be impossible : and vice versd,
ISO. Prove that the equations
a5+y + «=a + 6 + c,
X y z .
- + ^+- = 1,
a e
~+?^+-=0
are equivalent to only two independent equations, if 6c + ca + 06 = 0.
131. Obtain the several equations for determining a, )9, y so that
the equations
»* + 4cpa? + ^qof + 4ra; + 5 = 0, (of + 2paj+ a)* = {px-^y)\
may coincide : and in this manner solve the equation
(a:'+3a:-6)*+3«' = 72.
ALGEBRA. 19
132. Shew how to solve any biquadratic of the fonn
[by putting it in the form
(**+ "^ + ^y ={»*+« ^-^^}' j]
and hence solve the equations
(1) «*-8«»- 108 = 0,
(2) a* -10a!»- 3456 = 0.
133. Prove that the equation
aj* + 3aa:* + Zbx + — =
a
can be solved directly, and that the complete cubic aj* + 3/mc* + 3qx + r =
can be reduced to this form by the substitution a? = y + A.
Prove that the roots of the auxiliary quadratic are
(^-y)' + (y-»)'+(«-/3)'
^ Pty being the roots of the original cubia
134. The roots of the equation
(« + a - c)(a; + 5 + f)(a: + o - rf)(aj + 6 + c^ = «
will all be real if
16e<(a-6-2c)'(a-6-2(f)« and >-4(c-rf)*(^ + c + ^-«)*-
135. Determine X so that the equation in x
+ -+ =
x-¥a X x — a
may have equal roots ; and if X , X, be the two values of X, «,, a:, the
corresponding values of x, prove that
«,a?, = a«, \\^ = {A'-By.
136. Prove that, if two relations be satisfied, the expression
(ai^ + 005 + m) (a:* + &» + m) (x* + csB + m)
will contain no power of x except those whose index is a midtiple of 3.
Besolve «• - 20aj' + 343 and a;* + 36a!* + 1000 into their real quadratic
factors and identify the roots found from the expression in a^ with those
of the several quadratic factors*
2—2
20 ALGEBRA.
137. The equations
have the unique solution
X y g "(a A- by
o — 2 i- 6-2 =- ^ '
0+6 a+6
138. The four equations
aj + y - 2« _ a:* H- y* - 2g' __ xy-a^ _ o^ (og + y) - 2a" _ aV' - «*
a + 6 ~ a* + 6* *" a6 "" ab(a-^b) ~ a*6'
are consistent and equivalent to the three
x + y _^xy _ /a — 6\ • 2ah
1 39. The system of equations
«,-«, = a (a?, -a,),
a?3 - a;^ = a (aj^ - 05,),
ar4-a?, = a(ajj-aj,),
ajj-a?, = a(a;,-a;J,
will be equivalent to only two independent equations ifa(a— 1) = 1.
[This may also be proved from Statical considerations.]
140. The six equations
^9_ (cy+ bz){b y-^cz) ._ {by+cz)(bc-k'yz)
bc + yz ' ~ cy + bz '
,,_ (az + cx){cz-hax) j^^{cz + ax)(ca-hzx)
" ca-¥zx ' ^" az-^cx '
o _ (5a;-f ay)(qa;-f5y) - _ (ax + 5y)(a5 + ocy)
" ah-^xy ' ~ bx-^ay *
are equivalent only to the two independent equations
ax •¥ by -^ cz = 0, ayz + bzx + cocy + abc = 0.
[^GeometrioaUy these equations express relations between the six
joimng lines of a quadrangle inscribed in a circia]
ALQEBUA* 21
141. Having given the system of equations
« /v c'y + h'z a'z + <!x Ux + a'y
^ a c ^ '
prove that
a\bV + cc' - aa') + h\ce' -^ aa' - bb') -h c\aa' -^bb'^cc')^:^ 2aVc\
o'(M' + cc'-aa')+6(cc' + aa'-56') +«(«»' + W'-cc')«2a'^,
a (M' + cc'-oaO +6'(«' + ««' - W') + c(aa'+ M'-e(0 = 2a6'(?,
o(M' + cc'-aa') + *K +aa'- W) + <?'(««' +W'-oc') = 2aJc',
which are equivalent to the two-fold relation
corresponding terms in the two being taken with the same sign.
142. From the equations
0^ + 2y;s =s a, y*+2a» = &, a* + 2ay = c,
obtain the result
3(y» +«J+ asy) = a + 6 + c- ^a' + 6* + c'- 6c - ca-a6.
IV. Theory of Dimaan.
143. Determine the condition necessary in order that
af-^px-^q and af-k-p'x-^-q*
may have a common divisor a? + c, and prove that such a divisor will also
be a divisor of p^ + {q —p')x - q\
144. The expression
of + Sfluc* + 3&a:* + ex* + 3cfa* + 3ca: +/
will be a complete cube if
145. The expression «'-6x' + ca:' + cb-0 will be the product of a
complete square and a complete cube if
126 W 5d d"
5 ■" 6 "^ c "?•
146. Prove that aaf-^bx-^e and a+6a;* + c»* will have a common
quadratic factor if
6'c« = (c*-a« + 6')(c»-a*+a6)i
22 ALGEBBA.
and that (3MB* + 6a;* + c and o + 5iC* + caj* will have a common quadratic
factor if
(a*-c")(o«-c« + 6c) = oV.
U7. Prove that
a^* + a^a^ + ttgic" + OgO: + a^ and a^ + a^x+a^-^-ajxf •¥a^x*
will have a common quadratic factor if
148. Prove that das' + ftaj* + c and a-^bai^^ca? will have a common
quadratic factor if
149. The expression «' -{-pa:^ + qx + r will be divisible by x^ -h ax -k- b
if o'-2pa" + (p* + 5')a + r--jt>j' = 0, and 6*-j6'+»7>6-r* = 0.
150. The expression a^ -^px + ^^ will be divisible by as* + oo; + 6 if
a* - 4^a* = p* and (6» + ^) (6" - qY = pV.
151. The highest common divisor of ^9(0^- 1) -^'(af - 1) and
{q —p)af - qaf -h p is (a— 1)*, jt?, q being numbers whose gretjitest com-
mon measure is 1 and q being greater than p.
152. If n be any positive whole number not a multiple of 3, the
expression fic*' + 1 + («+ 1)** will be divisible by a* + 05+ 1 ; and, if n be
of the form Sr-l, by (a:' +aj+ 1)*.
153. Prove that
(aJ-.A'){a:(a:-X)+y{y-r)}'-5(a;-2r)« + 2;i(a;-X)(y-r)-a(y-7)«
will be divisible by (a: - X)' + (y - 7)' if
X'-Y' XY^ 1
a-6 "" h ^ h*-ab'
V. IdentUiea and Equdlitiea.
154. Prove that
(1) (a + 6 + c)*=a* + 5» + c' + 3(6+c)(c + a)(a + 6),
(2) J-+ J- , 2 (5- c)«.H(c-a)'^(a-6)'
"^ ' 6-c c-a 0-6 (6-.c)(c-a)(a-6)
(3) (^-6«)(^-c*) + (^-c')(>S'-a') + (5'-a')(^-6*)
= 4«(« — a) (« — 5) (« — c),
where 2^=a« + 6* + c", and 25=a+6 + c;
ALGEBRA. 23
(i) (6-c)(l+a6)(l+ac) + (c-a)(l+6c)(l4.&i)
+ (a-5)(l +ca)(l+c6)= (6-c)(c-a)(a-6),
(5) a (5 - c) (1 + ofr) (1 + oc) + the two Bimilar tenoB
= - abc {b — c)(c ''a)(a- 6),
(6) (6 - c) (1 + a*6) (1 + a'c) + the two similar terms
= - abe (a •^b'¥c){b -€){€- a) (a - .6),
(7) (6* - c^(l - a*6) (1 - aV) + the two similiLr temui
= (1 +a6c)(a*+6* + c"+6c+ca + a6)(6-c)(c-a)(a-5),
(8) 26V (c + a)« (a + 5)* + the two similar terms
= a*(6 + c)* + b*{c + a)* + c*(a + by + 16a*6V(fc + co + a6),
(9) (a' 4- 2fcc)* + (6*+ 2ca)*4. (c«+ 2a6)« - 3(a«:f. 26c) (6* + 2ca) (c*4^ 2a6)
= (a» + 6» + c" - 3a6c)',
(10) 8aVcV(6" + c'-a«)(c' + o*-6«)(o* + 6*-(0
= (a* + 6' + c^ (a + 6 + c) (- o + 6 + c) (a - 6 + c) (a + 6 + c),
(11) (a-6)*(a-c)*+(6-c)*(6-a)*+(c-a)*(c-6)'
= (a« + 6* + c'-6c-ca-a6)'
Hj{(6-c)*+(c-a)U(a-ty},
(12) a(6 - c)(b -^c — ay-h the two similar terms
= 16a6c(6 ~c)(a'- b)(a - c),
(13) a(6 — c)(5 + c — a)* + the two similar terms
= 16a6c (6 - c) (a - 6)(a - c) {(a + 6 + c)* - 4(a' -I- 6* + (0},
{14) {bed + «/« + dab + a6c)* - abed (a + 6 + c + J)*
= (6c-flk/)(ca-W)(a6-cc/),
(15) (a + 6+c + cO*-4(a + 6+c + d)(6c + a<^ + ca + M + a6+crf)
+ 8(6<jdf ■k-cda-\' dab -^-abc) =-(6+c-a-(i)(c+a-6 -(i) (a+ &-e - cQ,
n^x (6-Hc)'^(c4.a)'4-(a ^ 6)'^3(6^c)(c4-a)(a^6) .^
^^^' ^M^6' + c»-3a6c
(17) (»'-iB+l)<«*-«' + l)...(a*"-a*-*+l)
" a^ + « + 1~~ '
(18) {(6-c)'+(c-a)« + (a-6)*}{a*(6-c)V4'(c-a)« + c'(a-6)*}
= 3(6-c)*(c-a)*(a-6)«.
24 AXGEBRA.
155. Prove the following identities, where a, 5, e, d are the roots of
the equation
and the product of their differences
(6 - c)(c -a)(a- ft)(a -rf)(^- (i)(c- cO
is denoted by A,
(1) (5Va+7+a*cP6+7)(6-c)(a-ci)
+ (cV JTrf + h*<P c-ha) {e - a)(6 - d)
+ (a'b* c + d -^ <fd^ a+b){a-h){c -(£) = 0,
(2) {{b + c)* + (a + c0*}(6 - c)(a -rf) + the two similar terms = 0,
(3) (6V + a*(r)(ft-.c)(a-(f) + = -A,
(4) (6«-c")(a«-cr)(5(j + a<f) + = A,
(5) (6«-c^(a'-cr)(6 + c)(a + d?) + E-A,
(6) {6c(6 + c)* + arf(a + (f)«}(6-c)(a-.c0+ = -^>
(7) (6« + c«)(a*t(?)(ft-c)(a-(?) + = A,
(8) (6cF+? + a(faM^)(6-c)(a-i)+ =^^,
(9) (6c rrc'+a(£aT5) (6* -c") (»•-(?)+ = ZA,
(10) {{b ■¥ cY + (a + d)'] (b^c){a-d)-^ =--5L^,
(11) (6V + aV)(6-.c)(a-rf)+ = -ifA,
(12) (6c + (Kf)(6»-c«)(a»-cr)+ = if A,
(13) (6c + a^(6-c)"(a-ci)* + = ifA,
(U) (5 + c)"(a + ci)"(6-c)(a-(/)+ E-2irA,
(15) (6V + a*cr)(6»-c^(a»-cr)+ = (Zir-P)A,
(16) (6* + c»)(a' + cP)(6'-c«)(a"-d«) + = (P-ZiVr)A,
(17) (6c + a(i)»(6--0(a'~cr) + = (4P-ZA^A,
(18) (5 + c-a-(?)*(6-c)(a-cO + = A.
156. The expression
{aaf + bi^ + C2^ -f- {6c (y -«)*+ ca («-«)• + a6(a;-y)*}
will have a constant value for all values oi x^y^z which satisfy the
equation aa; + 6y + c« = 0.
157. If ary + « + y = 2, then will
g *~8a; _ y*-8y _ a y(4~ gy)
1+*' 1+y* ary-l
ALGEBRJL £5
158. H y, + mjn^ « y , + mjim^ = //, + miin,,
(',-y('.-(i)('i-0 «h«*.'*.
159. If -+f = 1 Mid- -»-^=— -.,
a b a a + 6
then win + - r- = ( - . I •
160. Hftvinir riven "aj -^^ = J, =c find the relAtion
between a» &» e ; and prove that
a:* y* «•
a—o^c b — abc c — abc'
161. Having given the eqnations
« -»- y + « = 1,
«u;-f6y+(»=rf, I
a\B + 6*y + ^« = «^> I
prove that a*a: + ft'y + ^* = ^*-(^-<*) (^ " *) (^'^ 0>
and o*«+ iV + ^** = ^*(^ ""*)(^~ *)(<'■"<')(«•♦" *+^ + ^«
162. If - + r + - = — 7 — • then, for aU integral values oi
1_ 1 1 1
163. If aj + y + « = a?y«, or if y5-h«5 + ay = 1,
1 -«» ^ 1 -y* "*■ 1 -V ■" 1 -a^ 1 ^ 1 -«-'
y + « z-^x x-hi/ y + « » + « « + y
and i 1- ^ + z = , - - -
1 — y« 1-«B 1-ary 1— y»l--«a;l-a:y
164. If y» + «; + ay = (y5)"' + («r)~* -k (ay)"*= ws then will
(1 +y»)(l+sa;)(l+a?|r) __!+♦»
7r+^)'(rTP'7(i +4 " (i -^ '
165. Having given the system of equations
bz-k-cy cx-i-az _ai/-i-bx ax-^by-^ez
b-c " c — a a-6 a + b-^c *
prove that (6 + c)x+(c + o)y+(a + 6)« = 0, (ca;4-eay4-a6«=^0,
and that cither a<<-6 + c = or aic «= (6 - e) (e - a) (a - 6).
26 ALGEBRA.
166. If Oy b, che real quantities satisfying the equation
o* + 6* + c* + 2a6c=l,
then will a\ &', c' be all less than 1, or all greater than 1.
167. J£ Xf f/j zhe finite quantities satisfying die equations
005 (y + z — x) + 6y (« + a;-y) +c«(a; + y— «) = 0,
o V + 6y + cV = 2bcyz + 2cazx + 2ahxy^
then will
X y z xyz
a{p-cy 6(c-o)" c(a-6)* abc{x^ ^y' -hs^-yz-zx-xy)'
168. If
yz + zx + xy = and (6-c)*a; + (c-a)"y + (a-6)*« = 0,
then win a; (6 - c) = y (c - a) = 2 (a - 6).
169. If a;(6-c)+y(c-a) + «(a-6) = 0, then will
bz — cy_ex'-az^ay — bx
b-c " c — a ~ a — 6
170. If 07, y, 2;, u be all finite and satisfy the equations
x=by + cz-^ du,
y = ax-hcz'¥ du,
z^ctx-\-by + duy
u =i ax -h by ■{- cz,
.- .,, a b c d ^
then will :;^ + = — y + = + = — ->= 1.
1+a 1 + 6 1+c l-\-d
iPfi -TB t^ -z^ yz .a? -of zx
171. If =T =— ,and
then will
6r-c X c—a y '
a — b z '
and if a + ^ = 6 + ,
6 — c c — a
X* - f/
then will each member of the equation be equal to c + — ^-
172. Having given the equations
y8 _ a* + ar" zx a'-^y*
x^ ^ xy a*-¥z* ,
prove that -T="i ««> y« + «a? + a;y = o",
and a'ajys = c* (« + y + z).
ALOEBBA. 27
173. Having given
X __ y + » y _ « + «
1 -SB* ""m + nys* l-y" "" w + nac'
prove Uiaty if x, y be tinequa],
1 — -i = , y5 + «aj + ry + m+ 1 =0,
1 -2* m + wicy ^ ^ '
and (y?)"* + (zx)"* + (ary)"* = n- 1.
yz zx
«-— y
X y
174. If u = A and a;, y be unequal, then will each member
1 ■"■ yz 1 "~ zx
111
of khis equation be equal to , , to « + y + «, and to - + - +
«
175. If 7 ---^» = ; V, = ; — ^-Ti , each member will be equal to
(y-«) {^-^) («-y) ^
o^ -f- (a* + 5* + c" - 26c - 2ca - 2a6).
176. If as, y be unequal and if
(2x-y-g)' ^ (2y-g-a:)*
« y '
each member will be equal to
— ~- ,to9(x* + y" + «*-y2-aB- ay),
and to -27 {« (y - «)' + y (s - a:)' + « (a; - y)'} -r (« + y + «).
177. Having given the equations
alx + hmy + c/w = aUx + 5w» y + cn'« = ox* + 6y* + cs* = 0,
prove that x {n%n — m'n) + y (?ii' - n'l) + « (/m' — Vm) = 0,
and that
(m-mz-ii-nyY (n - n'x - / - I'zY (l-l'y - m-mx)' ^
a b c '
1 78. Having given the equations
X V z
te + w»y + n«=0, (6-c) j + (c-o) -4 (a-6) - = 0,
prove that Tyar {mz - ny) + m'5j; {nx - ^«) + n'ay (ly - inx)
"^ fam (6-c)(c-a)(a-6).
28 ALGEBRA.
179. Having given
prove that i , =6 c , -ad , .
'- abed ad be
180. The equation
«. , «. , , ».
=
— ,.. .|. , -■■•r + r-
a: + 6j aj + i), ac + 6.
will reduce to
a Rimple equation if
aj + a,+ ... +a^
= 0,
Ojftj + a,5,+ ... +o,6.
= 0,
= 0,
a.6,- +«>;-+. ..+aA-
= 0,
and the
single
value of X will then be equal to
a, a, a
. •
181. Having given the equations
X y
l(mb-^nc-'la) m{nc-^la-mb) n(fo + m6-nc)'
^- ^ mz'\-ny nx-^lz ly-^-mx
prove that = — vr — = -^^ ;
and that
a
I m n
xfPy-k-cz — ax) y (cz -^ ax — by) z {ax -^by- cz) '
182. If a, 5, c, a;, ^, « be any six quantities, and
a^^bc—aff b^=^ca^y% e^ = db — s^j
x^^yz-aXf y^-zx'-by^ z^ = xy-ez;
and a,, 6„ c , a?,, y , «, be similarly formed from a^, 6^, c^, a^, y,, «,,
and so on; then will
a b e X y z
« (oa;* + Jy* + C8* - a6c • 2«y«) s .
ALGEBRA* 29
183. Proye the following equalities, liaving given that a + & + c =: 0,
5 " 3 2 '
7 " 5 2 " 3 2 •
g" ^- 5" -f c" ^ g' + 5Vc' g' + y-t-c* _ (g* -i- 6' -f c*)' g'4-5'4-c *
11 " 3 2 9 2 •
184. Prove that
(g + 6 + c)*-g*-6'-c" '^ '
185. Ifg-f& + e + J = 0, prove that
5 "* 3 2 '
186. Having given
a + J + c + g'+6' + c'-0, g* + 6* + o^ + g* + 6''+c'* = 0,
prove that
7 " 2 b
187. Prove that
188. !£ X-ax-^ey^-hZf 7 = ex + (y 4- g^^ J?=&c + ay + «,
then will
jr*+r"4.ir»-.3XrZ=(g* + 6* + c"-3gJc)(aJ* + y^ + s»-3a?y«),
Henoe shew how to express the product of any number of factors of
this form in a similar form.
[By means of the identity
tt" + 6' + c* - 3g6c =(g + 6 + c)(o + «6 + afo) (g + di^6 + «c),
where ••■ + ••+ 1 = 0.]
The same equation will be true if
X:^ax-{-hy-¥cZy Ymay-^bz + eXj J?=g« + &B+cy,
and these two are the only essentially different arrangements.
189. Jix-^u-^-z^xyz and af = yz, then will y and «be capable of
all values^ but x^ cannot be less than 3.
30 ALGEBRA.
190. Ifaj + y + «=ic"-fy* + «'=2, then will
also the greatest of the three x, y, z lies between ^ and 1, the next
between 1 and ^^ and the least between ^ and 0; and the diffei^ence
between the greatest and least cannot be less than 1 nor gi'eater than
191. Having given the equations
(i/ -h z)* = ia'yz, (z + xy = Ab'zx, (a: + y)* = 4c*a5y ;
prove that a* + 6' + c* «*» 2abc = 1.
192. Having given the equations
y z z X , X y
z y X z y X
prove that a* + 6* + c* = 26V + 2c V + 2a'6* + aVc\
193. Having given the equations
^(y'-«7+|(«'-«^)+J(*'-yO=o.
X y z
'\ >
x' (by' ^ cz' - ax') ^ {csf ^ ax' -by') z' {ax' -¥ by^ - cz')
prove that^ if a;, y, « be all finite,
194. Having given the equations
a? + y* +s^ = (7/-^z){z+x){x + y)y
a{3^^ii?^9f)^b{^'¥9?'-i^)^c{si?-^y''-7?))
prove that a" + 6' + c" = (6 + c)(c + a)(a + 6),
196. K a(6 + c-a)(6' + c«-a«) = 6(c + a-6)(c' + a«-6') and a, 6
be unequal, then will each member be equal to c (a ■\- b — c) {a* -^ V — c')
and to 2abc(a + b-hc); also 4a6c+ (6 + c-a) (c + a-5) (a + 6-c) = 0.
[This relation is equivalent to a" + 6" + c' = (fr+ c) (c + a) (a + 6).]
196. Ifaj = 6*+c»-a', y = c* + a*-6*, and a = a* + f * - c», prove that
y"«* + «*a^ + a^y*-«y«(y + «) (« + «) (ic + y)
is the product of four factors, one of which is
4a6c+(6 + c-a) (c + a-5)(a+6-c),
and the other three are formed from this by changing the signs of a, b, c
respectively.
ALGEBRA. 31
197. If - + f + -^-?+T' + -=^;
a c a be
then will
a\b e a/ o\e a b) c\a b cj
198. Simplify the fraction
a (6«-.c' )-i-5(c'- a ri-^c{c^-h ')
and thence the fraction whose numerator is
a^ (6 - c)* - 2Jc (a - h)* (a - <;)• + the two similar expressions,
and denominator
a? (6*-c^*- 2ic(a*- 6*) (a*-c^ + the two similar expressions.
[The numerator and denominator in the last case are each equivalent to
(5-.c)'(c-a)*(a-6)*.]
199. If ft'+6c + c*=3y* + 2y«+3«*, c* + ca + a* = 3«* + 2«b + 3x*, and
a^-l-a5-l-6*=3«'-l-2«y-h3y», then will
3 (6c + ca + ahy = 32 {y*«* + «•«• + o'y* + ocy* (« + y + 2) }.
VI. Tneqtialities.
[The symbols employed in the following questions are always sup-
posed to denote real quantities.
The fundamental proposition on which the solution generally de-
pends is ei' -I- 6' > 2a6.
Limiting values of certain expressions involving an unknown quan-
tity in the second degree only may be found from the condition that a
quadratic equation shall have real roots:— «.^. "To find the greatest
and least values of -^ — ^ i •" Assuming the expression = y, we
•11/ ~- ^x •+ 4
obtain the quadratic in x,
x'(l-y)-2(2-y)x+7-4y = 0,
and if a; be a real quantity satisfying this equation we must have
(2-y)'>(l-y)(7-4y),
or 3y«-7y+3<0,
80 that y must lie between ^~ — and ;, - — , which are accord-
6 6
ingfy the least and greatest possible values of the expression.]
32 ALGEBRA.
200. It Xy y, z be three positive qnantitioB whose sum is anify,
then will
(l-aj)(l-.y)(l-2r)>8xy«.
201. Prove that
4 (aV6*+c*+c£*) > (a+6+c+c?) (a"+5»+c»+cZ») > (a*+6*+c'+(f»)«> Uabed.
202. Prove that
{8a«6V+(6» + c'-a")(c» + a*-6')(a' + 6»-c«)}»
> 3 {26V -I- 2d'a' + 2aV - a* - 6* - c*}"
except when a=^h = c,
203. If a, 6, c be positive and not all equal
a'+6" + c"+ 3a6c>a*(6 + o) + 6*(c + a) +c*(a + 6).
204. H (a+6 + c)'<4(6+c)(c + a)(a + 6), then will
a' + 6* + c" < 26c + 2ca + 2a6.
205. J£ a,bf c be positive and not all equal, the expression
will be positive for all integral values of n, and for the values and - 1.
206. Prove that, if w be a positive whole number,
[-2-) >:?>«';
207. K a, 5, c be the sides of a triangle, then will
1 1 11119
6 + c — a c + a—b a-^-b — c a b o a-^b + c
and (6 + c-ay(c + a-6)«(a + 6-c)«>(6»+c"-a«)(c«+a'-.60(a*+6«-c»);
also X, y^ z being any real quantities,
«' (»- y) («-«) + h'{y^z) (y-flc) + c"(«-aj) («-y)
cannot be negative. If a; + ^ + « = 0, a'yz + b'zx-^tfxy cannot be posi-
tive.
208. If a^«=(l-»)(l-y)(l-«)
the greatest value of either of these equals is ^, x, y^ z being each posi-
tive and less than 1.
209. Prove that
{(oxb + c + bye + a + cza + b)' > 4aJc (a; +y + «) (oo; + Jy + cs) ;
a, b, 0, a;, y, z being all positive and a, 5, c unequal
ALQEUBA« SS
210. Prove HiBi, for real Talues of x,
211. Und the greatest numerical values without regard to sign
which the expression
(a?-8) (a:- 14) (a:- 16) (a:-22)
can have for values of x l>etween 8 and 22.
[When x lies betweto 8 and 14 the expression is negative and has the
greatest numerical value 576 when x = 10 ; when x lies between 14 and
16 the e xpre s s ion is positive and has its greatest value 49 when a; = 15 ;
and when x lies between 16 and 22 the expression is again negative and
has again the greatest numerical value 576 when x = 20.]
212. J£ a>b, and e be positive, the greatest value which the
expression
16 (x - a) (»- 6) (x - a - c) (a; - 6 + c)
can have for values of x between 6 - c and a + c is (a - 6)* (a - 6 + 2c)*.
213. J£p>m,
aj* — 2mX"4-©* p — m . p + m
-J— 5 ^, > ^- and < ^ .
ar + 2ma; -^ p' p-hm p-m
oi J mi €U^ + hx-^ e
214. The expression — , — i
car + 6a: + a
will be capable of all values whatever if
6*>(a4-c)*;
there will be two values between which it cannot lie if
6* < (a -I- c)' and > iac ;
and two values between which it must lie if
b* <4ac.
215. The expression ('^-^)(^-^
{x-c){x- a)
can have any real value whatever if one and only one of the two a, b
lie between c and d : otherwise there will be two values between which
it cannot lie.
216. The expression -5 — -. ^,
will always lie between two fixed limits if 5' < 4c* ; there will be two
limits between which it cannot lie if a* + c* ?• ab and 6* > 4c* ; and the
expression will be capable of all values if a' + c' < 06.
w. p, 8
34 ALGEBRA.
217. The expression -,-^ — ^j-, ,?
will be capable of all values, provided that
{ah' - a'by < 4 (a'^ - ah") {Kb - 7*6')
{or, which is equivalent, (2A// - a5' - a'6)« < 4 (^« - oft) (^'* - o'^')}.
Prove that this inequality involves the two
and investigate the condition (1) that two limits exist between which
the value of the expression cannot lie, (2) that two limits exist be-
tweep which the value of the expression must lia
[ (1) ipb' - a'by > 4 {a'h - oA') (bh' - b'h), V > ah,
(2) {ah' - a'by > 4 {ah - a'h) {bh' - b'h\ h' < ab.]
218. If (Tj, a;^ a;^ ... a;, be real quantities such that
then will 0, x^, x^, x^ ,.. x^ 1 be in ascending order of magnitude.
219. If x' + x/ + ... + «/ + aJ,aJ, + a^a^j -k-x^x^-k- ... = 1, then none of
2n
the quantities x^', a;/, ... xj can be greater than ; and their sum
2
must lie between 2 and r .
n+1
220. If a;,' + x,'+ ...a;/+2w(a;ga?3+a;3aj, + a;ja;^+ ...)= 1, m being
positive and < 1, then none of the quantities x^^ x^, ... xj can be
greater than — :=^ : and their sum must lie between
(1 ~m)(l +wi?i- 1)
1 jt 1
^1 and .
l-w l+wn-l
221. If a:i* + a;,*+ ... +a;/-aJia:!,-avc3- ... -aj^.jaj^ = — ^— > then
will a;/<r(n+l-r), and the greatest and least values oix^-k-x*+.,.+x*
.„,w + ln+ln+l n+1 , ,,
will be , , , ... , where cl, a_, ... a are the roots
Oj a, O3 o^
of the equation in 2;,
(« - 2)" - (n - 1) (« - 2)--' + (?^:iMLzA) (s « 2)--*
If
_(n-3)(.-4)(»-5)^^_^^.^^ ^^
ALQEDRA« 35
SO that if «,^i = «i, a?,,^, = «,_, : and in this case a;,^, = « (2' + 2*"'), and
h-iUl-« „...,■.„ ,4 or !■-*'.
233. If 2:B,-a!, = — »^= ?V^= ••• = -'^'^^• = 0, then wiU
2*a;,^, -05, = a(l +n- 1 2") : and if ac^^, = «, each will be oqnal to
afn-1 + ^^ j, and a:^^i = arr-l + 9=31) > and will haye ita least
value when r is the integer next below log, n.
VII. Proportionj Variation^ Scales o/yotafion.
224. If 6 + c + rf, c+rf + o, J+a + 6, a + 6 + c be proportionala,
then will
225. If y vary as the 8nm of three quantities of which the first ia
constant, the second varies as Xy and the third as ot?\ and if (a, 0),
(2a, a), (3a, 4a) be three pairs of simultaneous values of x and y, then
when X = no, y=(n- 1)* a.
226. A triangle has two sides given in position and a given peri-
meter 2« : if c be the length of the side opposite to the given angle, the
area of the triangle will vary as « - c.
227. The radix of the scale in which 49 denotes a square number
must be of the form (r + 1) (r+ 4), where r is some whole number.
228. The radix of a scale being 4r + 2, prove that if the digit in the
units' plaoe of any number X be either 2r + 1 or 2r + 2, iV^ will have the
digit in the units' place.
229. find a number (1) of tliroe digits, (2) of four di^ts, in the
denary scale such that if the first and last digits be interchanged the
result represents the same number in the nonary scale : and prove that
there is only one solution in each case.
[Tlie numbers are 445, 5567 respectively.]
230. If the radix of any scale have more than one prime factor
there will exist two and only two digits different from imity such that if
anj number N have one of those digits in the units* place, N* will have
the asme digit in the units' plaoe.
36 ALGEBRA.
231. Prove that the product of the numbers denoted by 10, 11, 12,
13, increased by 1, will bo the square of the number denoted by 131,
whatever be the scale of notation.
232. Prove that |2m-- 1 -r [m |m~ 1 is always an even number
except when m is a power of 2, and the index of the power of 2
contained in it = g — jc?, where q is the sum of the digits of 2m ~ 1 when
expressed in the binaiy scale and 2' is the highest power of 2 which is a
divisor of m,
233. The index of the highest power of q which is a divisor of
\pq -T (Ipy is the sum of the digits of /? when expressed in the scale whose
radix is q.
VIIL Arithmetical, Geametricalf and ffarmonical Progresaions,
234. If the sum of m terms of an a p. be to the sum of n terms as
m' : n'; prove that the m^ term will be to the n^ term as
2m- 1 : 2/1-1.
235. The series of natural numbers is divided into groups 1 ;
2, 3, 4 ; 5^ 6, 7, 8, 9 ; and so on : prove that the simi of the nimibers in
the n^ group is w" + (n - 1)".
236. The sum of the products of every two of n terms of an A p.,
whose first term is a and last term I, is
n{n^2){Zn-l)(a + t)'+in(n-hl) al
24(n-l)
237. The sum of the products of every three of n terms of an A p.,
whose first term is a and last term /, is
!5i^^l|M^{„(„_3)(a + 0'+4(n+l)aq,
and Ues between ^^^^^^^^^^o/ (a + and l^f^LzMtzD (a + 1)' ;
and the sum of the products of every three of n consecutive whole
numbers beginning with r is
4o
238. Having given that ^— , -— - , and — -r are in A. p. ; prove
that
a" + c' - 25" _ a + 6 + c
a' + c«-2y"' 2~ •
ALGEBRA. 37
239. J£ OybfC; &» e, a; or e, a, b he in A. p., then will
|(a + 6+c)'=a*(6 + c)-|.6«(<? + a)+c'(a + 6);
and if in o. P. y
240. If a, Z be the first and n^ terms of an a p. the continued
product of all the n terms will be
>(a/)'and<(*^y.
241. The first term of a o. p. is a and the n^ term /; prore that
the r^ term is
(a"-'r-»)--'.
242. If a, 6, c be in a. p., a, /S, y in u. P., and (uifbp,eym o. p.,
then will
. A . ..1.1.1
y p «
243. The first term of an h. p. is a and the n^ term /, prove that
the r"* term is
(n-l )a/
(n-r)Z + (r-l)a'
Prove that the sum of these n terms is < (a + /) ^- ; and their con-
tinned product < (a/)'.
244. If a, 6, c be in n. p., then will
14 111
i" + + j;=" •
o-c c-a a-o c a
245. If a, 5, r, J bo four positive quantities in H. p.,
a^d>b-\-c,
246. Prove that b + c, c + a, a + 6 will be in H.P., if a*, 6*, c* be
in A. p.
247. If three numbers be in o. P. and the mean be added to each
of the three, the three sums will be in h. p.
248. Prove that^ for all values of x except - 1,
l+x*
.♦1 <^
[For-j-j-^^^P<l ifN f:f^Pn ^beposiUve.]
38 ALGEBRA.
249. An A. p. , a o. p. , and an h. p. havo each the same firat and last
terms and the same number of terms (n), and the r*^ terms are a^, 6^, c/,
prove that
and thence that if ^, ^, C be the respective continued products of the
n terms
250. If n harmonic means be inserted between two positive
quantities a and b, the difference between the first and last of these
means bears to the difference between a and b a ratio less than
n— 1 : n+ 1,
251. K a^, a., a^ ... be an a p., 6^, ftj, 5,... a o. p., and -4, 5, (7
any three consecutive terms of the series ajb^y aj>i, ajb^, ... , then will
b,'C'-2b,b,B + bM=0;
and if A, B, (7, D be any four consecutive terms of the series % + b^y
tti + 6i, a, + ftg> ...» then will
Ab,-B{b, + 2bi) + e(b,-^2b^)^Db^ = 0.
IX. Permutations and Combinationa.
[The number of permutations of n different things taken r together
is denoted by ^P^ and the corresponding number of combinations by
252. Prove a priori that
A = ..^.+ 2r..^,., + r (r - 1) ._.P...,
= ...P.+ 3r...P,., + 3r (r - 1) ,.^,.. + r (r - 1) (r - 2) ...P,.„
H ..,/>^+p.,.,P^.,+^(^)r (r- 1),.^...+ ...
p being a whole number < r.
253. In the expansion of (a^ + a^ + . . . + Op)", where n is any whole
number not greater than p, prove that the coef&cient of any term in
which none of the quantities a^, a„ ... a^ appears more than once is \ti.
254. The number of permutations of n different letters taken all
together in which no letter occupies the same place as in a certain given
permutation is
ALQ£BEA« 39
S55, ProTe that
.C. H .^A- 2.,.(7,.. + 3.„C.., -... + (- 1)' (r + 1),
256. The namber of combinations of 2n things taken n together
when ft of the things and no more are alike is 2" ; and the number of
ocHobinationa of 3f» things, n together, when n of the things and no
more axe alike is
|2n
^"'■'2"(n)--
257. The namber of ways in which mn difTerent things can l>o dis-
tributed among m persons so that each person shall have n of them is
\itm
\ ar *
258. There are p suits of cardH, each suit coiisiHting of q cards
numbered from 1 to ^ ; prove that the numl>er of Hi^ts of q oLrds num-
bered from 1 to g which can be made from all the uuits is //*.
259. K there be n straight lines lying in one plane the number of
1
different n-sided polygons formed by tlicm i« .^ In - 1.
260. The number of ways in which p things may be distributed
among q persons so that everybody may have one at least is
«'-?(?- 1)' + ^;7-^7- 2)'-...
261. The number of ways in which r things may l>e distributed
among n-^p persons so that certain n of those persons may each have
one at least is {S^
(n +p)' - „ („ + ^ - 1 )' + » (''.;-L) („ +p - 2)' . . .
Hence prove that
5, =^. = ... = S,_, ~0,S,= [5 -S-.,, -= g + p) n+l.
X. Binomial Theorem,
262. Prove that
(1) l-ny-;^-+-^r
n(/t~ l) (n-2) l + 3a:
3 (l+^«-y
+ ... =0;
40 ALGEBRi.
,-, , -2n+l _/2»+lV
■^^2»-l)(^;)""="(2«-l).
n being a whole number.
263. Determine a, b, e,d,eia order that the n^ term in the ex-
pansion of
a + 6x + ca:* + da^ + ex*
may be na^~ ,
[The numerator is l + llx-^llaf-ha?,]
264. Prove that the series
l" + 2"aj-i-3V+...+r"a;'-*+ ...
is the expansion of a function of x of the form
a^ + a^x + a^ + . . . + ajxf'
also prove that
«- = 0» ».-i = «o = l> a,-t = ai = 2"-(w+l); «..,= «,.,.
26^. The sum of the first r + 1 coefficients of the expansion of
Iw + r
(1-a:)- is equal to ^^ .
266. If from the sum of n different quantities be severally sub-
tracted r times each one of the quantities and the n remainders be mul-
tiplied together, the coefficient in this product of the term which involves
all the n quantities is
267. Prove that
If l£
+ ... =n+ 1,
Lf
(n-3)(n-4)(n-5) . m- - 1
1^ ^ ' ra — l
(3) (p + y)--(„-l)^0> + y)-.+ <!LlMLz3)py(p + y).-.
(n-3)(n-4)(n-5) _P"*'-9"*'
n being a poedtiTO inieger.
ALQEBRA. 41
268. If p be nearly equal to g, then will ■ — "^ be nearly equal
-c^-
269. If © be nearly equal to q^ ; .; / i ! • i« » clo«« ai>-
1
PV. «.^A ic P
]Nt>ximati0n to ( -)"; wid if differ from 1 only in the r + 1** decimal
place, this approximation will be correct to 2r places.
270. K a, denote the coefficient of a^ in the expansion of ( _- — ^-)
in a series of ascending powers of x, the following relation will hold
among any three consecutive coefficients,
(1 +xY
271. If . L be expanded in ascending powers of a?, the coeffi-
cient oiaf'*^^ is (n + 2r)2""*, n, r being positive integers (including
zero).
27.2. If (1 +a?)" = ao + «i^+ ••• +«X + •••> ^^^^ ^ill
a,* + 2a/ + 3a/ -♦- . . . + na/ ?»
a/ -»-o*-»-a.'+ ... + a * 2 '
oil N
n being a positive integer.
273. Prove that
o« . ^i^'^) o.-. . n{n-l){n - 2)(n- 3)
-^+ — p — ^ + ^g-^^ -. +
~ UT" ••
274. The sum of the first n coefficients of the expansion in ascend-
ing powers of x of ^-.- ' is -'— 2 , 7i bemg a positive
integer.
275. Prove that, if n be a jKwitive integer,
3.4n(n-l) 4. 5 H(,t- !)(»- 2)
^■'^"■'r.2- 12 ■'r.2 13 "••••
(n-i- l)(7i + 2) , , . ov «. .
+ ^ '-; ^- = (/A» + 7?t + 8) 2""*.
276. The coefficient of a:'*"'* in the expansion of (1 + x)' (1 - ar)"*
18 2""* {(n + 2r) (n + 2r - 2) + n} ; and that of a;"*'"* in the expansion of
(1 + «)" (1 - «)-* iB J (n + 2r - 2) 2-* {(n + 2r) (n + 2r - 4) -i- 3ft}.
277. If the expansion of (1 + x)" (1 + «')" (1 + a;*)" be 1 + a^x + ajx^
+ ...+a^+... and /S^^E a^ +a, + a,3+ ..., *S^, = a^ + a, + a"+ ..., and ao
on to 5p then wiU ^,=^,= ^,= ... =^, = j (2*'- 1).
42 ALGEBRA.
278. If the expansion of (1 +a; + aj'+ ... -»-af)" be 1 + a,a5 + a^*+ ...
+ aX+--- and ^i = ai + a^^.i + ajp^j+ ..., '5t = at + »,+t + «s^^+t+ •••> and
so on to S,f then will S^= S^= ... =Sy.
279. Prove that ).. \ , ,,/ /, ,, \. ^
a (6 - c) + 6' (c - a) + cr (a - 6)
is equal to the sum of the homogeneous products of n dimensions of
280. Prove that the coefficient of x', in the expansion of
{l-axy'il'-bxy
in ascending powers of a; is
(r + 1) (g^^' - y ^') - (r + 3) fl& (g^^' - y "^Q
281. If a5 = -i 7 iT , r; zi will be equal to the sum of the
6 + n(g-5) (1-a;)*
first n terms of its expansion in ascending powers of a; ; a, b being anj
unequal quantities.
XL Exponewtial and Logarithmic Series.
[In the questions under this head, n always denotes a positive whole
number.]
282. Prove the following identities : —
(1) n''-(n + l)(n-l)"-t- ^'^|^^^'^ (n-2)"-... tonterms=l,
(2) (n-l)'-n(n-2)' + 5i^zi) (n-3)"-... to 7i^ terms Ejn-l,
(3) (n-2)--n(n-3)"+^^^^(n-4)--... to;r^terms=[n+n-2*,
(4)
r-n2" + — *^^3"-.... to n + 1 tenns = (- l)"[w.
[(1) is obtained by means of the expansion of €"'(€*- 1)"**, (2) from
that of €-'(€*- 1)", (3) from that of €-"(^-1)"*^, and (4) from that of
.-+».(c--- 1)-.]
283. Prove the following identities by the consideration of the
coefficients of powers of a: in the expansions of (e'-c"')" and of its
f 2aj" V
equivalent <2a; + -r«^+ .... | : —
ALGEBRJL 43
(1) »"'-»(,t-2r''+?^^^\n-4r--...=0.
(3) »'-n(n-2)- + ^*— ^(ii-4)--... = [»2-»,
(S) ^•♦•-»(n-2r' + ^!i^>(n-4r'-... = ^!» + 22--';
If [2 —
the number of terms in each series being ^ or - — , and r a whole
number < x •
S84. If S^ denote the series
(2i»+l)'-(2» + l)(2»-l)' + ^?~"!'^^^-(L>n-3)'-... ^ n+1 terms,
then win 'y, = fl; = -9.= ... =5^.,= 0, ^^^, = 2" 2n4.1 ,
2*(2n + l)'2n4.3
285. Fkxive that the sums of the infinite serios
^^ 1.2.3*3.4.5'*'5.6.7'*"' •
,. 1 1 _1
^^ 1.2.3.4*3.4.5.6"*"5.G.7.8^ ••
^^' 1.2.3.4^3.4.5.6'*'5.6.7.8'*"' •
are reqiectiTelj lpg2-^, I log 2-^, and tlog2-5Vi **^d that, if S^
denote the sum of the infinite series
1 1 1
"^ k' a T ■/.. '. n\ ■*" •••>
1.2.3...(n + 2) 3.4.5...(n + 4) 5. 6. 7...(n+ G)
(n+ 1)51 = 2/?..,--!-.
288. The coefficient of of in the expansion of (1 + x)* being denoted
b J a^ prove that
flj^-> -a,(p- 1)-' + a. (;, - 2)- -...+(- 1/" V.
=«.(»-pr'-».(»»-i>-ir'+«.(»-p-2r'-- +(-!)•"'"'».-,-..
^ being a whole numl)cr < tu
[From the expansion of c~'' (c' — I )* containing no lower power of x
than 0^, so that the coefficient of u:*'^ is zero. This result might be
used to prove (2G4).]
44 ALGEBRA.
287. By means of the identity
log (1 -0^)= log (1 -ic) + log (1 +{» + {»•),
prove that the sum of n terms of the series
n(w+l) (n~l)yi(n + l)(n-f 2)
3 (— 1)*~*
is if n be of the forms 3r or 3r- 1 ; and — ^ — ^ if n be of the form
3r + 1 ; also that the siun of n terms of the series
1 ^+ ^
is (- 1)""* if » be of the forms 3r * 1 and ^^-^ if » be of the form 3r.
288. By means of the identity
log (1 - X + ic") + log ( 1 + X + x") = log ( 1 + «* + X*),
prove that, if/(^) denote the sum of n+1 terms of the series
7^ n'(n'-P) n'(n'-l')(n'-2')
/(2n) = (- iy/(n) ; and that, if F(n) denote the sum of n+ 1 terms of
the series
1 ^(^•^^) (»*-l)w(n + l)(n+2)
(2n-i-l)/'(n) = (-l)- or 2(-iy-\
289. By means of the identity
, x"-3x + 2 - x + 2 ^, x+1
prove that
3-. + 2'3- <"-^)^"-^) + 2*y. An-6)(n-5)(n-4)(n-3)
^ o.,.-.« (n-9)(»-8)...(n-4) _ 2- - 1 .
+ 2d ^ + ...= __,
the series being continued so long as the indices of the powers of 3 are
positiva
290. Denoting by u^ the series
l" + 2"-f 7^ + ... + ^^! ' +... to infinity,
ALGEBRA. 45
prore thftt
fi(fi-l) n(n-l)(i»-2) , , .
s AX,
, ^, . n (n - 1
and thmt «*,♦,-«,= «, + »«,_, + -^j — «.-,+ .+ wii, + u^;
and hj means of either of these proTe that u,= 4140c
391. If «. denote the infinite series
then win 1 »t*^, +w + nu , + — - — - u , + ... + nw. + w^
9a« w _ 1 1 1 (- ir'
293. If ^. = :2-:3^i.t---^ HtTT'
then will 1 =« +r . +%^+ ... + - -'- + -.
• "^* T n-1 n + 1
293. Haying given
■ •—I' ■ • •—I'
prove that the limit of — ^ when n is indefinitely increased is
n + 1
u, - 2m^
w, + -J ?.
• c
294. If there be a series of terms n^, t^^ u,, ... u, ... , of which any
one is obtained from the preceding by the formula
w, = nw. , + (-!)•
and if tf^ = 1, then will
n(n — 1) n(n — 1)
[n = W.+ nw._, + g— / w,., + . . . + -^1^ — u, + nu^ + «^.
Prove also that .-" tends to become equal to - as n increases
|n ^ €
indefinitely.
295. Prove that
y^'-S ^ n-1 , (n>2)(n>3)
__.=2--^2 + ^ 2 -...
46 ALGEBRA,
and that
— - — 1^^^ — ->V(;>+^) +
[6j means of the expansions of the identicals
log{l-px)-^log{l-qx), log{l''X{p-\-q-pqx)];
or by expanding in ascending powers of x both members of the identity
1 . 1 ^ 2-(p + g)x ^
l-px I'-qx l-x{p-\-q—pqx)
296. If there be n quantities a,byC,.,, and 8^ denote their sum,
8^^^ the sum of any n — 1 of them, and so on, and if
prove that S,=S^ = S^=^ ... = S^^, S 0,
and 12/S;,^,H |n + 2a^...{22(a«) + 32(a6)}.
Also if a be any other quantity and if S^ now denote
then will S^ =S, = S^= .-. ='^.-, = 0,
S^= nahc..., 2aS^^j= ln+ 1 oic... (2a + a + 6 + c + ...),
and 12^,^, = |n-f2 ojc ... {22 (a«) + 32 (a6) + 6a2 (a) + 6a«}.
[These results are deduced from the identities
(1) (€««-l)(€^-l)... = €»-*-2(c*-**)+2(€«*-«»)-... *
(2) €« (€«»- IXc**- 1) (...) = €<«+••)' - 2 €(»+«--l)* + 2 {€(*+*-J*} - ...
by taking the expansions of every term of the form c"" and equating the
coefficients of like powers of a; up to x**\]
Xn. Summation of Series,
[If u^ denote a certain function of n and
the summation of the series means expressing S^bs b, function of 7i
involving only a fixed number (independent of n) of terms. The usual
ALGEBRA. 47
Artifioe bj which this is efToctcd oonsiBts in expressing u^ as the difTerence
c»f two quantities, one of which is the same function of n as the other is
c»f n— 1| (i7^— ^ii-i)' ^^ being effected we have at once
s,={u,.u:,^{U,-u^*...*(u,.u,.,)=u,-u,.
Thus if u be the product of r consecutive terms of a given A. p.,
beginning witn the n^\ we have
tt,= {a + n-16}(a4-n6) ... {a-\-(n + r''2)b\
_(a + fi-lft)(a + fi6) ... (a4-n4-r~1 6)~(a-*-n~26)...(a4-n4-r~26 )
wbenoe , = (r + 1) 6
and
1
'^.=/jr:f-jT^{(a4-n- 16)(a + n6)... (a + n-»-r-i 6)
-(a-.5)a(a4.6)... (a4.?^6)}.
The sums of many series can also be expressed in a finite form by
equating the coefficients of a^ in the expansions of the same function
of X effected by two different methods of which examples have been
already given in the Binomial, Exponential, and Ix^arithmic Series.
In the examples under this head, n always means a positive whole
number.]
297. Sum the series : —
(2) -i-+-ijL^+ -LAlL^ ^1.3.6...(2n-l)
^^ 2.4 2.4.6^2.4.6.8^ •■^2.4— 2;r72.7+2r
/3\ Li4.2-2* n2-
(4) iT+-77r-+...+
|4 ' [5 '^•••"|n + 3_'
2r» wr"
+ :r+ ... +
r-t-2 "* |n-fr'
<^^ (l+x)(l + 2*)-'(l + 2«)(l+&r)*-*(l+^)(l^.— 1,,.
(7) 1-5+1-3-T+- +
1.3 1.3.6 ^"^1.3.6... (2»+l)'
^' 5 S.7*3.7.11*"*3.7.11...(4«-l)'
48
ALGEBRA.
(9) j5 + |^+...+
n+2
(10) |-^+_+...+ __^,
(11)
(12)
1 5 11
+ > ,x + rs— m+ ... +
n* + n — 1
2,5 5,10 ■ 10.17
1
(l+n'Xl + n-l*)'
1 as ar •
(1 +x) (l+a*)'*'{l+ of) (1 +a^) "^ ••■ * (1 +!tr) (1 + a?*')
+ »\ >
o;
a;
a;
a^
-1
l+a:"l+a;*'"l+aj2"'
.- . . a; (1 - ax) ax{\ — a'x)
^ ^ (1 + a;) (1 +aa;) (1 +a'a;) "^ (1 + oo:) (1 +a'a;) (1 +a'a;)
a""*a: (1 - a*x)
+ ...
"+i/>.\ '
(1 + a'-'x) (1 + a"x) (1 + »"*'«)
(15) 1 , ''+1 , (r+l)(2r.H) ^
^ ' /? + r (p+r)(p + 2r) (p + r) (jt> + 2r) ( jt> + 3r)
^ (r + l)(2r+l)...(n-lr+l)
(/? + r) (p + 2r) ... (p + wr)
298. Prove that
wjfn-l) ^ w(7i-l)(yi-2) , ^ n(n-l)(n-2)(n-3)
^" 12 "" 13 14
+ (n-l)(-ir.
299. Prove that
(l+r + r»+/)(l+r« + r* + r^ ... (1 +r2"-* + r2- + r8.2"-»)
_ 1 «7^_7.2«+»+^.2«
1 -r-f^ + r^
300. Prove that
,., 1 1.3 1.3.5
(^> i+476+476:8-*--*^^ = ^'
/nv , 3 3.5 3.5.7
(2> ^■^8-*"8Tio-'8-To7r2-'-^^ = 2'
,^, - 11 11.13 11.13.15
(3) l+r4"'lT.-16-*" 11716. 18 "'•••^*
= 12;
ALQERRA. 49
and genefmllj that, if ^, 7, r and q-p-r be positive, the sum of the
infinite leriea
q'^r (q^r)(q^2r)^ (^ + r) (y + 2r) (9 + 3r) "^ -
wiUbe 2 — .
q-p-^r
[The lam of n terms of the last series is
_q_ (^ _ (p + r)(/>-h2r) .. .(;>■!■ nr) ^
301. Prove that
2 2L4 2.4. 6... 2n , 2 . 4 . 6 ... (2n-h 2 )
3'*'37i>"*" •''"3.5.7...(27i+l)"3.5.7...(2n+l)
302. Prove that, m being not less than n,
/iv 1 ** n(n-l) n(n-l)(7»-2) ^ ,^
TO+ 1
w- n+ 1 '
(2) , + 2!i + 3!L(!LliU4'-f^H^*
' m m{m-\) TO(m-l)(w-2)
= (m-f 1 ) {m - f 2)
"" (m - 7t + i ) (m - M + 2) '
^^ m m(w-l) m(m-l)(m-2)
(m+ l)(m + 2)(m+3)
— (w - 71 + 1 ) (m - n + 2) (w - w + 3) *
303. Prove that
J _ a n(n - 1) a(a> l) ^ n(n- 1) {n^ a(a-l)(a- 2)
'^S'^ [2 6(6-1)" |3" 6(6-l)(6-2)
+ ...
-""«™'K'-f)('-*°-.) •('-»-r^)-
304. Ptove that, if x be less than 1,
(r- 1) (r- 2) (r- 1) (r - 2) (r-3) (r- 4)
* ^ a;+ ^g X- ...
1 |3 ^«+ 1^ jf-...
W. P.
60 ALGEBRA.
[Obtained by expanding numerator and denominator of the fraction
(i-V-xr-a+v-x)-'--'
305. If ^^^^_-JP^^ nin-l)(n-2)(n-S) _^
n(«-l)(n-2) «(n-l)(n.-2 )( n-3)(«-4 )
and v, = n ^ + ^ .,
then •will uj + »/ = 2 («.»,_, + ».»._,)•
306. If «.31-^<^>^-h "<"-^)^;,-'>^"-^> ^-^...,
and
«(n-l)(n-2)
then will uj + v/ = (1 + a^" » (1 + a:*) (m,w,., + vv^_^).
307. Prove the identity
r»+l 7^ (2w+l)+ i^ (2m + !)•-...
= 2--^^l 2-» (tt+ 1) +5— ^|— ? 2"-«(u+ 1)»
j^ 2" •(ii + l)'+...
XIIL Hecumng Series,
[The series w^^ + Wj +m, + ... + w^ is a recurring series if a fixed
number (r) of oonsecutiye terms are connected by a relation of the form
in which n may have any integral value, but p^^p^j '"Pr-\ ^'^ inde-
pendent of n. It follows liiat the series a^ + a^x + a^' + . . . + a^o:" + ... is
the expansion in ascending powers of x of a function of x of the form
2 — * ^- — -^ ^-^ — -^n (the generating function of the series); and
if the 8cale of relation {A) and the first r - 1 terms of the series be given
this function will be completely determined ; when by scpai-ating this
ALGEBRA. 51
fnnctioii into its partial fractions z — '— + .- -^ — 4- ... and expanding
* 1 - a,« 1 - ajK '
eobh we obtain the n^ term of tlie series and the sum of n terms.
Thns the n** term of snch a series is /?!»/"* + ^,a ""* + ...where a,, a,, ...
are the roots of the auxiliary equation of' * +7>,af ~" +p^"* +...-♦- /> j = 0,
and j9p B^y... constants which con be determined from the first r terms
of the series. If however two roots of the equation be equal (saj a = a,)
we must write Bji instead of 7/,, if throe be equal (a, =a^^aj the
corresponding terms will be (/?, + nB^ + ri"j?j a^""' and so on.
If the scale of relation is not given we shall require 2 (r - 1) terms
of the series to be known to dct<^rmine completely the generating
fanction ; thus if four terms are given wo can find a recurring series
with a sode of relation between any three consecutive terms and whoso
first four terms are the given terms. J
308. Prove that every a. p. is a recurring series and that its
generating function is — jj r, - , a being the first term and h the
common difierence.
309. Und the generating functions of the following series : —
(1) l + 3a; + 5a:' + 7aj"+ ...
(2) 2 + 5x+13«'+ 35x"+ ...
(3) 2 + 4aj+14r«+52a:»+...
(4) 4 + 5x + 7a;* + llaj'+ ...
(5) 2 + 2a? + 8iB* + 20a;*+ ...
(6) l + 3x+12a:* + 54aj'+ ...
and employ the last to prove that the integer next greater than
(^3 + l]r is divisible by 2"**, n being any integer.
310. The generating function of the recurring series whose first
four terms are a^hfC^d^iB
ah'- eg* -fa; (a'd-2abc +^_
6' - oc f a: (flk/ - 6c) + ic* (c* - W) '
311. If the scale of relation of a recurring series l)e
and if w^ =x 2, u^ = 7, find u^ and the sum of the series i*^ + w^ + . . . + ?t^_, .
312. Prove that, if a„ a,, a, ...a^ be an A. p. and 6„ 6,, ...ft.acp.,
the series
^A» ^A» ••• ^A' •••
will be reoarring series.
4—2
52 ALGEBRA.
313. The series w^, ?«,, n^, ... , and v^, v^, v^, ... are both recurring
series, the scales of relation being
prove that the series u^v^, w^Vj, w,v,, ... is a recurring series whose scale
of relation is
^-+. -Pi^i^n^i + (pX + ?!>« - ^P.9,) ^n --P,Pa^i5',^.-i +pX\-m = ^•
[It is obvious that the series u^ + v^, u^-^v^, ^t + ^a> . .. ifl a recurring
series whose generating function is the sum of the generating functions
of the two series.]
314. Prove that the series
l« + 2«+3»+... + <
l> + 2» + 3'+...+n»,
r + 2'+3'+... + n'
are recurring series, the scales of relation being between 4, 6, ... r + 2
terms respectively.
315. Find the generating functions of the recurring series
(1) 1 + 2aj+ 5x'*+ 10x'+ 17a;* + 26a:*+ ...
(2) 1 + 3a: + 4a;" + 8ic" + 1 2a;* + 20a;* + . . .
(3) 3 + Gx-¥ 14a;» + 36u;" + 98a;* + 276a;*+ ...
(4) 3-a; + 13a;'- 9a;' + 41a;*- 53a;* + ...
and prove that the n*^ terms of the series are respe^ively (1) 1+n-l',
(2) i{4(-l)- + (29-3n)2— },(3) l-» + 2-' + 3-Sand (4) 2n-l-(-2)-.
316. Find the generating function of the recurring series
2 + 9a; + 6a;» + 45a;' + 99a;* + 1 89a;* + . . .
and prove that the coefficient of a;""* is one third the sum of the n^
powers of the roots of the equation s^-3z-9 = 0, and that the co-
efficients of a;*""* and of of" are each divisible by 3".
317. If the terms of the series a^ a^, a^... be derived each from
the preceding by the formula
_ pq
"•'^-p + q-a,'
prove that
^ ^^ (^o'P)p'^''K-g)g'''
" ^ {<^o-p)p'-'(ao-q)r •
[If we assume a^=_--^ ^e ge^ ^^ ^j^^ a scale of relation for
•♦1
A R
1^^.4. - (P + 7) W.4 , + w, = 0, 80 that M. -—, + —. .]
ALGKDRA. 53
XIV. ConvergetU Fractions.
[If ^ be the n^ conTergent to the continued fraction
?i ?. ?.
6, +6, + 6,+ ...
we haye the equations
P. = ^J'.-, + ^J'.-s* 7. = ^.^-i + ««?.-.;
and for the fraction
?j a, a,
6,-6,-6,-...
the equatioiM
The solution of each equation, a,, 6 being functions of n, must involve
two constants, since it is necessary tnat two terms be known in order to
determine the remaining terms bj this formula. These constants may
conveniently be taken to be p^^ ;>,, 7,, 7, respectively. The fraction
^ thus determined will not generally be in its lowest terms.
We will take as an example the question, ''To find the n^ con-
vergent to the continued fraction
1 1 4 1_2 2/i(n-l)
1 -3-6- 9 -... in -..."
Take u^ to represent either p^ or q^ (since the same law holds for
both), then w,+ , = 3«u, - 2?i (/» - 1 ) u^_, ;
or u.^,-2nw. = n{w,-2(n-l)tt,.,}.
So t..-2(n-I)u,.. = (n-l)K.,-2(n-2)n..,},
w, - 4w, = 2 (w, - 2m,), (== 2 or as m ^;> or q).
Hence, u^^ , - 2nu^ -^ [w or 0,
w ^. 2u - ^
or -*^» - 1 H - 1 or 0.
2u 2'm , « ^
So 1 — - - p-"-^ = 2 or 0,
«— 1 u-2
54 ALGEBRA.
and 2X = 2";
whence ^ = 2-+>-l or 2",
2"— 1
and the n"» convergent is o— i •]
111
318. The fif'^ convergent *02 + 2 + 2+ ^
(l-hV2r~(l-V2)-
(l+^2r'-(l-^2)"*'-
319. The n^ convergent to the continued £raction
XXX
x + 1 -flc+T-jc+T-...
i« equal to ,^.t^.. ; and that to
XXX
aj-1 +aj-l + x-l + ...
«""•'* — (— l)"aj
is equal to ^^ /_ i\» ^ *^^ *^® numerator and denominator of any
convergent to either fraction differ by unity.
320. Prove that the continued fraction
a. a^ a, a
a, +a, +aa+ ••• ^n
is equal to the continued fraction
1 1 a. a a
1 +aj +ag +a3+... a^_/
321. If -^ be the n^ convergent to the infinite continued fraction
-.-.-. 7 P,y Q^ will be the coefficients of af"* and x* respectively
in the expansion of ,i s .
* 1— oaj-or
322. Prove that^ — being the r^ convergent to the continued
fraction
1111
a + 6 + a + 6+ ...
ALO£BKA. 55
333. Prove that the products of the infinite continued fractionB
,-,1111 111
(1) - 7 - - , C + r - - ,
.^,11111 .1111
(2) - - - - - /£ + --.-
^' a + 6 + c + c/ +a+../ c -f 6 -fa + c/+ ...
324. Prove that the differences of the infinite continued firaotiona
1111 illil
'' a-k-i -^c -i-a-i- ...* 6 + a+ c + 6 + a+...'
2^ 1 1 1 1 1 11111
OAK TM a b a b J 6o6a
325. K»=i+T+T + T+...''^'*y=T+i+T+I + ...,
then will a:- y = a - 6, and a;y+ — = a + 6+l.
«A^ „a6aft .ftafta
a+6+a+6+...' ^ 6+a+6+a + ...
then will afz-Vy = a — bf and ay + — =a + 6 + a'6'.
flfy
on»T T* aJca J c6ac
327. If 05=7 T T T ^^y'=T,T^J^^ »
1+1+1+1 + .. . ^ 1+1 + 1+1...
then will a: - y =-=— r , andaf(l+y)= i .a '
««« Ti.a6crfa , d c b a d
328. K«=i+T + i + T + T+...'^^y=I + l+T + T + I+...'
XI. X /t \ l + c + d J ,- . ,l+a + 6
prove that a:(l+y)=ay-^^y;j-^, andy(l+a?)=d ^^^^^ .
329. The convergents to the infinite continued fraction
12 12 1
^ 5 -1-5-1 -5-...
recur after eight.
56 ALGEBRA.
330. The continued fractions
444 olii
* + 8+8+8+...' '*'*"4+4 + 4 + ...'
each to n quotients are in the ratio 2:1.
[This can be readilj proved without calculating either.]
331. Prove that
n n-1 71-2 2 1 l_n+l
n + n-1 +n-2"*" • ■*"2 + l+2"n + 2*
332. Proye that, if a> 1, the infinite continued fraction
1 a a+l 1
that '
a -a+l -a + 2-... a-1'
I a a a+l a+n-1
a + l+a+a+l+a + 2+... + a + n
_J 1 1
"a+l (a + l)(a + 2)'*'(a+l)(a+2)(a + 3)-... ^**'*'^*®™*'
and that on reducing this to a single fraction the bctor a + n + 1
divides out
333. The nf^ convergent to 1 - ^ j is equal to the (2n - 1)***
convergent to
1111
1+2 + 1 +2+...-
334. If ^ be the n^ convergent to r — ^ =- , then
5. ® r-1 +r+l +r-l + ...'
««i. -r/. n n+1 n + 2 ^ ...
335. Ifa; = - = ^ to x . prove that
w+ w + 1 + n+2 + ... ' *^
1111
+ —7 . IK / ^ - ... to 00.
n + x n n{n + l) w(n + l)(n+2)
336. Prove that the value of the infinite continued fraction
12 3
1 + 2 + 3 + .. .
IS . : and that €»2 + :r tt ■= -r . tox.
€-1' 1 + 2 + 3+ 4 + ...
ALQEBBA. 57
337. Prove that
1 n n (H4 1 ) n (n 4- 2 ) n (yn-r~l)
T+I+~"2 + 3 +...+ r +...
is equal to
1 :-T +
n+ 1 (w+ 1) (w + 2) (n + 1) (m + 2) (n + 3) " •' '
and tlience that the value of the fraction continued to infinity ia for
n = 1, 2, 3, 4, 5 respectively
€-1 €■ + ! 5c* -2 ITcVS ^ 3291* -24
"r'"^7''^9€' ' 32c*"' *^ ""625?" •
338. Any two consecutive terms of the series a,, a^ ... a^ ... satisfy
the equation
n(n-f 1)
"•*»"" 2»-a, '
find a, in terms of a, ; and prove that when n is indefinitely increased
the limit of
|n 1-a/
339. Having expressed J{n*-^a) as a continued fraction in the
form « + o- 9- » -- is the r^ convergent ; prove that
Pi>.^, - (w' + «) Wr^, = »» (- «)'» P.*, + «/>r-i = 2 (n* + a) y,.
340. Prove that ^(n* + a) can be expressed as a continued fraction
^1. - a n'-ha a n'-^a , ,1 . .- P , .,
in the form n + - - : and that if ^' be the
n+ M + n + n + ... q^
f^ convergent,
where a, fi are the roots of the equation in x
.i:* - 2 (?i" + a) x + a (n* + a) = 0.
1 + 3J
341. If — ^— — j= 1 + a,a; + a^+ ... +a,a:"+ ...,
and 1 -23'-a^ ^ ^ + 6,a; + 6,a:'+ ...+6,x"+ ...,
prove that a/ - 2ft/ - (- l)-^
58 ALQ£BiLi.
2 -X
342. If ^ -. i = 2 + a,a5 + a.a;'+ ... +ax" + ... ,
and -z — 5 -j= 1 + 6,a; + 6,a:*+ ... +bsi^ + ....
prove that a J - 36/ = 1.
343. If 1 _ 2rx + af ^ r -f a^a; + a^x'+ ... -naX -f ... ,
and ____= 1 + 6^a;+ 6,a» + ... + J.o" + ••• »
prove that a* - (r» - 1) V = 1.
344. In the equation
prove that a;=*vw*^w + «^
and find all the roots of the equation. Prove that
^7 - \/7 + \/7-7~to 00 = 2,
and express the other roots of the biquadratic in the same form.
345. Prove that
m' + mn + n*
\/p-\/p + Jp^...
2mn
m*
-mw + n*
2m7&
!»•
— wm-
-n'
2mn
n».
-wm-
m"
2i7in
where p = ^ — s-^ > *^d w > w.
346. Prove that
/iv r(r+l) r(r+l) . ^. ^
(1) -^-= — - -^ — ' to w quotients
_ r(r+l)'*' + (r+l)(-r)'*'
(r+l)"+'-(-r)"+' '
^ ' 1 + 2+ 2+ 2 + ...+ 2
- 3*5 •••*2»+r
ALGEBRA. fiO
..12612 n(n-f 1)
*^ 2+2+2+ 2 +...+ 2
1 1 (- 1)-
+ ST—: ... +
""1.2 2.3" 3.4" (m+1)(i»+2)'
... 1 1 r+1 r + 2 . .. ^
(4) r- - . =• — ^ to n quotionU
^' 1 + r + r + 1 + r+2 + ... ^
_ - 1 1 1
= ^"iTl"'(;^TI)7;nr2)"(r+l)(rT2)^
^^ T + 2^+3^2a-...+n+l-
«• aj' (- 1)V*»
(6) f ** ^^)' (2»-lx)'
1 + 3 - «• + 6 - 3**+ ... + 2»n- 1 - 2»- 1 as'
-"^ 3^6 -^ 2»+l '
r\ I L ^(^"^ ^) r(r + n-l)
^'^ 1 + 1+ 2 +...+ n
r + l"(r+l)(r + 2) - "(r+ l)(r+ 2) ... (r + n)'
^^ T+3+ 5 + 7 +... + 2n + l + ... 12'
(10) \ I I JIL to 00 .2j:i|.
^ ' 1 + 2+ 3 + ... +M-1 + ... €*+!
,,,, 3* 3.4 3.5 3» , <,2«»+l
<"> T+-2-+-r+...+;r:2+...*°*='^5?^2'
(12) ^'/^ i^ i(!Lii) to« = 12^,.
* ' 1+ 2 + 3 +...+ n +... 17€* + 3
347. Prore that
/i\ 1 1 1 1 * -*• * - 2i»
(2) i-T-3-T-...*""1"°*^*"-2l^i)'
,,,1149 n* _ 1 1 1
W i_3-5-7-...-2^m-**l + 3"^"'^n+r
60 ALGEBRA.
(4)
1 1 9 25 (2n-l)« ,11 1
1-4-8-12-...- 4w " 3 5 ^•"2n+l'
1 r' (r + l)' (r + n-1)'
^^ r-2r+l-2r + 3 •" 2(r + n)-l
1 1 1
= - + =-+ ... +
r r+1 r + n'
1 g' (a + 6)' (a + n-16) '
^^ a-3a + 6-2a + 36-...-2a + (2n-l)6
1 1 1
= - + r+... +
a a + 6 "* a + n6'
,^, «• (a-1)" a* (a-1)' ^ ^. ^
(7) y ^ ^ ^ Y ^ — =— ^ ton quotients
a(w + 2a-l) Tia . ,,
= _i i or 5 71 as n IS odd or even,
w + 1 n — Ja + J
.jj. a(a+l) a (a-1) a(a + l) . .. .
(8) — ^-Tj — - — ~ — ^ —^ — ' to n quotients
a 1
= a ;; , or a ^^ — ;
(a+l)(l-a-y-a (l-O* "1
^ ' T-2+aj-3 + 2a;-...-n + l+na;
«• «• «
A + l
= a!+-S- +Tr + ... +
2 "3"^"""n+l'
1 4 ^ (»'-!)' _ n(n+ 1)(2«+1)
( ^ T-6-13-... -»• + (» + !)•" 6
. 2 3 8 n*-l _ w(» + 3)
(^^/ 1-6-7-...-2ST1- 2 '
.11* 2* n*
^^^' T-F+^-2nT'-...-n» + ^+T'
1 1 1
= Ti "** ot ■*■••• "^
t >
!• • 2* (n + 1)
(14) =- =^ s to n + 1 quotients
^ ' 1 -a + 1 -a+2- ... ^
= l+a + a(a+l) + a(a+l)(a + 2) + ... +a(a+l)... (a + n-1),
,--.1135 2n-l
(15) T-2.4-6-....-i;r
= 2 + 1.3 + 1.3.5 + ...4.1 .3.5...2n-l,
ALGEBRA. C 1
nft\ ^ 2n-2 * 2_
^^*> 2«— l-2;r:3-...-3-T=2/.,
,,_. 112 3 n , ,
' 1 -r+2 -r+3 -r + 4- ... '
/lav r« (r + 1/ (r + 2)'
(*^) o i" ■« "o" o r tooo=r,
^ ' 2r+l - 2r+3 - 2r+5 - ... '
(20) - r ^ to 00 = jr
^ ' r -r+1 -r+2- ... r-2
XV. PorUtic St/»tem8 of Bquaitoni.
[Any system of algebraical equations
18 poristic if a certain relation holds between the coefficients a, b, e,
^ y, A : when n = 3, this relation is M= h* -ab-ch -^/y - ; when n = 4,
it is - JV= a6c + 2/gh - a/* -bg*~ ch' = 0, and when n = 5, it is
where Z = e — 4A. For any number of such equations, if there is one
solution for which x^J a;,, . . ., a;, are all unequal there is an infinite number
of snch solutions, but this cannot be the case unless a certain relation
hold, which relation involves L, Afy N only. See Proceedings o/ London
Math. Soe, Yol. it. page 312.]
348. If a;,, x^ be the roots of the equation
(l-m)(l-a5) mx
then will j-. ry^ ^ + 6 + ^0.
(1-«,)(1-«J «,«,
62 ALGEBRA*
349. If x^, x^ be the two roots of the equation
, / 1 \ a; w -
ar-hai mx + — )=— +— +1,
\ nix/ m X
then will a* + a (x,x^ + — j = -i + -« + 1,
\ ^ ' x^ic^ x^ ajj
and x,x„ + m (x. + xj = + — + = - a,
* ■ ^ * ■ XjX^ Tnx^ mx^
350. If a;,, x^ be the two roots of the equation
a*(l + rnf) (1 + a*) + a(w + aj) (mx - 1) = mx,
then will a" (1 + x^ (1 + a;^*) + a (a?^ + a:J {x^x^ - 1 ) = i»,a?,,
Ill 1
and w + ajj + ajg + —+—+ — = waija:^ +
m x^ x^ * * wiajja:^
351. If a;^ x^ be the roots of the equation
(l-m')(l-g') 47wa; (1 4- m*) (1 -f g^)
b — c c — a a-b '
6-c c-a a-6
352. If the quantities a;, y, « be all unequal and satisfy the
equations
a(y's? + 1) + y* + «• _ a{ii^aif + 1)4-2;' -ha;* _a{aifi/* + 1) + aj'y"
j/z zx ~' xy *
each member of the equations = a* - 1, and xyz (i/z + zx-^ xi/) = a;?/;:;.
353. Having given the equations
16 16 16
y«+ ax — = zx + av — =osi/+ az--:
" yz X zx *' y ^ xy z'
prove that, if a;, y, « be all unequal, a6 = 1, and each member of these
equations = 0.
354. Having given the equations
3/* + «* + ay8 = 25* + a* + aaa;=a:'+ y* + oa^,
prove that, if a;, y, « be all unequal,
a=l, and x + y+z = 0.
355. The system
(a'-a^)(b' + yz) = {a'^y'){b'-i-zx) = {a^-s^)(b'+xy)
is poristic if 6* = a" ; in which case each member will ho equal to
o' ^jz(y-\-z)(z-^x){x + y)
(x-^y-^zy
ALGEBRA. 63
356. Prove that the Bystem of equations
a; (a - y) = y (a - «) = 5 (a - ar) = 6*
can only be satisfied ilx=y = z; unless b* = a\ in whicli case the equations
are not independent
357. Prove that the system of equations
u {2a - a?) = a; (2a - y) - y (2a - s) = « (2a - w) = 6*
can only he satisfied if tt --= a; = y ^ c ; unless a* = 2b', in whicli case the
equations are not iudci>endent.
358. Prove that the system of equations
«i (1 - «•) = «• (1 - «^.) = • •• = ^. (1 - ^.* ,) = -^.^i (1 - --^i) = w
can only be satisfied if «, =^g = ^«,-^ ... ■ a?,^i» unless u bo a root of
the equation (\ -k-Jl-^u)'*' ={l -JiTiuy*' different from J, in
which case the equations are not indeiK>ndent.
py putting 1 - 4m = - tan" tf, it will ap|>ear that tlie roots of tlie
anxuiazy equation arc
359. Prove that the system of equations
a h a b
1 - ajj x^ I -x^ x^
can only be satisfied if x^ = a;,, unless r - a + A, in which case the equa-
tions are not independent.
360. Prove that the system of equations
a b a b a b
1-a?. a:. 1-a?, a;, 1-a?, a:,
18 poristic if (a-f 6 - c)* = a6, and the system
abababab
1-a:, a?, 1-^a;, a;, 1-aj, a;/l-a!^ a?,
if (a + 6-c)« = 2aA.
361. In general the system of equations
a b a b a b a b
+ - = - + -.= ..,r_ + = - +— =C
1-aj, aj, l^a5, a:, - l^aj, a!,^. l-a>.,, a?,
is poristic if ^ — = 0, where a, j8 are tlie roots of the quadratio
a:* + (a + 6 - c) a; + ai = 0.
64 ALGEBRA.
XVI. Properties of Numbers,
362. If n be a positive whole number, prove that
(1
(2
(3
(*
(5
(6
(7
(8
(9
2*" + 15w +1 is divisible by 9,
(2n+l)*-2n-l 240,
3«-+*-.8n-9 64,
3*'+* + 4aw-27 64,
3*'+* + 160n*-56n-243 512,
3«"*» + 2-+* 7,
3'"^* + 2*'+» 11,
3**^* + 4*'*' 17^
3.5«"^* + 2*'^» 17.
363. If 2p + l be a' prime number, (|^)* + (-!)' will be divisible
by 2p+l.
364. It p he A prime number, p-,C', + (- 1)*~* ^1 be divisible
hjp.
365. If j9 be a prime number > 3, , _,C, - 1 will be divisible by p\
366. If n - 1 and n + 1 be both prime numbers > 5, n must be of
one of the forms 30^, or 30^<fel2, and n*(n*+16) will be divisible
by 720.
367. If n - 2 and n + 2 be both prime numbers > 5^ n must be of
one of the forms 30^+15 or 30^ <fe 9.
368. Prove that there are never more than two proper solutions of
the question '* to find a number- which exceeds p times the integral part
of its square root by q'^ ; that if ^^ be any number between rp + f^ and
(r+l)/? + r* the two numbers p{p + r)-\-q and p{p + r-l) + q are
solutions ; but if q' be any number from rp + (r - 1)* to ry + r^ there is
only the single solution p{p + r''l) + q,
369. If n be a whole number, n + 1 and n' - n + 1 cannot both be
square numbers.
370. The whole number next greater than (3 + Jd)* is divisible
by 2-.
371. The integral part of -7s(^/3 + ^/5)•"*^ and the integer
next greater than (^3 + i^S)*", are each divisible by 2"**.
ALGEBRA. 65
>72. If n, r be whole nambern, the integer next greater than
(jjn-hl'^jn^iy is divisible by 2'**; as is also the integer next
than -J — r= (Jn-k- 1 + Jn - I)*'*' and the integer next lees than
^/n-¥ 1
373. The equation a^ - 2^ = <fe I cannot l)o satisfied by any integral
▼alaea of x and y different from unity.
374. The sum of the squares of all the numbers less than a given
namber y and prime to it is
TO-s)0-F)0-c-)--^f<^-«)<'-*)<i-«'> •'
the sum of the cubes is
and the sum of the fourth powers is
-^(l-«')(l-6')(l-c')... ;
where o^ 6, c, . . . are the different prime factors of N,
375. The product of any r consecutive terms of the series
is completely divisible by the product of the first r terms.
XVII. PrchabUities.
376. A and B throw for a certain stake, each one throw with one
die ; A'% die is marked 2, 3, 4, 5, 6, 7 and J^'s 1, 2, 3, 4, 5, 6 ; and equal
throws divide the stake : prove that J's expectation is i\ of the stake.
What will .4*s expectation be if equal throws po for nothing f
[l\ of the stake.]
377. A certain sum of money is to be given to the one of three
persons A, B, C who first throws 10 with three dice ; supposing them
to throw in the order named until the event happen, prove that A'b
chance of winning is (j^ , ^s ^ , and (7*8 (j^ .
378. Ten persons each write down one of the digits 0, 1, 2, ... 9
at random ; find the probability of all ten digits being written.
W. p. 5
66 ALGEBRA.
379. A throws a pair of dice each of wMch is a cnbe; B throws a
pair one of which is a regular tetrahedron and the other a regular octa-
hedron whose faces are marked from 1 to 4 and from 1 to 8 respectively ;
which throw is likely to be the higher) (The number on the lowest
face is taken in the case of the tetrahedron.) If A throws 6, what is the
chance that B will throw higher 1
[The chances are even in the first case ; in the second B\ chance
380. A^ B^C throw three dice for a prize, the highest throw win-
ning and equal highest throws continuing the trial : at the first throw A
throws 13y prove that his chance of the prize is '623864 nearly.
381. The sum of two positive quantities is known, prove that it is
an even chance that their product will be not less than three-fourths of
their greatest possible product.
382. Two points are taken at random on a given straight line
of length a: prove that the probability of their distance exceeding a
given length c (< a) is f j ,
383. Three points are taken at random on the circumference of a
circle : the probability of their lying on the same semicircle is f .
384. If q things be distributed among p persons, the chance that
every one of the persons will have at least one is the coefficient of 7? in
■
the expansion of [g (t' - I)'.
385. If a rod be marked at random in n points and divided at
those points, the chance that none of the parts shall be greater than - th
of the rod is -- .
386. If a rod be marked at random in /? - 1 points and divided at
those }X)int8, prove that (1) the chance that none of the parts shall be
< — th the whole is ( 1 — - ) i (wi > 2>) : (2) the chance that none of
the parts shall be > - th the whole is
to T terms where r is the integer next greater than /? - n, (n <;>) ; or the
equivalent
'-('-r*':V-'('-r'--
to V tenm where r is the integer next greater than tu Also (3) the
ALGEBRA* 67
chance that none of the parts shall be < — th and none greater than
- th of the whole is
" (> - s"' -(■ -'-^ - :r ^ ' V' (' -'-^' - =)"' -•
to r terms where r is the integer next greater tlian n \ provided
that - + ^~ - > 1. If - + ^ < If and none of the parts be > - th
the whole, it follows that none can be < - th the whole, so that the
m
case is then reduced to (2).
387. At an examination each candidate is distinguished by an
index number ; there are h Huccessful candidates, and the highest index
number is m-^n: prove that the chance that the number of candidates
exceeded m + n + r-l is
Im + r Iw + ;i - 1
\m ?/*+?< + r — I *
[It is assumed tliat all num1>ers are a priori equally likely.
388. There are 2m black balls and m white balls, from which six
balls are drawn at random ; prove that when m is very large the chance
of drawing four white and two black is ^r^^y and the chance of drawing
two white and four black is „**/ j.
389. If n whole numl)er8 taken at random be multiplied together,
the chance of the digit in the units' place of the product being 1, 3, 7,
or ^ is (I)", and the chances of the several digits are equal ; the chimce
4" — 2"
of its being 2, 4, 6, or 8 is — ^- — , and the chances are equal; the
5" - 4"
chance of its being 5 is tTuT" t and of its being is
10" -■ 8" - y^j^
10- •
390. If ten things be distributed among three persons the chance of
a particular person having more than five of them is yVvV^' '^^ ^^ ^
having five at least is ^VdVs*
391. If on a straight line of length a 4- 6 be measured at random
two lengths a, 6, the prolmbility that the common part of these lengths
c*
shall not exceed c is . , (c < a or 6) : and the probability of the smaller
b lying entii-ely within the larger a is .
5—2
08 ALGEBRA.
392. If on a straight line of length a + 6 + c be measured at random
two lengths a, b, the chance of their having a common part not greater
than d is 7^ v-7 tx » Id <a or b): the chance of their not having a
(c + a) (c + 6) ' ^ '^ ^
common part greater than d is , — . , ' ,v ; and the chance of the
^ ® (c + o)(c+6)'
smaller 6 lying altogether within the larger a is .
393. There are m+p-¥q coins in a bag each of which is equally
likely to be a shilling or a sovereign ', p + g being drawn p are shillings
and q sovereigns : prove that the value of the expectation of the remain-
ing sovereigns in the bag is — — — -^X. If m = 6, /? = 2, g' = 1, find the
chance that if two more coins be drawn they will be a shilling and a
sovereign, (1) when the coins previously drawn are not replaced, (2)
when they are replaced.
pn case (1) |, in case (2) ||.]
394. From an unknown number of balls each equally likely to be
white or black three are drawn of which two are white and one black :
if five more balls be drawn the chances of drawing five white, four
white and one black, three white and two black, and so on, are as
7 : 10 : 10 : 8 : 5 : 2.
395. A bag contains ten balls each equally likely to be white or
black ; three balls being drawn turn out two white and one black ;
these are replaced and five balls are then drawn, two white and three
black : prove that the chance of a draw from the remaining five giving
a white ball is -^g^.
396. From a very lai*ge number of balls each equally likely to bo
white and black a ball is c&awn and replaced p times, and each drawing
gives a white ball : prove that the chance of drawing a white ball at the
p+l
next draw is
p + 2'
397. A bag contains four white and four black balls ; from these
four are drawn at random and placed in another bag ; three draws are
made from the latter, the ball being replaced after each draw, and each
draw gives a white hall : prove that the chance of the next draw giving
a black ball is 33.
398. From an unknown number of balls each equally likely to be
white or black a ball is drawn and turns out to be white ; this is not
replaced and 2n more balls are drawn : prove that the chance that in
the 2n + 1 bcJls there are more white than black is , -, . If the first
4/1 + 2
draw be of three balls and they turn out two white and one black and
2n more balls be then drawn from the remainder, the chance that the
ALGEBRA. 69
nuyority of tlie 2h + 3 bolL* are white in --— f—rr ;rr : aud the
that in the 2ra balLs there are more white than black is
lln* + 13/i
4(2/4+1)(2h + 3)'
399. From a Ui^ number of balls eaoh equally likely to be white
or Uadc p + g being drawn turn out tohe p white and q black : prove
that if it in an even chance that on three more balla being drawn two
will be white aud one black
^^-1 + ^2
nearly, p and q being both large.
400. A bag contaiiiH m white balls and n black balls and from it
balls are drawn one by one until a white ball is drawn ; A bets B at
each draw x : y that a black ball is di-awn ; prove that the value of A*b
expectation at the beginning of the drawing ia — -^ — x. If balls be
drawn one by one so long as all drawn arc of the same colour, and if for
a sequence of r white balls A is to ])ay B rx£^ but for a sequence of r
black balls J9 is to ]>ay A ry£, the value of ^'s expectation will be
— ^ ;, : and if A i>ay B x for the first wliite ball drawn, rx for
m+1 M+1 * "^ ^
r(r+l)
the second, x for the third, and so on, and B jiay A y for the
r(r+l)
first black ball dra>»^ ry for the second, —^^ — y for the third, and
on, the value of A^^ expectation at tlie beginning of a drawing will be
|m + ?i + r- 1 |m w
-^^ ^ {n(/4 + 1) ... (n + r)y-m(m+l) ... {m-^rSx}.
m-¥n \m + r \n -f r * ^ ' ^ ' ^ ^
80
401. From an unknown number of balls each eqiudly likely to be
red, white, or blue, ten are drawn and turn out to be five red, three
white, and two blue ; prove that if three more balls be drawn the chance
of their Ijeing one red, one white, and one blue is ^{^ ; the chance of
three red is ^^ \ of three white is ^^ \ and of three blue is /^ ; and the
chance of tliere being no w^liite ball m the throe is J J.
PLANE TRIGONOMETRY.
L Equations,
[In ihe solution of Trigonometrical Equations, it must be re-
membered that when an equation has been reduced to the forms
(1) sin a; = sin a, (2) cosa^^cosa, (3) tana: = tan a, the solutions are
respectiyely (1) a; = n» + (— l)*o, (2) a? = 27i» ± o, (3)a; = n» + a, where
n denotes a positive or negative integer.
The formube most useful in Trigonometrical reductions are
2 sin ^ cos ^ = sin (il + J?) + sin (^ — B)^
2 cos A cos B = cos (^ - jB) + cos (A + B),
2 sin ii sin ^ = cos (A—B)- cos (A + B),
and (which are really the same with a different notation)
. • T> « • A + B A — B
Bin A +8m^= 28m— ^ — cos —jz — ,
cos ^ + COS J0 = 2 cos -^ COS — - — ,
A n ^ e% ' ^~A , A + B
COS il - COS iff = 2 sin —^ — sm — - — ;
which enable us to transform products of Trigonometrical functions
(sines or cosines) into sums of such functions or conversely sums into
products. Thus to transform
sin2(^-C) + sin2(C--4) + Bin2(^-jB).
We have
sin2(C-.-4) + sin2(^-J5) = 2sin(C-J5)cos(i5 + (7-2ii)
and sin2(i5-C) = 2sin(i5-C)co8(5-C),
whence the sum of the three
= 2sin(i5-C){co8(5-C)-co8(5+C-2ii)},
= -4 sin(^ - C)Bin (C- i4)sin (.1 - jB).
PLAXE TRIQONOMETRT.
71
Again, to transform
cos (j5 - C) cos (C - -4 ) cos (-4 - j5),
we have
2cob{C'-A)cob(A-'B) = cob(C'-B) + cos(B + C''2A),
whence
4co8(i?-C)co8((7--4)cofl(-4-5)=l + cos2(i5-.C) + coe2(C--4)
+ cos 2 (^ - J?).]
402. Solve the equations
2 sin a; sin 3x= 1,
cos X cos 3x = cos 2x cos 6x,
sin 5x cos 3a: = sin 9a; cos 7a?,
sin 9a; + sin 5a; + 2 sin* ac = 1,
cos mx cos TUB = cos (m 4-/?) a; cos (n — />) oe,
sin vix sin no; = cos (m +/>) a; cos (n + j?) x^
tan* 2a; + tan* a; =10,
cos a; + cos (a; - a) = cos (a; - j8) + COB (a; + /3 - a),
2 sin* 2a; cos 2a; s sin* 3x,
2 cot 2a; - tan 2a; = 3 cot 3a5,
ft V^ ^ 1
8 cos a; = . - + .
sm a; cos x '
sin 2a; + cos 2a; + sin x - cos a; = 0,
(1 +sina;)(l - 2 sin a;)* =(1 -cos a) (1 + 2co«o)*,
sin a cos (fi + x) _ tan p
sin P cos (a + x) tan a '
cos 2a; + 2 cos a; cos a - 2 cos 2a = 1,
sin a cos 3a; — 3 sin 3a cos a; + sin 4a + 2 sin 2a = 0,
cos* a sin* a ,
— + -. — = 1,
cos X sm X
(cos 2x - 4 cos a; - 6)* = 3 (sin 2a; + 4 sin x)\
(cos 5a; -10 cos 3a; + 10co8a;)*=3(sin6a5- lOsina;)*.
403. If
and
then will
cos (x + 3i/) = sin (2a; + 2y),
sin (3a; + y) = cos (2a; + 2y) ;
a;=(5m-3n)|+^
y-(5»-3m)^+-^j
*; or a;-y = 2nr+H,
wi, «, r being integers.
72 PLANE TRIGONOMETBT.
06
404. The real roots of the equation tan' x tan ^ =: 1 satisfy the
equation cos 2aj = 2 - ^5.
3/3
405. Given cos Sa; = - j^ , prove that the three values of cos a; are
72
3s;
10'
/3 . IT /3 , «• /3 .
Vz'^IO' V2^6' -V2«?^
406. If the equation tan^r = 7 =- have real roots, a* > 1.
^ 2 tana; + a+l '
407. Find the limits of - — 7-^ — I for possible values of x.
tan (oj - a) '^
Ttx X t V X 1 -sin 2a , 1 + sin 2a]
It cannot lie between , ; — ^r- and , ; — — - .
L l+sm2a l-sin 2a J
408. The ambiguities in the equations
cos ^ + sin^=«fcyi +sin-4, cos ^r -sin -^ = ±,71 -sin -4,
may be replaced by (- 1)"*, (- 1)", where w, n denote the greatest
. ^ + 90* ^ + 270' ^. ,
integers in ^^y > "360^~ respectively,
409. The solutions of the equation sec* a; + sec* 2a; = 12 are
V 27r
x = nv^-^f a: = nir«fc-r-, 05 = j^cos"* (-^),
410. The roots of the equation in ^, tan— 5- tan fl = m, are all
roots of the equation
sinasin0{l-cos(a + 0)} + m'cosaoos0{l + cos(a + 0)} = m sin* (a + $).
411. The equations
tan(tf + j8)tan(tf + y) + tan(tf + y)tan(tf + a) + tan(tf + a)tan(tf + )8) = -3,
cot {6 + p) cot (^ + y) + cot {6 + y) cot {0 + a) + cot (d + a) cot (tf + /3) = - 3,
will be satisfied for all values of if they are satisfied hy 6=0,
A^n TU COS (a + tf ) COS (/3 + tf ) COB (y + tf ) ^ , . ,
412. If ^-j — ' = — /, ^ ' = — ^^ — -f a, ft y being unequal
sin* a sin* fi sin* y ' "^' ' ^ ^
and less than ir, then will a + j3 + y = ir, and
^ 3-cos2a-co82i8-cos2y 1 + cos a cos )8 cos y
tWlC' ; — jr ; pr-TT ; x = : : 5 — ; •
sin 2a + sm 2/3 + Sin 2y sin a sin p sin y
413. Ifa + j8 + y = » and be an angle determined by the equation
sin (a - tf) sin (p-O) sin (y - tf) = sin* tf,
then will
sin (g - 6) _ sin {^-0) _ 8in(y-tf) _ sin tf cos tf
sin* a 8in*/3 ~ sin'y "~ sin a sin j8 sin y l+cosacosjSoosy '
PLANE TRIQONOMETRY. 73
[These equatioim occur when a, j8, y are the angles of a triangle ABC
and is a point such that i OBC = t OCA = i OAB = e.]
414. If a, )9, y, 8 be the four roots of the equation
sin 20 - wi cos0~nsin0 + r = Oy
then will
a + j8 + y + 8 = (2/1 + 1) a- (/> integral);
also
8ina4-sinj8 + siny + 8in8=sf7i, cos a + cos )9 + cos y + cos 8 => it,
sin 2a + sin 2j3 + sin 2y + sin 28 ^ 2mn - 4r, cos 2a + cos 2/3 + cos 2y
+ COS 28 = ?*• - m*.
415. If two roots 0,, ^, of the equation
cos(tf-o)-ccos(2tf-o) = w(l -tfcostf)'
6 9 /l— s
satisfy the equation tan » tan J- a/ = — , then will me sin' a = cos a.
416. Find x and y from the equations
x(l +sin*tf-co8tf)-ysintf (l + costf)=c(l +cos0),
y (1 4- cos* 6)-xRUi6 cos = c sin ;
also eliminate from the two equations.
1 The results are 07 = c cot' ^, y = <5co<iQi y*=cx.
417. The equation
2 8in(a-fff4-y-g) + 8in2g-sin(ff4-y)-Bin(y-f a)-sin(a4'ff )
2 cos (a + /9 + y - ^) + cos 2^ - cos (/? + y) - cos (y + a) - COB (a + ^)
2sin2g-f ain(a-f)3-fy-g)~sin(a + g)-8in(ff + g)~sin(y + g )
"2cO82^ + COs(a + j3 + y-6')-CO8(a + tf)-CO8(j3 + ^)-C0B(y + tf)
is satisfied 'd O^^a, P, or y, or if
$^a 0-B O-y ^
cot —^r— + cot — 7p- + cot -Q-^ = 0.
J J i2
418. The equation
2sin( a + /3 + y-Hg)-fsing-6in(ff + y-f g)--sin(y + a + tf)-sin(a + ff + tf)
2cos(a + /3 + y + ^) + cosd-co8(j3 + y+6l)-cos(y + a + 6^)-co8(a + /3+6l)
_ 2 sin tf + sin ( a + ff 4- y -f g) - sin (a -h tf ) - sin (/3 4- tf ) - sin (y + 0)
""2cos6 + co8(a4-/3 + y + tf)-C0B(a + ^)-cos()8 + tf)-C0B(y + 6')
is independent of $ ; and equivalent to
sm ^ sm ^ sin ' ' -"^^ * ^'^^
nn^^cot^ + cot| + cot|jaO.
74 PLANE TRIGONOMETRY.
419. Having given the equations
x + y cos c + z cos h _y -^-z cos a + a5 cos c _^z-vx cos 6 + y cos « _ «
cos (« - a) ~ cos {a — 6) cos (« - c)
where 2# = a + 6 + c j; prove that
a; y 2; m
sin a sin 6 sine sins'
420. If a, j8, y, 8 be the four roots of the equation
acos2tf + 6sin2tf-ccostf-cf8intf + « = 0,
and 2d=a + j8 + y + 8, then will
ah c
cos 8 sin 8 cos (« - a) + cos (« - )3) + cos (* - y) + cos (• - 8)
d
"" sin (« - a) + sin (« - j3) + sin (« — y) + sin (• — 8)
e
"" cos (« - a - 8) + cos (« — /3 - 8) + cos (« — y - 8) *
421. Reduce to the simplest forms
( 1 ) (a; cos 2a + y sin 2a - 1 ) (a; cos 2)9 + y sin 2)3 - 1 )
- {x cos a + )8 + ysina + )3- cos a - )8}*,
(2) (a:coso+)3+y sina+)3-cosa-)3) (arcosy +S+y sin y +8- cos y -8)
-(a;cosa + y + ysina + y- cos a - y) (a; cos )3 + 8 + ysin)3 + 8 — cos )3 - 8).
[(1) (a:« + y«-l)8in'(a-)3), (2) (a^' + y'- l)sin(J8-y)8in(a^8).]
422. If )9, y be different values of x given by the equation
sin (a + a;) = m sin 2a
cos ^-5-^ ± m sin ()8 + y) = 0.
423. The real values of x which satisfy the equation
8in(^cosa;j = co8(^sina;J ore 2nir or 2nv * « > ** being integral
424. If 0?, y be real and if
sin'a: sin*y + sin' (x + y) = (sin x + sin y)*,
a; or y must be a multiple of v. ,
[The equation is satisfied if sin a; = 0^ sin y = 0, or
cos i^'\-y)'\- cos X cos y = 2 :
and this last can only be satisfied if
co6a;BC06y = ifcl,and co8(a: + y) = l.]
PLANE TBIGONOMETRT. 75
425. li a, P, y he three angles unequal and less than 2w which
itisfy the equation
a h ^
COBX BUIX
then will sin (jS + y) + sin (y + a) + sin (a + /3) = 0.
426. If j9, y be angles unequal and less than w which satisfy the
equation
cos a cos X sin a sin x 1
— + ==_
a be
then will
(6* + c* - a') cos j3 cos y + (c* + o' - 6") sin /3 sin y = a' + 6* - c*.
427. If a, /3 be angles unequal and less than w which satisfy tlie
equation
a cos 2x+b sin 2jr = 1,
and if {I co8»2a + m sin'2a) (l co«» 2/3 + m sin'2^)
= {/ cos* {a + 13) + m sin* (a + )8)}',
then will either / = m, or a* - 6* = -; .
m+l
428. If a, )9, y, S be angles unequal and less than ir, and if /3, y be
roots of the equation
a cos 2a; + 6 sin 2a; = 1,
and dy 8 roots of the equation
a' cos 2a; + 6' cos 2a; = 1 ;
and if {/cos' (a + )3) + m sin* (a +)3)} {/cos* (y + 8) + m sin* (y -i- 8)}
= {/cos* (a + y) + w sin* (a + y)} {/ cos* (j9 + 8) + m sin* ()Gr + 8)},
then will either / - m, or oo' - 66' = ^ .
m + l
II. IderUiliea and EqttalUies.
429. If tan'4 = 1 + 2 tan*^, then will co82i?= 1 +2co8 2i.
430. Having given that sin {B+C-A)^ sin (C+A-B), sin (A +B^C)
are in a. p. ; prove that tan Ay tan B, tan C are in a. p.
431. Having given that
1 + cos ()3 - y) + cos (y - a) + cos (a- j3) = ;
prove that /3-y, y-a, or a-j3 is an odd multiple of ir.
432. If cota, cotjS, coty be in A.P. so also will cot(/3-a), cot)8,
cot (j3 - y) ; and
— St. — V — y a - i — ^ — — \ be respectivelT in A.P.
sina ' 8inj3 ' any ' '^ ''
76 PLANE TRIGONOMETRY.
433. Having given
cos fl = COB a cos P, cos ff = cos a' cos j8, tan ^ tan ^ = tan ^ ;
prove that sin* /3 = (sec a - 1 ) (sec a - 1).
434. Iftan2a = 2 , f/^ ^ , and tan 2)8 = 2 , ^"'^ ,,
then will tan (a- fl) be equal to r or to -^^,.
, c-b a + d
435. If a, )3, y be all imeqnal and less than 2ir, and if
cos a + cos j3 + cos y = sin a + sin j3 + sin y « ;
then will cos'a + cos*j3 + cos'y = sin'a + sin'/S + sin'y = f ;
cos (j8 + y) + cos(y + a) + cos (a + j3) = sin (j3 + y) + ... = ;
cos 2a + cos 2)3 + cos 2y = sin 2a + sin 2)3 + sin 2y = ;
and generally if n be a whole number not divisible hj 3,
cos na + cos n)3 + cos ny = sin wa + sin nfi + sin ny = 0.
436. Prove that
a
(l - tan'0 (l-tan'J.) (l-tan«J,) ... to oo =
tana
437. If C=2cosfl-5cos'e + 4cos''tf, ^=2sine-5sin»e + 4sin*tf;
then will
(7 cos 30 + ^sin 30= cos 20, and (7 sin 30- S' cos 30= sin cos 0.
438. Having given
a;cos^ + y8in^sa;co6^' + ^sin^' = 2a| 2cos^oos^ = l;
prove that y* = 4a (a + x) ^ ^, ^' being unequal and less than 2ir.
439. Prove that
1 1 a + 6
a QOA*6 + 2A sin cos + 6 6in'0 a cos' ^ + 2h sin ^ cos ^ -i- 6 sin'^ ~~ ah-h*
if a -f A (tan + tan <^) + 6 tan tan ^ = 0,
= 2^^^J5(l+tan0tan<^).
or
440. Having given the equations
t*- 1 1 +2cco8)3 + 6'
l + 2ecoBa + e''~ 6*-l '
prove that each member = — = * -; : and that
* e + cos a sin a
^ a^ 3 1+6
tan Titan^ = * , .
2 2 1 -e
PLANE TRIGONOMETRT. 77
441. Having given the equations
co8a + coB)3 + co8y + co8(^ + y-a) + C08(y + o-j3) + C08(a + j3-y) = 0,
ftin a + sin j3 + sin y + sin (^ + y - a) + sin (y + a - ^) + sin (a + )8— y) = ;
prove that
cos a + cos P + cos y = sin a -f sin j8 + sin y = 0.
442. Having given the equations
sin {P^ +y ) + sin (y + a') + sin (a' + /?') - sin (^ + y) - sin (y +a)^in (a+^)
cosa'+co8)3'+co8y' — cosa— cos j8— cosy
sina' + sin/^ + siny' — sina — sin/^-siny '
and a' + /8' + y' = a+)8 + y ;
prove that either
a' )3' y' a 6 y
cos — cos — cos jr = cos cos ^ cos f: ,
Z Z ^ M z z
a . P , y , a , B , y
or Rin - sin ^ sin ^ =s sin - sin ^ sm ' .
z z z z z z
443. Eliminate $ from the equations
cos (a - 3$) sin (a - ZO)
cos'*^ cos'd
[Tlie resultant is wi* + m cos o - 2.]
444. li Xy y satisfy the equations
- . _ X cos 3tf + y sin 3tf _ y cos 3^ — a; sin 3^
^^~ ^'0 " ^?$ •
then will x' + y* + x = 2.
445. Having given the equations
T COS — -~ « COS d COS COS — ^ ,
z z
y cos —^ - sm fsintp sin ^ ,
z z
r^a cos 6 cos ^ 4* 6 sin ^ sin ^,
= 6c + ca + ai ;
prove that a V + 6y = c'.
Also eliminate 0, ^ from the first three equations when the fourth
equation does not hold«
[The resultant in general is 4a5*y' = «• (aj* + y* + «• - 1)*, where
^ ^ (6-^c)(c^a)(a^•6) | J :^ y|_ j ,
b€ + ca-^ab \a + b 6 + c e-^a) '
or
78 PLANET TRIGONOMETRY.
446. Having given
(x - a) cos 6 + yHiD.O= (x-a) cos d' + y sin ^ = a,
tan-- tan^ =2e;
prove that y' = 2aa; - ( 1 - c*) a:*,
0, ^ being unequal and less than 2^.
447. If (1 + sin ^) (1 + sin <^) (1 + sin i/f) = cos cos <f> cos ^,
then will each member = (1 - sin ^) (1 — sin ^) (1 - sin ij/), and
sec'tf + sec'<^ + sec*i^ - 2 sec tf sec <^ sec ^ = 1.
448. Eliminate a from the equations
cos sin
= 7n.
e + cos a sm a
[The resultant is 1 = 2e m cos ^ + r/i* (1 - c*).]
449. If tfj, 6^ be the roots of the equation in 0,
(a* - b') sin (a + fi) {a'cos a (cos a - cos ^) + 6* sin a (sin a - sin tf)}
= 2a'6'sin()3-^),
then will
,. a + ^. . a+tf, „ a + e. a + tf_ ^
a'sm — jr— * sm - ^ + 6 cos — jr— ^ cos — ^ = 0.
450. The rational equation equivalent to
(m'- 2m/i cos ^4 + n*)^ + (w" - %il cos ^ + ^^ + (^ - 2^m cos C + w")* = 0,
where ^ + ^ + C= 180", is
(mn sin -4 + 7i/ sin ^ + Im sin C)' = 4/m7» (/ cos -4 + m cos J5 + n cosC^.
451. Having given the equation
cos A = cos B cos C* sin -ff sin Ccos A,
prove that
cos -ff = cos C CO? -4 «*=.sin Csin A cos -ff,
cos C = cos ii cos B ife sin ^ sin B cos C7 ;
and that
sec*-4 + sec*5 + sec*(7 - 2 secii sec -ff sec C= 1,
tan (^5^* ^) tan (45" ±1) tan (45"± ^ = 1.
452. If
tanM tan ^' = tan'^ tan jB' = tan*Ctan C = tan ii tan^tan C,
and cosec 2-4 + cosec 2B + cosec 2C = ;
then will
tan(.4-^') = tan(^-i5^=tan(C-CO = tan^ + tan^ + tajiC7.
PLANE TRIQONOMETRT. 79
453. Having given
a: sin 3 (^ - y) + y sin 3 (y - a) + 2J sin 3 (a - /3) = ;
prove that
X sin (i3 - y) + y sin (y - a) + s sin (a - P)
X cos Ifi-y) + ^ cos (y - a) + « cos (a - fij
sin 2 ()3--y) + sin 2 (y - a) -f sin 2 (a - )3) ^
cos 2 ()3 - y) + cos 2 (y - a) + CDS 2 (a - /3) ~
454. Having given the eq^uations
a;«^^* + y»-2/?ycos^ \
. I a: + y + 2 = )
j^^y + a'-2yaC0S<^ V, ^ ^ , n >
^•=a'+j5*-2a)3cos^J ^ V Y }
prove that /3y sin ^ + ya sin ^ + a/3 sin ^ = 0.
455. Reduce to its simplest form the equation
{x cos (a + j8) + y sin (a + fi) - cos (a - )8)}
{x cos (y + 8) + y sin (y + 8) - cos (y - 8)}
= {x cos (a + y) + y sin (a + y) - cos (a - y)}
{a; cos (13 +S) +y sin (^ + 8) -cos (j9- 8)}.
[The reduced equation is sin (^ - y) sin (a - 8) (1 - a:* - y*) = 0.]
456. Having given the equations
yz' - i/'z + zx'- z'x + a-y - a'y = 0, il + J5 + C --= 1 80',
TKx sin'il + yy sin'^ + zz' 8in*C = (yz' + %/z) sin ^ sin C cos ^
+ {zx + «'j;) sin (7 sin -4 cos -5 + (jy + a/y) sin il sin ^ cos (7 ;
prove that either a; = y ^^ s ; or a/ == y - s'.
457. Having given the equations
(yz - y';:) sin A + (sa;' - 2'a;) sin -5 + (x/ - a;'y) sin C = 0,
xsf -^yy' + «-' - (y^' + y -) cos -^ + (^^a;' + zx) cos J5 + (ay + a/y) cos C,
xsin (S-A) + ysin (*S'- 5) + S8in (^- C) -
= x' sin (.S^ - il) + y sin (6'- 2?) + s' sin (^- C),
where 2^=^ + ^ + (7;
prove that
^ _ y _- ^ . a:' _ y 2'
sin .4 sin B sin 6' ' sin A sin ^ sui (7 *
[It is assumed that cos S is not zero.]
458. Having 'given the equations
y* + s* - 2y3 cos a 2' + «• - 2«aj cos fi o^ -¥ jf - 2xy cos y
sin'a sin'^ " sin'y '
80 PLANE TRIOONOMETRT.
prove that, if 28 denote a + )3 + y, one of the following systems of equa-
tions will hold :—
X y z
cos (8 - o) cos (« - fi) cos (« - y) '
X y z
COB 8 cos (« - y) cos (« -p) '
X y z
cos (a - y) cos 8 cos (« - a) '
a; ^2
cos (« - ^) "" cos (8 - a) cos 8 '
459. Having given the equation
cos a sin a
+ = — 1 *
cos sin '
^, ^ cob"^ sin'fl ,
prove that + -: = 1.
cos a sin a
4 GO. Eliminate 6, ^ from the equations
X V X t/
-cos(? + fsinfl=-cos^+ rsin^ = 1,
a a ^
^ — ^ a — O a — ^ _
4 cos —^ cos — jr- cos Q = 1.
z ^ A
[The resultant equation is (--cosoj +r^-8ina) =3.]
461. Having given the equations
a* + 6*- 2a6cosa = c*+ c^- 2c<i cosy,
6* + c*-26ccosj3 = a* + (?-2(K£cos8,
a5sina + ccf8iny = &esin)9 + a(fsin8;
prove that cos (a + y) = cos (j3 + 8).
rrhese are the equations connecting the sides and angles of a
quaorangle.]
462. Having given the equations
sinfl + sin^=a, cos0 + cos^=6;
prove that
(1) ten| + tan|=^,^^j,
(2) tan6l + tan^=^^f^-^-^.,
(3) co8gcos^= ^ 4(aV6') " '
PLANE TRIOONOmTRr. 81
(4) ^o«^^=y f
(«' + ft*)
(5)coe2g.coB2^=<^'-'*'>/^Y/'-^>.
^ a" -I- 6"
(6) co8 3^-».co8 3<^=6»-.3a*6-36-»-4-?..
a" + 6"
463. Eliminate from the equations
xcos^vsintf, .^ ^ / , . ,^ — n y^
[The resultant equation i8- + ^=a + 6, provided that a ain* 9 + hcaifB
does not vanish, in which case the two equations coincida]
464. Prove that
cos* + cob' (a + ^) - 2 008 a cob 0008 (a + ^)
is independent of 0,
[It is always equal to sin' a.]
465. Prove that
sin 2a cos /3 cos y sin ()3 - y) + sin 2)3 COB y 008 a sin (y - a)
+ sin 2yoosaooB/38in(a~/3)E 0,
cos2acos)3co6ysin()3-y) + ... + ...
= sin ()8 - y ) sin (y - «) sin ( a - /8).
466. Prove that
„ . 30 r . , 50 . , 30)
28in^|sm y-am'-^l
= cos* + cos* 20 + oob' 30- 3ooB0 ooB 20008 30 ;
A o • 30 f. .30 ,50)
and 2 sin -^ •! cos* -^ - cos* -^ >
= sin' + sin' 20 + sin' 30 - 3 sin sin 20 sin 30.
467. Prove that
.- . sin 2a sin ()3 - y) + sin 2)3 sin (y - a) -»- sin 2y sin (a - /9)
^ ' sin (y - ^) -I- sin (a - y) -I- sin 03 - o)
= sin ()3 + y) + sin (y + a) + sin (a + )8) ;
.^. cos 2a sin ()3 - y) + cos 2)3 si n (y - a) + cos 2y sin (a - )3)
^ ^ sin (y - )8) + sin (a - y) + sin ()3 - a)
= cos ()3 + y) + 008 (y + a) + cos (a + )3) ;
.^. sin 4a sin ()3 - y) -I- sin 4)3 sin (y ~ a) -I- sin 4y sin (a - )3)
^ ^ sin(y-)3) + sin (a-y) +8in()3-o)
= 2S8in(2a + /3 + y) + S8in2 03 + y)-i-San(3)3 + y);
w. p. 6
82 PLANE TRIGONOMETRY.
... 008 4a,sin ()3 - y) + cos 4/3 ain (y — a) + cos 4y sin (a - j8)
sin (y - /3) + sin (a - y) + sin (^ - a)
= 2Soos(2a + ^+y) + Scos2(/8 + y) + Scos(3i94y).
468. Prove that
... sin 3a sin ()8 - y) + two similar terms , , ^ ^
i^) 5 — ^—7^ — -{ — I =— n — I = tan(a + /8+y),
' cos 3a sin (p - y) + two similar terms \ i- f/^
.^. sin 5a sin ()3~y) -I- ... -I- ... _ sin (3a -f- )3 -f y) + ... + ...
^ C085asin(/8-y) + ... + ..." cos(3a + ^ + y) + ... + ... '
.^. sin 7a sin ()3 - y) +... + ... _ sin (a + 3)3 + 3y) + sin (5a + /3 4- y) + . . .
^ ^ cos7o8in(j3-y) + ... + ... " co8(a + 3^ + 3y)+cos(6a + )3 + y)+ ... '
,,. sin' a sin (/3-y) + ... + ... _ . / /> \
469. Prove that, if n be any posilive whole number,
sin na sin (j3 - y) + sin n/3 sin (y — a) + sin ny sin (a - /3)
sin (y - /3) + sin (a - y) + sin (/8 - a)
= 2S sin (pa + ^'/S + rj3) ;
where p, q, r are three positive integers whose sum is n, no two vanisli-
ing together, and the coe£ELcient when one vanishes being 1 instead of 2.
V 4P 4P
[K we write a -i- ^ , )3 + o- > y + ^ f or a, )3, y we get a similar
equation with cosines instead of sines as the functions whose argument
is the sum of n angles.]
470. Prove that, if n be any odd positive whole number,
sin na sin ()3 - y) + ... + .. . ^ . , ^ .
. ^ '-^ = 2 sm (pa + gj9 + ry) ;
8m2(y-/3)+ ... + ... ^ ^'^ ''^
where je>, g, r are odd positive integers whose sum is n.
[The same remark as in the last.]
471. The resultant of the equations
ecos^ -i-y sin^ = a;coB^ + yain^= 1,
o cos ^ cos<^ + 6 sin^sin<^+c+/(cos tf +COS <^) + ^(sin d+sin<^) + A sin (^ + <^)
is (c-6)a*+(c-a)y» + a + 6 + 2/y + 25ra:+2Aa:ys=0.
472. Prove that
^ ^ 1 1 _Binntt-fj>8in(n~l)a
2coBa-2co6a~2oo6a- ...-2oosa+/?" 8in(n+ l)a+j9sinna '
there being n quotients in the left-hand member.
PLANE TRKaoNOicrntT. 83
473. Prove that
sin'— ^ am
aec'a sec'a sec'a nnna
—\ — -= — -. — to n quotients a :i — 7 sr -, .
4- 1-4- ^ am (n -I- 2) a -I- sin na
rif we call the dexter u , 1 - u . = ^ . — , whenoe the equa-
*• •» •-! 2 8mnaco8a ^
tion«.(l-«...) = j-L_ = !^-.]
474. Prove that, if sin a 4- sin /3 4- sin y = Oy
{P-y)^^(^-')^y rin(«>y)
475. From the identity
, (x- 6) («^) (g-c)(g-a ) . (a;- a)(a?-6) . .
(a-ftKa-c)"*"^ (6-c)(6-a) (c-a)(c-6)"" '
deduce the identities
oo82(o^a)^-;*:i|J4^;i::>> + ... + ...5oo.4*,
^ '8in(a-j9)8m(a-y)
. ^,: ,sin(^-)3)Bin(^-y) _ . .^
Bin 2 (^ + a) . -; — ^— .-> — v + ... + ... = sin 4^.
^ '8m(a-/3) sin (a-y)
476. Prove the identities
(1) cos2(i3 + y-o-8)8in03-y)sin(a-8)
+ cos2(y + a-/3-8)8in(y-a)sin()S-8)
+ oos2(o + /8-y-8)8in(o-^sin(y-8) S-SJT,
(2) cos3()3'»-y-a-8)8in()3-y)8in(a-8)+two similar terms=-16JrZy
(3) cos(/J + y-o-8)sin'()8-y)sin'(o-8) + ... + ... sJTX,
(4) cos05-y)cos(o-8)8in*()8-y)sin*(a-8) + . .. + ... ^KL,
(5) coB()9 + y-a-8)sin"08-y)sin»(o-S)+. .. + ... =-16A'Z;
where K denotes
sin(/J-y)sin(y-a)sin(a-)8)sin(a-8)sin(/8-8)sin(y-8);
and L denotes
oos()5 + y-a-8) + oos(y + a-)8-8)+cos(a + /8-y-8).
[The first is deduced from the identity ^6V + a*<i^ (6 - c) (a - rf)
4- two similar terms 2 (6-c) (c-a) (a-6) (rf-a) (J-6)(ii-c) by
putting cos 2a 4- f sin 2a lor a and the like: the same substitutions iu
other identities of (155) give (2), (3), (4), (5).]
f " " *
-u->fci:^u- r«c*i.— '.
li: - ^^
^atfbw.
-U-p*C*U- r«C.J.-t
j;^r r--is ^-^
"^^ /r ^ 1-r r--^ ^--*
^ ^ ^ *
>y 'it^.iz::'^:''^ v^ ^-^ ^r ~c=o, bat ud. k «!•«
^^ >#/ /r, /, iM4Fia.0^^^£BttrtetriMigle. It «jit be
PLANE TRIOONOMETBT. 85
482. Prove that
(3 + 2 COB 2^ + 3 cos 2i? + 3ooB 2C + oo« 2^- C)*
+ (3 sin 2i? - 3 sin 2(7+ sin 2i? - C)*
~ lb cos A 008 B GOB C - I).
483. Prove that
sinM (sin* ^ + sin* (7 -sin* -4) + ... + ... = 2 sin* il sin* ^ sin* C7
(l+4ooBiloo6^oosC7);
(sin il + sin-S+sinC^(-sini(+sinJ5 + sinC?) (sin il - sin ^ + sin C7)
(sin ii + sin ^ - sin (7)
= 4 sin* A sin* B sin* C.
484. Prove that
sin -4 (sin il - sin ^) (sin il - sin (7) + ... + ...
= sin il sin ^ sin C7 (3 - 2 cos A -2 cos £-^2 cos C);
sin* A (sin A - sin B) (sin ^4 - sin (7) + ... + ...
= (cos ^ + cos ^ + cos (7-1)' (cos* ^ + ...-co6^oos(7- ...);
sin* i? sin 2(7 - 2 sin ^ sin (7 sin (5 - (7) - sin* (7 sin 25 = ;
sin* j9 cos 2(7 - 2 sin i? sin (7 cos (5 - (7) + sin* (7 cos 2i? a sin' il.
III. Poriaiic SyaUms of Equatiom.
[In all the examples under this head, solutions arising from the
equality of any two angles are excluded, and all angles are supposed to
lie between and 2ir.
A system of n equations of which the type ia
acos(a -o^^,)+6coe(a +a,J + c + ^(sino^+sino^,)
+ 2g (cob a^ + cos a^^,) -)- 2A sin (a^ + o^^,) = 0,
where r has successively integral values from 1 to n, and a^^, = 0^, is a
poristic system, when solutions in which angles are equal are exduded :
that is, the equations cannot be satisfied unless a certain relation hold
connecting the coefficients <h\ ^^f 9yK <^^ ^ ^^ relation be satisfied
the number of solutions is infinite, the n equations being equivalent to
n - 1 independent equations only so that there is one solution for each
value of Oj. All the examples here given are reducible to this type.]
485. Having given
/« y + a a + iS
tan /Scot '—^ =tanycot —~,
iS + y
prove that each « tan a cot ^ ' ; and that
siu (ft + y) + sin (y + a) + sin (a + /9) 3 0.
86 PLANE TRIGONOMETRY.
486. Having given
a + ^ B-^O
tan — s— tanp=tan*— ^tana,
a-^ B
prove that each = - cot —^ tan $, and that
Bin (o + tf) + sin (/8 + ff) = sin (a + /8).
487. Having given the equations
tan^- — ^ tana=tan'^^ — ^ tan B = tan £ tanv- :
2 2s 2 ' n
prove that
sin08 + y) + Bin(y + o)+8in(a + ^=:0,
cos )3 cosy sini^siny 1
— ^ + — ' + = 0,
n m m-^-n
and that 0»ir.
488. Having gi^en the system of equationa
aoos/Scoay + ftsin^siny-a cob yoosa + ^sinynna
saco8acos/3 + 68ina8in)3=£Cy
prov^ that &c ->• oa + oi = Oy
and that the given STstem is equivalent to any one of the following
systems :
(1) tan^cota = tan^«cot^ = tan"-±^coty = ^;
P + y
COS^-jr-'
2 a
<2) ^= - %-'
COS a COB ^—5-'
(3>
2 b
' P-y
*
(4) cot|cot|+cot|cot|+cot|cot|
R y ,26
= tan^tan^ + ... + ... = 1 + — ;
(5) sin (j8 + y) + sin (y + a) + sin (a + )8) = 0,
a — h
coe08 + y) + cos(y + a) + cos(a + j8)= — r ,
l+oos(^-.y) + co«(y-a) + c08(a-^) = -^jj^,;
(6)
(7)
PLANE TBIOONOMBTBT. 87
COB a + 008 P + COS ysina+sinjS + ffliiy a — 5
co8(a + ^ + y) ~ 8iii(a + )3 + y) "~a+6'
coe a COS /3 cos y — sin a sin /3 sin y _^ 1
6''cos(o+ )3 + y) " a*sin(a + )3 + y) "" (oTft/
^t 9
tan^-htan| + tan| cot? + cot| + cot| ^6
tan ^— t cot ^ — ^
/Q\ ^ (^^ °' -•■ **^ )^ + **^ y) _ * (<^* o + cot j8 + cot y) _ - ^
^^ (2o + 6)tan(a + j8 + y) " (a+ 26) cot(a + /3+ y)" '
(10) a'(tanj9tany + tanytana + tanatan/3~ 1)
= 6*(cot)Scoty-»-cotycota + cotacot/8-l)«-(a + 6)*;
and that a, )3, y are roots of the equation in 0,
6ooa(a+ff-fy) g sin (a-t-^ + y) ^ ^ ^ ^^^
cos^ sin^
489. Having given the equations
e cos (j8 + y) + COB (/8 - y) = « cos (y + o) + co« (y - a)
= eooB (a+ )9) + coe (a~ /3) ;
prove that each member of the equations is equal to — ^ , and that
sin(/8 + y) + 8in(y + o) + 8in(a + /8) = 0,
cos (/3 -I- y ) + cos (y + a) + cos (a -I- )8) = «.
490. Having given the equations
8in(tt-j8)-l-sin(a-y) _ gin (/9 - y) -f rin (j8 -> •)
8in)3 + siny-2sina "" siny + 8ina-2 8in^ '
prove that
Bina + sin/3 + siny = 0y co8a + oo6/34-coey = — Se.
491. Having given the equations
■in(a-to) ^{p-t±.')
prove that eaok member is equal to
88 PLANE TRIGONOMETRY.
and that cos (^ -»- y) + cos (y + a) + cob (o + )8) = 0,
sill ()3 + y) + sin (y + a) + 8in(a + /3) = - «.
492 If coB(tt-fff-^) ^ coB(a + y-^)
sin (a + P) cos'y sin (a + y) cos'jS '
each member will be equal to
oos()8-l-y-^)
sin ()3 + y) cos'a '
and cottf- sin (ff + y) sin (y -t- a) sin (a + ff)
" cos (j8 4- y) cos (y + a) cos (a + )3) + sin' (a + )3 + y) *
493. Having given the equations
a* COB a cos P-\-a (sin a + sin )3) = a' cos a cosy -f a (sin a -f sin y) = - 1,
prove that a* cos j3 cos y + a (sin /8 -f sin y) + 1 = 0,
that cos o + cos j3 + COB y - cos (a + )3 + y) ,
2
8ina + sin)Sf + siny=sin(a + )3 + y) :
and that
tan — — -' cos a = tan —^ — cos a = tan _ cos y = - .
2 2 '^ 2 ' a
a + y . + /9
— / "*» •^ - wi tan — r^ ;
i94. Having given the equations
CI ^ V
sin ^ = oi tan — ^ , Biny =
prove that
sin a = m tan ^ ■ ; Bina + sinj8 + siny + sin(a+/5-»-y) = 0,
oosa -»- cos/8 + coBy+ cos(a + j3 + y) = - 2«i,
and sin amn fi sin y = m' sin (a + )3 + y^*
= - -8in{(j8 + y)+sin(y + a)+sin(o + )3)}.
495. The system of equations
cos (j9 + y) + m (sin j3 -»- sin y) 4- w =* 0,
cos (y + a) + m (sin y -»- sin a) + n = 0,
cos (a + j3) + m (sin a + sin )3) + n = 0,
is equivalent to
f»*=l, ms8in(a + ^4-y)y mn + sin a + sin /3 + sin y = 0.
496. Having given the equations
8in(^ + y) + ifcsin(a-»-fl) = sin(y + a) + A;sin(j3 + tf)
= sin (o + )3) -»- A; sin (y + tf ) ;
prove that A:* «: 1, and that each member = 0.
PLANE TRIOONOMETRr. 89
497. Having given the equations
acoa(fi-y) + b (cos j8 + cos y) + c (sin /3 + sin y) + c?= 0,
a cos (y — a) + 6 (cos y + COS a) + e (sin y + sin a) + cl = 0,
a cos (o - )3) + 6 (cos a + cos j3) + c (sin a + sin/3) + «f = ;
prove that
a* + 6* + c* = 2aJy a (cos a + cos/3 + oosy) + & = 0y
a (sin a -»- sin j3 + sin y) + c = 0.
498. Having given the equations
(m + cos ^) (m + cos y) + n (sin /3 + sin y) s (m + cos y) (m + oos a)
+ n (sin y + sin a) = (m + cos a) (m + oos)3) + n (sin a + sin/3) ;
prove that each = — n' ; and that
cos (a + ^ + y) - cos a - COS P - COS y = 2m,
sin(a + /3 + y)-sina-Bin^~siny = 2it.
499. Having given the equations
cos (^ - /8) + cos (^ - y) + cos (i3 - y) = cos (fl - y) + cos (^ - a) + COS (y - o)
= cos ( tf - a) + cos (fl - ^) + cos (a - /8) ;
prove that each =cos(j8-y)-»-cos (y-o) + cos(a-^) = -1 ; and that
cos a + cos)8 + cosy + cos ^ = = sina + sin/5 + siny + sin A
500. Having given
m cos a + n sin a - sin {fi -t- y) _ m cos )S + nsin)S-sin(y + o)^
cos(j3 + y) " cos(y + a) '
prove that each
m cos V + n sin y - sin (a.+ 3) / /» \ . • / « \
= ^ , ' ^, — \—r^^ = fiicos(a + ^ + y) + n8m (a+ B + y).
cos(a + p) \ f ff \ t- ri
501. The three equations
i3 + y . )8 + y
wcos-^^' nsm^=-^' ^
^-■^■— ^ — . = ( m — n) cos — ;r- ,
cosa sina ^ ' 2
y + a . y + a
mcos^-jr— nsin^-TT-
f_ 2 , . y-a
^ - i X = (m-n) cos ^-;r-- ,
cos/^ sinjS ^ ' 2
wcos--^ nsin— — ^
_^ — I = (fi» - n) cos — j~ ,
cosy siny ^ ' 2
are equivalent to only two independent equations.
90 PLANE TBIGONOMETRT.
502. Haying given
cos (a + d) _ cos ()8 4. ^)
sin (^ + y) ~ sin (y + a) '
prove that each = . y — ^ = ± 1.
^ 8m(o+)8)
503. Having given the equations
m (cos J8 + cos y) - w (sin )8 + siny)+sin()5 + y)
= m (cos y + cos a) — w (sin y + sin a) + sin (y + a)
= m (cos a -f cos)8) - n (sin a + sin )3) + sin (a + )3) ;
prove that masin(a + )3 + y), w«cos(o + )8 + y).
504. Having given the equations
B+y B — y Y + a y— o a + B a — B
CCOS^-^' + COS^— ^' e cos i-^- + cos '-^ eCOB— 5-^ + C0S — —-
[ B + y ! fl — y . y+a . y — a \ a + B ] cT—jS
esin^— ^ + sm^— ^ ^ sin -^-— + sin i-^— «sm — ^ +Bm — ^
•5 — J.
505. Having given the equations
g*sinjgsinyf 6(cQeff + cosy) + l _ g*sinysintt-i- g(oo8y -f cosa) + 1
cos'^Zl 008-2^^
2 2
c'sin tt sin ff 4- g (cos a + cos jS) -f 1
cos" — ap-
prove that each = or 4. If each =: 0,
sina + sin)3+siny + sin(a + j3 + y) = 0,
and 6{cosa + coe)8 + coBy + cos(a + )3 + y)}3:-<- 2;
and, if each = 4,
e* 2e
8ina+sin)8 + 8iny~'Bin(j34-y) + 8in(y+a) + sin(a + )8)
8
sina + 8in^ + siny + Bin(a + )8 + y)*
Also the given cfystem is equivalent to the system
and the two corresponding equations.
PLANE TBIQOKOMETBY. 91
506. Having given the system of three equations whose type is
a cos )3 cos y-f6sin)3siny + c +/(8in j3 + sin y)
+ ^ (cos j3 + 0OS y) + A sin ()3-f y) = 0,
prove that oj - A* = ftc -/• + co - ^.
507. Prove that any system of three poristic equations between
Oj P, y IB equivalent to two independent equations of the form
/ (cos a+ cos )3-l- <X)S y)-l-iiiC08(a-»-)3-f-y)4-n (oos/^-t-y-f cosy-f a-f C08a-|-/^a|i,
/(sina+8in/3+8iny) + iit8in(a-f-)3+y)+n(8in^+y+siny-»>a+Bina+^)=^.
[The equation between jS, y will be
2 (n"-/«)co8(/3-y) + 2 (/w + np) cos 09-i-y) + m" + n*-p"-^-/»
+ 2^(8in^ + siny) + 2(mii + ifp)(oos/3 + oo6y) + 2i»9sin08 + y) = 0.]
508. The condition for the coexistence of four equations of the
type in (506) between o, ^ ; ft y ; y, 5 ; and 8, a respectively is
A = a6c+ 2/^A- <^- V- c^' = 0.
509. Having given the system of five equations
- cos a cos 6+ T^o,ajiB=^-,
a '^ e
and the like equations between ft y; y, £; S,c;c^a respectively; prove
that
a* + 6* -»- c* = (6 + c) (c + a) (a + b).
[An equivalent form is Mfc 4- (6 -f c - a) (e + a - 6) (a •»> 6 - c) s 0.]
510. Prove that the system of n equations
a +a a. -f a a +a
tan-i^-oota. = Un^'oota. = tan-Y.oots=...
CL 4* a h
= tan -3 — -^^-^ cot o = — .
2 • a'
is equivalent to only n — 1 independent equationsi
IV. InequalUUs,
511. Prove that coi ^ > 1 + cot 0, for values of $ between and w :
and that, for all values of 9, — ^ — < 2 -l- cos A
92 PLANE TRIGONOMETBY.
512. Prove that, tor real values of Xj -j — ^ 5 — r ^®s between
SCr "" ^X OOS p n" X
1— COSa - 1 4-0080
1-008)8 1+C08/3*
513. J£ X, y, z he any real quantities and A, B, C the angles of
a triangle, prove that
»* -I- y* -f 2* > 2yz 008 A + 2»e cos jS + 2xy cos C,
unless (v cosec il = y cosec jS = « cosec C
514. Under the same conditions as the last, prove that
(x sin" -4 + y sin" B-^z sin" C)" > 4 (y« + »b + asy) sin" A sin" jB sin" C,
unless X tan il = y tan B^z tan C7.
515. JI A^ B,Che the angles of a triangle, prove
sin jS sin (7 + sin (7 sin il -f sin il sin jS
Sn" -4 + sin"i + sin" C
lies between the values ^ and 1 : and
sin^ sin jS sin (7
l+cosilcosi?cosC7
between and -.^ .
516. Having given the equation
sec )8 sec y 4- tan )8 tan y = tan a,
prove that, for real values of P and y, cos 2a must be negative ; and that
tan j3 -I- tan a tan y cos^
tan y 4- tan a tan )8 cosy*
517. Prove that, A, B, C being the angles of a triangle,
ABC
|>sin^Bin-5 8in-^>(l-oosil)(l-oos^)(l -oo6C)>oo6i(cosjScos(7;
and that
ABC
cos^cos-^cos-^>sinilsinjS8inC7>sin2ilsin2jSsinC7;
except when A = B-C.
518. Prove that, A^ B, C being the angles of a triangle,
A B
COSgOOSg
Ssin^sinjSsin"'
»^2'°^2'^2
1 4-co6iico6J3co6Cp^/y3sinii8ini?8inC7.
PLANE TEIGONOMCTRT. 93
519. On a fixed straight line AB la taken a point C such that
AC^2CB and anj other point F between A and C; prove that, if
CF = CA sin A, AF . BF' will vary as 1 + sin 30 and thenoe that
AF, BF* has its greatest value when F bisects AC,
520. If a + ilyj3 + ^, y+(7be the angles subtended at a point bj
the sides of the triangle ABC^ then will
(sin'o sin'jS sin*y\* 2 sin' a sin' ff sin' y
5S:i"*"n5J? ^^Ij) "" . A . h . C'
sin -^ sm — sin --
2 2 2
except when the point is the centre of the inscribed circle.
521. Having given the equations
cot P cot y 4- cot y cot a + cot a cot p ■■ tan j3 tan y + tan y tan a + tan a tan j3 ;
prove that cos 2 (^- y) + cos 2 (y- a) + cos 2 (o - j8) > - |.
522. If a, )3y y be angles between and ^ , and if tana tan /3 tan y= 1,
then will
iinosin^siny<^-^,
unless a = j3 = y.
523. Prove that
/cos'(a-tf) sin'(a-^ ) fcos'(tt 4-tf) sin'(tt-t-tf) )
\ a' "" 6^^~/\ S^ -" 6' /
cannot be less than — jr^ ; and can never be equal to it unless tan' a
lie between -i and r. .
a" 6"
624. If «, a^ y, z be any real quantities, and a, 6, c, a', 6', c' cosines
of angles satisfying the condition
-1, c', 6', a
</, - 1, o', 6
6', a', - 1, c
a, i, c, - 1
=0;
prove that
a;'-»-y' + 2' + w'> 2a! yz + 2V%x + 2c'ajy + 2aaice -i- 2hiay + 2eaKe,
except when
«• y*
l_a''-6'-c'-2a'*c 1 -a'-6''-c'-3a6'c
1 - o' - 6* - c" - 2abc' 1 -a"- 6" -c''-2a'6V '
94 PLANE TBIQONOMETRT.
V. Properties of Triangles,
[In these questions a, b, e denote the sides and A^ B, C the respec-
7 opposite angles of a triangle, R is the radius of the circumscribed
circle, and r, r^, r,, r, the radii of the inscribed circle and of the escribed
circles respectivelj opposite A^ B^ C]
525. Prove that if
l + cosii = cos^ + cosC7, secii— l-sec-fi+secC.
526. Prove that
a>6cosi? + ccos(7, 6>ccos(7+aco8il, and oacosil +6cos^.
527. If a triangle A'BC be drawn whose sides ar66 + c, c + a,a + fr
respectively, and if the angle A' ^Ay then will ^a lie between 5 + c and
2 (6 + c), and
B^C . . A , ZA
ooB — g~ = 4ain^ -sin-^.
528. If 0, ^, ^ be acut^ angles given by the equations
then will tan»5 + ton'f + taii'^=l ;
MAM
and tan ^ tan ^ tan ^ = tan ^ tan -^ tan ^ •
529. If sin il , sin ^, sin (7 be in harmonical progression so also
will be
1 - cos A, 1— cos By I - cos C
530. From the three relations between the sides and angles given
in the forms
a" = 6* + c* - 2b€ cos A, &c.
deduce the equations
a
assuming that each angle lies between and 180^
531. In the side BC produced if necessary find a point F such that
the square on PA may be equal to the sum of the squares on PB, PC ;
and prove that this is only possible when Ay B, C are all acute and
tan A < tan B + tan C, or when ^ or C is obtuse. When possible, prove
that there are in general two such points which lie both between B and
(7, one between and one beyond, or both beyond, according as il is the
greatest, the mean, or the least angle of ihe triangle.
PLANE TfilGONOMETRY. 95
532. The sides of a triangle are 2pq +p*, p' +pq + q*, and p' - g* ;
prove that the angles are in A. p., the common differenoe being
2tan-«^^-^-^^
533. The line joining the middle points of BC and of the perpen-
dicular from A on BC makes with BO the angle cot~* (cot i? - cot C).
534. The line joining the centres of the inscribed and Gircumscribed
circles makes with BC the angle
_j /cos -5 + cos C— r
-1 /cos -o + cos c; - 1\
\ sini^-sinC? /
535. The line joining the centre of the circumscribed circle and the
centre of perpendiculars makes with BC the angle
, / tan^tanC~3 \
\tan^-tanC/'
536. The line joining the centre of the inscribed circle and the
centre of perpendicuIarB makes with BC the angle
C-B C08-4
, ( ^C-B cosii \
\ 2^2^2^"-2")
537. In a triangle, right-angled at A, prove that
r, = r, + r, -I- r.
538. If BA^AC^^ \BC and BC be divided in in the ratio 1 : 3,
then will the angle ACQ be double of the angle AOC.
539. If the sides of a triangle be in a. p. and the greatest aag^e
exceed the less by (1) 60^ (2) 90^ (3) 120^ the sides of the triangle will
(1) ^13-1:^13:^13 + 1,
(2) ^7-1 : V7: n/7 + 1,
(3) ^/5-l : V5: ^5 + 1.
[In general if the sides be in a. p. and the greatest angle exoeed the
least by a, the sides ik411 be as
1^7 - cos a + ^1- cos a : ^7 ^ooaa : ^7-cosa — ^1 -cos a.]
540. If be the centre of the circumscribed circle and AO meet
^in A
OD : AO^oobA : cos(jB-C).
541. Three parallel straight lines are drawn through the angular
points of the triangle ABC to meet the o^^xwite sides ia A\ B', C :
prove that
A'B.AV BC^A CAJO^B
AA* ^ BF* ^ ccr* "^ '
96 PLANE TRIGONOMETRY.
the segments of a side being affected with opposite signs when they fall
on opposite sides of the point of section.
[The convention stated in the last clause ought always to be attended
to, but it is not yet so sufficiently recognised in our elementary books as
to make the mention superfluous. BC -k-CA -k-AB^O ought always to
be an allowed identity.]
542. The perimeter of a triangle bears to the perimeter of an
inscribed circle the same ratio as the area of the triangle to the area of
the circle, which is
cot -^ cot -^ cot ^ : IT.
[The first part of this proposition is true for any polygon circum-
scribed to a circle, and a similar one for any polyhedron circumscribed to
a sphere.]
543. A triangle is formed by joining the feet of the perpendiculars
of the triangle ABC^ and the circle inscribed in this triangle touches the
sides ia A\ B^, C : prove that
B^C GA' A'B! ^ . „ ^
-^^ « -j^ = -j^ = 2 cos il COB jd cos C
544. A circle is drawn to touch the circumscribed circle and the
A
Bides ABf AC ; prove that its radius is rsec'-^ : and if it touch the cir-
cumscribed circle and the sides AB, AC produced ita radius is r^ sec' -^.
li BssC and the latter radius => R, cos A = l.
545. EEaving given the equations
c'y + 6'« = a'z + c'x = h*x + a"y ;
1 X V z
prove that • ^ a = •\ij = - on -
'^ sm '2A Bm2B sm2C
546. Determine a triangle having a base c, an altitude h, and a
given difference a of the base angles : and if $., $. be the two values
4/»
obtainable for the vertical angle, prove that cottfj + cottf, = — r-j- .
Prove that only one of these values corresponds to a proper solution ;
and if this be $^, that
^ e, JW + c« sin'o - 2h
tan-^ =^^^ y^ r .
2 c(l-co8a)
Account for the appearance of the other value.
547. Determine a triangle in which are given a side a, the opposite
angle A, and the rectangle m' under the other two sides : and prove
that no such triangle exist, if 2« Bin I > «.
548. Find the angles of a triangle in which the greatest side is
twice the least, and the greatest angle twice the mean angle. Prove
that a triangle whose sides are as 17156 : 13395 : 8578 is a very
Mpproxim&te solution.
PLANE TRIOONOMETRT. 97
549. A tariangle A'B^C has its angles respectivelj complementaiy to
the half angles of the triangle ABC and its side BC equal to BC : proTe
that
b.A'BC' .bABC^f&n- :2sin|BinJ.
550. Two triangles ABC, A'SC are such that
cot -4 + cot -4' = cot -5 + cot J?' = cot (7 + cot C" ;
a point P is taken within ABC such that its distances from A, B, C aire
as B'C : C'A' : A^R : prove that the angles subtended at F by the sides
of the triangle ABC are J?' + C, C" + ^', ^' + J?' ; also that
C0tii' + C0tjB' + C0tC^=C0tii 4-cot J? + cotC.
551. If A\ B', C he the angl^ subtended at the oentroid of a
triangle ABC by the sides,
cot il - cot -4 ' = cot ^ - cot jB' = cot C - cot C
= f (cot A-k-cotB + cotC).
552. If 8 cos A cos B cos C = cos'o, each angle of the triangle ABC
xl ^U 111 lit, \
must lie between the acute angles cos~* ( ) , and the difference
between the greatest and least angles cannot exceed a.
553. If a straight line can be drawn not intersecting the sides of a
triangle ABC, and such that the perpendiculars on it from the angular
points are respectively equal to the opposite sides, then will
. ,A ^ ,B ^ ,C ^ A ^ B ^ C «
tan"^ + tan" ^ + tan' -=tan- + tan^+tan2 = 2,
A B C ,
cos" -r + cos' -^ + cos' ^ = siu il + sin ^ + siu C
[Of course these equations are equivalents.]
554. With A, Bf C as centres are described circles whose radii are
aoosA, BcobB, ccobC respectively, and the internal common tangents
are drawn to each pair : prove that three of these will pass through the
centre of the circumscribed circle and the other three through the
centre of perpendiculars.
555. The centroid of ABC is (r, a triangle A!BC is drawn whose
sides are GA, GB, GC, and circles described with centres A, B, C and
radii a' sin A\ h' sin J?', e' sin C respectively : prove that three of the
internal conmion tangents to a pair of circles intersect in G, and the
other three in the point of concourse of the lines joining A, B, C to the
corresponding intersections of the tangents to the circumscribed circle
at A, B, C.
W. P. "1
M PLASE
A flwi!ss K tt X, fiM a sDaiciEt line drmwn ihio M gfc A\ perpeM&idv
<fV ^Mr wMnMi ItMetrjt oi tW an^ ^ viD meet tfe circip in two pointB
/>^^ ^^ -wi^m ^Mt^rnre from ^ k a meu jvopGraoBal b g tp ee ii the
ffiitMiLi^a faci jg, C f«p«tiT€ir ; and ^ P^Lg- ^ CJ^ = | rJ^-Q. Alao
^ ^ ^ 4l< tfe icniiBkt fine ifaawn thrcN^ A' paralM to die bisector
19^ ttrtfSS dw: <ireie in two otiier points lia^in^ tlie same propertr.
Jl C ft^ fsHi points and J anj point sodi thai a' = 4&v ^^^ <^^
Ims |#mmi viO eoindde^ and its locus will be a rectangular bvperbola
iPlmi!: Imi are A a
%&l. U O l)0t ike point witbin the triangle ABC at wbicb the sum
4if A^ 4at/Uakteii ai A^ B^ C i^tL miniminn, strsigbt lines drswn tbroogb
A^ B, C at riii^ si^^ to OA^ OB, OC ratpcctiielT will form tbe
tMOMMns <«|«ikteral triang^ wbidi can be drcomsciibed to ABC', and
M AlfC ht tlds mazimmn trian^ then wffl
OA' : O^ : OC-^BC : Cil : JA
Pfr/r« abo tbai
OAmf Am(B^C)^OBwa^ B^il{C -A)^OCw^^Cwi{A-^B)^i^
t^hfi. It XfPfZ he perpendiculars from the angular points on any
ilnuglii Hne; prore that
taty Mrpeodicolar bcong reckoned negative which is drawn from its
t u if mlur point in Uie opposite sense to the other twa
f^ff9. If perpendiculars 0Z>, OF, OF be let fall from anj point O
€m tha sides of the triangle ABC, and as, y, s be the radii of the circles
A£F, BFD, CDF^ respectively ; prove that
l«a'(«'-j^(«»-«^+ ... - 8a^(ax«cosii + ...) + a*yc'=0.
HO. If be the centre of the circle inscribed in ABC, OD, OF, OF
p^Tpcmdiculars on the sides, and x, y, z radii of the circles inscribed in
ifa« quadrilaterals OFAF, OFBD, ODCE\ prove that
(r- 2a;)(r- 2y)(r- 2«) = f^- 4a:ya.
561, A triangle A'BC is circumscribed to the triangle ABC, prove
tbai when its perimeter is the least possible BC = BC'Jl - sia J?' sin C',
uiA, if x^y,z\)^ the sides of the triangle A'BC\ that
fl^-g* y*-y s*-c* (x + y + z){y'{'Z''X){z'k-x-y)(x-^y-'Z)
X y z 4xyz
562. If />,! Pff p^he the perpendiculars of the triangle
1111 COSii COS^ COB (7 1
Pi P. Pm ^ Px Pn Pt ^
also/)^ is a harmonic mean between r,, r,.
PLANE TRIQONOMETBY. 99
563. The distances between the centres of the escribed circles
being a, P, y; prove that
a* /P /
4ff =
r + r r +r r -hr
V. + Vi + ^/. V<^ (^ - a) (<r - )8) (<r - y)
where 2<r = a + )8 + y. Prove also that
x/ V^ 4- v, 4- r,r, Jr^r^ + r,r, + r,r.
564. The distances of the centre of the inscribed circle from those
of the escribed circles being a, fi\ y ; prove that
47? =
a' _ p^ ^ Y
T —T T —7* r — r
that 32i?' - 2/? (a" + jS'* + y'^ - a jS'y = 0.
565. Prove that the area of the triangle
rr.r.
= V-.V^=-7=^
566. Taking p to be the radius of the polar circle of the triangle,
prove that the area
and that
567. The cosines of the angles of a triangle are the roots of the
equation
4i?*x"-4/?(/? + r)iB'+(2r*-|.4/?r-p«)a:+p« = 0;
and the radii r^, r^, r, are roots of the equation
a:"(4i? + r-a;) = (a;-r)(2^ + r -p").
568. Prove that
St t t
— * ■-* » sin il sin i? sin (7,
and ^>^?^ = (l + cosi4)(l + cos-ff)(l+cosq.
n— ^
-£ * ^ * * •
^V Tm; skfdE tf rvfr ff -ait tardiB -vioA vnufti ttr bos tf a
</X : <X/ = r<flr r., r,, rj : Jt
^ ' <i/^ Uii^ ittMiJUi4 <mie ViD pas tiuti^ tfe centre of
f luf «^«#rii Otf; ittKiibi&d circle vill pas dmng^ die centre of ]
f 4« > 4 i ^W »lfg«ui^«»ifc«C = (l~c»wJ>(l--co«^>(l~cwO.]
M%, If O^ i9 |pe the centres of tlie dmnnBcribed ad i na crib c d
4a;I««, mi4 it i)m centre of perpeodkoLuv
mA Vi^^ \m (fa«c^lre of tLe escribed ciide opposite J,
/)74/ If Ui4} ts^niira //f ibe in«cribed circle be equidistant fran the
tmtirn //f ^m i;mmMmri\ttid circle and from the centre of perpendiculars,
4ffm m$$idm i4 tint iriarjgle ntust be 60* ; and with a similar property for
mi mtifiSM isiriiUt, oiui angle must be 60* or 12Q*.
/>75# TlMt tamUui fd tlie angle at which the drcumacribed circle in-
i4»rmu!lM Um mfri\M drcl« ofijiosite A is
1 -f cosil_--coft^-oos(7
~ 2 '
and If «, /?, y U Ui#) tbr«e nuch cosines
ftTfl, If /' \m Miy fKjIiit on the circumscribed circle,
PA nln A >/7iiiin B + PC sin (7= 0;
a tMti'Ulii oonvttfiilon li«iliig niailo In rospoot to sign : also
PA • Mill '2 A ^ /*//• sin 2JB + PC sin 2C = 4Ai4i?C.
PLANE TBIOONOMETBT. 101
577. If P be any point in the plane of the triangle, and the
centre of the circumscribed circle,
PA^mn 2A + ... + ... - 40P* 8in AanBanC^^ 2£lABC.
578. If P be any point on the inscribed circle
Pii* Bin il + P^* Bin ^ + PC* sin (7
will be constant ; and if on the escribed circle opposite A,
- P^" sin ii + P^sin^ + PCsin (7
will be constant.
579. Prove that, if P be anj point on the nine points' circle,
Pii' (sin 2i? + sin 2(7) + P^ (sin 2C+ sin 2il) + PC* (sin 2i< + Bin 2P)
= 8i^sini(Bini?BinC(l +2cosiicoB^coBC).
580. If P be any point on the polar circle,
P^'tanil+P^tan^ + P^tanC
will be constant. If p be the radius of this circle and 8 the distance of
its centre from the centre of the circumscribed circle, then will
and if S' be the distance of its centre from the centre of the inscribed
circle, then will
S'" = p"+2r».
581. The straight line joining the centres of the drcomflcribed and
inscribed circles will subtend a right angle at the centre of perpen-
diculars if
l+(l-2cosil)(l-2cosJ?)(l-2oosC^ = 8coBi(cosJ?coBC. *
582. If P be a point within a triangle at which the sides subtend
angles -i + o, -5 + )8, C + y respectively,
sm a sin p siny
583. Any point P is taken within the triangle ABC and the angles
BPC, GPA, APC are A\ B, C" respectively; prove that
A^PC(cotii - cot -4') = t^CPA (cot -B- cot B) = AiLPj5 (cot C - cot C).
584. Having given the equations
cos* -i (p sin' tf + gees' tf) = cos* jB (p sin* C + tf + g cos* C + tf)
= cos'C(pBin*-B-tf+gcos*(7-e');
prove that each ~ (p + g) cos A cos B cos C ; and that
(jP4-g)' _ icos'iicos'^cos'C
Tpq " {cob A - COB B COB C) (cob B - cob C cob A) {cob C -^ cob A cob B) *
102 PLANE TRIGONOMETRY.
VI. ffeights and Distances. Polygons.
ft
585. At a point A are measured the angle (a) subtended by two
objects (points) P, Q in the same horizontal plane as A and the distances
h, c at right angles to AP, AQ respectively to points at which PQ sub-
tends the same angle (a) ; find the length of PQ.
586. An object is observed at three points -4, jB, C lying in a hori-
zontal straight line which passes directly underneath the object; the
angular elevations at A^ B, C are $, 26, 36, and AB = a, BC = 6 ; prove
that the height of the object is
J n/(« + *) (36 - a).
K cot 0-3, a : 6 = 13 : 5.
587. The sides of a rectangle are 2a, 26, and the angles subtended
by its diagonals at a point whose distance from its centre is e are a, fi :
prove that
1 6a*b'i^
2a being that side which is cut by the distance c.
588. The diagonals 2a, 2b of a rhombus subtend angles a, /3 at a
point whose distance from the centre is c : prove that
h' (a" - c*)" tan" a-^a' (6" - c*)" tan" )8 = ia'h'c\
589. Three circles A, B, C touch each other two and two and one
common tangent to A and B is parallel to a common tangent of A and
C : prove that if a, 6, c be the radii, and />, q the distances of the centres
of i? and C from that diameter of A which is normal to the two parallel
tangents
pq- 2a* = %hc.
590. Three circles A, B, C touch each other two and two, prove
that the distances from the centre of A of the conmion tangents to B and
C are equal to
25c (6 + e)-'a{b- cy^ihc^a (a + 6 + c)
and that one of these distances =Oifa(5 + c^2 J2hc) = 2hc.
591. Circles are described on the sides of the triangle ABC as
diameters : prove that the rectangle under the radii of the two circles,
which can be described touching the three, is
4ig'(l •¥oo&A){\ -f co8^)(l -i-cos(7) cosiicos^cosC
(l+cos-4)(l+coBJ5)(l+co8C)-(cos5-cosC)*-(cosC-cosii)'-(cosJ-cos-fi)' '
592. Four points A, P, Q, ^ lie in a straight line and the distances
AQy BPj AB are 20^ 26, 2c respectively; circles are described with
diameters AQ, BP, AB : prove that the radius of the circle which touches
the three is
c(C''a){c- 6)
PLANE TRIGONOMETBT. 103
593. A polygon of n sides inscribed in a circle is such that its sides
subtend angles 2€l, Acl, So, ... 2na at the centre ; prove that its area is to
the area of the inscribed regular Ti-gon in the ratio
sinna : nsina.
594. A point F is taken within a parallelogram A BCD ; prove that
the value of
^APC cot APC - A^P2> cot BPD
is independent of the position of P.
595. The distances of any point P on a circle fix>m the angular
points of an inscribed regular n-gon are the positive roots of the
equation
^2n(2n-l)(2n-2)(2»-3)^^ ..,. ^^^^.
d being the diameter of the circle and the angle subtended at the centre
by any one of the distancea Prove that if we take d=2, the equation
may alao be written
[2 [5
+ (-!)■ 2(1 -cos nfl) = 0.
596. The sides of a convex quadrilateral are a,h,e^d and 2$ is their
sum : prove that
j8(8''a-d)(8-b-d)(8''C-d)
cannot be greater than the area.
597. The equation giving the length x of the diagonal joining the
angles {a, d), (b, c) of a quadrilateral whose sides taken in order are
a, bf c, df is
{a:* (oi + erf) - (flkJ + M) (6c + od)}' sin* o
+ {as" (a6 - erf) - (oc-M) (6c-a;0}"cos'a
=:4aVc'rf*sin'2a;
where 2a is the sum of two opposite angles.
[This equation, being a quadratic in cos 2a, leads to the equation
giving the extreme possible values of a; ; which can be reduced to the
form
3(aj«-a'-6')'(«'-c*-rf')'
- 4 {x* (oi + crf)- (oc + M)(6c + arf)}'+ IGa'^Vrf'^ 0.]
104 PLANE TRIOONOMETBT.
698. In any quadrangle ABCD, the yertices are B, F, G, (the
intersectiona of BC, AD ; CA^ BD ; and AB, CD respectively) ; prove
that
(EB.EC-BA.ED)' (FC .FA- FB.FD )'
FA.FB.FC.EDrin'B" FA.FB.FC.FDan'F
(GA.GB-GO.GDy
"GA.GB.GC.GDem'G'
Vn. Bs^Mmriana of Trigonameirieal Functions. Inverse FuncUone.
599. By means of the equivalence of the expansions of
2<'sina;x<'coBa:, and c^Bin2a;;
prove that
. , » » nr „-_! . nir
sin (n-r) 7 008 -J- 2 sm-j-
4 4 4
71 — r Ir In
600. Prove by comparing the coefficients of d**"* that the expansions
of sin and cos in terms of $ satisfy the identity
2 sin cos = sin 20.
601. Prove that
sin (n + 1) -r-
2--(«-l)2-4-^^^::^y^^^^2-'-...= f , (n integral).
L- (Bin,)
602. Prove, from the identity
1 1 2tsin0
1-aa l-a»"* l-2aJCO8 + a:"'
that
^^^^^(2coser^{n^l){2cos0r'
^<"-y-^>(2cosg)--.,.,
J , J ^, . . ^/ sin(n+l)0-sin(n-l)0\ .
and deduce the expansion of cos w ( = — ^ <> ' a I "^
terms of cos when n is a positive integer.
60S. From the identity
log (1 + flw"*) + log (1 - a») = log (1 - 2*05 sin 0- x^
or from the identity
1 1 ^ 2(l-ia;8ing)
l-a» i +«»"*'" l-2i«Bin0-«*'
PLANE TBIGONOMETRT. 105
deduce the equations
00Bn$=l- ^Bm'0 + — 5_ — iwa.*0 i — 'an'$ + ...,
2 (4 |6
n being an integer, even for the first and odd for the second. Also
IT IT
prove that, if lie between — ^ and ^ , both are true for all values of n ;
and thence deduce the true expansions of cos n6 and sin n0 in terms of
sin for any value of 0,
604. From the expansion of (sin $)***^ in terms of sines of multiples
of $, prove that
A-1 /o IX 2n(2n-3) 2n(2n-l)(2n-5) ^ ,^
0= l-(2n-l)+ — \^ ^ ^ ^ ^-...ton+1 terms.
[f if
605. Prove that
- n-1 ^ n-1 2n-l „^ n-12n-13n-l «^
1 COsfl + s C0S2fl s 5 cos 30 + . ..to 00
n n 2n n 2n on
n-ie
cos
n 2
if lie between — ir and «-.
606. Prove that
1 (n ix, (^-2)(n-3) (n-3)(n-4)(n-5)
l-.(n-l)+ ^2 13 + ...
= (-l)"sin2(n+l)^+Bin-^.
607. Prove that, if tan (= <) be less than 1,
• a ./» s w(n+l)(n+2) . ^w(n + l)...(n + 4) .
sin nfl COB 0=71/ ^ i4^ ^r+— ^ y=— ^ ^r-...
[3 [5
and cosn0cos-0=l.!^tl),.^!^(!L^^
[These results are obviously true when n is a negative whole number.]
608. The sum of the infinite series
1 1 13
1 + ^ cos 20 - TT— T cos 40 -f ^ ' ^ cos60~ ...
is /i/cos (1 + cos 0), if lie between - ^ and ^ .
106 PLANE TBIGONOMETBY.
609. Prove that the identity
may be deduced from the identity
2cosnfl=(2co8fl)"-n(2costf)-« + ^^%^(2co8^-*-...
If
when n is an even integer, by writing -^-0 ior $ and taking the terms
in reverse order ; and similarly for sin nO when n is odd.
610. If
prove that ^(2n) = (- iyF{n).
[^(n) = COB -5- for all values of n.]
611. If the constants a^, a^ a^,,, a, be so determined in the ex-
pression
Oj sinx + a, sin2x+ ... + a^ sin no; + sin (n + 1) x
that the coefficients of Xy a?y a;',...a^'* shall vanish, the value of the
expression will be 2" sin x (cos x-l)* ] and if, in
a, cos a; + a, cos 2a; + ... + a, cos rio; -f cos(n + \)x
the coefficients of all the powers of a; up to a?*""* inclusive vanish, the
value will be
2"(cosa;-l)" rcosa;+ y)*
612. Prove that
-*-" sin* ra cos**"Va _ (2n + 1 ) «
r-i a:' + tanVo = (1 +»)•"*' -(1 -«)•"*»
where (2n + l)a = ir.
613. From the identity
-o /I -* ^ . 0-ix . + ix
r - 2 cos ^ + c = 4 sin — ^ — sm — - — ,
resolve the former into its quadratic factors.
[The ««ult is 4Bin.|(l . ^ (1 . ^.) (l .^) ... all
factors of the form 1 +73 j-r^ being taken where r is a positive or
. .
negative integer.
PLANE TRIGONOMETRY. 107
Similarly a = the product of all factors of the form
A f "
4C08 -
2
re*
1+ "^
. ^ where r is an odd positive or n^;atiye integer.]
614. From the result in the last question deduce
^^ sin'^" '-- (nr + C)"
(2) ^:.?.r.''.c_.-zr^;
38in*d -^— m(nr4-^/
zero being included among the values of r.
[By equating coefficients of ac*" in the results, it appears that
2»— »
J — r-j-^ = sum of the products n together of all expressions included
in ^ for integral values of r from - oo to oo including zera]
615. Prove that
1 1 _1 1 1_ 1 1
8in^"^"^ir-tf ir + ^ 2ir-(^ "*■ 2fl- + d "*■ 3*— ^"" • ' '
B ir' (2ir)' (3ir)«
28ind ir*-^ (2ir)'-e' (3ir)'-^
616. Prove that
tan0 1 1
Se ir* - 46>* (3ir)' - 4^' (5fl-/ - 4e'
+ ... to 00 .
617. Prove that, if be an angle between — -j and j ,
^ . ,/, 2 sin*^ 2.4 sin'^
* •/) A . l\tan*^ A 1 l\tan"^
= tan'e?-(l+3J--^+(l+g+g)-^-...tooo.
[The former is true for all valaes of $.]
618. Prove that
n\ 1* 2* 3' ^ 2»
<*' rTT2rri3riri"*<>*=737i'
1* 2* 3* 4r*
108 PLANE TRIGONOMETEY.
619. In a triangle the sides a, b, and the angle ir — O opposite b are
given, and is small : prove that, approximately,
= l+r7r+-^^ n w
b-a 6 2 6" [4
620. Prove that the expansion of tan tan . . . tan x is
a; + 2n-5 + 47i(5n-l)^ +-^(175w*-84»+ 11)^+ ...;
when the tangent is taken n times.
621. Prove that the expansion of sin sin. . . sin a? is
fl.-w?J + n(5n-4)^-j(175n' + 336n+162)|J+...;
the sine being taken n times.
622. Prove that the expansion of tan~' tan'\.. isnT^x is
aj-2njJ + 4n(5n + l)|^-^(175n' + 84/i + ll)|^ + ...
[That is, the expansion of tan*" a; might be deduced from the ex-
pansioii of tan" a; by patting —n for n, the index applied to the function
denoting repetition of the ^mctional operation.]
623. Prove that
(1) tan-»TV = tan->^ + tan-'^,
(2) tan-'^ = tan-'^ + tan-',|7,
(3) = tan-'^-tan-»3^-tan-*^ + tan"*T5T,
(4) j=:tan-4 + tan-'| + tan-'| + 2tan"'^ + tan-*^,
= tan-' J + tan"'| + tan~*i + tan"*^ - tan"*^,
= 3 tan-4 + tan-» 3^ + tan-» 3T^.
624. The convergents to #72 are 1, f, |, ...^ ... ; prove that
tan"* tan*"* — = tan"*
1
and that tan'
1 -i tan-* — = tan-* / ^—t\
' 4
PLANE TRIGONOMETBT. 109
625. Find x from the equation
cot~*a; + cot"* (n* -« + 1) = cot"*(n- 1);
and find the tangent of the angle
tan"* 3 + 3 tan"* 7 + tan"* 26 - 7 .
626. Prove that, if tan (a + 1)3) = t, a, fi being real, that a will be
indeterminate and fi infinite.
627. Prove that if cos (a + ifi) = cos ^ + f sin ^, where a, )3, ^ are
real, sin ^ = ^ sin' a, and that the relation between a and P is
i^-c~^ = * 2 sin a.
628. Prove that, if tan (a + i)3) = cos ^ + f sin ^ and a, ^, ^ be real,
IT TT .^ , /l + sin^\
629. Prove that, if tan (a + ip) = tan ^ + f sec ^, and a, ^, ^ be real,
VIII. Series.
[In the summation of many Trigonometric series in which the r*^
term is of the form a^ cos rO, or a^ sin rd, a^ being a function of r, it is
convenient to sum the series in the manner exemplified by the following
solution of the question : —
** To find the sum of the series l + 2cos0+3co6 20 + ... +noosn — 10."
Let C denote the proposed series and S the corresponding series with
sines in place of cosines, namely,
S= 2sin0 + 3sin20+ ... + nsinn-l 0,
the first term being sin . or 0, then, if cos $ + isiji0 = z^
C + t-S^= l + 2s + 3s*+ ... +714^ = 7^ Ti '-
_ 1 -(cosn^ + t sinn0)(n+ 1) +n (cosn 4- i g-ftsinn-t- \S)
same numei-ator
f2sin-j (sin--tcos^j =-f2sin^j rcostf + tsintfj
_ cosg-tsin0~(n4- 1) (cosn-l^ + tsinn- 1 0) 4- n (cos n^ -f t sin ntf )
' -(2-1)'
110 PLANE TRIGONOMETRY.
whence, equating possibles and impossibles on the two sides,
_ (n + 1) cos (n — 1)^ — cos 6-nooBn0
= 2(l-cos^) '
and also
^__ (w + 1) sin (n — 1) d + sin d — n sin w^
= 2(1- cos 6) •
It is obvious that in* general if
/{x) = a^ + 0^05 + ajx^ + ...,
and z have the same meaning as above,
/{xz) +/(xz~^) = 2(a^ + a^x cos fl + . .. + ajx^ cos ntf + . . .),
/(xz) —/{xz"^) = 2i (a^x sin fl + . . . + ajxT sin nd + . . .).
Some doubt may often arise as to the limiting values of the angle $
beyond which results found by this method may not be true, but this
can always be cleared up by the use of the powers of a; as coefficients as
in the forms just given. Thus, to take a very well-known case, to sum
the infinite series sin — ^ sin 20 + ^ sin 30 ... . Take
C = a5CO80--s-cos20 + ..., and ^= asintf — ^sin 20+ ...
and we have
C + t/S^^ a»- Jac*«" + |a^«'- ...
= log (1 + xz)
= logp (cos ^ + 1 sin ^) = logp + 1^
where p = */ 1 + 2a; cos + a:*, and tan 6 = pj .
'^ ^ ^ 1 + a; cos
Thus S = tan"* ( = ^ ) , meaning by this the angle between - -
\i "T" a; cos 0/ a
It X sin
and ^ whose tangent is tij '^liich is free from ambiguity,
w 1 + a; cos V
(since the series manifestly vanishes when x = 0), and when a: = 1, the
result will be ^ if ^ lie between those limits, or between — ir and w.
So also the corresponding series in cosines
cos0-Jco8 20 + |cos30-... = log72(l + co8 0) = Jlog/^4co8»|V
which is sometimes written log [2 oos^ ] without the proper limitation
that cos ^ must be positive. The series will be convei^nt only if
j^ he Jesss than 1 ; and this will be generally the case.
PLANE TRIGONOMETRY. Ill
Many series also may be summed by the same method as was
explain^ under the corresponding head in Algebra : that is by obtaining
the r*** term (u^) of the proposed series in the form CT ^^ — U^, Thus, to
sum the series
cosec X + coaec 2a; + . . . + oosec 2""'aj,
we have
Bin2'-*a! sin (2'- 2'"" ») a; ^„^,
sin Z X sin z x sin J x sin iS x
so that i/;^, = - cot 2'-*aj, and S^ = i7^^^, - ^, = cot f - cot 2-'«.
Such being the method, it is clear that givim^ the answer would, in
these cases, amount to giving the whole solution.]
630. Sum the following infinite series, and the corresponding series
in sines,
(1) cosd + Jcos2fl + Jcos3d + ...,
y«v /I cob3$ cos 50
(2) costf— j3-+-^3^ ...,
(3) cos tf - 1 cos 3tf + ^ cos 5d - . . .,
/^\ 1 A w(n+l) „^ n(n+l)(n+2) «^
(4) 1 -ncostf+-^j — ^cos2tf ^^ ^ ^ cos 3*4-...,
/I.V -, ^ cos 20 cos 30
(5) l-coe0+-72 13- + --'
//•v /I ,008 30 1.3 COS 50
(6) COS0 + J— ^+274 -5— -^ ••'
,p,x /I t COS 30 1.3 cos 50
(7) COBfl-i-^+g-J-g -.
COS 20 2 . 4 cos 30
"t" . • •«
(8) COSO + l^p + g-g g
(9) acos0 + Ja:*cos20 + Jai*cos30+..., when x = cos 0,
/^rvx /I t .cos 30 1.3 .cos 50 , ^^
(10) acoe0 + J«* — 5 — + o~l*^ — 5 — + •••» when « = COS 20.
[(1) C = -ilog(4sin-|). 5=tan-(j^^)=''--i'il tfUebe-
tween and ir, '
(2) C= J sin (cos 0){€^» + €-•*«•), ^ = J cos (cos0){f^ •-€-■»»•),
(3) C = *j being ofthe sign of COS0, 'S = i ^<« (r7^:fl 1 '
112 PLANE TRIGONOMETRT.
2co8^j cos 71^, S'=(2co8^j sinw^j being be
tween — ir and ir,
(5) C=€-«»«co8(sintf), 5^= -€-<»•« sin (sin tf),
cos ^ + sin ^ + ^sin $\y if sin 0. be
positive,
(7) C = log r^cos 5 + ^2 coB^j , and cos*Ay=co8fl, if cos0 b€
positive,
(8) (7 = p*cos2^, S=p*Bm2<f}, where pcos^ = cos"* T^^sin^V
'/ B B / B\
psia4>- log (cos J + sin J + ^^ ''^ 9 ) » ^ being between and 2ir,
(9) C=Jlog(cosec*tf), /S^=tan"'(cotfl),
(10) cos* . = sin fl (sin + cos fl), if cos 2$ be positive.]
^cos20
631. Sum the series
. ,^ sin* 3^ sin*3-»d
8m*fl+ — =— +... +
■-1 9
3 3
,. 008* 3g . ,,._. C0B'3-'g
0OS» g +...+(-1) g.-t ,
008 COS 30 COS 3'~'d
T" '^ Qt/l + ••• + •
m/i 9
sin3fl 8in3V sin3"d
sin 2B 3sin6g 3""* sin 2 (3"-*fl)
l + 2cos2d l4-2cos6d l + 2cos2(3-»tf)'
2co8g-co8 3g 2*cos3-^g-2-'coa3'g
sin3^ ^ •*•■*■ 8in3"d
1 -f 2 cos 2^ 1 + 2 cos 2'^"'$
smie ■^•••■^ sin2«-d '
1 - 2 cos 2g l~2cos2"g
sin26^ +..-+^ gij^2-d '
gsin3^-3sin5g 5 sin 3 (4-'g) - 3 sin 5 (4- 'g)
cos3d-cos5fl •*••••■*■* co8 3(4"-»^)-co8 5(4"-»d) '
l4-4 8ing8in3g «^, 1 4- 4 sin 4'^ ^g sin 3 (4-^^)
3 sin g - sin 3g 3 8in3"*g-sin 3'g
^^^"35 "*■'•■*■ 3"-"*cos3"d '
PLANE TRIGONOMETRY. 113
632. Prove that
sec6 + sec(^ ^6j + sec (— +^j + ...+8ec<2(w-l)— +tf>
M-l
is equal to 0, or to (- 1 ) * m sec mOf according as m is even or odd :
also that
8ec'^ + fiec*^^%^^-h...4.sec'|2(wi-l)-+fl|
VI
is equal to , or to w* sec* mO, according as m is even
l-(-l/cosm^
or mid.
[The equation which expresses coHfnO in terms of co8 will be
satisfied by cos^, cos (- - + dj , ..., coS'{(w- 1) — + ^[; the results of
the question follow on finding the sum of the reciprocals of the roots,
and the sum of their squares.]
633. Prove that
sec" - + sec — + . .. + sec
n n
cos ^ + cos (a + ^) + cos(2a + tf) + ... + COS (w»- lo + tf) = 0,
if ma = 4ir, m being any positive integer except 2 ; that
.(«.,)r.iK-4,,
if n be any even positive integer except 2 ; and that
, IT ,2w .3w ,n— l* n*— 1
sec" - + sec* — +sec* — +... + sec*— jr = — s- ,
n n n 2 n 2
if n be any odd positive integer except 1.
634. Prove tliat, if n be a whole number > 4,
-cos * — Bin" — cos * — + Bin" — coe * — ...=0,
fl
sm
n n n n n n
the number of terms being — - — , or - — 1, as n is odd or even.
[The roots of the equation (1 + a;)" = (1 - a;)" are the values of
t tan — , where r may have all integral values from to n— 1, omitting
^ if n be even. Hence ^ v.^., — -rs = 3 ^ ^ , where r has
n
all integral values from 1 to — s— or ^r -1 and A is (-1)^* sin* — cos""* —.1
° 22 ^ ^ ' n n -*
W. P. S
114? PLANE TRIGONOMETRY.
635. Prove that
n sin ruf^ sin ^ sin ^
008 rnfi — cos n$ cos ^ — cos $ cos ^ - cos (a + $)
^ sin <f> sin ^
•" ... "»
cos «^ - cos (2a + ^) ••• 008<^-COS(7l-la+6)
nBmn$ anO sin(a4-0)
and . ii = : 7. +
cos n^ — COS 710 cos ^ — cos cos^— cos(a+0)
sin (2a 4-0) sin(n-la + 0)
cos«^-cos(2a + 0) *" cos<^-ooa(n^o + 0)'
where n is a positive integer, and na = 2ir.
CONIC SECTIONS, GEOMETRICAL.
I. Parabola,
[The focus and vertex are denoted always by S and A respectively.]
636. Two parabolas having the same focus intersect: prove that
the angles between their tangents at the two points of intersection are
either equal or supplementary.
637. A chord PQ of a parabola is a normal at P and subtends a
right angle at the focus : prove tliat SQ is twice SP^ and that PQ
subtends a right angle at one end of the latus rectum.
638. A chord PQ of a parabola is a normal at P and subtends a
right angle at the vertex : prove that SQ is three times SP.
639. Two circles each touch a [mrabola and touch each other at the
focus of the parabola : prove that the angle between the focal distances
of the points of contact is 120^
640. Two parabolas have a common focus and axes at right angles,
a circle is drawn touching both and passing through the focus: prove
that the points of contact are ends of a diameter, or subtend an angle of
Z{f at the focus.
641. Two parabolas have a common focus, a circle is described
touching both and passing through the focus : prove that the angle
between the focal distances of the points of contact will be one third
of the angle between the axes, or one third of the defect from four right
angles of this angle.
642. Two parabolas Ay B have a common focus and axes at right
angles : prove that any two tangents drawn to il at right angles to each
other will be equally inclined to the tangents drawn to B from the same
point.
643. In a parabola AQ is drawn through the vertex A at right
angles to a chord AP to meet the diameter through Pm Q: prove tJ^t
Q lies on a fixed straight line.
116 CONIC SECTIONS, GEOMETRICAL.
614. Thi-ough any point /* of a parabola is drawn a straight line
QPQf perpendicular to the axis, and terminated by the tangents at the
end of the latus rectum : prove that the distance of P from the latus
rectum is a mean propoi-tioual between QP, PQ\
645. A circle touches a pai*abola at a point whose distance fram
the focus exceeds the latus rectum, and passes through the focus : prove
that it will cut the parabola in two points, and that the common chord
will cut the axis of the parabola in a fixed point at a distance from the
focus equal to the latus rectum.
646. A parabola is described touching a given circle, and having
its focus at a given point on the circle : prove that if the distance of the
point of contact from the focus be less than the i*adius of the circle, the
circle and parabola will have two other common tangents whose common
point will lie on a fixed straight line which bisects the radius drawn
from the focus.
647. With a given point as focus is described a parabola touching a
given circle : prove that the point of intersection of the two other
common tangents lies on a fixed circle, such that the polar of the given
point with respect to it passes through the centre of the given circle.
[If the given point lie on the given circle, the locus degenerates into
the straight line bisecting at right angles the radius through the given
point.]
648. On the tangents drawn from a point are taken two points
Py Q such that SP, SO^ SQ are all equal : prove that PQ is perpen-
dicular to the axis and its distance from is twice its distance from A.
649. Two equal parabolas have a common focus S and axes opposite,
and SPQ is any straiglit line meeting them m P, Q ', with centres P, Q
are drawn circles touching the respective tangents at the vertices : prove
that these circles will have internal contact, and that the rectangle under
their radii will be fixed.
650. On a focal chord PQ as diameter is described a circle which
meets the parabola again in P'yQ'', prove that the circle FSQ' will touch
the parabola.
651. A circle touches a parabola in P, passes through S and meets
the parabola again in Q, ^; a focal chord is drawn parallel to the
tangent at P : prove that the circle on this chord as diameter will [muss
through Q, Q, and that the focal chord and QQ' will intersect on the
directrix.
652. Two parabolas whose foci are S^ S' have three common
tangents, and the circle circumscribing the triangle formed by these
tangents is drawn : prove that SS' will subtend at any point on this
circle an angle equal to that between the axes of the parabolas.
653. From any point on the tangent at any point of a parabola
perpendiculars are let fall on the focal distance and on the axis : prove
that the sum, or the difference, of the focal distances of the feet of these
/>ei7)endicular8 is equal to half the latus rectum.
CONIC SECTIONS, OEOMETBICAL. 117
654. The normal at a point P is produced to so that PO is
bisected by the axis : prove that any chord through subtends a right
angle at P; and that the circle on PO as diameter will have double
contact with the })arabola.
655. From a fixed point is let fall OQ perpendicular on the
diameter through a point P of a parabola : prove that the perpendicular
from Q on the tangent at P will pass through a fixed point, which
remains the same for all equal parabolas on a common axis.
656. A circle is drawn through two fixed points i?, S^ and meets a
fixed straight line through R again in P : prove that the tangent at P
will touch a fixed parabola whose focus is S.
657. Two fixed straight lines intersect in : prove that any circle
through and through another fixed point S meets the two fixed lines
again in points such that the chord joining them touches a fixed
parabola whose focus is S,
658. The perpendicular AZ on the tangent at P meets the parabola
again in Q : prove that the rectangle ZA^ AQ \a equal to the square on
the semi latuis rectum and that PQ passes through the centre of curva-
ture at A.
659. Two parabolas have a common focus and axes at right angles :
prove that the directrix of either passes through the point of contact of
their common tangent with the other.
660. Through any point P on a parabola is drawn PK at right
angles to ^Z' to meet the axis in K: prove that AK \a equal to the
fucal chord parallel to AP, £xplain the result when P coincides with A .
661. A circle on a double ordinate to the axis PP* meets the
pirabola again in Q, ^ : prove that the latus rectum of the parabola
which touches PQ, PQ\ PQ, FQ[ is double that of the former, and its
focus is the centre of the circle.
662. l*hree points A, B,C kc^ taken on a parabola, and tangents
drawn at them forming a triangle A!BC'\ a, 6, c are the centres of the
circles BCA\ CAH^ ABC: prove that the circle through a, 6, e will pass
through the focus.
663. Two points are taken on a parabola, such that the sum of the
parts of the normals intercepted between the points and the axis is equal
to the part of the axis intercepted between the normals : prove that the
difference of the normab is equal to the latus rectum.
664. The perpeodiciilar SY beiog drawn to any tangent, a straight
line is drawn through Y parallel to the axis to meet in Q the straight
line through S |)arallel to the tangent : prove that the locus of (^ is a
parabola.
665. K X be the foot of the directrix, SY perpendicular from the
focus on a chord PP', and a circle with centre S and radius equal U> XY
meet the chord in QQ^: prove that PJ^, QQ' subtend equal angles at S,
118 OOHIC SECnOKSy GEOMETRICAL.
666. A given ttraigfat line meets one of a aeries of coaxial circles in
A, B: prove that the parabola which touches the given straight line,
the tangents to the circle at A, B, and the common radical axis will have
another fixed tangent
[If JT be a point circle of the system, L the intersection of the given
straight line with the radical axis and KO drawn at right angles to ICL
to meet the radical axis in O, the fixed tangent is the straight line
through perpendicular to the given straight line.]
667. Two tangents TP, TQ are drawn to a parabok, OP, OQ are
tangents to the circle TPQ : prove that TO will pass through the focus.
668. A triangle ABC is inscribed in a circle, AA' \a k diameter, a
parabola is described touching the sides of the triangle with its directrix
passing through A' and >5^ is its focus : prove that the tangents to the
circle at B^ C will intersect on ^^1'.
669. The normals at two points P, Q meet the axis in /?, ^ and the
chord PQ meets it in : prove that straight lines drawn through 0, />, q
at right angles respectively to the three lines will meet in a point.
670. Normals at P, P" meet the axis in (r, G\ and straight lines at
right angles to the normab from G, G' meet in Q : prove that
PG.GQ = FG''.G'Q.
671. The tangent to a parabola at P meets the tangent at (^ in T*
and meets HQ in R\ also the tangent at Q meets the directrix in K\
prove that PT^ TR subtend equal or supplementary angles at K.
672. Two equal parabolas have a common focus and axes inclined
at an angle of 120®: prove that a tangent to either curve at a common
IX)int will meet the other in a point of contact of a common tangent.
673. The chord PR is normal at P, is the centre of curvature at
P and U the pole of PR : prove that OU will be pei-pendicular to SP,
674. From a fixed point is drawn a straight line OP to any point
P on a fixed straight line : prove that the straight lines drawn through
P equally inclined to PO and to the fixed straight line touch a fixed
jiaraDola.
675. A parabola whose focus lies on a fixed circle and whose
directrix is given, always touches two fixed parabolas whose common
focus is the given centre, and whoso directrices are each at a distance
from the given directrix equal to the given radius; and the tangents at
the pointtf of contact are at right angles.
676. Tlie centre of curvature at P is 0, PO meets the axis in G and
OL is drawn (lerijendicular to the axis to meet the diameter through P :
prove that LG is parallel to the tangent at P.
677. The straight lines Aa^ Bb, Cc are drawn perpendicular to the
sides BCf CA^ AB of a triangle ABC : prove that two parabolas can be
drawn touching the sides of the triangles ABC, ahc respectively, such
that the tangent at the vertex of the former is the axis of the latter.
CONIC SECTIONS, OEOMETRIGAU 119
678. A right-angled triangle is described self-conjugate to a given
parabola and with its hypotenuse in a given direction : prove that its
vertex lies on a fixed straight line parallel to the axis of the parabola and
its sides touch a fixed parabola.
679. Two equal parabolas have their axes in the same straight line
and their vertices at a distance equal to the latus rectum ; a choid of the
outer touches the inner and on it as diameter is described a circle : prove
that this will touch the outer parabola.
680. Tangents are drawn from a fixed point to a series of con-
focal parabolas : prove that the corresponding normals envelope a fixed
parabola whose directrix passes through and is parallel to the axis of
the system, and whose focus ^S" is such that OS^ is bisected by S.
681. A point on the directrix is joined to the focus S and SO
bisected in F\ with focus F is described another parabola whose axis is
the tangent at the vertex of the former and from two tangents are
drawn to the latter parabola : prove that the chord of contact and the
corresponding normals' all touch the given parabola.
682. Prove the following construction for inscribing in a parabola a
triangle with its sides in given directions : — Draw tangents in the given
directions touching at A^ B, C7, and chords AA\ BBf^ CC parallel to
BC,CA,AB\ A' EC wiU be the required triangle.
[The construction is not limited to the parabola, and a similar
construction may be made for an inscribed polygon.]
683. Two fixed tangents are drawn to a parabola : prove that the
centre of the nine points' circle of the triangle formed by these and any
other tangent is a straight line.
684. At one extremity of a given finite straight line is drawn any
circle touching the line, and from the other extremity is drawn a tangent
to this circle : prove that the point of intersection of this tangent with
the tangent parallel to the given line lies on a fixed parabola, and those
with the tangents perpendicular to the given line on two fixed hyperbolas.
685. Two parabolas have a common focus and from any point on
their common tangent are drawn other tangents to the two : prove that
the distances of these from the focus are in a constant ratio.
686. Two tangents ai*e drawn to a parabola equally inclined to a
given straight line : prove that their point of intersection lies on a fixed
straight line passing through the focus.
687. Two parabolas have a common focus S^ parallel tangents drawn
to them at P, Q meet their common tangent in /^, (^ : prove that the
angles FSQy FSQ are each equal to the angle between the axes.
688. Two parabolas have parallel axes and two parallel tangents
are drawn to them : prove that the straight line joining the points of
contact passes through a fixed point.
[A general pix)perty of similar and similarly situate figures. |
120 CONIC SECTIONS, GEOMETRICAL.
689. On a tangent are taken two points equidistant from tlie focus :
prove that the other taugents drawn firom these points will intersect on
the asds.
690. A circle is described on the latus rectum as diameter and a
straight line through the focus meets the two curves in P, Q : prove that
the tangents at P^ Q will intersect either on the latus rectum, or on a
straight line parallel to the latus rectum and at a distance from it equal
to the latus rectum.
691. A chord is drawn in a given direction and on it as diameter a
circle is described : prove that the distance between the middle points of
this chord and of the other commpn chord of the circle and parabola is
of constant length.
692. On any chord as diameter is described a circle cutting the
parabola again in two points: prove that the part of the axis of the
parabola intercepted between the two common chords is equal to the
latus rectum.
693. Two equal parabolas are placed with their axes in the same
straight line and their vertices at a distance equal to the latus rectum ;
a tangent drawn to one meets the other in two points : prove that the
circle of which this chord is a diameter touches the parabola of which
this is a chord.
694. A parabola is described having its focus on the arc, its axis
parallel to the axis, and touching the directrix, of a given parabola :
prove that the two curves will touch each other.
695. Circles are described having for diameters a series of parallel
chords of a parabola : prove that they will all touch another parabola
related to the given one in the manner described in the last question.
696. A circle is described having double contact with a parabola
and a chord QQ' of the parabola touches the circle in P : prove that
QPy Q'P are respectively equal to the distances of $, Q' from the com-
mon chord.
697. The locus of the centre of the circle circumscribing the tri-
angle formed by two fixed tangents to a parabola and any other tangent
is a straight line.
698. The locus of the focus of a parabola touching two fixed straight
lines one of them at a given point is a circle.
699. Two equal parabolas Aj B have a common vertex and axes
opposite : prove that the locus of the poles with respect to ^ of tangents
to B\& A,
700. Three common tangents PF, QQ\ RR' ai-e drawn to two
Darabolas and PQ, FQf intersect in L : prove that LR, LR' are parallel
the axes. Also prove that if PP* bisect QQ' it will also bisect RR^
I PF will be divided harmonically by QQ\ RR.
CONIC SECTIONS, GEOMETRICAL. 121
701. Two equal parabolas have a coramon focus and axes opposite ;
two circles are described touching each other, each with its centre on one
parabola and touching the tangent at the vertex of that parabola : prove
that the rectangle under their radii is constant whether the contact be
internal or external, but in the former case is four times as great as in
the latter.
702. Two equal parabolas have their axes jmrallel and opposite,
and one passes through the centre of curvature at the vertex of the other :
prove that this relation is reciprocal and that the parabolas cut at right
angles.
703. From the ends of a chord PP* are let fall i)erpendicular8
PJ/, FM' on the tangent at the vertex : prove that the circle on PP*
as diameter and the circle of curvature at the vertex have PP' for
radical axis.
[The analytical proof of this is instantaneous.]
704. A parabola touches the sides of a triangle ABC in A\ B'y C^
BC meets BC in P^ another parabola is drawn touching the sides and
P is its point of contact with BC : prove that its axis is parallel
to B'C\
705. The directrix and one point being given, prove that the para-
bola will touch a fixed parabola to which the given straight line is
tangent at the vertex.
706. The locus of the focus of a parabola which touches a given
parabola and has a given directrix parallel to that of the given parabola
is a circle.
707. A triangle is self-conjugate to a parabola, prove that the
straight lines joining the mid points of its sides touch the parabola ; and
that the straight line joining any angular point of the tiiangle to the
point of contact of the corresponding tangent will be parallel to the
axis.
708. Four tangents are drawn to a parabola : prove that the three
circles whose diameters are the diagonals of the quadrilateral will have
the directrix as common radical axis.
709. A circle is di-awn meeting a parabola in four points and
tangents drawn to the parabola at these points : pix>ve tliat the axis of
the parabola will bisect the diagonals of the quadiilateral so formed.
710. The tangents at P, Q meet in 7*, and is the centre of the
circle TPQ : prove that OT subtends a right angle at S and that the
circle OPQ passes through S,
711. Three parallels are drawn through A, B, C to meet the
opposite sides of the triangle ABC in A\ B", C : prove that a i)arabola
can be drawn through A'B'C and tlie middle points of the sides, and
that its axis will be in the same direction as the three parallels.
122 CONIC SECTIONS, GEOMETRICAL.
712. A chord LL of a circle is bisected in 0, and // is its pole;
two parabolas are described with their focus at 0, their directrices
passing through Hy and one of their common points on the cii-cle : prove
that the angle between their axes is equal to LUL\
II. CevUrcd Conies,
[In these questions, unless other meanings are expressly assigned,
S, tS" are the foci of a central conic, C the centre, A A', Blf the major
and minor axes, T, t and 6r, g the points where the tangent and normal at
a point F meet the axes, and CD the semi-diameter conjugate to CP.^
713. If SYj SZ be perpendiculars on two tangents the straight line
drawn through the intersection of the tangents perpendicular to YZ
will pass through S^.
714. If SYy SZ be drawn perpendicular respectively to the tangent
and normal at any point, YZ will pass through the centre.
715. A common tangent is drawn to a conic and to the circle whose
diameter is the latus rectum : prove that the latus rectum bisects the
angle between the focal distances of the points of contact
716. If a triangle ABC circumscribe a conic the sum of the angles
subtended by BC at the foci will exceed the angle A by two right
angles.
717. Two conies Uy V have a common focus S, the tangents to ^ at
two common points meet in P and to F in Q : prove that PQ passes
through S (the common points being rightly selected when there are
four).
718. Perpendiculars aST, S'Y' are drawn on any tangent and FP,
F'P' are the other tangents from 7, T' : prove that SP, S'F will inter-
sect on the conic.
719. A circle touches the conic at P and passes through S, PQK
drawn perpendicular to the directrix meets the circle in Q : prove that
QSK is a right angle.
720. On a tangent are taken two points 0, 0' such that SO = SO*
= major axis : prove that the radius of the circle OSO' is equal to the
major axis.
721. Tangents OP, OQ, O'F, UQ! are drawn to a certain circle :
prove that the foci of the conic which touches the sides of the two tri-
angles OPQ, O'FQ He on the circle.
722. In an ellipse in which BB = SS^ a diameter PF is taken and
circles drawn touching the ellipse in P, F and passing through ^S' : thoir
second common point will lie on the latus rectum.
CONIC SECTIONS, GEOMETRICAL. 123
723. Prove that when SG = FG, SP is equal to the latua rectum ;
and if PK drawn always at right angles to JSP meet the axis major
in K, SK has then its least possible length.
724. A common radius CPQ is drawn to the two auxiliary circles
of an ellipse and tangents to the circles at P, Q meet the corresponding
axes in U^ T: prove that TU will touch the ellipse.
725. A circle has double contact with a conic : prove that the
tangent from any point of the conic to the circle bears a constant ratio
to the distance from the chord of contact.
726. The foot of the perpendicular from the focus on the tangent
at the extremity of the farther latus rectum lies on the minor axis.
727. The common tangents to an ellipse and to a circle through the
foci will touch the circle in i)oints lying on the tangents at the ends of
the minor axis ; and the common tangents to an ellipse and to a circle
with its centre on the major axis and dividing SiS^ harmonically will
touch the circle in points on the tangent at one end of the major axis.
[These two cases are undistinguishable analytically.]
728. The tangent at P and normal at Q meet on the minor axis :
j>rove that the tangent at Q and normal at P will also meet on the minor
axis and PQ will always touch a confocal hyperbola.
[The data ^-ill not be )>ossible for the ellipse unless SS' > -B2^.]
729. Prove that at the point P where SP = PG,
SP : IIP = BC : ACT.
730. A tangent meets the auxiliary circle in two points through
which are drawn chords of the circle parallel to the minor axis : prove
that the straight line drawn from the foot of the ordinate parallel to the
tangent will divide either chord into segments which are as the focal dis-
tances of the points of contact.
731. Two diagonals of a quadrilateral intersect at right angles :
prove that a conic can be inscribed with a focus at the intersection of the
diagonals.
732. Given a focus S and two tangents, the locus of the second
focus is the straight line through the intersection of the tangents per-
pendicular to the line joining the feet of the perpendiculars from S on
the tangents.
733. Given one focus, a tangent, and a straight line on which the
centre lies, prove that the conic has a second fixed tangent
734. From the foci *V, S' are drawn perpendiculars SPT, S'P'Y' on
any tangent to the auxiliary circle meeting the conic in P, P* : prove
that the rectangle SY, S^P' = the rectangle S^T, SP = BC*.
735. The length of the focal perpendicular on any tangent to the
auxiliary circle is equal to the focal distance of the corresponding point
on the ellipse.
124 CONIC SECTIONS, GEOMETRICAL.
736. Through C is drawn a straight line parallel to either focal dis-
tance of P, and CD is the radius parallel to the tangent at P : prove
that the distance of D from the former straight line is equal to BC.
737. Prove that, if an ellipse be inscribed in a given rectangle, the
points of contact will be the angular points of a parallelogram of con-
stant perimeter : and investigate the corresponding theoi*em when the
conic is an hyperbola.
738. A straight line is drawn touching the minor auxiliary circle
meeting the ellipse in P and the director circle \xi Qy Q : prove that
QPy PQ' are equal to the focal distances of P,
739. Given a focus and two tangents of a conic, prove that the
envelope of the minor axis is a parabola with its focus at the given
focus : also a common tangent to this parabola and any one of the conies
subtends a right angle at the given focus.
740. A perpendicular fi*om the centre on the tangent meets the
focal distances of the point of contact in two points : prove that these
})oints are at a constant distance SG from the feet of the focal perpen-
diculars on the tangent
741. The tangent at a point P meets the major axis in T ] prove
that
SP : ST '.: AN : AT.
742. The circle passing through the feet of the perpendiculars from
the foci on the tangent and through the foot of the ordinate will pass
through the centre ; and the angle subtended at either end of the major
axis by the distance between the feet of the perpendiculars will be equal
or supplementary to the angle which either focal distance makes witli
the corresponding perpendicular.
743. Given a focus and the length and dii-ection of the major axis,
pi*ove that a conic will touch two fixed parabolas whose common focus is
the given focus and semi latus rectum along the given line and of the
given length.
744. A conic is described having one focus at the focus of a given
parabola and its major axis coincident in direction with and equal to
half of the latus I'ectum of the parabola : prove that this conic will touch
the parabola.
745. A conic touches two adjacent sides of a given parallelogram
and its foci lie on the two other sides one on each : prove that each
directrix touches a fixed parabola. If A BCD be the parallelogram, S^ S'
the foci on BC, CD respectively, and on AS, AS' be taken AL=AB,
AL' = AD, the excentricity of the conic will be the ratio LL' : SS\
746. Three jwints A, B, C are taken on a conic such that CA, CB
ai-e e<jually inclined to the tangent at C : pix)vc that the normal at C
win pasa through the pole of A H,
CONIC SECTIONS, GEOMETRICAL. 125
747. Given one focus, a tangent, and the length of the major axis,
prove that the locus of the second focus is a circle : and determine the
(Mrtions of the locus which correspond to an ellipfe, and those which
correspond to an hyperbola in which the given focus belongs to the
branch which touches the given straight line.
748. Given a focus, the excentricity, and a tangent : prove that the
directrix will touch a fixed conic having the same focus and excentricity,
and the minor axis of this envelope will lie along the given tangent.
749. A conic described with its foci at the centres of two given
intersecting circles and touching a tangent drawn to either circle at a
common point will touch the other tangents at the common points ; its
auxiliary circle will pass through the common points, and any tangent
to the conic will be harmonically divided by the circles.
750. Through any point are drawn two tangents to a conic, and
on them are taken two points F, Q ao that 0, P, Q are equidistant from
S : prove that S'O is perpendicular to FQ, and, if tS'O, FQ meet in i?,
that twice the rectangle S'O, S'Bj together with the square on SO, is
equal to the square on SS\ sign being attended to.
751. In any conic if FO be taken along the normal at F equal to
the harmonic mean between FG, F(j, will be the point such that any
chord through it subtends a right angle at ; and if from F perpendi-
culars be let fall on any two conjugate diameters the straight line join-
ing the feet of these perpendiculars will bisect FO,
752. The triangle ABC is isosceles, A being the vertex, and conies
are drawn toucliing the sides AB, AC and the perpendiculars from B, C
on the opposite sides : prove that the foci of these conies lie on either a
fixed circle or a fixed straight line, and trace the motion of the foci as
the centre moves along the straight line which is its locus.
753. Two diameters FF, QQf of a conic are drawn, and FR, FR
let fall perpendicular on PQ, P'Q'] prove that the chord intercepted
by the conic on RR subtends a right angle at F,
754. A triangle ABC circumscribes a conic, and <Sa, Sb, Sc are
drawn perpendicidars on the sidej : prove that
ASbc : aSBC=^ ASca: aS'CA= ASah: aSAB
= Aahc : A ABC,
755. From a point on an ellipse perpendiculars are let fall on the
axes and produced to meet the corresponding auxiliary circles : prove
that the straight line joining the two points of intersection passes
through the centre.
756. Two conies have common foci S, S' and any straight line being
taken another straight line is drawn joining the poles of the former
with respect to the two conies : prove that the conic whose focns is S
and which touches both these straight lines and the minor axis will be
a parabola and that its directrix will pass through S',
126 CONIC SECTIONS, GEOMETRICAL.
757. An ellipse and hyperbola are confocal, a straight line is drawn
parallel to one of their common diameters and its poles with respect to
the two conies joined to the centre : prove that the joining lines are at
right angles, and that the polars of any point on a common diameter are
also at right angles.
758. Tangents (or normals) are drawn in a given direction to a
series of confocal oonics : prove that the points of contact lie on a rect-
angular hyperbola having an asymptote in the given direction ai^d
passing through the focL
759. An hyperbola and an ellipse are confocal, and from any point
jT on an asymptote are drawn TOy TPy TQ touching the hyperbola in
and the ellipse in P, Q respectively, prove that OT is a mean proportional
between OF and OQ.
760. The tangents drawn to a series of confocal conies at the points
where they meet a fixed straight line through S all touch a fixed para-
bola whose focus is S' and directrix is the given straight line, and which
touches the minor axis.
761. From a point on any ellipse are drawn tangents OF, OQ to
any confocal : prove that the chord of curvature at in direction of
either tangent is double the harmonic mean between OF, OQ ; tangents
drawn outside the ellipse being considered negative.
762. An ellipse is described touching two given confocal conies and
having the same centre : prove that the tangents at the points of con-
tact will form a rectangle. For real contact one of the given conies
must be an ellipse.
763. An ellipse is described having double contact with each of two
confocals : prove that the sum of the squares on its axis is constant,
and that the locus of its foci is the lemniscate in which SF . HF = dif-
ference of the squares on the given semi-major axes.
764. From a point on a given ellipse are drawn two tangents
OF, OQ to a given confocal ellipse and a diameter parallel to the
tangent at meets OF, OQ in the points P', Q' : prove that the har-
monic mean between OS, OS' bears to the harmonic mean between OF,
OQ the constant ratio OS^-OS\OF-^ OQ'.
m
765. On any tangent to a conic are taken two points equidistant
from one focus and subtending a right angle at the other : prove that
their distance from the former focus is constant.
766. The perpendicular CY on a tangent meets an ellipse in F, and
Q is another point on the ellipse such that CQ = CY : prove that the
perpendicular from C on the tangent at Q is equal to CF,
767. A tangent to a conic at F meets the minor axis in T, and TQ
is drawn perpendicular to SF : prove that SQ is of constant length ; and,
FM being drawn perpendicular to the minor axis, that QM will pass
through a fixed point.
CONIC SECTIONS, GEOMETRICAL. 127
7G8. Three tangents to a conic are such that their points of inter-
section are at equal distances from a focus : prove that each distance is
equal to the major axis; and that the second focus is the centre of
perpendiculars of the triangle formed bj the tangents.
769. A conic is inscribed in a circle and is concentric with the nine
points' circle of the triangle : prove that it will have double contact with
the nine points' circle.
770. An ellipse is inscribed in an acute-angled triangle ABC with
its foci S^ S" at the centre of the circumscribed circle and the centre of
perpendiculars respectively; SA, JSB, SC meet the ellipse again in a, 6, c :
prove that the tangents at a, 6, e are parallel to the sides of ABCy and
form a triangle in which S^, S are the centre of the circumscribed circle
and the centre of perpendiculars respectively.
771. With the centre of perpendiculars of a triangle as centre arc
described two ellipses, one inscribed in the triangle the other circum-
scribing it : prove that these ellipses are similar and their major axes at
right angles ; and that the diameters of the inscribed conic parallel to
the sides are as the cosines of the angles.
772. With a focus of a given conic as focus and any tangent as
directrix is described a conic similar to the given conic : prove that it
will touch the minor axis. If with the same focus and directrix a para-
bola be described it will intercept on the minor axis a segment subtending
a constant angle at the focus.
[In general if the described conic have a given excentricity e not less
than that of the given conic it will intercept on the minor axis a
segment subtending at the given focus a constant angle = 2 cos~* (->]•]
773. With a focus ^ of a given conic as focus and any tangent as
directrix is described a conic touching a fixed straight line perpendicular
to the major axis, another fixed straight line is drawn parallel and
conjugate to the former : prove that the segment of this latter straight
line intercepted by the variable conic subtends at S the same angle
as the segment intercepted by the given conic.
774. With the vertex of a given conic as focus and any tangent as
directrix is described a conic passing through one of the foci of the given
conic : prove that the major axis is equal to the distance of either focus
of the given conic from its directrix.
775. An ellipse is inscril)ed to a given triangle with its centre at
the circumscribed circle of the triangle : prove that both auxiliary
circles of the ellipse touch the nine points' circle of the triangle ; and
that the three perpendiculars of the triangle are normals to the
ellipse.
776. A conic touches the sides and passes through the centre of
the circumscribed circle of a triangle : prove that the director circle of
the ellipse will touch the circumscribed circle of the triangle.
128 CONIC SECTIONS, GEOMETRICAL.
777. A diameter FF* being fixed, QYQ! is any ciiord parallel to it
bisected in F, and FY intersects CQ or CQ! in R : prove that the locus
of jS is a parabola.
778. A chord is drawn parallel to the major axis and circles drawn
through S to touch the conic at the ends of tJie chord : prove that the
second common point of the circles is the intersection of the chord with
the focal radius to its pole ; and that the locus of this point is a parabola
with its vertex at S.
779. With anj point on a given circle as focus and a given
diameter as directrix is described a conic similar to a given conic : prove
that it will touch two fixed similar conies to which the given diameter
is latus rectum, its points of contact lying on the radius through the
focua
780. Given the side BG of a triangle ABC and that
cos A -m cos B cos (7,
prove that the locus of ^ is an ellipse of which BC is the minor axis
when m is positive, an ellipse of which BC is major axis when m is
negative but 1 + m positive, and an hyperbola of which BC is transverse
axis when 1 + m is negative.
781. Given a focus S and two tangents to a conic, prove that.the
envelope of the minor axis is a parabola of which S is focus.
782. A circle is drawn touching the latus rectum of a given ellipse
in S the focus on the side towards the centre and also touches the
tangents at the ends of the latus rectum : prove that the two other com-
mon tangents will touch the ellipse in points lying on a tangent to the
circle.
783. From the iod Sy SC are let fall perpendiculars ST, ST on any
tangent to an ellipse : prove that the perimeter of the quadrilateral
SY TS' will be the greatest possible when YY' subtends a right angle
at the centre.
[This is only possible when SS' is greater than BB'; when SS' is
equal to or less than BB* the perimeter is greatest when the point of
contact is the end of the minor axis.]
784. A conic is described having one side of a triangle for directrix,
the opposite vertex for centre and the centre of perpendiculars for focus :
prove that the sides of the triangle which meet in the centre are con-
jugate.
785. The angle which a diameter of an ellipse subtends at an
extremity of the minor axis is supplementary to that which its conjugate
subtends at the ends of the major axis.
786. Two pairs of conjugate diameters of an ellipse are FF*, DI/;
pp'y ddT respectively ; prove that Fp, Fp' are respectively parallel to
V'd, Ud\
CONIC SECTIONS, GEOMETRICAL. 129
787. Tangents TV, TQ are drawn to a conic and chords Qq^ Pp
parallel to TV, TQ respectively : prove that pq is paitdlel to PQ Also
prove that the diameters parallel to the tangents form a harmonic
[)encil with CT and the diameter conjugate to CT.
788. A chord QQ^ of an ellipse is paraUel to one of the equal
conjugate diameters and QxV, Q:N' are perpendiculars on an axis : prove
that the triangles QCN, QfCN' are equal and that the normals at Qy Q'
intersect on the diameter which is perpendicular to the other equal con-
jugate diameter.
789. Any ordinate NP of an ellipse is produced to meet the
auxiliary circle in Q and normals to the ellipse and circle at P, Q meet
in J? ; RKj PL are drawn perpendicular to the axes : prove that A", P, L
lie on one straight line and that KP^ PL are equal respectively to the
semi-axes. (The point Q may be eitlier point in which NP meets the
auxiliary circle.)
790. On the normal to an ellipse at P are taken two points Q, Q'
such that QP = PQ' = CD : prove that the cosine of the angle QC(y is
CP* — CD*
Mjyt _ p/rt } w^^ i^ froD^ Q or Q' be drawn a straight line normal to the
ellipse at 7?, the parts of this straight line intercepted between P and
the axes will be equal respectively to BC and AC,
791. Through any point Q of one of the auxiliary circles is drawn
QPP perpendicular to the axis of contact meeting the ellipse in P, i** :
prove that the normals to the ellipse at P, P* intercept on the normal
to the circle at Q a length equal to the diameter of the other auxiliary
circle.
792. Tangents to an eUipse at P, D ends of conjugate diameters
meet in 0, any other tangent meets these in P, D' respectively : prove
that the rectangle under OP, OD' is double that under PP*, DD\
[The ratio of the two rectangles is constant for any two fixed
points P, 2>, having a value depending on the area cut off by the seg-
ment PD.\
793. A chord PQ is drawn through one fociis, Z is its pole and
the centre of the circle LPQ\ prove that the circle OPQ will pass througli
the second focus.
794. Through two fixed points A^ B of a conic are drawn chords
i4P, BQ parallel to each other: prove that PQ always touches a
concentric similar and similarly placed conic.
795. A parallelogram A BCD circumscribes a given conic and a
tangent meets AB, AD in P, Q, and CB, CD in P, Q': prove that the
rectangles BP^ DQ, and BP\ DQ' are equal and constant.
796. An equilateral triangle PQR is inscribed in an auxiliary circle
of an ellipse and P', Q\ R ai*o the corresponding points on the ellipse :
prove that the circles of curvature at P, ^, R meet in one point lying
on the ellipse and on the circle PQ'R.
W. P. ^^
130 CONIC SECTIONS, GEOMETRICAL.
797. A conic, centre 0, is inscribed in a triangle ABC and through
B, C are drawn straight lines parallel to the diameter ooi^ugate to OA :
prove that these straight lines will be conjugates.
798. A chord EF of a given circle is divided in a given ratio in S]
oonstruct a conic of which E is one point, S a focus, and the given circle
the circle of curvature at E,
799. A point P is taken on an ellipse equidistant from the minor
axis and a directrix; prove that the circle of curvature at F will pass
through a focus.
800. An ellipse is drawn concentric with a given ellipse, similar to
it, and touching it at a point F ; prove that the areas of the two are as
CF* : SF.S'F; and their cuiTatures at P in the duplicate ratio of
SP.STiCF'.
801. Any chord FQ of an ellipse meets the circle of curvature at F
in Q^ : prove that FQ' has to FQ the duplicate ratio of the diameters of
the ellipse which are respectively parallel to the tangent at F and to the
chord FQ.
802. Two circles are described with S, S' as centres and intersecting
in F, F; prove that with any point on l^e conic, whose foci are S^ S'
and which passes through F, as centre, can be described a circle touching
both the former, and tibat all these tangent circles cut at right angles a
fixed circle touching the conic in P, F\
803. Given a focus, a point, and the length of the major axis ;
prove that the envelope of either directrix is a conic having its focus at
the common point and excentricity equal to the ratio of the local
distances of the common point.
804. Given a point and the directrices; prove that the locus of each
focus is a circle, and the envelope of the conic is a conic having the
given point for focus and the distances between the directrices for
major axis.
805. A circle is described having internal contact with each of two
given circles one of which lies within the other, and the centre F of the
moving circle describes an ellipse of which A A' is the major axis; through
A is drawn a diameter of the moving circle ; prove that the ends of this
diameter will lie on an ellipse similar to the locus of P, and having a
focus at A and centre at A\
806. A conic has one focus in common with a given conic, touches
the given conic and passes through its second focus: prove that the
major axis is constant.
807. Two similar conies U, V are placed with their major axes in
the same straight line, and the focus of ^ is the centre of V : prove that
the focal distance of the point of contact with 6^ of a common tangent is
equal to the semi-mnjor axis of Y.
CONIC SECTIONS, GEOMETRICAL. 131
808. In a conic one focns, the ezoentricity, and the direction of
the major axis are given, and tangents are drawn to it at points where it
meets a given circle having its centre at the given focus : prove that
these tangents all touch a fixed conic having the given excentricity and
whose auxiliary circle is the given circla
809. The tangent to a conic at P meets the axes m Tyt and the
central radius at right angles to CP in Q : prove that the ratio of QT to
Qt is constant.
810. Through a given point on a given conic are drawn chords
OP, OQ equally inclined to a given direction: prove that PQ passes
through a fixed point
811. A chord PQ is normal at P to a given conic and a diameter
LV is drawn bisecting PQ\ prove that PQ makes equal angles with
LP, L'P and that LP + L'P is constant
812. A conic is described through the foci of a given conic and
touching it at the ends of a diameter : prove that the rectangle under
the distances of a focus of this conic from the foci of the given conic
is equal to the square on the semiminor axis of the given conic ; and
that the diameter of this conic which is conjugate to the major axis of
the given conic is equal to the minor axis of tiiat conic.
813. A conic is inscribed in a triangle ABC and has its focus at ;
the angles BOC, COA, AOB are denoted by A\ R, C; prove that
OAmnA _ OBsiaB OCamC .
Bin(^'-i<)"8in(i5'-^)-8in(C"-C)"""^'^^''"^
Under what convention is this true if be a point without the triangle?
814. Two conies are described having a common minor axis and
such that the outer touches the directrices of the inner; MPF is a
common ordinate ; prove that MF is equal to the normal at P.
815. Two tangents OA, OB are drawn to a oonic and a straight
line meets the tangents in Q^ Q\ the chord AB va, R, and the conic
in P, P'; prove that
QP . PQ' '.RF^QF . FQ' : RF*,
and that for a given direction of the straight line each of these ratios
is constant.
816. With B the extremity of the minor axis of an ellipse as centre
is described a circle whose diameter is equal to the major axis, and the
tangents at the end of the major axis meet the other Qommon tangents
to the ellipse and circle in P, P, Q, Q': prove that B, P, F, Q, Q' lie
on a circle whose diameter is equal to the radius of curvature of the
ellipse at B,
817. A chord of an ellipse subtends at S an angle equal to the
angle between the equal conjugate diameters : prove that the foot of the
perpendicular from C on this chord lies on a fixed circle whose diameter
is equal to the radius of the director circle.
132 CONIC SECTIONS, GEOMETRICAL.
818. A parabola is drawn with its focus at ^S^ a focus of a given
conic and touches the conic : prove that its directrix will touch a fixed
circle whose centre is S\ and that the tangent at the vertex of the
parabola touches the auxiliary circle.
819. A parabola is drawn through the foci of a given ellipse with
its own focus F on the ellipse ; px>ve that the parts of the axis of the
parabola intercepted between P and the axes of the ellipse are of con-
stant length, and if through the points where the axis of the parabola
meets the axes of the ellipse straight liues be drawn at right angles to
the axes of the ellipse their point of intersection will Ue upon the
normal to the ellipse at P.
820. A given finite straight line is one of the equal conjugate
diameters of an ellipse ; prove that the locus of the foci is a lemniscate
of Bernoulli.
821. A parallelogi*am is inscribed in a conic and from any point
on the conic are drawn two straight lines each parallel to two sides :
prove that the rectangles under the segments of these lines cut oShj the
6ides of the parallelogram are in a constant ratio.
822. Two central conies in the same plane have two conjugate
diameters of the one parallel respectively to two conjugate diameters of
the other ; and in general no more.
823. In two similar and similarly placed ellipses are drawn two
parallel chords PF, QQ'; PQ, FQ' meet the two conies in R, Sy Ry S'
respectively : prove that RR\ SS' are parallels : also that QQ\ RR and
PPy SS' intersect in points lying on a fixed straight line.
824. A circle described on the intercept of the tangent at P made
by the tangents at Ay A' meets the conic again in Q] prove that the
ordinate of ^ is to the ordinate of P as the minor axis is to the sura of
tlie minor axis and the diameter conjugate to P, (As BC : BC + CD,)
825. A point P is taken on a conic and is the centre of the cii-clft
SPS'y PO is divided in 0' so that PO' . PO^BC \ AC*: prove that the
circle with 0' as centre and O'P as radius will touch the major axis at
the foot of the normal at P,
826. With a fixed point P on a given conic as focus is described
a parabola touching a pair of conjugate diameters; prove that this
parabola will have a fixed tangent parallel to the tangent at P and that
tliis tangent divides CP in the ratio CF : CD*,
827. Through a point are drawn two straight lines conjugates
with respect to a given conic ; any tangent meets them m Py Q\ prove
that the other tangents drawn from Py Q intersect on the polar of 0.
828. A parabola is described having S for its focus and touching
the minor axis ; prove that a common tangent will subtend a right
angle at S and that its point of contact with cither conic lies on the
directrix of the othor.
CONIC SECTIONS, GEOMETRICAL. 133
829. Prove that the two points oomraon to the director circles
of all couiui iiiscribed in a given quadrilateral may l>e constructed as
follows : take aa'y bb\ ec the three diagonals of the quadrilateral forming
a triangle ABC and let be the centre of the circle ABCy then \£ F^ Q
lie the required i)oiut8, 0, P^ Q lie in one straight line perpendicular to
the bisector of the diagonals, PQ is bisected by this bisector and the
rectangle OP^ OQ is equal to the square of the radius of the circle ABC,
830. At each point P of an ellipse is drawn QPQ' parallel to the
major axis so that QP = PQ' = SP: prove that Q, Q' will trace out
ellipses whose centres are A^ A' and whose areas are together double the
area of the given ellipse. If QPQ' be drawn parallel to the minor axis
instead of the major, the loci are ellipses whose major axes are at right
angles to each other and they touch each other in S and touch tlie
tangents at A^ A',
831. A given ellipse has its minor axis increased and major axis
diminished in the ratio Jl-e : 1, its centre then displaced along the
minor axis through a length equal to a and the ellipse then turned about
its centre through half a right angle: prove that the whole effect is
equivalent to a simple shear parallel to the minor axis by which the
major axis is transferred into tiie position of a tangent at one end of the
latus rectum.
832. A point P is taken on an hyperbola such that CP = CS:
prove that the circle PTG will touch CPtit P, and, \iQ,Q'he two other
IK>int8 such that the ordinate of P is a mean proiwrtional between those
of Qf Qi ^^t the tangents at ^, ^ will intersect on the circle whose
radius is CS,
833. An hyperbola is described through the focus of a parabola
with its own foci on the parabola ; prove that one of its asymptotes is
parallel to the axis of the parabola.
834. A parabola passes through two given points and its axis is in
a given direction : prove that its focus lies on a fixed hyperbola.
835. Two tangents of an hyperbola U are asymptotes of another V\
prove that if Y toucb one of the asymptotes of U it will touch both.
836. In an hyperbola whose excentricity is 2, the circle on a focal
chord as diameter passes through the farther vertex. Any chord of a
single branch subtends at the focus S interior to that branch an angle
double that which it subtends at the farther vertex A', \iRSR be a
chord, ^Pp, tiQq chords inclined at 60* to the former, the circles qPP!^
pQR will intersect in S and A\ and if the former intersect the circle on
the latus rectum in U, F, the angle A'SU is three times A' JSP, and UV
is a diameter of the last-mentioned circle.
837. The straight line joining two points which are coi\jngates with
respect to a conic is bisected by the conic : prove that the line is parallel
to an asymptote.
838. A conic is drawn through two given points with asymptotes
in given directions : prove that the locus of its foci is an hyperbola.
134 CONIC SECTIONS, QEOMETRICAL.
839. A straight line is drawn equidistant from focus and directrix
of an hyperbola, and through any point of it is drawn a straight line at
right angles to the focal distance of the point : prove that the intercept
made by the conic will subtend at the focus an angle equal to the angle
between the asymptotes.
840. Two hyperbolas 27, F are similar and have a common focus,
and the directrix of T is an asymptote of TJ \ prove that the conjugate
axis of ^ is an asymptote of F.
841. In an hyperbola LL is the intercept of a tangent by the
asymptotes : prove that
SL.SL^CL.LL\ and SL . S'L ^CL .LL\
842. To an h3rperbola the concentric circle through the foci is
drawn : prove that tangents drawn from any point on this circle to the
hyperbola divide harmonically the diameter of the circle which lies on
the conjugate axis ; and if OP^ OF the tangents meet the conjugate axis
in Uy U' and PMy FM' be perpendiculars on the conjugate axis, UM\
TTM will be divided in a constant r^tio by C
843. A circle is drawn touching both branches, prove that it inter-
cepts on either asjrmptote a length equal to the major axis ; the tan-
gents to it where it meets the asymptotes pass through one or other of
the fod, and those meeting in a focus are inclined at a constant angle
equal to that between the asymptotes ; and the straight lines joining tiie
points where it meets the asymptotes (not being parallel to the trans-
verse axis) will touch two fixed parabolas whose foci are the foci of the
hyperbola.
III. EecUmgtilar Hyperbola.
[In the questions under this head, b. h. is an abbreviation for
rectangular hypetbola.]
844. Four points A, B, C, D are taken on a R. h. such that BC is
perpendicular to AD : prove that CA is perpendicular to BD and AB
to CD.
845. The angle between two diameters of a B. h. is equal to the
angle between the conjugate diameters.
846. A point P on a r. h. is taken, and PJT, PIT drawn at right
angles to PA, PA' to meet the transverse axis; prove that PK^PA\
and PK' = PA^ and that the normal at P bisects KK'.
847. The foci of an ellipse are ends of a diameter of a r. H. ; prove
that the tangent and normal to the ellipse at any one of the common
points are parallel to the asymptotes of the hyperbola : and that tan-
gi^ntB drawn from any point of the hyperbola to the ellipse are parallel
^ a pair of conjugate diameters of the hyperbola.
CONIC SECTIONS, QEOMETRICAL. 135
848. Prove that any chord of a R. H. subtends at the ends of a
diameter angles either eqaal or supplementary : equal if the ends of the
chord be on the same branch and on the same side of the diameter, or
on opposite branches and on opposite sides ; otherwise supplementary.
849. A circle and r. el intersect in four points, two of which axe
ends of a diameter of the hyperbola : prove that the other two will be
the ends of a diameter of, the circle. Also, if ^^ be the diameter of the
hyperbola, P any point on the circle, and PA^ QB meet the hyperbola
again va Q^ R\ prove that BQ, AR will intersect on the circle.
850. With parallel chords of a b. H. as diameters are described
circles ; prove that they have a common radical axis.
851. The ends of the equal conjugate diameters of a series of con-
focal ellipses lie on the confocal r. h.
852. The ends of a diameter of a R. H. are given ; prove that the
locus of its foci is a lemniscate of Bernoulli, of which the given points
are f ocL
853. From any point P of a R. H. perpendiculars are let fall on a
pair of conjugate diameters of the hyperbola : prove that the straight line
joining the feet of these perpendiculfu*8 is parallel to the normal at F.
854. The tangent at a point P oi a r. h. meets a diameter QCQf in
T : prove that CQ, TQf subtend equal angles at P,
855. Through two fixed points are drawn in a given direction two
equal and parallel straight lines, and on them as diameters circles are
described : prove that the locus of their common points is a r. H. Also
if segments similar to a given segment be described on the two lines on
opposite sides, the locus of their common points is a r. H.
856. If PF, QQ' be diameters, the angles subtended by PQ\ FQ
at any point of the r. h. will be equal or supplementary ; and similarly
for PQ, Fq.
857. Two double ordinates QQ\ RR are drawn to a diameter PF of
a R. H. on opposite branches : prove that a common tangent to the circles
of which QQf^ RR are diameters will subtend a right angle at P and F.
858. Prove that a circle drawn to touch a chord of a R. h. at one
end and to pass through the centre will pass through the pole of the
chord.
859. Two R. H. are such that the asymptotes of one are the axes of
the other : prove that they cut each other at right angles, and that any
common tangent subtends a right angle at the centre.
860. Two points are taken on a R. H. and on its conjugate such that
the tangents are at right angles to each other : prove that the straight
line joining them subtends a right angle at the centre.
861. Tangents to a r. h. at P, meet in T and intersect CQ, CP
reepectiveiy in F, (^ i prove that a circle can be described about
CFTQf.
13G CONIC SECTIONS, QEOMETRICAL.
862. A fixed diameter PR being taken, and Q being auj other
point on the curve : prove that the angles QPF^ QFP differ by a con-
stant quantity.
863. On opposite sides of any chord of a B. EL are described equal
segments of circles : prove that the four points in which the completed
circles again meet the hyperbola are the angular points of a parallel-
ogram.
864. A circle and rectangular hyi^erbola meet in four points : prove
that the diameter of the hyperbola which is perpendicular to a chord
joining two of the points will bisect the chord joining the other two.
805. A point moves so that the straight lines joining it to two fixed
lK>ints ure er|ually inclined to a given direction : prove that its locus is
a a. H. of wldch the two fixed points are ends of a diameter.
866. Circles are drawn through two given points and diameters
di-awn in a given direction : prove that the locus of the extremities of
these diameters is a R. u. whose asymptotes make equal angles with the
line of centres of the circles and with the given direction.
867. Pix>ve that the angles which two tangents to a r. h. subtend at
the centre are equal to the angles which they make with their chord of
contact.
868. A parallelogram has its angular points on a R. H. and from
any point on the hyperbola are drawn two straight lines parallel to the
sides : prove that the four points in which these straight lines meet the
sides of the parallelogram lie on a circle.
869. Two circles touch the same branch of a r. h. and touch each
other in the centre : prove that the chord of the hyperbola joining the
lK)int8 of contact subtends an angle of 60° at the centre.
870. Two unequal parabolas have a common focus and axes oppo-
site ; a R. H. is described with its centre at the common focus touching
both : prove that the chord of the hyperbola joining the points of con-
tact subtends an ansle of 60*^ at the centre.
"O"
871. Through any point on a r. h. are drawn two chords at right
angles to each other : prove that the circle through the point and the
middle points of the chords will pass through the centre.
872. A chord of a r. h. subtends a right angle at a focus : prove
that the foot of the perpendicular on it from the focus lies on a fixed
straight line.
873. A circle meets a r. h. in four points 0, P, $, R and 00\ PP*^
QQ^y RR are diameters of the hyperbola : prove that 0' is the centre of
l>erpendiculars of the triangle PQR, and similarly for the others.
874. Two equal circles touch a R. H. in and meet it again in
/*, Qy P\ Q respectively, 0^ P, Q being on one branch : prove that
"^r, Q(/ arc diameters of the hyiKjrbola, Pq[, P'Q parallel to the
CONIC SECTIONS, GEOMETRICAL. 137
normal at 0, and that the straight lines joiuiug (/, F to the centre of
the circle OFQ will cut off one-third from OP^ OQ respectively.
((/, P will be the centres of perpendiculars of the vanishing tri-
^ es 0, 0, P; 0,0, Q respectively.]
875. The length of a chord of a r. h. which is normal at one
extremity is equal to the corresiK)nding diameter of curvature.
[Take on the hyperbola three contiguous points ultimately coincident
and consider the centre of the circumscribed circle, centroid, and centre
of perpendiculars of the infinitesimal triangle]
876. A diameter PR being taken, a circle is drawn through P*
touching the hyperbola in P ; prove that this circle is equal to the circle
of curvature at P, and that if PI be the diameter of curvature at P, PR
the common chord of the hyperbola and circle of curvature, RI will be
equal and parallel to PP'.
877. A triangle is inscribed in a circle, and two parabolas drawn
touching the sides with their foci at ends of a diameter of the circle :
prove that their axes are asymptotes of a rectangular hyperbola passing
through the centres of the four circles which touch the sides.
878. Three tangents are drawn to a r. h. such that the centre of
the circle circumscribing the triangle lies on the hyperbola : prove that
the centre of the hyperbola will lie on the circle ; and that at any com-
mon point tangents drawn to the two curves pass through the points of
contact of a common tangent
879. A circle meets a b. h. in points P, F, Q, Qf and P, P are ends
of a diameter of the hyperbola : prove that the tangents to the hyper-
bola at P, P and to the circle at Q, Q^ are parallel, and the tangents to
the circle at P, P* and to the hyperbola 9XQ,f/aML meet in one point
880. The tangent at a point of a b. h. and the diameter perpendi-
cular to this tangent being drawn ; prove that the segments of any
other tangent from its i>oint of contact to these two straight lines sub-
tend supplementary angles at the point of contact of the fixed tangent
881. The normal at a point P meets the curve again in Q ; RR
is a chord parallel to this normal : prove that the points of intersection
of QR, PR and of QR, PR lie on the diameter at right angles to CP.
882. A triangle ABC is inscribed in a b. H. and its sides meet one
asymptote in a, 6, o and the other in a', h\ c respectively : through cijh^c
are drawn straight lines at right angles to the corresponding sides of the
triangle : prove that these meet in a point 0, and, 0^ being similarly
found from a\ h\ c\ that OO is a diameter of the circle ABC.
CONIO SECTIONS, ANALYTICAL.
CARTESIAN CO-ORDINATES.
I. Straight Liney Linea/r Trimaformationj Circle.
[In any question relating to the intersections of a curve and two
straight linesy it is generall j convenient to use one equation representing
both straight lines. Thus, to prove the theorem : " Any chord of a
given conic subtending a right angle at a given point of the conic passes
through a fixed point in the normal at the given point f we may take
the equation of the conic referred to the tangent and normal at the given
point
aa? + 2kxy + 6y*= 2a; ;
the equation of any pair of straight lines through this point at right
angles to each other is
and at the points of intersection
(a + 6) as" + 2 (A + \6) ay = 2a: ;
or, at the points other than the origin,
(a + 6)aj + 2(A + \6)y = 2,
which is therefore the equation of a chord subtending a right angle at
the origin. This passes through the point y = 0, (a + 6) a; = 2 ; a fixed
point on the normaL
If two points be given as the intersections of a given straight line
and a given conic the equation of the straight lines joining these points
to the origin may be formed immediately, since it must be a homogeneous
equation of the second degree in a;, y. Thus the straight lines joining
the origin to the points determined by the equations
aa? + 2hocy + 6y* = 2a;,
px'¥qy=\,
are represented by the equation
<M? + 2hxy ■^hy' = 2x (px + qy),
CONIC SfiCnONS, ANALYTICAL. 139
and will be at right angles if a + 6 = 2p, or if the straight line
px-^qy^l pass through the point (-^— <» 0], a somewhat different
mode of proving the theorem already dealt with. In general the equa-
tion of the straight lines joining the origin to the two points determined
by the equations
««■ + fty* + c + 2/y + 2gx + 2hxy ■ 0,
2 e
va aaf-\- 2hxy -^ h^^ - - (px + qt/) (ffz +y^) + -s (/>« + ^^^ 0.
The results of linear transformation may generally be obtained firom
the consideration that^ if the origin be unaltered, the expression
a:' + 2a:yooB«» + y'
must be transformed into
X« + 2Xrco8 + r*,
if (x, y), (Xy T) represent the same point and m, O be the angles
between the co-ordinate axes in the two systems respectively. Thus if
tt= aa;" + 6y' + c+ 2/^ + 2^ + 2^iy
be transformed into
tr = i4X» + ^7* + c + 2/^7+ 2ffX + 2i7Zr,
then X (fls^ + ^ + 2a!y cos (i>) + u must be transformed into
X(X«+7« + 2X7co80)+Cr,
and if X have such a value that the former be the product of two linear
£actorBy so also must the latter ; hence the two quadratic equations in X
c(X + a)(X + 6) + 2^(Xcos<i> + A)
= (X + a)/* + (X + 6)^ + c(Xoo8« + A)*,
and c(X + i<)(X + ^) + 2^ff(XcosO + Zr)
= (X + i4)^« + (X + ^)e^ + c(XoosO + i7)*
must coincide ; and thus the invariants may be deduced. Also, by the
same transformation,
X(a:* + y* + 2q^ cob ia) + ao^ -k- h^ + 2hxy
must be transformed into
X (X* + 7* + 2X7 cos O) + ilX* + ^7* + 2HXT,
and if X have such a value that the former is a square, so must the
lattw; hence the equations
(X + a)(X + 6)a(Xco6« + A)',
(X + ^)(X + jB) = (XcosO + i?)',
140 CONIC SECTIONS, ANALYTICAL.
must coincide, ivlience
a + ft-2Aoo8(a_ Aa- B- 2llcosil
sin'cD "" sin'O '
sin' Q> sin' Q
One special form of the equation of a circle is often useful : it is
where (oj^, y^), (a;^, y^) are the ends of a diameter, and the axes rect-
angular. The corresponding equation when the axes are inclined at an
angle a> is obtained by adding the terms
{(«-«i)(y-y.)+ («-«.) (y-yi)}<508ca;
each equation being found at once from the property that the angle in
a semicircle is a right angle. In questions rating to two circles, it is
generally best to take their equations as
a:* + y"-2aa; + A: = 0,
a:' + y*-2&B + A; = 0,
the axis of x being the radical axis, and k negative when the circles
intersect in real points.]
883. The equation of the stra^ht lines which pass through the
origin and make an angle a with the straight line a; + y » is
»■ + 2x1/ sec 2a + y* = 0.
884. The equation ha^ - 2kc2/ + ay* = represents two straight lines
at right angles respectively to the two whose equation is
ofic" + 2hxy + 6y*= 0.
If the axes of co-ordinates be inclined at an angle oi, the equation will be
(a + 6 - 2A cos w) (a:* + y* + 2a^ cos «) = {oaf + 2hxi/ + 6y*) sin' ci».
885. The two straight lines
a:»(tan'^ + cos'^)-2a^tantf + y'sin'tf=0
make with the axis of x angles a, p such that tan a '^ tan fi- 2.
886. The two straight lines
{a? + y") (cos' sin' a + sin' 0) = (x tan a - y sin $)'
include an angle a.
887. The two straight lines
05* sin* a cos' 6 + ix^/ sin a sin + y' {4 cos a - (1 + cos o)' cos' tf} =
include an angle a.
CONIC SECTIONS, ANALYTICAL. 141
888. Form the equation of the straight lines joining the origin to
the points given bj the equations
and prove that they will be at right angles if V + ife* = c*. Interpret
geometricallj.
889. The straight lines joining the points given by the equations
aa:" + 6y*-»-c + ^ + 2yac+2Aajy = 0, /»;•»• ^^=1,
to the origin will be at right angles if
a + 6 + 2 (/g' + ^/>) + c (/>• + 5^ = ;
and the locus of the foot of the perpendicular from the origin on the
line /KB + yy = 1 is (a + 6) (a:* + y*) + %fy + ^gx + c = : also the same
18 the locus of the foot of the perpendicular from the point
/ 2y _ 2/\
V a + 6' a + V*
890. The locus of the equation
aj*-lx«-l
y = 2 +
2+ 2+ to 00
is the parts of two straight lines at right angles to each other which
include one quadrant
[The equation gives y = 1 + a; when x is positive and y = 1 — a; when
X is negative.]
891. The formulie for effecting a transformation of co-ordinates, not
necessarily rectangular, are
prove that 0^ - pq') (pq' -p'q) = qq' - pp.
892. The expression
oa:" + 6y • + c + 2/^ + 2gx + 2hxi/
is transformed into
ii«' + -»/ + c + 2/V + 2G^« + SiTajy,
the origin being unchanged : prove that
/'•»-/"2/Sr<»«<* _^ + Q^-2F gcosO
sin' CD " sin'O '
and
2/yA-a/'-V^ 2FGff'AF *-BG * ^
sin'(i> "" sin'O '
M, Q being the angles between the co-onlinate axes in the two caAOH.
142 CONIC SECnONS, ANALYTICAL.
893. Prove that, ABC being a giyen acute-angled triangle and P
any point in its plane, the three circular loci
PB' + PC=:n.PA'; PC-^PA' '^n.Pffy PA' + PB'^n.PC,
have their radical centre at the centre of the circle ABC^ each locus cuts
the circle ABC at right angles, and the centre of any locus lies on the
straight line joining an angular point to the middle point of the opposite
sida
894. A certain point has the same polar with respect to each of
two circles; prove that a common tangent subtends a right angle at
this point.
895. A chord through A meets the tangent at B, the other end of
a diameter of a given circle, in P and from any point in the chord pro-
duced are drawn two tangents to the circle : prove that the straight Imes
joining A to the points of contact will meet the tangent at ^ in points
equidistant from P.
896. The radii of two circles are a, h and the distance between
their centres J 2 (a* + b*) ; prove that a common tangent subtends a right
angle at the point which bisects the distance between their centres : and
that if through the point which divides the distance between the centres
in the ratio a' : b' be drawn two straight lines at right angles to each
other equally inclined to the line of centres these straight lineB will pass
through the points of contact of the common tangents.
897. From a point on a fixed straight line are drawn two tan-
gents to a given circle meeting in P, Q the tangent at A which is parallel
to the tangent at either point where the fixed straight line meets the
circle : prove that AP + AQ is constant.
898. Three circles U^ F, W have a common radical axis and from
any point on U two straight lines are drawn to touch F, W respectively:
prove that the squares on these tangents will be in the ratio of the dis-
tances of the centre of U from those of T, W.
899. Tangents drawn from a point P to a given circle meet the
tangent at a given point A in Q^Q] prove that if the distance of P
from the fixed tangent be given, tiie rectangle QA, AQ' will be constant,
900. Given two circles, a tangent to one at P meets the polar of P
with respect to the other in P ; prove that the circle on PP^ as diameter
will pass through two fixed points which will be imaginary or real as the
given circles intersect in real or imaginary points.
901. One circle lies entirely within another, a tangent to the inner
meets the outer in P, P' and the radical axis in Q : prove that, if ^S' be
PSP* SOP
the internal point-circle of the system, the ratio sin — ^— : cos -^ is
constant.
OOKIC SlCnONS, ANALTTICAL. 143
902^ On two circles are taken two points such tliat the tangents
drawn each from one point to the other circle are equal : prove that the
points are equidistant from the radical axis.
903. The equation of a circle in which (x^, y,), (x^ y^ are ends of
the chord of a s^pnent containing an angle is
-(«-»,) (y-y»)}-o.
904 On the sides AB, AC of a triangle ABC are described two
segmentt of circles each containing an angle 6 ( > ^ j and on the side BC
a segment containing an angle -^r ^0: prove that the centre of the last
circle lies on the radical axis of the other two; each segment being
towards the same parts as the opposite angle.
905. Hiere are two systems of circles such that any circle of one
system cuts any circle of the other system at right angles ; prove that
the circles of either system have a common radical axis i^ch is the line
of centres of the other system.
906. On a fixed chord AB of a given circle is taken a point such
ihat^ P being any point on the circle, OA . OB = ^ PA . PB : prove that
the straight line which bisects PO at right angles will pass tlm>ugh one
end of the diameter conjugate to AB ; and, if Q be the other point in
which the straight line meets the circle, that QCy = QA . QB.
907. A circle U lies altogether within another circle V ; prove that
the ratio of the segments intercepted by U, V on any straight line can-
not be greater than
where cs 6 are the radii and c the distance between the centres.
908. An equilateral triangle is drawn with its sides passing through
three given points A,B, C : prove that the locus of its centre is a circle
having its centre at the centroid of ABCy and that the centres of two
equili^eral triangles whose sides are at right angles will be at the ends of
a diameter of the locus.
[The radius of the locus is the difference of the axes of the minimum
ellipse about ABC, the altitude of the maximum equilateral triangle is
eqiud to three-fourths the sum of the axes of the minimum ellipse,
and is also equal to the minimum sum of the distances of any point from
A, B, C]
909. Prove that the equation
{aP008(a + )3) + y nn (a + )3) -acoB(a-)3)}
{a:cos(y-»-8) + y 8in(y+ 8) -aco8(y-8)}
= {« cos (a + y) + y sin (a + y) - a cos (a - y) }
{« cos (^ + 8) + y sin (^ + 8) - « cos (^ - 8) }
144 CONIC SECTIONS, ANALYTICAL.
is equivalent to the equation a? + f^=-a': and state the property of the
circle expressed by the equation in this form.
910. Four fixed tangents to a circle form a quadrilateral whose
diagonals are aa\ hb\ caf^ and perpendiculars p^p' ; q^q' I r,f^ are let fidl
from these points on any other taiigent : prove that
, B-y a-8 , y-a ^-8 , a- fi y-S
;?p cos^- ooB—^r- = qq cos ^— — cos ^-^— = rr cos— ^cos ' ^
, , . a^e . )8-0 . y-0 . 8-0.
= 4a' sm -^ sin ^-^ Bin ^—^ sin — ^ ;
the co-ordinates of the points of contact being {a cos a, a sin a), and the
like in j3, y, 8, 0.
911. The radii of two circles are R^ p, the distance between their
centres is JB? + 2p' and p < 2/? : prove that an infinite number of tri-
angles can be inscribed in the first which are self-conjugate with respect
to the second ; and that an infinite number can be cii*ciimscribed to the
second which are self-conjugate to the first.
[In general, if 8 denote the distance between the centres, and the
polar of a point A on the first circle with respect to the second meet the
first in j5, C, the chords ABy AC will touch the conic
y'(2/?' + 2p«-8*) + (i?" + p"-8')(2a:«-.2&c + S*-i?'-p«) = 0,
and BC will touch the conic
and these two will coincide if 8* = i?* + 2p'.]
912. A triangle is inscribed in the circle a:* + ^ = /^, and two of its
sides touch the circle (a: — 8)* -i- y* = p' : prove that the third side will
touch the circle
/ 4^r^V ^ p, f 2r« (Jg' + 8*) \«
r (/2--8V/ -^y-^x (i?«-8«)- -7 '
which coincides with the second circle if 8* = i?* * 2Rr. Also prove tliat
the three circles have always a commou radical axis.
913. Two given polygons of n sides are similar and similarly
situated : prove that in general only two polygons can be drawn of the
Hame number of sides circumscribing one of the two given polygons and
inscribed in the other ; but that if the ratio of homologous sides in the
two be cos* jr- : cos' -^ sin' - , where r is any whole number less than
In, In n
- , there will be an infinite number.
CONIC SECTIONS, ANALYTICAL. 145
II. Parahola referred to iU aoeU,
[The equation of the parabola being taken if = 4ax, the oo-ordinatea
of any point on it may be represented by (— f i —-) i and with this
notation the equation of the tangent is y = mx + — ; of the normal
my + a: = 2a + —5; and of the chord through two points (m^, m,),
2m m^ - y (m, + m,) + 2a = 0. The equation of the polar of a point
(JTj) is yF= 2a(a; f uY), and that of the two tangents drawn from
(X, 7) is (y - iaX) (y« - iax) = {y7- 2a (a; + X)\\
As an example, we may take the following, *' To find the locus of the
point of intersection of normals to a parabola at right angles to each
other."
If (X, F) be a |)oint on the locus, the points on the parabola to
which normals can be drawn from {X, Y) are given by the equation
m*F+ m*(X- 2a) -a = 0;
BO that, if m^ m^ m, be the three roots of the equation
2a-X ^ a
fn^ + m^ + m^ = — rj? — , tn^tn^ + m^m^ + fn^tn^ = U, tn^ni^^ *^ "^ >
and since two normals meet at right angles in (X, T) the product of two
of the roots is - 1 ; let then m^m^ = - 1. Then
a 3a-X 7
or the locus is the parabola ^ = a(x — 3a).
Again, '' The sides of a triangle touch a parabola and two of ita
angular points lie on another parabola with its axis in the same direc-
tion, to find the locus of the third angular point."
Let the equations of the parabolas be y* = 4aa;, (y — k)* = 46 (a; — A),
and let the three tangents to the former be at the points m^, m^, m^
The point of intersection of (1), (2) is , a ( — + — ), and this will
lie on the second parabola if
( \m, m^ J \m^m^ /'
and similarly for m^ m,. Hence m,, m, are the roots of the quadratic
hence, if (X, F) be the point of intersection of the tangents at m^ m^
\w, »»,/ \ m,/ TO,
W. P. Aft
140 coxic SEcnos^ axaltthtai^
(*-;)'-
ni^Mj
80 that (ar-i&t)' = 4(lft-a)*(«X-4UX
or the third pomt liea on aziothcr ^mnhc^ ^vifih xts axii in the
direction as the two g^en pArabolas^ msd which cnmndf with the
second if a = 46.]
914. Two parabolas have a common vertex A and a common azia^
an ordinate NPQ meets them, the tangent at P meets the oater parabola
in J?, i?" and AR, AR meet the ordinate in Z, JT; prore that NF^ NQ
are respectively harmonic and geometric means between SL^ JTiT.
915. A triangle is inscribed in a parabola and a similar and
similarly placed trian^e circomacribes it : prove that the sidea of the
latter triangle are respectively four times the corresponding sides of the
latter.
916. Two tangents />, q being drawn to a given parabola £\ through
their point of intersection are drawn the two parabolas oonfocal with U^
and A'f A" are their vertices : prove that
/P-^V SA\A"A r^,,|„n
Aj A' being taken on opposite sides of S,
917. An equilateral triangle is inscribed in a parabola : prove that
the ordinates y^, y,, y^ of the angular points satisfy the equations
3(y,+y,)(y,+y,)(y,+yJ-^32a'(y, + y,+y,) = 0,
and that its centre lies on the parabola 9^*= 4a (x— 8a).
918. An equilateral triangle circumscribes a parabola: prove that
tho ordinates y^^ y^ y^ of its angular points satisfy the equations
(yi + y. + y/ = -^ (y^ya + y»yi + yiy. + 3a«),
^«'(y,+y.+y,) + 3(y,+ y,-yj(y3 + y.-y,)(y,+y,-y,) = 0.
[Tho simplest way of expressing the conditions for an equilateral
triangle is to equate the co-ordinates of the ccntroid and of the centre of
|K;r])cndicularH.]
919. The pole is taken of a chord PQ of a parabola : prove that
t hn peq)ondiculars from 0, P, Q on any tangent to the parabola are in
geometric progression.
920. Four fixed tangents are drawn to a parabola, and from the
lingular |K)ints taken in onler of a quadrangle formed by them are let
fall por|NMi(liru1arH p^, p^^ p^, p^ on any other tangent : prove that
CONIC SECTIONS, ANALYTICAL. 147
921. The perpendiculars from the angular points of a triangle ABC^
whose sides touch a parabola, on the directrix are />, q^ r, and on any
other tangent are x,yfZ: prove that
p tan A _ q tan B _^ r tan C
ar(y-«) ~ y{z-x) ~~ z\x-y)'
[Of coarse the algebraical sign must be r^^arded.]
922. The distance of the middle point of any one of the three
diagonals of a quadrilateral from the axis of the inscribed parabola is
one-fourth of the sum of the distances of the four points of contact from
.the axis.
923. Through the point where the tangent to a given parabola at P
meets the axis is drawn a straight line meeting the parabola in Q^ (/
which divides the ordinate at P in a given ratio : prove that PQy PQ^
will both touch a fixed parabola having the same vertex and axis as the
given one.
[If the ratio of the part cut off to the whole ordinate be i; : 1, the
ratio of the latus rectum of the envelope to that of the given parabola
willbe2ife: l+)fc.]
924. Two equal parabolas have axes in one straight line, and from
any point on the outer tangents are drawn to the inner: prove that they
will intercept a constant length on any fixed tangent to the inner equal
to half the chord of the outer intercepted on the fixed tangent
925. A tangent is drawn to the circle of curvature at the vertex
and the ordinates of the points where it meets the parabola are y,, y^ :
prove that
926. On the diameter through a point of a parabola are taken
points Py F w} that the rectangle OP, OP' is constant : prove that the
four points of intersection of the tangents drawn from P, F lie on two
fixed straight lines parallel to the tangent at and equidistant from it.
927. The points P, P' are taken on the diameter through a fixed
point of a parabola so that the mid-point of PF is fixed : prove that
the tangents drawn from P, P' to the parabola will intersect on another
parabola of half the linear dimensions.
[In general if tangents to the parabola y' = 4aa? divide a given
segment LL' on the axis of x harmonically, their point of intersection
lies on the conic
^ ' a
where OL + OX' = 2c and LL' = 2m.]
928. A chord of a parabola passes through a point on the axis
(outside the parabola) at a distance from the vertex equal to half the
latus rectum : prove that the normals at its extremities intersect on the
parabola.
148 CONIC SECTIONS, ANALYTICAL,
929. The sum of the angles which three normals drawn from one
point make with the axis exceeds the angle which the focal distance of
the point makes with the axis hy a multiple of tt.
930. Normals are drawn at the extremities of any chord passing
through a fixed point on the axis of a parabola : prove that their point
of intersection lies on a fixed parabola.
[More generally, if a chord pass through (X, F), the locus of the
point of intersection of the normals at its ends is the parabola
2{2ay+7(x-.X-.2a)}'+(r'-.4aZ){ry + 2Z(x-X-2a)} = 0.]
931. Two normals to a parabola meet at right angles, and from the
foot of the perpendicular let fall f i*om their point of intersection on the
axis is measured towards the vertex a distance equal to one-fourth of
the latus rectum : prove that the straight line joining the end of this
distance with the point of intersection of the normals is also a normaL
932. Two equal parabolas have their axes coincident but their
vertices separated by a distance equal to the latus rectum ; through the
centres of curvature at the vertices are drawn chords PQ, F'Q equally
inclined in opposite senses to the axis, P, F^ being on the same side of
the axis: prove that (1) FQ^ P^Q are normals to the outer parabola;
(2) their common point R lies on the inner; (3) the normals at P', ^, R
meet in a point which lies on a third equal parabola.
933. From a point are drawn three normals 0-P, OQ^ OR and
two tangents OL^ OM to a parabola : prove that the latus rectum
OP.OQ.OR
= 4
OL.OM
934. The normals to the parabola y* = iax at points P, Qy R meet
in the point (X, Y) : prove that the co-ordinates of the centre of perpen-
diculars of the triajigle PQR are X- 6a, -\Y*
935. Three tangents are drawn to a parabola so that the sum of
the angles which they make with the axis is ir : prove that the circle
round the triangle formed by the tangents touches the axis (in the focus
of course).
936. The locus of a point from which two normals can be drawn
making complementary angles with the axis is the parabola
y" = a(x — a).
937. Two (equal) parabolas have the same latus rectum and from
any point of either two tangents are drawn to the other : prove that the
centres of two of the four circles which touch the sides of the triangle
formed by the tangents and their chord of contact lie on the parabola to
which the tangents are drawn. Also, if two points be taken conjugate
to each other with respect to one of the parabolas and from them tangents
drawn to the other at points L, M ; Ny 0, respectively, the rectangle
under the perpendiculars from any point of the second parabola on the
chords LNy MO will be equal to that under the perpendiculars from the
aame point on MN^ LO.
CONIC SECTIONS, ANALYTICAL, 149
938; Prove that the two parabolas y* = oos, y's4a(x + a) are so
related that if a normal to the latter meet the former in F^ Q and A be
the vertex of the former, either AF or AQ is perpendicular to the
normal.
939. The normals at three points of the parabola y* = iax meet
in the point (X, Y) : prove that Uie equation of the circle through the
three points is
and that of the circle round the triangle formed by the three tangents is
(a;-a)(a:-2a + Z)+y(y+ r) = 0.
[Hence if be the point from which the normals are drawn and 0(y
be bisected by Sy SO' is a diameter of the circle round the triangle formed
by the tangents.]
940. In the two parabolas y'=2c(x'^c) a tangent drawn to one
meets the other in two points and on the chord intercepted as diameter
28 described a circle : prove that this circle will touch the second parabola.
941. On a focal chord as diameter is described a circle cutting the
parabola again in P, Q : prove that the circle FSQ will touch the parabola.
942. On a chord of a given parabola as diameter a circle is described
and the other common chord of the circle and parabola is conjugate to
the former with respect to the parabola : prove that each chord touches
a fixed parabola.
943. Two tangents OL, OM to a parabola meet the tangent at the
vertex in P, Q : prove that
*FQ = OLcoB QFL = Oif cos FQM.
944. Two parabolas have a common focus and direction of axis, a
chord QVQf of the outer is bisected by the inner in F, VF parallel to
the axis meets the outer in F : prove that Q F is a mean proportional
between the tangents drawn from F to the inner.
945. Prove that the parabolas
y* e 4aa;, y* + 4cy + 4ax = 8a*
cut each other at right angles in two points and that each passes through
the centre of curvature at tlie vertex of the other. If the origin be
taken at the mid-point of their common chord their equations will be
y* - c* - 4a* = * (2cy + 4aj;).
[The general orthogonal trajectory of the system of parabolas
y*+2Xy + 4aa:=8a*
for different values of X is y* — 4aa; = Ci*".]
946. On a focal chord FSQ of a parabola are taken points p^ q an
opposite sides of S so that qS .Sp= QS . SF, and another parabola is
drawn with parallel axis and passing through q, p: prove that the
common chord of the two parabolas will pass thix)ugh S,
150 CONIC SECTIONS, ANALYTICAL.
947. A chord PQ of a parabola meets the azk in T, CT is the mid-
point and O the pole of the chord, a normal to PQ through U meets the
axis in O and OK is perpendicular from on the directrix : prove that
SO is paraUel to TJTand SKtoGU.
948. Through each point of the straight line x = my + A is drawn a
chord of the parabola y* = ictx, which is bisected in the point : prove that
this chord touches the parabola
(y-2am)*=8a(a:-A).
949. Prove that the triangle formed by three normals to a parabola
is to the triangle formed by the three corresponding tangents in the
ratio
(^ + ^. + 0" ' h
where t y t , t are the tangents of the angles which the normals make
with the axis.
950. Three tangents to the parabola y* « 4a (re -h a) make angles
Oy Pf y with the axis : prove that the co-ordinates of the centre of the
circle circumscribing the triangle formed by them are
. 1 ^ 8in(a + ff-f-y) ^^ cos(a + ff + y)
^ sin a sin p sin y ^ sinasmpsiny
951. Three confocal parabolas have their axes in a. p., a normal is
drawn to the outer and a tangent perpendicular to this normal to the
inner : prove that the chord which the middle parabola intercepts ou
this tangent is bisected in the point where it meets the normal
952. Two normals OP, OQ are drawn to a parabola, and a, fi are
the angles which the tangents at P^ Q make with the axis: prove
that
OP OQ a
8ina + 8inj3cos(a-/3) ^ sinj3 + sinacos()3 — a) ~ sin'asin'^'
953. From any point on the outer of two equal parabolas with a
common axis tangents are drawn to the inner : prove that the part of
the axis intercepted bears to the ordinate of the point from which the
tangents are drawn a constant ratio equal to that which the chord
intercepted on the tangent at the vertex of the inner parabola bears to
the semilatus rectum.
954. Prove that the common tangent to the two parabolas
af cos" a = 4a (a; cos a + y sin a),
y* sin* a = — 4a (a; cos a + y sin a),
subtends a right angle at the origin.
955. Two parabolas have a common focus S and axes in the same
straight line, and from a point P on the outer are drawn two tangents
PQ, PQ* to the inner : prove that the ratio
coB^QPQ' : oo8^ ASP
Qstant, A being the vertex of either parabola.
CONIC SECTIONS, ANALYTICAL. 151
956. A parabola circamscribes a triangle ABC and iU axii makea
with CB an angle (measured from CB towards CA) : prove that its
latus rectum is
2^8intf8in(C-tf)8in(5 + tf);
and that for an inscribed parabola the latus rectum is four times as
laige.
957. A triangle ABC is inscribed in a given parabola and the fociur
is the centre of perpendiculars of the triangle : prove that
(1 — co8i4)(l -co8i5)(l -cos (7) = 2 cos ii cos ^006 (7;
and that each side of the triangle touches a fixed circle which passes
through the focus and whose diameter is equal to the latus rectum.
958. A parabola is drawn touching the sides AB^ AC iA a triangle
ABC at B^ C and passing through the centre of perpendiculars : prove
that the centre of perpendiculars is the vertex of the parabola and that
the centre of curvature at the vertex is a point on BC,
959. The latus rectum of a parabola which touches the sides of a
triangle ABC and whose focus is /S^ is equal to SA , SB . SC-^ jR*.
960. A chord LL' of a given circle has its mid-point at and its
pole at P ; a parabola is drawn with its focus at and its directrix
passing through P : prove that the tangent to this parabola at any
point where it meets the circle passes through either L or L\
961. A triangle, self- conjugate to a given parabola, has one angular
point given : prove that the circle circumscribing the triangle passes
through another fixed point Q such that OQ is parallel to the axis and
bisected by the directrix.
962. A triangle is inscribed in a {larabola, its sides are at distances
X, yyZ from the focus and subtend at the focus angles 0, ^ ^ (always
measured in the same sense so that the sum is 2ir) : prove that
A dt
«-«iJ «•«-!. ^r. I ttntf+sin^+sin^ + 2tan = tanjtanj
sin u sin 9 sm w ^ ^ 2 2 3
*• * y* * a* ?
where 21 is the latus rectum.
963. Two points L, L are taken on the directrix of a parabola
conjugate to each other with respect to the parabola : prove that any
other conic through LSL' having its focus on LU will have for the cor-
responding directrix a tangent to the parabola.
2X
964. An ellipse of given excentricity -j — ^ is described passing
through the focus of a given parabola y* = ^ok and with its own foci on
152 CONIC SECTIONS, ANALYTICAL.
the parabola : prove that its major axis touches one of the parabolaSi
confocal with the given parabola,
y' = 4a(l-X«)(a;-aX«),
and that its minor axis is normal to one of the two
y" = 4a (1 + X«) (a: + aX').
965. An ellipse is described with its focus at the vertex of a given
parabola; its minor axis and the distance between its foci are each
double of the latus rectum of the parabola : prove that the pole with
respect to the ellipse of that ordinate of the parabola with which the
minor axis in one position coincides always lies on the parabola and also
on an equal parabola whose axis coincides with that of the ellipse.
966. A parabola touches the sides of a triangle ABC in the points
A\ F, C and is the point of concourse of AA'^ BBy CCi prove that,
under a certain convention as to sign,
OA cosec BOC + O^cosec COA + OCcosec i40^ = :
also, if P be a point such that PA' bisects the angle BPC and PB^ PC
respectively bisect the external angles between PC, P-4, and PA^ PB^
PA=^PB-\'PC.
967. A triangle circumscribes the circle a^ + y* = a", and two
angular points lie on the circle {x - 2a)' + y* = 2a" : prove that the third
angular point lies on the parabola y* = a(x- fa). Prove also that the
three curves have two real and two impossible common tangents.
968. Two parabolas have a common focus, axes inclined at an angle
a, and are such that triangles can be inscribed in one whose sides touch
the other : prove that ^^ = 2/^ (1 + cos a), l^^ l^ being their latera recta.
969. A circle is described with its centre at a point /* of a parabola
and its radius equal to twice the normal at P: prove that triangles can
be inscribed in the parabola whose sides touch the circle.
970. Two parabolas A^ B have their axes parallel and the latus
rectum of A is four times that of B : prove that triangles can be inscribed
in B whose sides touch A, If the axes be in the same straight line the
normals to j9 at the angular points of such a triangle will all meet in
one point, as will the normals to A at the points of contact, and the loci
of tiiese points of concourse are straight lines perpendicular to the
axis.
[Taking the equations of the parabolas to be
y* = 16a;r, y* = 4a (a; + A),
the straight lines will be a: = 2a, a; = 8a + A.]
971. The circle of curvature of a parabola at P meets the parabola
again in Q and QL, QM are drawn taiigents to the circle and parabola
at Qj each terminated by the other curve : prove that when LM subtends
a right angle at P, PL is parallel to the axis, and that this is the case
when the focal distance of 7 is one-third of the latus rectum.
CONIC SECTIONS, ANALYTICAL. 153
972. If tLe tangent at P make an angle with the axis, the tangent
to the circle at Q will make an angle ir - 3^ with the axis ; also the
angle between the tangent at P and the other common tangent to the
]iarabola and circle will be 2 tan~* (^ tan 6)y and if ^ be the angle which
tliitt common tangent makes with the axis
973. From a point on the normal at P are drawn two tangents
to a ])arabola making angles a, fi with OP: prove that the radius of
curvature at /* is 20 P tan a tan p.
974. The normal at a point of a {larabola makes an angle 9 with
the axis : prove that the length of the chord intercepted on the normal
bears to the latus rectum the ratio 1 : sin $ cos' 6, and the length of the
common chord of the i>arabola and the circle of curvature at the point
bears to the latus rectum the ratio 2 sin $ : cos' 0,
975. At a point P of a parabola is drawn a circle equal to the
circle of curvature and touching the parabola externally; the other
common tangents to this circle and the {larabola intersect in Q : prove
that, if QK be let fall {perpendicular on the directrix,
SQj^QK _ AS
:SQ -^ QK " AJS + :SP'
III. EUipBe referred to Ua cuoes.
[The equation of the ellipse in the following questions is always
supposed to be -i + n- => 1» and the axes to be rectangular, unless other-
wise stated. The point whose excentric angle is ^ is called the point d.
The excentricity is denoted by e. The tangent and normal at the ix>int
are respectively
-coe^-ffsintf = l, - -v,--r^ = a'-6';
a b ' COS0 sin^ '
the chord through the two points a, )9 is
and the intersection of tangents at a, fi, (the |)ole of this chord)
a + fl - . a-hP
a cos — — Dsm — -r -
2 2
cos — ^^ cos -^-
154 CONIC SECTIONS, ANALYTICAL.
xX vT
The polar of a point (X, 7) is —f + ^= 1; and the equation of the
two tangents from (X, 7) is
It follows from the equation of the tangent that if the equation of
any straight line be lx-\-my= 1, and I, m satisfy the equation a'l^+b*m*= 1,
the straight line touches the ellipse -> + ^ = I9 a result often usefuL
The equation of the tangent in the form
xcoad + ysm$ = J a* cos* 6 + b* sin*
may be occasionally employed with advantage.
The points a, ft will be extremities of conjugate diameters if
IT
a'^psa^. Any two points are called conjugate if either lies on -the
polar of the other, and any two straight Unes if either passes through
the pole of the other.
If {X, J) be the pole of the chord through (a, fi) it will be found that
sin a sin )9 cos a cos P sina + sinj5 cos a + cos ff 1
— T^ = — 7="= — 2? — = — 3T — ^'r~T'
a' b' b a a" "^ 6''
which enable us to find the locus of (X, Y) when a, P are connected by
some fixed equation. Thus, "If a triangle be circumscribed about an
ellipse -i + Tf = 1 and two angular points lie on the ellipse -^ + ?7i = 1>
to find the locus of the third angular point.''
If a, )3, y be the three points of contact and (a, P), (a, y) be the
pairs of points whose tangents intersect on the second ellipse, we have
a' ,a + )8 b* . ,a + P ,a-i8
_.co8'-^+gr.sin'-2-=co« _2^,
and the like equation with y in place of p. Hence p^ y are the two
roots of the equation
where A=-^,+^,+ l, B = ^,-^+l, C = J.+|1-1;
and we have therefore
P^y . p-hy p-y
cos - ^ sin ^ cos ^
2 2 2
^cosa "" ^sina *
so that the co-ordinates of the third angular point are
Aa Bb .
^coso, -^smo,
CONIC SECTIONS, ANALYTICAL. 155
ao tliat its locus is the ellipse
This locus will be found to coincide with the second ellipse if
— i^ r> ik 1 = 0, and if we so choose the signs of a\ V that the relation is
a b
-? +rr = l> -^ = 2;-,, ^t=2", C = -2— .;,80 that the co-ordinates of the
ah a ab
third point are —a' cos a, —i/mncLy or its exoentric angle is ir + a, and
similarly the excentric angles of the other points are w-k- P, ir + y.
Hence the ellipses
«• y* , ^ y" 1
a' ^ 6' ' a" 6" '
will be such that an infinite number of triangles can be inscribed in the
second whose sides touch the first, if with any signs to a', 6' the relation
-7 + x> = 1 ^ satisfied and the excentric angle of any comer of such a
a
triangle exceeds that of the corresp<»iding point of contact by ir.
If this condition be not Satisfied the two given ellipses and the locus
will be found to hare four common tangents real or impossible.
Again, for the reciprocal problem, "If a triangle be inscribed in the
ellipse —, + ^ = 1 and two of its sides touch the ellipse -i + ?-, = 1| ta
find the envelope of the third side.''
Taking a, )3, y for the angular points and (a, fi), (a, y) for the sides
which touch the second ellipse, we have
and a like eqiuition with y in place of p. Hence, as before,
^' .i4-y''5^ ./5-y=^'
cos*— r-' cos"^-^
which, since the third side is
P-hy . B-^y
«~" 2 y-^ 2 ,
proves that the envelope is the ellipse
A ~7t + "^ T#i = C^t
CI
156 CONIC SECTIONS, ANALYTICAL.
wHch coincides with the second ellipse if — a ^ a 1 = 0, or if with any
signs of a', 6', — + v> = 1. The excentric angles of the points of contact
will bea — IT, j5 — ir, y — ir (or a + ir, /9 + ir, y -f ir, which are practically the
same). If this condition be not satisfied the three conies intersect in
the same four points real or impossible.
The relations between the excentric angles corresponding to normals
drawn from (X, Y) may be found from the equation
aX bY , .,
cos 6 sin
a biquadratic whose roots give the excentric angles of the points to which
normals can be drawn from (X^ Y), If tan '^ = Zy this equation becomes
£*bY-\- 2Z* {aX -fa'- b') + 2Z(aX~ a* + 6«) - 67= 0.
This equation having four roots, there must be two relations inde-
pendent of X, Y between the roots, as is also obvious geometrically.
These relations are manifest on inspection of the equation ; they are
and the relation between Z^, Z^ Z^ is therefore
111
which is equivalent to sin ()3 + y) + sin (y + a) + sin (a + /5) = 0,
if a, )9, y be the corresponding values of 6,
Since 1 - {Z^^ +...) + Z^Z^Z^Z^ = 0, it follows that
tan ^ = 00,
ora + )3 + y + Sisan odd multiple of ir.
The following is another method of investigating the same question.
If the normal at («, y) to the ellipse pass through ( JT, F),
a^xY^ b^yX= («• - fc*)*^. (A)
Now if — + -r^ = 1, and — + — i^= 1, be the equations of two lines
a b o, b
joining the four points to which normals can be drawn from (X, F), the
can be made to coincide with (A). The identification of the two gives
X=l, Zr + 1=0, mm' + 1=0,
whence it follows that normals at the points where the two straight lines
a b * cU bm
meet the ellipse all meet in a point. The point is given by
ax -by a' — b'
i (1 - m') ^ m(l-/") " FVm^ '
CX)NIC SECTIONS, ANALYTICAL, 157
If a, )3 be the two points on the former, and y one of the points on
the latter,
COB— ^ srn-^^
, ^ ^ cosy siny , ^
COS — -- COS ^
2 2
whence
Bin— - cos—^ -i-(co8y8m-2^ + 8myco8--~-jcoe--^ = 0,
or sin ()8 + y) -»- sin (y + o) + sin (a + /8) = 0.
The equation formed from this by replacing y by S must also hold, whence
cos ^ *^ 2 ^^ 2~
sin o + sin ^ ~ TOSoTcos^ sin (a + ^) '
and tan^^^ = cot— Y", or a + /3 + y-i-S is an odd multiple of -.]
976. A chord AP is drawn from the vertex of an ellipse of excen-
tricity «, along PA is taken a length PB equal to PA -?- «*, and BQ is
drawn at right angles to the chord to meet the straight line through P
parallel to the axis : the locus of Q is a straight line perpendicular to the
axis. Similarly if BP be a chord through a vertex on the minor axis
and along BP be taken a length BB equal to BP -f e", and BQ be drawn
at right angles to BB to meet the straight line through P parallel to
the minor axis, the locus of Q is a straight line parallel to the major
axis.
[The equations of the loci, with the centre as origin, are
977. Tangents drawn from a point P to a given ellipse meet a given
tangent whose point of contact ia in Q, Q*: prove that if the distance
of P from the given tangent be constant, the rectangle OQ, OQ will be
constant. Also if the length QQ' be given the locus of Q will be a conic
having contact of the third order with the given ellipse at the other end
of the diameter through ; and the conic will be an ellipse, parabola, or
hyperbola according as the given length QQ is less than, equal to, or
greater than the diameter parallel to the given tangent.
978. Two ellipses have the same major axis and an ordinate NPQ
is drawn, the tangent at P meets the oUier ellipse in points the lines
joining which to either extremity of the major axis meet the ordinate in
Z, M : prove that NP is a harmonic and NQ a geometric mean between
156 CONIC SECTIONS, ANALYTICAL.
979. The equation giving t the length of the tangent from (X, Y) to
the ellipse -^ + ^ = 1 is
a'U* ^V ■*"ft*V^ ?/ ■*" t \b' a') ~ '
where ^= -r + tt - !•
980. The major and minor axes of an ellipse being AA\ BB*,
another similar ellipse is described with BB' for its major axis, F is any
point on the former ellipse and L the centre of perpendiculars of the
triangle FBB' : prove that L will lie on the second ellipse and that the
normals at Z, F will intersect on another ellipse whose minor axis is 46,
, . . „a* + 6*
and major axis 2 .
981. A given ellipse subtends a right angle at 0, and 00' is drawn
perpendicular to and bisected by the polar of : prove that 00' is
divided by the axes in a constant ratio, CO' is a constant length, the
middle point of 00' is the point of contact of the polar of with its
envelope, and the rectangle under the perpendiculars from 0, C on the
polar of is constant.
982. The rectangle under the perpendiculars let fall on a straight
line, from its pole with respect to a given ellipse and from the centre of
the ellipse, is constant (= A) : prove that the straight line touches the
confocal -= — r- + 7=^ = 1.
983. The rectangle under the perpendiculars drawn to the normal
at a point F from the centre and from the pole of the normal is equal
to the rectangle under the focal distances of F.
984. The sum or the difference of the rectangles under the pei'pen-
diculars upon any straight line (1) from its pole with respect to a given
ellipse and from the centre, (2) from the foci of the given ellipse, is
constant (= 6') ; the sum when the straight line intersects the ellipse in
real points, otherwise the difference; or with proper regard to sign
in both cases, the rectangle (2) always exceeds the rectangle (1) by 6*.
985. Through a point are drawn two straight lines at right angles
to each other and conjugate with respect to a given ellipse : prove that
the arithmetical difference between the rectangles under the perpen-
diculars on these lines each from the centre and from its own pole is
equal to the sum of the rectangles under the focal perpendiculars, and to
the rectangle under the focal distances of the point.
986. On the focal distances of any point of an ellipse as diameters
are described two circles : prove that the excentric angle of the point is
equal to the angle which a common tangent to the circles makes with
tAe minor axis.
CONIC SECTIONS, ANALYTICAL. 159
987. The ordinate XF at a point /* of an ellipee is produced
to Q 80 that HQ : NP :: CA : CNy and from Q two tangents are
drawn to the ellipse : prove that they intercept on the minor axis pro-
duced a length equal to the minor axis.
988. A circle of radius r is described with its centre on the minor
axis of a given ellipse at a distance er from the centre : prove that the
tangent te this circle at a point where it meets the ellipse will touch the
minor auxiliary circle.
989. A point P on the auxiliary circle is joined to the ends of the
major axis and the joining lines meet the ellipse again m Q^ Q^\ prove
that the equation of QQ is
(a' + 6») y sin tf + 26* a; cos tf « 2ab\
where is the angle ACP, and if the ordinate to P meet QQf m R, R ]b
the point of contact of QQ with its envelope.
990. From a point P of an ellipse two tangents are drawn to the
circle on the minor axis: prove that these tangents will meet the
diameter at right angles to CP in points lying on two fixed straight
lines parallel to the major axis.
991. Two tangents are drawn to an ellipse from a point P: prove
that the angle between them is
^^^ I, SP.S'P )'
992. If 2^i Q ^^ ^^^ lengths of two tangents at right angles to each
other
993. It p, q he the lengths of two tangents and 2ma, 2nib the axes
of the concentric similar and similarly situated ellipse drawn through
their point of intersection
P
P'^9' i^^'-^V
994. The lengths of two tangents drawn to an ellipse from a point
ou one of the equal conjugate diameters are p^ q : prove that
995. If /?, ^ be the lengths of two tangents drawn from a point on
X* y* . X* y"
the hyperbola ^ = a-6 to the ellipse zi + n = 1| ^^'^ ^ tte central
distance of the point, then will
and 0>+9)=4m (p, Va6). •
100 CONIC SECTIONS, ANALYTICAL.
996. If two tangents be drawn from any point of the hyperbola
T- = a — 6 to the ellipse -s + ?• = li the difference of their lengths
ah ^ a
will be 2 (a — 6) (1 - -j — -. j^A , where r is the central distance of the
point : and if a parallelogram be inscribed in the hyperbola whose sides
touch the ellipse and r , r be the central distances of two adjacent
angular points, then will
the lengths of the sides of the parallelogram will be
Vr,' + r,'-(a-6)'*(«-6),
and the point of contact on any side will divide that side in the ratio
r' + ab-'a'-'b' : ab.
997. A circle is described on a chord of the ellipse lying on the
straight line p-' + qj-^l as diameter : prove that the equation of the
straight line joining the other two common points of the ellipse and
circle is
a; y a* + 6"
998. In an ellipse whose axes are in the ratio J2 + 1 : 1, a circle
whose diameter joins the ends of two conjugate diameters of the ellipse
will touch the ellipse.
999. Normals to an ellipse at P, Q meet in and CO, PQ are
equally inclined to the axes : prove that the part of PQ intercepted
between the axes is of constant length and that the other normals drawn
from will be at right angles to each other.
1000. If be the point in the normal at P such that chords drawn
through subtend a right angle at P, and 0' be the corresponding point
for another point F, 00\ PF will be equally inclined to the axes and
their lengths in a constant ratio.
1001. A circle is described having for diameter the part of the
normal at P intercepted between the axes, and from any point on the
tangent at P two tangents are drawn to this circle : prove that the chord
of the ellipse which passes through the points of contact subtends a right
angle at P.
1002. The normals at three points of an ellipse whose excentric
angles are a, P, y will meet in a point, if
sin ()8 + y) + sin (y + a) + sin {a + fi) = 0,
which is equivalent to
B-^y 7+a ^ a + B
tan'^„— cota = tan ^-^— cot )3 = tan - -coty.
CONIC SECTIONS, ANALYTICAL. 161
1003. If four normals to an ellipse meet in a point the sum of the
corresponding excentric angles will be an odd multiple of w. Also two
tangents drawn to the ellipse parallel to two chords through the four
points will intersect on one of the equal conjugate diameters.
1004. The normals to the ellipse at the points where it is met by
the straight lines
?!?+^y=i * + y = _i
a b ^ ap bq '
will all intersect in one pointy
(ax _ by _ *** ~ ^*\
1005. From a point P of an ellipse PM, PN are let fall perpendi-
cular upon the axes and MN produced meets the ellipse in. Q^ qi prove
that the normals at Q, q intersect in the centre of curvature at p^ Pp
being a diameter.
•
1006. From a point are drawn normals OPy OQy OJ?, OS, and />,
q, r, 8 are taken such that their co-ordinates are equal to the interoepta
on the axes made by the tangents at /*, Q^ R, S\ prove that />, q, r, $ lie
in one straight line. Also, if through the centre C be drawn straight
lines at right angles to CPy CQy CH, CS to meet the correspondlmg
tangents, the four points so determined will lie in one straight lina
[If Xf Y he the co-ordinates of 0, the two straight lines will be
xX ^t/Y=a'~ 6«, a'Xx + b'ry + aV = 0.]
1007. The normals to an ellipse at P, Q, R, S meet in a point and
the circles QRS, RSP, SPQ, PQR meet the ellipse again in the poinU
Fy Qy R, S' respectively : prove that the normals at P*, (^, Ry JSt meet
in a point.
1008. Normals are drawn at the extremities of a chord parallel to
the tangent at the point a : prove that the locus of their intersection is
the curve
2 (ax sin a 4 6^ cos a) (ax cos a + &y sin a) = (a* - b*)' sin 2a cos' 2flk
.1009. Normals are drawn at the extremities of a chord drawn
through a fixed point on the major axis : ptov^that the locus of their
intersection is an ellipse whose axes are
a \ cj b \ c/
the distance of the given point from the oenti-e being ca,
W. P. W
162 CONIC SECTIONS, ANALYTICAL.
1010. The normal at a point P of an ellipse meets the curve in Q
and any other chord PF is drawn ; QF and the straight line through P
at right angles to PF meet in R : prove that the locus of R is the straight
line
« . y . . «* + 6"
- cos © — ?• sin = —i rj ,
a b a—h
where ^ is the excentric angle of P. The part of any tangent intercepted
between this straight line and the tangent at /* is divided by the point
of contact into two parts which subtend equal or supplementary angles
at P.
1011. A chord PQ is normal at P, PF is a chord perpendicular to
the axis, the tangent at F meets the axes in T, T*, the rectangle
TCTR is completed and CR meets PQ in U : prove that
CR.CU=a'-h'.
1012. Along the normal at P is measured PO inwards equal to
CDf and the other normals OL, OM, ON are drawn: prove that the
parts of LP, MP, NP intercepted between the axes are equal to a + 6 ;
the tangents at L, M, N form a triangle whose circumscribed circle is
fixed ; and if r^, r,, r, be the lengths LP, MP, NP,
r, + r, + r,= 2(a-6),
^•i Va (a - ^) = 2a* {ah - P(/) ;
any of the three r^, r,, r, being reckoned negative when drawn from a
point whose distance from the major axis is greater than f^ . .
Corresponding results may be found when PO is measured outwards, but
in that case two of the normals will always be impossible unless
a>26.
1013. The chord PQ is normal at P, and is the pole of PQ: prove
that
where p is the perpendicular from the centre on the tangent at P,
1014. Perpendiculars jp^ p^ are let fall from the ends of a given
chord on any tangent, and a perpendicular />, from the pole of the
chord : prove that
a-^
;>^,=;?3'cos»-^
where o, j5 are the excentric angles of. the given iwints.
CONIC SECTIONS, ANALYTICAL. 163
1015. Two circles have each double contact with an ellipse and
touch each other : prove that
*^ " ? **" 1-^ '
r^ r^ being the radii ; also the point of contact of the two circles is equi-
distant from the chords of contact with the ellipse.
[Only the upper sign applies when the circles are real; the corre-
sponding equation for the hjrperbola is formed by putting - 6' for l*, as
usual, when the circles touch only one branch, but for circles touching
both branches the equation is
1016. Two ellipses have common foci S, Jff, and from a point P on
the outer are drawn two tangents PQ, PQ' to the inner: prove that
QPq SPS'. ^ , ,.
cos — j7 — : cos — TT- IS a constant ratio.
1017. The sides of a parallelogram circumscribing an ellipse are
parallel to conjugate diameters : prove that the rectangle under Uie per-
pendiculars let fall from two opposite angles on any tangent is equal to
the rectangle under those from the other two angles.
s
1018. The diagonals of a quadrilateral circumscribing an ellipse are
oa', bb\ cc\ and from 6, b\ c, c' are let fall perpendiculars /?,, /?,, Pg, p^ on
any tangent to the ellipse : prove that the ratio pj)^ : pjD^ is constant
and equal to X, where
and (Xj, yX (x ^ y^ are the points a, a'. If the points of contact of the
tangents from o be Z, L\ from h' be M^ M\ from c be Z, iT, and from e'
be L\ Mj the value of X is equal to the ratio of [LL'M'M] at any point
of the ellipse to its value at the centre.
1019. Prove that the equation
/?cos(a-/8)+|sin(a-/8)-l|{^cos(a+i8) + |sin(a4./5)--l|
= <- cos a 4-^ sin a -cos/8>
is true at any point of the ellipse -§ + !• = 1 ', and hence that the locus
1 64 CONIC SECTIONS, ANALYTICAL.
of a point from which if two tangents be drawn to the ellipse the centre
of the circle inscribed in the triangle formed by the two tangents and the
chord of contact shall lie on the ellipse is the confocal
1020. Two tangents are drawn to an ellipse from a point (X, F) :
proTC that the rectangle under the perpendiculars from any point of the
ellipse on the tangents bears to the square on the perpendicular from the
same point on the chord of contact the ratio 1 : X ; where
^^''^K-^^w)-[b^-a^A-^-'y''^)[^
1021. Four points A, B^ C, D are taken on an ellipse, and perpen-
diculars p^ p^ p^ p^ let fall from any point of the ellipse upon the
chords AjBj CD ; ACy BD respectively : express the constant ratio
p^p^ ' PJp^ in terms of the co-ordinates (Xj, F,), (X,, F,) of the polea
of he J AD, and prove that the value of the ratio will be unity if
a' b* a- + 6**
1022. A tangent is drawn to an ellipse and with the point of con-
tact as centre is described another ellipse similar and similarly situated
but of three times the area : prove that if from any point of this latter
ellipse two other tangents be drawn to the former, the triangle formed by
the three tangents will be double of the triangle formed by joining their
points of contact.
1023. Two tangents TP, TQ meet any other tangent m F,Qf i
prove that
PF,Qq^ TF. TQ^ cos' "^ ;
where a, )3 are the excentric angles of /*, Q,
1024. Two sides of a triangle are given in position and the third in
magnitude : prove that the locus of the centre of the nine points' circle
of the triangle is an ellipse ; which reduces to a limited straight line if
the acute angle between the given directions be 60^. If c be the given
length and 2a the given angle, the axes of the ellipse will be equal to
c sin 3a c cos 3a
4 sin' a cos a ' 4 sin a cos' a '
1025. The tangent at a point P meets the equal conjugate diameters
vclQ^Q \ prove that tangents from Q, Q will be parallel to the straight
line joining the feet of the perpendiculars from F on the axes.
1026. The excentric angles of the comers of an inscribed triangle
are a, /3, y : prove that the co-ordinates of the centre of perpendiculars
are
CONIC SECTIONS, ANALYTICAL. 166
"2^ (c(wa4.008^ + 008y)--2^c(w(a + /3 + y),
and those of the centre of the circumscribed circle are
a* — &*
—J— {co8 a + co6)3-foo8y + cos(a + /3 + y)}
- ^— {sin a + 8in)34-8iny-Bin(a + /3 + y)}.
■
The loci of these points when the triangle is of maximum area are
respectiyelj
4(aV+5y) = (a*-6X
16(aV + 6«y^ = (a«-M)«.
1027. The centre of perpendiculars of the triangle formed by tan*
gents at the points a, /3, y is the point given by the equations
A P^y 7""* a-j8
MX COS ^— s— COS -^-^ COS — ^
= a' {cos a + cos/3 -I- COS y - COS (a •»- /3 + y)} + 2 (a* 4- &*) COS a 008/3 oosyy
46y cos ^^— ^ cos *-^ cos —^
:=6'{8ina + sin/3 + 8iny + sin(a + i34- y)} + 2(a' + &^nna8in/38iny.
1028. Two points H, IT eae conjugate with respect to an ellipse,
P is any point on the ellipse, and PH^ PE' meet the ellipse again in
(?, Q : prove that QQ passes through the pole of HIT.
1029. The lines
form a triangle self-conjugate to the ellipse : prove that
and that the co-ordinates of the centre of perpendiculars of the triangle
are
1030. A triangle is self-conjugate to a given ellipse and one comer
of the triangle is fixed: prove that the circle circumscribing the
triangle passes through another fixed point (X, that C, 0, (f are in one
straight line, and that CO . CO = a' + V.
166 CONIC SECTIONS, ANALYTICALl
1031. In the ellipses
or o a
a tangent to the former meets the latter m P^Q: prove that the tan-
gents at Pf Q are at right angles to each other.
1032. Two tangents OP, OQ are drawn at the points a, P : prove
that the co-ordinates of the centre of the circle circumscribing the tri-
angle OPQ are
cos
cos
sm
2 a* + (a* - 6*) cos a cos j8
7^ 2a
~2^
g + jg
2 6" + {V - a«) sin a sin iS
a-jS 26
cos-2-
If this point lie on the axis of x, the locus of is a circle (or the axis
of a;).
1033. Two points P, Q are taken on an ellipse such that the per-
pendiculars from Qy P on the tangents at P, Q intersect on the ellipse :
prove that the locus of the pole of PQ is the ellipse
aV + 6y = (a'+67,
and that if 2? be another point similarly related to P, the same relation
will hold between Q, R; the centre of perpendiculars of the triangle
formed by the tangents at P, Q, R will be the centre of the ellipse, and
the centre of perpendiculars of the triangle P, Q, R lies on the ellipse
aV + 6y = (a'-67.
1034. Three points (aj^, y^, (a?,, y,), (aj^, y^ on an ellipse are such
that a5j + a;, + ajj = 0, ^i + y, + yj = : prove that the circles of curvature
at these points will pass through a point on the ellipse whose co-
ordinates are
^ar.g^a;, 4y,y^,
a' * 6« •
1035. At a point P of an ellipse is drawn a circle touching the
ellipse and of radius equal to n times the radius of curvature, and the
two other common tangents to the circle and ellipse intersect in (X, Y)
and include an angle : prove that
a«-X«"^6'-X'
1=1,
and 4naV tan« ?^ - (^•^)'-(^'±^)'
ana ^nab tan ^ - («' - nX') (nV - 6«) '
n being reckoned negative when the circle has external contact and X
being the semidiameter parallel to the tangent at P.
CONIC SECTIONS, ANALYTICAL. 167
1036. A triangle of minimnm area circumscribes an ellipse, is its
centre of perpendiculars and OM, ON perpendiculars on the axes : proTe
that MNia a normal to the ellipse at the point of concourse of the three
circles of curvature drawn at the points of contact of the sides.
1037. The tangent at the point whose excentric angle is ^ touches
the circle of curvature at the point whose excentric angle is $ : prove
that
sin
sin
2 ^ 1 - «W^
'^TS 2cosd(l-<j'co6'd)*
H P be the point ^, and T the pole of the normal at P, FT will be the
least possible when the |X)int lies on the normal at P.
1038. The hyperbola which osculates a given eUipee at a point and
has its asymptotes parallel to the equal conjugate diameters meets the
ellipse again in the same point as the common circle of curvature ; and if
P be the point of osculation and the centre of the hyperbola, PO is the
tangent at (^ to the locus of and is normal to the ellipse
1039. A rectangular hyperbola osculates a given ellipse at a point P
and meets the ellipse again in the same point as the common circle of
curvature : prove that, if be its centre, PO will be the tangent at
to the locus of and will be normal to the ellipse
1040. An hyperbola is described with two conjugate diameters of a
given ellipse for asymptotes : prove that, if the curves intersect, the tan-
gent to the ellipse at any common point is parallel to the tangent to the
hyperbola at an adjacent common point, and the parallelogram formed
by the tangents to the hyperbola will be to that formed by the tangents
to the ellipse as m' sin' ^ : 1, the equation of the hyperbola being
^ 2a?y ^ ^ y*
-^ + — .^ cot ^ - f, = w.
a ao
If the common points be impossible the points of contact of the common
tangents will li>{ on two diameters, and the parallelograms formed by
joining the points of contact will be for the ellipse and hyperbola respec-
tively in the ratio w* : sin* 0.
1041. A triangle circumscribes the ellipse and its centroid lies in
the axis of a; at a distance c from the centre: prove that its angular
points will lie on the conic
1G8 CONIC SECTIONS, ANALYTICAL.
1042. A triangle is inscribed in the ellipse and its oentroid lies in
the axis of a; at a distance e from the centre : proye that its sides will
toach the conic
_4ay ,(2a;-3c)'
[In this and the preceding question the axes need not be rectangular.]
1043. A triangle is inscribed in the ellipse and the centre of perpen-
diculars of the triangle is one of the foci : prove that the sides of the
triangle will touch one of the circles
1044. A triangle circumscribes the circle a* + y* = a* and two of its
angular points lie on the circle (as - c)* + y* = 6* : prove that the locus of
the third angular point is a conic touching the common tangents of the
two circles ; that this conic becomes a parabola if (c ^ a)' = 6' - a' ; and
that the chords intercepted on any tangent to this conic by the two
circles are in the constant ratio
2a' : J{:lab + 6* - c')(2a6 - 6* + c%
1045. . A triangle circumscribes an ellipse and two of its angular
points lie on a confocal ellipse : prove that the third angular point lies
on another confocal and that the perimeter of the triangle is constant.
1046. Two conjugate radii CPy CD being taken, PO is measured
along the normal at P equal to k times CD : prove that the locus of
O is the ellipse *
Q^_ y
and this ellipse touches the evolute of the ellipse in four points which
are real only when k lies between - and j- : k being negative when PO is
measured outwards.
1047. The ellipses
are so related that (1) an infinite number of triangles can be inscribed
in the former whose sides touch the latter ; (2) the central distance of
any angular point of such a triangle will be perpendicular to the
opposite side ; (3) the normals to the first e]li|)se at the angles of any
such triangle, and to the second at the points of contact, will severally
meet in a point.
CONIC SECTIONS, ANALYTICAL. 169
1048. The ellipses
are such that the normals to the latter at the comers of any inscribed
triangle whose sides touch the former meet on the latter.
1049. The semi-axes of an ellipse U are CA, CB; LCL is the major
axis and C the focus of another ellipse F, LC = Bt\ CL' — CA : prove
that the auxiliary circle of V touches both the auxiliary circles of C;
one of the common tangents, FP^, of U and V is such that P lies on the
auxiliary circle of V; and PL, PL' ar » parallel to CA^ CB\ CP'^ CL ai'e
isquaUy inclined to CA^ CB; if the auxiliary circle of V meet U also
in, Q, B, tS, the triangle QRS has the centre of its inscribed circle at 6',
and the straight lines bisecting its external angles touch V and form a
triangle whose nine points' circle is the auxiliary circle of F, and whose
circumscribed circle has its centre at the second focus of V; aloo if the
three other common tangents to L^, V form a triangle Q'B'S', the centt«
of its circumscribed circle is C and its nine points* circle is the auxiliaiy
circle of U ; the sum of the excentnc angles of Q^ B, S is equal to that
of the points of contact of the triangle (i'B!S\ and if this sum be
S the excentric angles of Q, B^ /S are the roots of the equation
^ — A h
tan — -- = - tan $, and those of the ix)ints of contact of Q'R^ are the
S — d a
roots of the equation tan --— = j tan 0; the three perpendiculars of the
triangle QRS' are normals to U and meet in the second focus of F,
OP is normal at P and a circle goes through P and the other three
points of contact. The straight lines through Q, B, S at right angles to
CQ, CB, CS will touch V in points y, r, s such that Cq, Cr, Cs make
with CA angles respectively equal to the excentric angles of Q, B, S. The
normals at Q, B, S meet in a point C from which, if the fourth normal
ffp be drawn, Pp ia a diameter of U ; and the normals at the points of
contact of QiB'S' meet in a point o on the same normal O'p such that
op ; 0/? = a6 : a* - a6 + 6*.
1050. A triangle LMN is inscribed in the ellipse -= + Ti = 1 so that
or b
the normals at Z, J/, N meet in a p<iint 0, and from the fourth
normal OP is drawn : prove the following theorems.
(1) OP will bear to the semi-diameter conjugate to CP the ratio
k : 1 where k is given by either of the equations
JT = (A* - a) cos (a + /3 + y), T = (6 - Jfca) sin (a + /3 + y),
where JT, Y are the co-ordinates of and a, /3, y the excentric angles
of Z, M, N.
(2) The sides of the triangle LMN will touch the ellipse
— + il- .i 1
170 CONIC SECTIONS, ANALYTICAL.
in points whose excentmc angles are ir + a,ir + /3, ir + y, if
ci 1/ 1
a\a-kh) V^ka'-b)" c^-V
(3) The tangents at Z, if, IT will form a triangle whose comers
i? V*
lie on the ellipse -j-, -!- -^ = 1, at points whose excentric angles are ir + a,
IT + /3, IT + y ; whei-e -4a' = a*, -56' = 6*.
(4) An infinite number of such triangles LMN can be inscribed in
the ellipse — , + ^, = 1 and circumscribed to the ellipse — ?i + ^ = 1, the
excentric angles a, j9, y satisfying the two independent equations
cos a + cos P + cos y * (t ) cos (a + jS + y),
sino + Binj8 + siny = f ^ j sin (a + jS + y),
and the relation between the axes beinir — + t = !• The ratio k : 1
"a b
remains the same for all such triangles, and if L\ M\ N' be the points
of contact of the sides, the ratio of the areas of the triangles L' M' N\
LMN is always the &ame, being afb' : ab, the ratio of the areajs of the
corresponding conies.
(5) Four points related to each triangle LMN : (a) the centroid,
(fi) the centre of perpendiculars, (y) the centre of the circumscribed
circle, (8) the point of concourse of the normals, lie each on a fixed
ellipse co-axial with the original, and the excentiic angle is always the
excess of the sum of the excentric angles of Z, My N above ir, while the
several semiaxes are
, , /a' b\a /a' b\b ,^, aa'^-W aa' -W
(^> "2^ 6' -26" a' (^) a^^""-^)^ ^.(«'-^')-
[The results here given include all cases of triangles inscribed in the
ellipse — , + rf = 1 with sides touching a co-axial ellipse.]
1051. Triangles are circumscribed to an ellipse such that the
normal at each point of contact passes through the opposite angular
point : prove that the angular points lie on the ellipse
aV 6y
— + — = 1
(\-a7 (X-67
CONIC SECTIONS, ANALYTICAL. 171
X being the greater root of the equation
a* 6*
• + \ A* ~ ^ >
X-a' X-6-
the locus of the centre of perpendiculars of the triangles is the ellipse
(«'-\)'J+(6'-X)'g = («'-6%
and the perimeter of the triangle formed by joining the points of contact
18 constant.
1052. The two similar and similarly situated conies
will be capable of having triangles circumscribing the firat and inscribed
in the second, if
tn * 2nh = — - + T« •
a
1053. A circle has its centre in the major axis of an ellipse and
triangles can be inscribed in the circle whose sides touch the ellipse :
proTe that the circle must touch the two circles
1054. A triangle LMN is inscribed in a given ellipse and its sides
touch a fixed concentric ellipse : prove that the excentric angles a, )3, y
must satisfy two equations of the form
sin (j3 + y) + sin (y + a) + sin (o + j3) := m,
cos (j8 + y) + cos (y + a) + cos (a + j3) = »*,
where m^ n are constant ; and that the equation of the ellipse touching
the sides is
^(f/»' + n+l)+|5(m« + r*-l) + 4m^=(^ ^ j-
Also prove that the area of the triangle LMN bears to the area of the
triangle formed by joining the points of contact a constant ratio equal
to that of the area of the ellipses. If (x^ y^ be the centroid, (a?^ y^ the
centre of perpendiculars, and (x^, y^ the centre of the circumscribed
drcie, and a + )3 + y = ^,
— ? = m sin + n cos tf, — = wsin ^ - w» cos ^ :
.a
2aa;, =m(a' + 6*) sin tf + {n(a« + 6*) -a* + 6*) cos ^,
26y, ={n(a« + 6*) -a* 4. 6'} sin d-w(a« + 6*) cos ^;
4aa?,= (a* - 6'){w sin tf + (n + 1) cos d},
4*y,= (a* - 6'){w cos tf - (n- 1) sin tf};
from which the loci can easily be found.
172 co5Hr
1053. Triangles are inacriiked ia m prat eQ%ae waA Iksl their mdea
tonch a fixed taoceatnn cDxpae of gh^en. area j- ^^ — =^ : prove that this
ellipse win have doable ooatakct wish ench of the elfipees
^^^~\ t J'
1056u A trun^ LMS is dfltmuatr ibgd aboot a giTen ellipee of
focus ^sodi that the ai^ks SMS^ :fSL.SLMmn aH «{aal (=^: prove
that sin ^= ^- , and that Z^ JT, JT lie oq one of two fixed cirdfis whose
J. . 2a* , ^ sEnZsinJTsinjr , .
comnKm radms is - - : aiao tan 9 = , ^ — _ — ^y and the point
6 1 4> eos Lt cos Jv cos Jf
of contact of Jf JT lies on the straight line joining L to the point of
intersection of the tangents at Jf, S to the cxrcle LMS.
1057. A triangle is fonned l^ tangents to the dlipse at points
whose excentric an^es •, P*y sttJdafy the poristie sjstem
cos(/3-i>7)^(sin)3-l>8in7)-i'»=:0, ke.:
prove that the locos of the angnho* points is
the envelope of the sides of the triangle formed bj j<Hning the points of
contact is the parabohi
2x\
J=(«-i)('.+i*-);
and the centroid of this latter triangle, its centre of perpendiculars, and
centre of circumHcribed circle lie on three fixed straight lines parallel to
the axis of x.
1058. A triangle is formed by tangents to the ellipse at points
whose excentric angles a, /3, y satisfy the poristie system
cos P cos y + m (sin j8 + sin y) + m* == 0, Ac. :
prove that the locus of the angular points is
the envelope of the sides of the triangle formed by joining the points of
contact is the hyperbola — , -, - (? + wi ) +1 = 0; the locus of the cen-
9a:* /3v \*
troid is the ellipse — j +(x'*'*^) ~^> *^** ®^ *^® centre of perpendi-
culars the ellipse aV+ (6y + tna* + 6*)* = 6*; and that of the centre of
the circumscribed ciix;lc the straight line 2bi/=^m (a* -6').
m"
CONIC SECTIONS, ANALYTICAL. 173
1059. A triangle is formed by tangents drawn to a giyen eUipse
at points whose excentric angles satisfy the equations
ic08(a + j8 + y) + m(oos/^ +y+e08 y+a+COBa + /3)+n(c06a+C08/3+0OSy)=/i,
/sin(a + j9 + y) + m(8inj3 + y+ ... + ... ) + w(8iD a+ ... ) = 5' :
proye that the angular points will lie on the conic whose equation is
-¥ iqnj-+ i {Im + up) - +4^w-^=0;
(which, since its equation inyolves four independent constants, will be
the general equation of the conic in which can be inscribed triangles
whose sides touch the given ellipse -, + "f, = 1.) The loci of the oen-
troid, &c. of the triangle whose angular points are a, )3, y can easUy be
formed, and it wiU be found that the centroid lies on an ellipse similar
and similarly situated to the given ellipse ; that the locus of the
centre of perpendiculai*s is similar to the given eUipse but turned through
a right angle; and that each locus reduces to a straight line when fn*=n*f
in which case a + j3 + y is constant
1060. The maximum perimeter of any triangle inscribed in a given
ellipse is
2/3 a'-^b'-^Ja^^'b' + b*
•J a' + 6" + 2 v/a* - a*b' + b* '
and if 2T, 2 F, 2'Z be diameters parallel to its sides
X'^Y' + Z'^a'-hb' + Ja'~-^b^b\
1061 . A parallelogram of maximum perimeter is inscribed in a given
ellipse and 2 A, 2 Y are its diagonals : prove that
1^ 1 Ji^ 1^
and that the perimeter is 4 J a* + 6*.
1062. A hexagon ABGA'BC of maximum perimeter is inscribed
in an ellipse : prove that its perimeter is 4 v — ; the tangents at
Ay B, C and A\ B^ C form triangles inscribed in the same fixed circle
of radius a f 6 ; also, if a triangle be inscribed in the ellipse vrith sides
each parallel to two sides of the hexagon, the sides of this triangle will
touch a fixed circle of radius - — , and its area will be half that of the
a ¥b
174 CONIC SECTIONS, ANALYTICAL.
hexagon. Also, if X, Y^ ZY)e radii of the ellipse each parallel to two
fades of the hexagon,
and if X\ Y\ Z' be radii each parallel to the tangents at two oomers of
the hexagon
J_ J JL - ^ 1 1 2 _ 1 /I i\
1063. A hexagon AECA'BC is inscribed in the elUpee ^ -^ ^= 1»
and its sides touch the ellipse -^ + r^= 1 ; a triangle ahc is inscribed in
the former ellipse so that he is parallel to BC^ and BC, &c. : proTO that
a, A will be at the ends of conjugate diameters, the area of the triangle
will be half that of the hexagon, the tangents at ABC and those at
A'RCT form triangles inscribed in the ellipse
and the sides of the triangle ohc touch the ellipse ,_ „ + .,^.^ =1.
[The relation ^ ^^' + ^ -^-^ = 1 must be satisfied.]
1064. The tangent to a conic at P meets the directrices in JT, K\
and from JT, iT' are drawn two other tangents intersecting in Q ; proTO
that PQ is normal at P and is bisected by the conjugate axis.
1065. Two straight lines are drawn parallel to the major axis at a
distance ht"^ from it : prove that the part of any tangent intercepted
between them will be divided by the point of contact into two parts
subtending equal angles at the centre.
1066. The part of any tangent intercepted between the two straight
lines
<^^^^\-'^-k^^y-<^-^r
is divided by the point of contact into two parts subtending equal angles
at the point (a, 6).
1067. Two tangents to the ellipse -, + ^ = ~t irii ii^^ersect in a
point T on the axis of x\ prove that the part of any tangent to the ellipse
-- + 1^ = 1 intercepted between them is divided by the point of contact
ah
into two parts subtending equal angles at the point on the axis of x
which is conjugate to T with respect to the latter ellipse.
COinC SECTIONS, ANALYTICAL. 175
1068. The value of X is so determined that the equation
repTesents two straight lines : prove that the part of any tangent to the
eUipse intercepted between these two straight lines is divided by the
point of contact into two parts which subtend equal (or supplementary)
angles at (Z, T). H be the point (Z, Y) the two values of ~ are
^ • Discuss the case when O coincides with
SorS\
1069. Two conjugate diameters of a given ellipse meet the fixed
straight line p- + gj-'=^ in F, F, and the straight lines drawn through
P, P' respectively at right angles to these diameters intersect in Q :
prove that the locus of Q is the straight line
apx + hqy = a* + 6* ;
and the locus of the intersection of straight lines drawn through P, F
perpendicular respectively to CF^ CF is the straight line
qax - phy = 0.
1070. A parallelogram circumscribes a given ellipse, and the ends
of one of its diagonals lie on the given straight lines /'~ + 9^ = ^l :
prove that the ends of the other diagonal lie on the conic
^ y"
-Hu-'t)'-
1071. In the ellipses
CPQ is drawn to meet the curves, and QQf is a double ordinate of the
outer : prove that FQf is normal at Q\
1072. From any point on the normal to a given ellipse at a fixed
point (a cos a, b sin a) are drawn the three other normals to the ellipse
at points F^Q^JR: prove that the centroid, the centre of perpendiculars,
and the centre of the circumscribed circle of the triangle FQE lie re-
spectively on the straight lines
ox sin a -ft^ cos a =0, 6a; sin a + ay cos a <= 0, 2 (o^c sin a - 6y cos a)
+ (a* - 6*) sin o cos o = 0.
176 CONIC SECTIONS, ANALYTICAL.
1073. A triangle is inscribed in an ellipse and its centre of perpen-
diculai-s is at the point (X, Y) : prove that the locus of the poks of its
sides is the conic
1074. A fixed point is taken within a given circle, a pair of
parallel tangents drawn to the circle, and ^0^' is a straight line meeting
the tangents at right angles. An ellipse is described with focus and
axis AA\ and the other two common tangents to this ellipse and the
circle meet in F : prove that F lies on a fixed straight line bisecting at
right angles the distance between and the centre of the circle.
1075. With the focus of an ellipse as centre is described a circle
touching the directrix ; two tangents drawn to the circle from a point
F on the ellipse meet the ellipse again in ft ^: prove that QQ' is
parallel to^ the minor axis, and that tangents drawn from Q, Q' to the
circle will intersect in a point F' on the ellipse so that PF^ is also
parallel to the minor axis. The tangents to the circle at the real
common points pass through the further extremity of the major axis,
and the points of contact with the ellipse of the (real) common tangents
are at a distance from the focus equal to the latus rectum.
1076. At the ends of the equal conjugate diameters of an ellipse
whose foci are given are drawn circles equal to the circle of curvature
and touching the ellipse externally : prove that the common tangents to
the ellipse and one of these circles intersect on the rectangular hyperbola
which is confocal with the ellipse.
1077. From any point F on the ellipse -i + ^g = 1, tangents are
drawn to the ellipse -s + 7i =" 1 • prove that they meet the former
ellipse in i)oints Q, Q' at the ends of a diameter, and that the tangents at
Q, Q will touch ^e circle which touches the ellipse externally at F and
has a diameter equal to the diameter conjugate to CF,
1078. A circle is drawn through the foci of a given ellipse and
common tangents drawn to the ellipse and circle : prove that one pair of
straight lines through the four points of contact with the circle will
envelope the hyperbola
a'- 26* 6'" '
confocal with the ellipse.
1079. From a fixed point (X, Y) are drawn tangents OP, OQ to a
conic whose foci are given : prove that the locus of the centre of the
circle OFQ is the straight line
2xX_^ l yY
iuad tlie locus of the centre of perpendiculars is a rectangular hyperbola
CONIC SECnONS, ANALTTICiXi. 177
ol which one aaymptote is parallel to CO, redncing to two straight lines
if lie on the lemniscate of which the given foci are vertices.
[The equation of the rectangular hyperbola is
(Xaj+ry)(Xy-y«) = c'(Xy+7«-2X7);
and if (/ be its centre, C&.CO^CS*, and the angle SCff is three times
the angle SCO.]
1080. Two tangents OPy OQ being drawn to a given conic, prove
that two other conies can be drawn conf ocal with the given conic and
for their polars of the normals at P, Q.
1081. Two conies have common foci Sj 8*^ a point is taken such
that the rectangle under its focal distances is equal to that under the
tangents to the director circles : prove that the polars of will be
normals to a third conf ocal conic at points lying on the polar of with
respect to that conic.
1082. A diameter FP' of a given ellipse being taken, the normal at
F^ intersects the ordinate at F in Q: prove that the locus of ^ is the
ellipse
and that the tangents firom Q meet the tangent at P in points on the
auxiliary circle.
1083. A chord FQ of an ellipse is normal to the ellipse at P, and
p, q are perpendiculars from the centre on the tangents at P, ^ : prove
that
p'-q'-(a^-p'){p'^b^)'
1084. The locus of the centre of an equilateral triangle inscribed in
a given ellipse is the ellipse
^ (a* + 3by + 1^' (3a« + by - (a* - by.
1085. From two points on the polar of a point are drawn two
pairs of tangents at right angles to each other to a given ellipse : prove
that the four other points of intersection of these tangents lie upon the
tangents at to the confocals through : and the tangents drawn from
a pair of these points to the corresponding confocal will be parallel to
each other.
[The latter proposition is more readily proved geometrically.]
1086. An ellipse is described passing through the foci of a given
ellipse and having the tangents at the end of the major axis for direc-
trices : prove that it will have double contact with the given ellipse, and
that its foci will lie on two circles touching the given ellipse at the ends
of its major axis and having diameters equal to half the latus rectum«
W. P. 12
176 come SBcmoKs, ahalttical.
1087. The least distanoe between two points Ijing nqpeetirelj on
the fixed ellipses
k
y
i?"*"?'^' ^^6^"^'
(a''6*-a'6'0 (o'-a'*- ft* + n
Explain how it comes to pass that this vanishes for confocal and for
similar ellipses,
1088. Prove that if the ellipse -^ — t + ^^—\ = 1 — touch a parallel
to the ellipse ;3 + ^ =" !> ^® distance between the ellipse and its parallel
will be Jyi^ and the ratio of the curvatures at the point of contact will be
V(«'-A)(6'-X):(a'6V-X«)(X-/t).
IY« Hyperbola^ referred to its axet or asymptotes,
[The equation of the hyperbola, referred to its axes, only differing
from that of the ellipse by having — V instead of 6', many theorems
which have been stated for the ellipse are obviously also true for the
hyperbola. It is convenient still to use the notation of the excentric
angle and denote any point on the hyperbola by a cos a, 6t sin a, and all
the corresponding equations, but the excentric angle is imaginary.
A point on the hyperbola may be denoted by a sec a, 2>tana, but
the resulting equations are not nearly so symmetrical as the corre-
sponding equations in terms of the excentric angle are for the ellipse.
The angle a so used is sometimes called the excentric angle in the case
of the hyperbola. When referred to its asymptotes the equation of the
hyperbola is ^ocy = a' + 6', but the axes are not generally rectangular,
and questions involving perpendicularity should not be referred to such
axes. The equation is often written a:^ = c' : in this form the equation
of the polar of (JT, Y) is xY-s-yX-Wy and that of the two tangents
from (JT, r) is 4 («y - c») (ZF- c*) = («7 + yZ- 2c')'.]
1089. Prove that the four equations
6 (05*^3^ -a*) =a (y ± Vy* + ^*)
represent respectively the portions of an hyperbola refeiTod to its axes
which lie in the four quadrants.
1090. The equation of the chord of an hyperbola referred to its
axes which is bisected in the point (JT, Y) is
6'Jr(a;-Jr)^aT(y-r);
and the corresponding equation when referred to the asymptotes is
CONIC SECnONS, ANALYTICAL. 179
1091. Hie equation of the chord of the hyperbola xy^tf whose
eztremitieB are the points (a;,, y^), {x^ y^ is
'=1.
• ^^
1092. The locos of points whose polars with respect to a giyen
parabola touch the circle of curvature at the vertex is a rectangular
iiTperbola.
1093. . The normals to any hyperbola 2;y = c* at any point where it
is met by the ellipse a^ + ^ = c'(l + sec w), w being the angle between the
asymptotes, are parallel to one of the asymptotes.
1094. A circle described on a chord AB oi 201 ellipse as diameter
meets the ellipse again in (7, />, and AB^ CD are conjugates with respect
jc* •/■ a* - 6*
to the ellipse : prove that AB touches the hyperbola — « - jtj = -i — is f
and CD the hyperbola -- - ^ = , — .1 .
^'^ or b* ar-o
1095. A doable ordinate PP* is drawn to the ellipse -« + a " ^t
and the tangent at P meets the hyperbola -^ - || = 1 in Q, Q': prove
that P'Qf FQ are tangents to the hyperbola; and, if i?, i^ be the
points in which these lines again meet Uie ellipse, that RR divides PP
in the ratio 1 : 2.
1096. Two circles are drawn, one having double contact with a
single branch of a given hyperbola and the other having single contact
with each branch, and their chords of contact with the hyperbola meet on
an asymptote : prove that the pole of either asymptote with respect to
one circle \a the pole of the other asymptote with respect to the other
circle, and that its locus is a rectangular hyperbola passing through the
foci of the given hyperbola and having one asymptote in common
with it.
1097. A circle is drawn with its centre on the transverse axis to
touch the asymptotes of an hyperbola : prove that the tangents drawn to
it at the points where it meets the hyperbola will also touch the
auxiliary circle of the hyperbola. If the circle have its centre on the
conjugate axis and common tangents be drawn to it and the hyperbola,
the locus of their points of contact with the circle is the curve
(6V-ay)(«' + y' + 6')" = 4(a" + 6«)a:'(6V-ay-a'6').
1098. The tangent to an hyperbola at P meets the asymptotes in
Z, L : prove that the circle LCL' passes through the points where the
normal meets the axes, that the points where the tangent meets the axes
are conjugate with respect to the circle, and the pole of LL is the point
through which pass all chords of the hyperbola subtending a right angle
at P.
180 CONIC SECTIONS, ANALYTICAL,
1099. Two hyperbolas have the same asymptotes, and NPQ is drawn
parallel to one asymptote meeting the other in N and the curves in P, Q;
a tangent at Q meets the outer hyperbola in two points and the straight
lines joining these to the centre meet the ordinate NQ ia L, M: prove
that NQ is a geometric mean between NL, NM and that NP is a
harmonic mean between NQ and the harmonic mean between NL^ NM,
1100. The axes of an ellipse are the asymptotes of an hyperbola
which does not meet the ellipse in real points : prove that the difference
of the excentric angles of the points of contact of tangents to the ellipse
drawn from a point on the hyperbola will have the least possible vaJue
when the point is on one of the equal conjugate diameters ; also the
locus of the points of contact with the hyperbola of the common tangents
is the curve
1101. The locus of the equation
c* c* c*
y=x +— — —
^ 05 + a; + 05 + ... to 00
is that part of the hyperbola ^'-xy = c* which starting from the axis of y
goes to infinity along the line y = x,
1102. The locus of a point from which can be drawn two straight
lines at right angles to each other, each of which touches one of the
rectangular hyperbolas xy — ^t?^ is also the locus of the feet of the per-
pendiculars let fall from the origin on tangents to the hyperbolas
•
1103. An ellipse is described confocal with a given hyperbola, and
the asymptotes of the hyperbola are the equal conjugate diameters of
the ellipse : prove that, if from any point of the ellipse tangents be
drawn to the hyperbola, the centres of two of the circles which
touch these tangents and die chord of contact will lie on the hyperbola.
1104. The centre of perpendiculars of the triangle whose angular
points are \cm^, ^j , ^cm„ ^ , Tern,, ^j is the point Tc/x, H ,
where yum^jm,^ = — 1.
1105. Denoting by the point m the point whose co-ordinates are
cm, — , prove that if a circle meet the rectangular hyperbola a^ = c* in the
«
four points m^, m,, m,, m^,
and i^ of four points m^ m , m,, m , any one is the centre of perpendi-
culars of the triangle formea by joining ^e other three
fn^tn^jn^ = — 1 •
CONIC SBCnONS, ANALTTICAL. 181
1106. The rectangular hyperbola a:* - ^ = a* is cut orthogonaUj hy
all the dlipses represented by the equationa
aB*+3/ + Xy-3a* = 0, 3a:' + y' +Xa;+ 3a*-0;
and the rectangular hyperbola xy=^a'hy the ellipses
«^-iry + y' + X(aj + y)-3a*=0, a:* + a:y + y' + X(aj-y) + 3a* = 0.
1107. Normals are drawn to a rectangular hyperbola at the ends of
a chord whose direction is given : the locus of their intersection is
another rectangular hyperbola, whose asymptotes make with the
as3rmptotes of die given hyperbola angles eqiial and opposite to those
made by the given direction.
1108. The normal to a rectangular hyperbola at P meets the curve
again in Q, and 6, ^ are the angles which the central radii U> F, Q make
with either asymptote : prove that
ton' ^ tan «^ = 1 , tan PCC + 2 tan CPC = 0,
and that the least value of the angle CQP is sin'*^. Also if the diameter
PP be drawn, QP will subtend a right angle at -r .
1109. In a rectongular hyperbola the rectangle under the distances
of any point of the curve from two fixed tangents is to the square on the
distance from their chord of contact as cos ^ : 1, where ^ is the angle
between the tangents.
1110. A circle is described on a chord of an ellipse as diameter which
is parallel to the straight line - cos a + r sin a = : prove that the locos
of the pole with respect to the circle of the straight line joining the two
other common points ia the hyperbola
- -j :—p- = a — ,
COS a sin a
[If the diameter be the straight line - cos a + ^ sin a s cos )3, the pole
of the other common chord is the point {X, F), where
= 5.+(a' + 6')cosA
cosa cosp ^ '
-. — H + (a* + 6*) cos A]
Bin a cos p
1111. An ellipse and an hyperbola are so related that the
asymptotes of the hyperbola are conjugate diameters of the ellipse : prove
that by a proper choice of axes their equations may be expressed in the
forms
«• . y* 1 «• y*
i»
■^P^^' a«"S-«=^
182 CONIC SECTIONS, ANALTTICAX.
1112. An hyperbola is described with a pair of eonjugate diameters
of a given ellipse as asymptotes: prove that the angle at which the
curves cut each other at any common point is
/ 2a'ysin((?-y) \
V8intfsin^ + 6*costfoos^/'
where 6, & are the angles which the common diameters make with the
major axis of the ellipse. The equation of the hyperbola will be
-i + 2A -^ - ^ = m, the axes of the ellipse being the coK)rdinate axes,
and if tangents be drawn to both curves at the common points, the
parallelogram formed by the tangents to the hyperbola will bear to that
formed by the tangents to the ellipse the ratio
m«:l + A«.
1113. From two points (oj^, y,), (oj^, y,) are drawn tangents to the
rectangular hyperbola xy = c* : prove that the conic passing through the
two points and through the four points of contact will be a circle if
aj,y, + «^, « 4c*, and «,«, = y,y,.
1114. A triangle circumscribes a given circle and its centre of
perpendiculars is a given point : prove that its angular points lie on a
fixed conic which is an ellipse, parabola, or hyperbola, as the fixed point
lies within, upon, or without the given circle.
[For the co-ordinates of the centre of perpendiculars of a triangle
formed by tangents to an ellipse at points whose excentric angles are
a, j3, y, see Question 1027].
1115. Three tangents are drawn to the rectangular hyperbola rey = a'
at the points (a;,, y,), (a;,, y ), (a;,, y,) and form a triangle whose circum-
scribed circle passes through the centre of the hyperbolar: prove that
a;,4-a;,-ha;, ^ y,4-y,-f y, ^^ ,
and that the co-ordinates of the centre of the circle are ' ^* , ^'^f^* , a
point on the hyperbola.
a
1116. A fixed point whose coordinates are X, Y being taken,
a chord PQ of the hyperbola osy = c* is drawn so that the centroid of the
triangle OPQ lies on the hyperbola : prove that FQ touches t^e conic
{xY^ yX + Xr- 9c')«= 36c»ay,
which is an ellipse when XY:>9(^, and an hyperbola when XY<9<f,
but degenerates when XY^^ 9c*. If 0(/ be bisected by the centre of the
given hyperbola, the centre of the envelope divides OO' in the ratio 3:1.
y
qoiao SEcnoNfi^ amalttioal, 18S
1117* A triangle is inscribed in the hjperbola oeys^P so tbat its
oentroid is the fixed point f ca,-K (a point on the hyperbola): prove
that its sides will touch the ellipse
whiok touches the asymptotes and the hyperi>ola (at the fixed point), the
cnrvatares of the two curves at the point of contact are as 4 : 1, and the
tangents to the ellipse where it again meets the hyperbola are parallel to
the asymptotes.
1118. The circle of curvature of the rectangular hyperbola at the
point (a cosec $^ a cot 6) meets the curve again in the point (a cosec ^
a cot ^) : prove that
tan^tan';r» 1.
1119. Circles of curvature are drawn to an hyperbola and its con-
jugate at the ends of conjugate diameters : prove that their radical axis
Is parallel to one of the asymptotes.
[If X, F be the co-ordinates of one of the points, the equatum of
the radical axis will be
the upper sign being taken when the straight line joining the two points
is parallel to the asymptote - = ^ .]
1120. Triangles are inscribed to the circle »* + y* = 2a« whose sides
touch the rectangular hyperbola a:* — y' = a': prove that the locus of the
centres of perpendiculars is the circle
»■ 4- y* + 4aa; = 0.
1121. On any hyperbola P, Q, E are three contiguous points and L
the centre of perpendiculars of the triangle PQR; find the limiting
position of L when Q, R move up to P; and prove that its locus for
different positions of P is
4 {isr+ (a« - 6*) uy + 27 »•&• («• - by s= o,
where ^=6V-ay-aV, and U = aif -^ y' - a' + b\
V. Polar Co-crdinatei.
1122. The equation of the normal drawn to the circle r>2aoos9
at the point where » a is
a sin 2a s r sin (2a - 0).
184 dONIC SECTiONS, ANALYTICAL.
1123. The equation of the straight line which joins the two points
of the circle r = 2a cos at which = a, = j3, is
2a cos a cos)3 = r cos (a + j3 - 0).
1124. A chord AP of a conic through the vertex A meets the
latus rectum in Q, and a parallel chord FSQ" is drawn through a focus
S: prove that the ratio AF. AQ : Q'S. SF is constant.
1125. Prove that the equations
c = r («costf »fel)
represent the same conic. If (r, 6) denote a point on the curve when
the upper sign is taken, (— r, ir + 0) will denote the same point when the
lower sign is taken.
1126. A chord PQ of a parabola is drawn through a fixed point on
the axis, and a straight line bisecting the angle PSQ meets the directrix
in ; from perpendiculars OF, OQ' are let fall on SP, SQ : prove
that SF and SQ^ will be of constant length.
1127. Two circles are described touching a parabola at the ends of
a focal chord and passing through the focus : prove that they intersect
at right angles and that their second point of intersection lies on a fixed
circle. Also prove that the straight line joining the centres of the
circles touches an ellipse whose ezcentricitj is ^ and which has the same
focus and directrix as the parabola.
fThe equation of the parabola being 2a='r(l i-cosO), those of tiie
circles may be taken to be
r cos' a = a cos (^ — 3a), r sin' a = a sin (tf - 3a).]
1128. A conic is described having a common focus with the conic
c = r (1 + « cos tf), similar to it, and touching it where 6 = a: prove that
its latus rectum is ■= — ^^^ — «« and that the anirle between the axes
l+2«cosa + e" ^
of the two conies is 2 tan"* ( — ; j . If «> 1, the conies will inter-
V sina /
sect again in two points lying on the straight line
- (1 + « cos a) + c sin a {« sin tf + sin (tf - a)} == 0.
1129. Two chords QPy PR of a conic subtend equal angles at the
focus : prove that the chord QR and the tangent at P intersect on the
directrix.
1130. Two conjugate points Qy Qf are taken on a straight line
tlirough the focus S of a, conic, and the straight line meets the conic in
P : prove that the latus rectum is eqiud to
2SP.SQ 2SP.S(y
JSQ'-SP^ iSQ'-SP'
CONIC SECTIONS, ANALTTICAL. 185
1131. Throng a point on the axis of an ellipse at a distance
ellipse in F and a tangent at right angles in Y : prove that the rectangle
FO . OY is equal to the square on the semi latus rectum.
y-
1132. The points of an ellipse at which the circle of curvature
through the other ends of the respective focal chords are given bj
the equation
2r'-r(3a + c) + 2ac = 0,
where 2a is the major aauB, r the focal distance, and 2c the latus rectum*
1133. The two circles which are touched hy any circle whose
diameter is a focal chord of a given conic have the directrix for their
radical axis and the focus for one of their point circles.
[The equations of the two circles are
r" (1 * c) + cer cos 6 = c\
that of the conic being e = r (1 + 6 cos 6),]
1134. The radii of two circles are a, b and the distance between
their centres is c, where c (a + 6 -f c) = 2 (a - 6)' ; the centre of a circle
which always touches them both traces out an ellipse whose vertex (the
nearer to the centre of the smaller circle) is A : prove that the ends of
the diameter of the moving circle drawn through ^ lie on a fixed ellipse
with its focus at A,
1135. Prove that any chord of the conic
c ■« r (1 + « cos ^)
which is normal at a point where the conic is met by the straight lines
£^(5 + ^) = *sintf + (B'-l)C08tf
will subtend a right angle at the pole.
1136. A conic with given excentricity and direction of axes is
described with its focus at the centre of a given circle : prove that the
tangents to this conic at the points where it meets the circle touch a
fixed conic of which the given circle is auxiliary circle.
1137. Two parabolas have a common focus and axes opposite, a
circle is drawn through the focus touching both parabolas : prove that
3r*=<i*-a*6* + 6*,
a, h being the latera recta and r the radius of the circle.
1138. Four tangents to the parabola 2a = r (1 + cos ^) are drawn at
the points 2^ , 20^ 2$^ 2$^ : prove that the centres of the circles circum-
scribing the tour triangles formed by them lie on the circle
2rcoBtf,costf,costf,costf^»acos(tf, + tf, + tf, + tf^-tf).
186 OOXIC SECTIONS, AKALYTIOAL,
1139. Through a fixed point is drawn any straight line, and on it
are taken two pointd such that their distances from the fixed point are
in a constant ratio and the line joining them subtends a constant angle
at another fixed point : prove that their loci are circles.
1140. Twa circles intersect, a straight line is drawn through one of
their common points, and tangents are drawn to the circles at the points
where this line again meets them : prove that the locus of the point of
intersection of these tangents is the cardioid
cr = 2ah {1 + cos (^ + a - ^)} ;
the second common point of the circles being the pole, the common
ehord (c) the initial line, a, b the radii, and a, fi the angles subtended by
a in the segments of the two circles which lie each without the other
circle.
1141. The equation of the circle which touches the conio
c = r (1 + c cos ff)
at the point where » a, and passes through the pole, is
- ( 1 + « COS a)* = cos (a - tf ) + e cos (2a - tf ) ;
c
and the equation of the chord joining their points of intersection is
- (1 4- 2c 008 o + c^ = e* cos tf + e" cos (tf - o).
1142. Two ellipses have a common focus S, a common excentricity
0, axes in the same straight line, and the axis of the outer (^) is to that
of the inner (D as 2 - 6* : 1 — 6* ; on a chord of Uy which touches T, as
diameter is described a circle meeting U again in the points P, Q : prove
that the circle PSQ will touch U and that PQ will touch a fixed similar
ellipse having the same focus S and its centre at the foot of the directrix
of ^.
1143. Two similar ellipses ?7, V have one focus 8 common, and the
centre of T is at the foot of that directrix of U which is the polar of S'y
a tangent drawn to T at a point P meets U in two points : prove that
the circle through these points and S will touch Z7 at a point Q such that
SPj UQ are equally inclined to the axis, H being the second focus of U.
1144. A conic is described having the focus of a given conic for its
focus, any tangent for directrix, and touching the minor axis ; prove that
it will be similar to the given conic.
[Also easily proved by reciprocation.]
1145. Any point P is taken on a given conic, A is the vertex,^ the
nearer focus, and on ^P is taken a point Q such Uiat PQ exceeds SP by
the sum of the distances of A^ S from the directrix : prove that the
locus of Q is a conic whose focus is A^ similar to the given conic and
having its centre at the farther vertex.
CX)XIC SECTIONS, ANALYTICAL. 187
1146. The point ^ is a focus of an ellipse c - r {I + e cos 0), O^ 0' are
two points on anj tangent such that SO = S(/ = me, and SOy SO' meet
the ellipse m Fj Q: prove tliat FQ touches the conic
e = mr \l + « (1 + t») cos 6] ;
and the tangents at F, Q intersect on the conic
m*(l— e*)f — tfcoetfj +2emcos^( — «cos^j = l.
[The latter conic is a circle with its centre at the second focus and
radius equal to the major axis, when ni{l - e*) = 2.]
1147. With the vertex of a given conic as focus and any tangent as
directrix is described a conic passing through the nearer focus : prove
that its major axis is of constant length equal to the distance between
the focus and directrix of the given conic, and that the second directrix
envelopes a conic similar to the given conic and having a focus in
common with it.
1148. Two straight lines bisect each other at right angles: prove
that the locus of the points at which they subtend equal angles is
r* a cos O-b sin 6
ah b cos ^ - a sin ^ '
2a, 26 being the lengths of the lines, their point of intersection the pole
and the initial line along the length 2a.
1149. The focal distances of three points on a conic being r,, r,, r
and the angles between them a, j3, y, prove that the latus rectum (2/)
is given by the equation
4 , a , 8 , y 1. 1.^1.
I sm - sm - sm I = - sin a + - sm j8 -I- - sm y ;
the angles a, ^, y being always taken so that their sum is 2v,
1150. An ellipse circumscribes a triangle ABC and the centre of
perpendiculars of the triangle is a focus : prove that the latus rectum
will be
2/? cos A COB B cos C
. A , B , a '
sin^sm^sm^
i? being the radius of the circle ABC.
1151. Two ellipses have a common focus and axes inclined at an
angle a, and triangles can be inscribed in one whose sides touch the
other : prove that
c/ A 2e^c^ = «/c/ + «/c/ - 2«,«/,<j, 008 a,
0j, e being the latent recta, and e^, e^ the exoentricities. Also if 0, ^ ^
he the angles subtended by the sides of any such triangle at the focus
c^ = 4<j, COS ^ cos ^ cos ^ , or 4Cj 006 ^ sm -^ sin ^ , &c.
188 CONIC SECnOXS, ANALYTICAL.
1152. Two ellipses hare a oommon fooas and axes inclined at the
angle
cos
where 2c, 2(/ are the latera recta and c, a' the excentricities : prove that
any common tangent subtends a right angle at the focus.
VT. General Equation of the Second Degree.
[The general equation of a conic, in Cartesian Co-ordinates, being
w= aa*+6y' + c+^ + 2^a; + 2Aay = 0,
the equations giving its centre are
du ^ du ^
_. = ■— = 0.
dx * dy
The equation determining its excentricitj may be found at once from
the consideration that
a + &-2Acoflo> ab-h'
sin'cD ' sin' CD
are unchanged by transformation of co-ordinates ; and therefore that
(a->-6-2Acosa))' ^ (a' + jg')'
(a^-A*)sin«ai "" a'/J* '
where m is the angle between the co-ordinate axes, and 2a, 2fi the
axes.
The excentricitj e is thus given by the equation
c* . _ (a + 6 - 2A cos <!))•
r=V"^ (a6-A«)sin«ai '
The area of the conic ti = is
IT A sin o>
A being used to denote the discriminant
a, A, g
A, ^ /
or abc-hygh-af''bg'''Ch\
The foci may be determined fr*om the condition that the rectangle
under the perpendiculars from them on any tangent is constant. Thus,
taking the simple case when the origin is the centre and the axes
rectangular, if the equation of the conic be
aa;* + 6y* + 2Aajy + c = 0,
CONIC SECTIONS^ ANALYTICAL. 189
and (X, T), (- X, — 7) two conjugate foci, we must have in order that
the atraight line px + qy — 1 may be a tangent
or ;^(/i + Z«)+^(/i+r^ + 2/)5r2:r-l=0 (A).
Bat the atraight line will be a tangent if the quadratic equation
ox' + 6y* + c {px + qi/)*+2hxt/=: 0,
found hy combining the equations, have equal roots ; that is, if
(a + cp*) (6 + o/) = (A + cp9)«,
or p'bc -^ q'ae - 2pqch -^ ab - h' = (B).
Now (A), {B) expressing the same geometrical fact must be coinci-
dent, so that
/i-fX'^ ^-f 7' ~xr _ 1
6c ac eh h* -ab'
The equations for X, Z are then
X'-r« XF c
a — b h ah—h*'
Also we can obtain for /a the equation
/ he \/ ae \ c^h'
equivalent to
whose roots are the squares of the semiaxes. To each root correspond
two foci, real for one and unreal for the other.
The same method applies to aU cases ; and, the foci being found, the
directrices are their polars.
The more useful special forms of the general equation are
(1) ar* + fty* + 2Aay = 2a;,
where a normal and tangent are co-ordinate axes ;
which, for different values of A, represents a series of conies all passing
through four fixed points, a pair of joining lines being the co-ordinate
<') M-" -"(!)'
the equation of a conic touching the co-ordinate axes at distances h, k
from the origin. It is sometimes convenient to use this as the equation
190 CONIC SECTIONS, ANALYTICAL.
of a conic touching four given straight lines^ A, k, X being then para-
meters connected by two equations
the equations of the two other given straight lines being
When (3) represents a parabola, X = 1 ; and the equation may be written
The equation of the polar of (X, 7), when the equation is in the
most general form it = 0, is
P= « (aZ + AF + ^) + y (AX+ 67+/) +^X+/r4- <? = 0,
and this may be adapted to any special case. The equation of the two
tangents from (X, 7) is
(a«* + V + c + 2/y + 2^a; + 2/w^) (aX' + 67' + c + 2/r + 25rX + 2AXr)
= {a:(aX+ Ar+^) 4-y (/tX+ftr+Z) +^X+/r + c}»,
or uU==F'.
The equation of a tangent at (X, Y) to the parabola
X y _.
(AX)4 (AF)* '
the signs of the radicals in the equation of the tangent being determined
by those of the corresponding radicals in the equafion of the cutrd at the
point (X, F). Of course the equation of the polar cannot be expressed
in any such form.
The condition that the straight line px-^qy-^r^O may touch the
conic u = is
where A =^bc-f*^ F-gh- af, &c. The systems (a, 6, c,/, g, A), (-4, J5,
C, F, G, H) are of course reciprocal.]
1153. Trace the conic, any point of whifeh is (1) asin(tf-a),
6sin(^+o); (2)|(^+— ), 2^'^m)' ^^^"^ «» ^» *»> *^ * *^ «^^®^
1154. Prove that all conies represented by the equation
aj« (a« + 6* - 2a6 cos tf ) + y* (a* + 6* + 2ah cos 6) - ixyab sin 9 = (a' - &•)•,
*-^">tever the value of 0, are equal and similar, the lengths of the
ibein^ 2{a^b).
CONIC 8ECTI0XS, ANALTOPAL. 191
1155. On two parallel fixed straight lines are taken points A, F;
B, Q respectively, A^ B being fixed and P, Q variable, subject to the condi-
tion that the rectangle under A P, BQ is constant : prove that PQ touches
a fixed conic which wUl be an ellipse or hyperbola according aa P, Q are
on the same or opposite sides of AB,
1156. One side AB of a rectangle ABCD slides between two
rectangular axes : prove that the elliptic loci of C7, D have equal areas
independent of the length of AB^ and that the angle between their axes
1157. If in any position AB make an angle 6 with the axis of «
and a, P he the angles which the tangents at (7, Z> to their loci make
with the axes of y, x respectively,
AB
cot a + cot tf = cot j8 + tan 6 = j^^ .
1158. A straight line of given length slides between two fixed
straight lines and from its extremities two straight lines are drawn in
given directions : prove that the locus of their intersection is an ellipse.
1159. A circle being traced on a plane, the locus of the vertex of
all cones on that base whose principal elli|>tic sections have an excen-
tricity e is the surface generateil by the revolution about its conjugate
axis of an hyperbola of excentricity e"*.
1160. Trace the conies
2a5*-2a:y + 2ay-a* = 0, 2y* - 2a^ - 2ay + a* = 0, 2a?y-2ay + a' = 0;
proving that they touch each other two and two.
1161. Trace the following conies:
(1) 5a:* + 20y' + 8xy-35x-80y + 60 = 0,
(2) 36x* - 20y" + 33xy - 105x + 7y - 23 = 0,
(3) Z^T? + 29y" + 2ixy - 72a: + 126y + 81 = 0,
(4) 144x*- 144y»-120a:y+ 120x- 24y+ 1 =0,
(5) 369a:» + 481/ - 384a^ - 2628« + 3654y + 2484 = 0,
(6) 16a:« + 9y»-243cy-96a;-72y4-144 = 0,
(7) 4a:« + y»- 4a?y- 24x+ 22y+ 61 =- 0,
(8) ?«:• - 7y' - 48a:y + 1 75a: + 1 75y - 1050 = 0,
(9) 8^x« + 403/* - 116ay- 5460a: + 3780y + 88641=0.
[To reduce the equation tt = 0, if ab — V be finite first move the
origin to the point whose co-ordinates satisfy the equations
^=0 - = 0-
192 CONIC SECTIONS, ANALYTICAL,
the equation will become
and if the axes be then turned through the acute angle 6 determined by
the equation (a — h) tan 20 = 2h^ the equation becomes
where a\ h' are the roots of the equation (z ^a){z-h)=^ h\ and the sign
of a' — 6' is the same as that of h. When ab = A', we may suppose u to
be {ax + py)* + 2/y + 2gx + c, and arranging it in the form
(aa; + j8y- A;)' + 2 (/+ ifcjS) y + 5 (gr + Aki) aj + c- A:*,
if we determine k by the equation
the straight line ax + fy = k will be the axis and the straight line
2 (/+ A?j8) y + 2 (^ + A;a) « + €-*•= 0,
the tangent at the vertex.]
1162. Prove that in general two parabolas can be drawn through
the points of intersection of the conies
u = aa5* + 6y* + c + 2^+ 2^ + 2Aa;y = 0,
w' = aV + 6y + c'+ 2/'y + 2g^x+ 2h'xy = 0;
and that their axes will be at right angles if
h K
1 1 63. The equation of the director circle of the conic u «• is
C(a:' + y')-26^-2J5ry + ^+^ = 0,
A^ B, &c. being the reciprocal coefficients,
11 64. The equation of the asymptotes of the conic u » is
1165. The foci of the conic u = are given by the two equations
Fx-^Gi/-H=Cxi/,
2Gx-'2Fy'A-¥B=C{x''i/').
1166. The equation of the chord of the conic u = which is bisected
by the point (X, Y) ia
or (a:-jr)^+(y-r)^=0.
CONIC SECTIONS, ANALTTICAL. 193
11 67. Prove that the origin of co-ordinates lies on one of the equal
conjugate diameters of the conic u = 0, if
a + 6 ab-hf
1168. The rectangle under the distances of theorigin from two con-
jugate foci of the conic ««=» is
1169. The equation of the asymptotes of the conic u = is
d^u /du\' d^u /du\* ^ dSi, dudu_^
^ Wy/ <^y* \dK) dxdy dxdy" *
that of the axes is
d^u f /^Y « (^^^ - /<^u _ d?u \ A* da
dxdy\\di) " \d^) j W dp) dxdy*
and the foci are determined by the equations
(du\' fdu\* du du
dx) \dy) dxdy
d^u (Pu tru
daf d\f dxdy
[The co-ordinates are supposed rectanguhur in the two latter, but not
in the first.]
1170. The equation of the equal conjugate diameters of any conic
u-0 is
d^ "*■ dj^J W \^/ '^ d^\dy) " d^^ dy)
- 9 f^ ^ « "?tr j '\ (/du\' /du\'\
^ W dy' dxdy\ j\\dx) ■*• \dyj ]'
1171. The rectangle under the distances of any point {x, y) from
two conjugate foci of the conic u » is
V ( \dSaf di^/ \dxj \dyj ) { dxdy dxdy)
d'ud'u /d'uy
da^ di^" \dxdy)
1172. The equation of a conic, confocal with u^ 0, and having its
asymptotes along the equal conjugate diameters, is
l\(fttr rfy*/ dxdyl J
° \^ " dp) \S| ~ dy\) dxdy dx dy *
W. P. A&
194 CONIC SECTIONS, ANALYTICAL.
1173. Frore that the general equation of a conic confocal with
UaQ ifl
If X = 16, this is the confocal whose asymptotes lie along the equal
conjugate diameters of u ; and, if X » 8, it is the confocal about which
can be described parallelograms inscribed mu.
1174. The equation
represents a pair of parallel tangents to the conic i« = 0, and t^e equation
. du du ^
represents a pair of conjugate diameters of the conic
u = aa? + 6y* + 2Aa5y = c. .
1175. The axes of the conic aa^-^hj^-k-^hxy^c make with the
lines bisecting the angles between the co-ordinate axes angles 0', prove
that
tan2g = , (a--6)sina>
(a + 6) cos 0) - 2a
1176. If -^+ |^= 1 be the equation of a conic referred to con-
jugate diameters, the condition that the circle a^+i^-k- 2ocy cos cd = r* may-
touch the conic is
/I _1_\ /]: _ J_\ cob' CD
Hence determine the relations between any conjugate diameters and
the axes.
1177. The locus of the foot of the perpendicular let fall from a
point (X, Y) on any tangent to the ellipse 005* + 2Aay + 6y* - 1 = t7 = 0,
6(aj-X)'-2^(aj--r)(y-r) + a(y-.r)'
the axes being rectangular. Prore that tliis reduces to a circle and a
point-circle when
Z'-7' X7 _ 1
a- 6 "^ h 'A^TS*
CONIC SECnONS, ANALYTICAL, 196
1 1 78. The equations determining the foci of the conic v = are
y(x-k-f/ cos 0)) _ 05 (y + 05 cos co) _ 1
a cos m- h h cos ctf — A ab — h^'
[The equation — -~- = ^ — ■= is true whatever be the inclination of
the axes.]
1179. The general equation of a conic conf ocal with the conic v ^ is
and the given conic is also cut orthogonally by any conic whose equa-
tion is
1180. The equations of the equal conjugate diameters of the conic
v^O is
ah-h^" a + 6 '
when the axes are rectangular ; or
V 2 (a5* + y* + 2xy cos «)
ah-h* a-¥b-2h cos oi '
when the axes are oblique.
1181. The tangents to the two conies
aba ^ a + 6
at any common point are at right angles ; and if both curves be hyper-
bolas they will have four real common taiigent&
1182. An ellipse and an hyperbola are confocal and the asymptotes
of one lie along the equal conjugate diameters of the other : prove that
any conic drawn through the ends of the axes of the ellipse will out
the hyperbola orthogonally.
1183. Each common tangent drawn to the two conies
will subtend a right angle at the centre.
1 184. Two common tangents to the circle a;* + y* = 2aa5 and the conic
ar*+(y-Xa;)*+2a» =
subtend each a right angle at the origin : also the tangents are parallel
to each other, and the straight lines joining the origin to the points of
contact with either curve are parallel to the axes of the conia Hence
prove that, if at a point on an ellipse where the rectangle under the focal
distances is equal to that under the semi-axes a circle equal to the circle
of curvature be drawn touching the ellipse externally, and FF'f QQf be
196 CONIC SECTIONS, ANALYTICAL.
the other common tangents, FQ!^ PQ will pass through the point of
contact and be parallel to the axes.
1185. Two parabolas are so situated that a circle can be described
through their four common points : prove that the distance of the centre
of this circle from the axis of one parabola is equal to half the latus
rectum of the other.
1186. An hyperbola is drawn touching the axes of an ellipse and
the asymptotes of the hyperbola touch the ellipse : prove that the centre
of the hyperbola lies on one of the equal conjugate diameters of the
ellipse.
1 1 87. On two fixed straight lines are taken fixed points A^B\ C^Di
prove that the parabola which touches the two fixed straight lines and the
asymptotes of any conic through A, B, C^ I) will also touch the straight
line which bisects AB and CD.
1188. With two conjugate semi-diameters CPy CD of an ellipse as
asymptotes is described an hyperbola, and pdia a common chord parallel
to FD and bearing to it the ratio nil: the curvatures of the two curves
at any common point will be as 1 : 1 — n*.
1189. Five fixed points are taken, no three of which are in one
straight line, and five conies are described each bisecting all the lines
joining four of the points, two and two : prove that these conies will
have one common point.
1190. A conic is drawn touching a given conic at F and passing
through its foci S, S' : prove that the pole of SS' with respect to this
conic will lie on the common normal at F, and will coincide with the
common centre of curvature when the conies osculate.
1191. A parabola is drawn having its axis parallel to a given straight
line and having double contact with a given ellipse: prove that the locus
of its focus is an hyperbola conf ocal with the ellipse and having one
asymptote in the given direction.
[If the given direction be that of the diameter of the ellipse through
the point F (a cos a, 6 sin a) and the latus rectum of the parabola be
2ka'b^'i-CF*f the co-ordinates of the focus are
} a cos a I -J ^ ,i . i + T ) > i 6 Bin a ( -5 ^ , . , \ , + y ) >
* \a* cos* a + 6* sin" a */' * \a* cos' a + 6* sin* a kj '
and the equation of the directrix is
- . a*cos*a + 6'8in*a k(a*^b') ^
axcosa-^bj/sma^ sr -H ^ — ^.J
1191*. An hyperbola is drawn touching a given ellipse, passing
through its centre, and having its asymptotes parallel to liie axes:
prove that the centre of curvature of the ellipse at the point of contact
lies on the hyperbola, and that the chord of intersection of the two
curves touches the locus of the centre of the hyperbola at a point whose
distance from the centre of the hyperbola is bisected by the centre of
the ellipse. At the point of contact the curvature of the hyperbola is
two-tlurds oi that of the ellipse.
CONIC SECTIONS, ANALYTICAL. 197
1192. Taking the equation of a conic to be u»0, if X be so deter-
mined that the equation u + X (x* + ^ + 2aBy cos cd) » represent two
straight lines, the part of any tangent to the conic intercepted between
these straight lines will be divided by the point of contact into two parts
subtending equal (or supplementary) angles at the origin. If the co-
ordinates be rectangular [ta = -\ and px-¥qi/ = l be one of the two
straight lines, then will
cpq+fp + gq + h^Of a + 2^ + <y* = 6 + 2/y+cg*.
1193. The part of any tangent to the ellipse a^t^ + hV^tfV
intercepted between two fixed straight lines at right angles to each other
is divided by the point of contact into two parts subtending equal
angles at the point (X, T) : prove that
and that the two straight lines intersect in the point ( JT, F^, where
X" " r""a*-6*'
1194. The tangent at a point F to the parabola t^^s^iaos meets the
tangent at the vertex (A) in Q and the straight line a; + 4a = in (^: prove
that the angles QAF, Q[AP are supplementary; and, generally, that the
two straight lines y-mx-v—. t/ + m(a;+4a) + — «0, have a similar
property with respect to the point \—^ , — j .
1195. An ordinate MP is drawn to the ellipse ^ + n^* ^ <^<^ ^^
tangent at F meets the axis of a; in ; from are drawn two tangents
to the ellipse -7 ''"Tfl = ^ > prove that the parts of any tangent to the
first ellipse intercepted between these two will be divided by the point
of contact into two parts subtending equal angles at if. For the two
lines to be real, P must lie between the latent recta.
1196. The straight lines AA\ BR, CC are let fall from A, B, C per-
pendicular to the opposite sides of the triangle ABO, and conies are
described touching the sides CA, AB and the perpendiculars on them :
prove that the locus of the foci is
(ar' + y*){(6* + ac)a:+6(a-c)y-(6* + ac)(a-o)}
= etc {(6* + ac)x^b(a'-c) y},
reducing to
« (a5* + y* — a*) = 0, when = ck
(The origin is A\ A' A the axis of y, and the lengths A'B, A' A, CA'
are denoted by a, 6, c.) In the last case trace the positions of the fod
for all different positions of the centre on A' A,
198 CONIC SECTIONS, ANALYTICAL.
1197. Two conies have four-point contact at 0, their foci are S, JET,
S'f H' respectively, and the circles OSH^ OS'W are drawn : prove that
the poles of SH^ S'H' with respect to the corresponding circles lie on
the common chord of the two circles.
1198. Two conies osculate at a and intersect in 0, the tangents at
meet the curves again in 6, c, the tangents at 6, c meet the tangent at a
ia Of B and each other in A : prove that A a, Bb, Cc, meet in a point
and that Ay 0, a lie on one straight line.
1199. Two conies osculate at and intersect in P, any straight line
drawn through F meets the conies again mQ,Q': prove that the tangents
at Q, Q' intersect in a point whose locus is a conic touching the other
two at and also touching them again, and the curvature at of this
locus is three-fourths of Uie curvature of either of the former, and that
the straight lines joining with the other two points of contact form
with OP and the tangent at a harmonic pencil If one couio be a
circle and the angle POQ a right angle, OP, OQ will be parallel to the
axes of the other.
[The equations of the two conies being
ic* + 6y* + hxy = aXf
mf-^h^^ + h'xy^axy
that of the locus is
{(A - A') a + 6y}* = 46 (a;* + fty" + Aay - aa).]
1200. A given conic turns in one plane about (1) its centre, (2) a
focus : prove that the locus of the pole of a fixed straight line with
respect to the conic is (1) a circle, (2) a conic, which is a parabola when
the minor axis can coincide with the fixed straight line.
[The locus in general is to be found from the equations
^ . - a'co8*tf + 6*sin*tf
a5 = »cos^-o'sm^ + ^^ ^ r-^,
. ^ ^ (a* - 6") sin tf cos tf
the fixed point about which the conic turns being origin, the fixed
straight line being x = h; a, 6 the semi-axes, and p, q the co-ordinates of
the centre when its axis (26) is parallel to the fixed straight line.]
1201. A conic has double contact with a given conic : prove that its
real foci lie on a conic confocal with the given conic, and its excentricity
is given by the equation
where — , + tf = 1 ^ *he given conic, (X, Y) the pole of the chord of
contact, and a* — c', h^-c? the squares on the semi-axes of the confocal
through the foci The foci are given by the equations
a«-c«"*"6'-c«"^' a«-c'"*'6'-c«"^'
CONIC SECTIONS, ANALTTICAL. 199
the equation of the oonio of doable contact being
1202. Tangents are drawn to the conic oaf + hf^ + 2hxy = 2x from
two points on the axis of x equidistant from the origin: prove that
their four points of intersection lie on the conic bf^ + hxy = x,
1203. On any diameter of a given ellipse is taken a point such that
the tangents from it intercept on the tangent at one end of the diameter
a length equal to the diameter : prove that the locus of the point is the
curve
(s-a'-(^:-)'(^»-
1204. On the diameter through any point P of a parabola is taken
a point Q such that the tangents from Q intercept on the tangent at F
a length equal to the focal chord parcel to the tangent at F : prove
that the locus of Q is the parabola
3y' + 4a(a; + 4a) = 0.
1205. Tangents are drawn to the conic cuc^ + ^y* + 2hxy = 2x from
two points on tiie axis of x^ dividing harmonically the segment whose
extremities are at distances p, q from the origin : the locus of their
points of intersection will be the conic
{p (oo; + Ay - 1) - «} {q (oa; + Ay - 1) - x]
= (opy - jP - y) (a«* + 5y* + 2Aajy - 2a;}.
1206. From F, F ends of a diameter of a given conic are drawn
tangents to another given concentric conic : prove that their other
points of intersection lie on a fixed conic touching the four common
tangents of the given conies ; so that if the two given conies be confocal
the locus is a third confocal and the tangents form a parallelogram of
constant perimeter. In this last case, if Qy Q' be the points of intersec-
tion and tangents be drawn at F, F", Q, Q\ their points of intersection
will lie on a fixed circla
[The equations in the latter case are -i — ^ + t?—z = 1> -i + ^ = ^i
^+^ = 1, ^he™ V = «'6*. ^ + y. = (^(^l±^), «,d the
perimeter is twice the diameter of the circle. If X be negative, one of
the conies must be an hyperbola for real tangents, in which case the
locus will be an hyperbola and the difference of the sides of the paral-
lelogram will be constant]
1207. From two points 0, O are drawn tangents to a given conic
whose centre is (7: prove that if the conic drawn through the four
points of contact and through 0, CX be a circle, CO, CCf will be equally
inclined to the axes and 0, O will be conjugate with respect to the
rectangular hyperbola whose vertices are the foci of the given conic.
200 CONIC SECTIONS, ANALYTICAL.
1208. The area of the ellipse of minimum ezcentricity which can be
drawn touching two given straight lines at distances h^ k from their
point of intersection is
xf /»t ,.v (V + ^*-2Woos«)* .
vhk (h* ■¥ i^) ^ ^smtt:
(A* + A!*+2AA;co8co)*
and, if be the miTiirrmTn ezcentricity,
l-c«""A'*'sin'a»*
rci. a+6-2Aco8a> ^ah-h* ...
I Smce r-i and — r-s — are invariants, so is
■■ sm'a> sin'o) ^
(g 4- 6 - 2A cos ci>)*
(a6 - A«) sin^o) *
which is thus equal to ^^ ^^' if a, j3 be the semi-axes, or to \ — -4-
if be the excentricity. This function of e continually increases with 6*
so long as 0* is less than 1, hence in a system of ellipses the excentridty
will be a minimum when z = , and when therefore
1 — 0"
(<n-6-2Acosci))*
has its least value.]
1209. The conic of four-pointic contact with a given ellipse at the
point (a cos tf, 6 sin 0) has its minimum excentricity (0') given by the
equation
and the locus of its centre for different values of is the curve
<^*^-@4)-(?4)"("--'-)'
1210. The axes of any conic through four given points make with
the bisectors of the angle between the axes of the two parabolas
through the four points an angle ^ : prove that the excentricity is given
by the equation
0* 4 cos'fl ,
r^'"Bin«tf-sin«2<^^
and that the minimnm excentricity ia a / -i ^ 9 when sin 2^ = 0, so
that either axis of the ellipse of least excentricity is equally inclined to
the axes of the two parabolas.
1211. Three points AyB,C are taken on an ellipse, the circle about
ABC meets the ellipse again in F, and FF' is a diameter : prove that of
all ellipses through A, B^ C7, F the given ellipse is that of least excen-
tricity.
OOKIC SECTIONS, AKALTTICAL. 201
1212. Of all ollipeeB circnmBcribing a parallelogram, the one of
least excentridty luis its equal conjugate diameters parallel to the sidea.
1213. The ellipse of least exoentricity which can be inscribed in a
giyen parallelogram is such that any point of contact divides a side into
segments which are as the squares on the respective adjacent diagonals.
1214. Four points are such that ellipses can be drawn through
them, and e is the least excentricity of any such ellipse, e' the exceutri-
dty of the hyperbola on which the centres of the ellipses lie : prove that
also that the equal conjugate diameters of the ellipse are parallel to the
asymptotes of the hyperbola.
1215. The equation of the conic of least excentridty through the
four points (o, 0^ {«', 0), (0, b), (0, b') is
«• 4a»^cosci> y' /I 1\ /I 1\ . i a
aa aa +bo bo \a a/ ^\b bj
and its axes are parallel to the asymptotes of the rectangular hyperbola
through the four points.
1216. The axes of the conic which is the locus of the centres of all
conies through four given points are parallel to the asymptotes of the
rectangular hyperbola through the four points.
1217. The equation of the director circle of the conic
M-'-^(S)'
18 (X*-l)(a5* + y* + 2«ycosw) + ^ + Ay+(it« + Ay-W)cosw = 0.
■
1218. The equation of a conic, having the centre of the ellipse
a'y* + 6 V = a'b^ for focus and osculating the ellipse at the point 0, is
(«• + yO (a* cos* tf + 6« sin'fl)' ^ {(a* - 6*) (ox cos' tf - 6y sin*^ + a'b']\
1219. A rectangular hjrperbola has double contact with a parabola :
prove that the centre of the hyperbola and the pole of the chord of con-
tact will be equidistant from the directrix of the parabola.
1220. A conic is drawn to touch four given straight lines, two of
which are parallel : prove that its asymptotes will touch a fixed hyper-
bola and that this hyperbola touches the diagonals of the quadrilateral,
formed by the given lines, at their middle points.
1221. A parabola has four-pointic contact with a conic : prove that
the axis of the parabola is parallel to the diameter of the conic through
2a'b'
the point of contact, and that the latus rectum of the parabola is —^ ,
where a, b are the semi-axes of the conic and r the central distance of
the point of contact. If the conic be a rectangular hyperbola, the
envelope of the directrix of the parabola is
a* = (2r)*cosy.
202, CONIC SECTIONS, ANALYTICAL.
1222. The locus of the centre of a rectangular hyperbola having
four-pointic contact with the ellipse a'^ + 6 V s= aV is Uie curve
W + ftV "a'"*"*-'
1223. The locus of the foci of all conies which have four-pointic
contact with a given curve at a given point is a curve whose equation,
referred to the normal and tangent at the given point, is of the form
{mx + y) (ic" + y^ =aa^.
1224. The excentricity e of the conic whose equation, referred to
axes inclined at an angle co, is t^ = 0, satisfies the equation
fl* _ (a - by sin'o) + (a + 6 cos a> - 2A)*
1 -c* (a6-A')sin*ci) '
1225. The coordinates of the focus of the parabola
are given by the equations
and the equation of its directrix is
X{h-i-k cos io) + f/{k+h cos ui)=hk cos (di
1226. In the parabola (j) + f ^ j = 1, a tangent meets the axes of
co-ordinates in P, Q and perpendiculars are drawn from P, Q to the
opposite axes respectively : prove that the locus of the point of intersec-
tion is
x-^y cos ta y-^x cos to
-j + 1 = cos OK
k h
1227. The asymptotes of the conic
always touch the parabola
1228. One angrular point of a triangle self-conjugate to a given
conic is given : prove that the circles on the opposite sides as diameters
will have a common radical axis which is normal at the point of contact
to the similar concentric and similarly situate conic touching the polar
of the given point.
1229. Two circles of radii a^h {a> h) touch each other, and a conic
is described having real double contact with both : prove that, when the
Dotinto of contact are not on different branches of an hyperbola, the ex-
come SECTIONS, ANALYTICAL. 203
centricity '^ H ^ '*' r) > ^^^ ^^® latus rectum is >, =, or < a — 6, aooording
as the conic is an ellipse, parabola, or hyperbola. If the contacts are on
n. A- h
different branches, the excentricitj < r , and the asymptotes always
touch a fixed parabola.
1230. A triangle ABC circumscribes an ellipse whose foci are S, S'
and SA=-SB = SO : prove that
S'A^S'B.S'G.^.
SA.SB.SC^ '
and that each angle of the triangle ABC lies between the acute angles
-1 1 *« •
008 • -g- .
[When the conic is an hyperbola, 0* - 1 replaces 1 — «', and one angle
of the triangle will be obtuse and > ir — cos"' — ^ .]
1231. On eveiy straight line can be found two real points
conjugate to each other with respect to a given conic and the distance
between which subtends a right angle at a given point not on the straight
line.
1232. Prove that the axis of a parabola, which passes through the
feet of the four normals drawn to a given ellipse from a given point,
will be parallel to one of the equal conjugate diameters of the ellipse.
[If (X, y) be the given point and a'y* + 6V = a*6* the given ellipse,
the equation of the axis of the parabola will be
1/ a>X\\( 6*F\ ^1
1233. A conic is drawn through four given points lying on two
parallel straight lines : prove that the asymptotes touch the parabola
which touches the other four joining straight lines.
[The equation of the conic being taken to be
that of the asymptotes will be found by adding ^ — — ^ - m to the
sinister so as to give for the asymptotes the equation
and for the envelope the equation
{(=-0"-<'-)e*?)}"-*Ic-)-
The student should observe, and account for, the factor f — ^ j .]
204 CONIC SECTIONS, ANALYTICAL.
1234. An ellipse of constant area vc* is described Having four-
pointic contact with a given parabola whose latus rectum is 2m : prove
that the locus of the centre of the ellipse is an equal parabola whose
vertex is at a distance ( — ] from the vertex of the given parabola ; also
that when c = m the axes of the ellipse make with the axis of the para-
bola angles
} tan"' (2 tan <^),
where ^ is the angle which the tangent at the point of contact makes
with the axis.
1235. An ellipse of constant area irc' is described having four-
pointic contact mth a given ellipse whose axes are 2a, 2b : prove that
the locus of its centre is an ellipse, concentric similar and similarly
situate with the given ellipse, the linear ratio of the two being
-T j : 1. Also the described ellipse will be similar to the given
ellipse when the point of contact F is such that
CP':CI)'^ci:{ah)K
YH. Envdapea {of the second dose),
[The equation of the tangent to a parabola, iu the form
a
gives as the condition of equal roots in m, ^ = iax ; and the equatibn of
&e tangent to an ellipse
-C0Ba + Tsma=s 1,
a
written in the form
gives as the condition of equal roots in z
So in general if the equation of a line in one plane, straight or
curved, involve a parameter in the second degree, it follows that through
any proposed point can be drawn two lines of the series represented by
the equation. These two lines will be the tangents (rectilinear or
curvilinear) from the proposed point to the curve which is the envelope
of the system. If the proposed point lie on this envelope the two tan-
gents will coincide, hence the equation of the envelope may be found as
the condition of equal roots.
Thus, '<To find the envelope of a system of circles each having for
Its dJameter a focal chord of a given conic."
CONIC SECTIONS, ANALTTICAL. 205
IILSL' be a focal chord, ASL^a^ the mid point of LL\ and P
any point on the circle, we shall have
SL sz •= , SL' = 4 , OL = ■= — -= — -J- , SO = = — -s — -J— f
l+0OO6a' 1 — «C08a' l-«"co8"a' l-«"coe*a
and 0F'=^S(y-^SI^~2S0.SF COB FSO, whence the equation of the
circle
c" = r*(l -c*oo6*a)-2cercoBaooB(tf-a),
or, if tanas A,
(r*- cO(l +X") - e"r'- 2ctfr (coe fl + XBintf) = 0,
whence, as the condition for eqnal roots in A,
{r« (1 -e^-c'-2c«- cos ^}(r'-c') = c«««r' Bin* fl,
equivalent to
(r* - c* - cer cos ^* = «*r*,
or r*(l *«) -c«r cos tf = c',
80 that the envelope is two circles, one of which degenerates into a
straight line when 6=1, being the directrix of the parabola; and one
degenerates into a point when 6=2, being the farther vertex of the
hyperbola. In general, ii A, A' he the nearer and £u-ther vertices, the
two circles will have for diameters the segments AJSJi, A'SM'^ where
M'S^SM^cike semi latus rectum. In this case every one of the
system of curves has real contact in two points with the envelope, but it
frequently hi^pens that the contact becomes impossible for a part of the
series.
A method which is often the best is exemplified in the following :
*' To find the envelope of a chord of a conic which subtends a right angle
at a given point."
Move the origin to the given point, and let the equation of the
conic be
u=aa* + fty* + c + yy + 2gx + 2hxy = 0,
and let jXB+^=l be the equation of a chord. The equation of the
straight lines joining to the origin the ends of this chord will then be
oo* + 6^*+ 2Airy + 2 (gx-^fy) (px-k-qf/) + c (;»: + gy)*= 0,
which will be at right angles if
a-».6+2(jpy + 3/) + c(;?'+g^)=0.
Hence the equation
(a + 6) (iM; + 5y)'+ 2 (jpy + 3/) (;»: + ^ry) + c(;>' +^ =
represents two parallel chords of the series and involves the parameter
p : q in the second degree ; whence, for the envelope,
(a + 6a*+2^ + c)(a + 6y*+ 2^y + c) = (a + 6 ajy +>Sb + ^)',
or (/• -^ + / -ca) (ic» ^-y*) = (^+/y + c)*,
or the envelope is a conic haviug a focus at the given pointy directrix the
polar of the given point with respect to the given conic, and excentricity
y
—Jilt—
206 CONIC SECTIONS, ANALYTICAL.
The envelope of any series of lines is to be found from the condition
that the equation shall give two equal values of the parameter, but in all
the following examples it will be found that the equation of the line can
be written in the form
where X is the parameter, so that the envelope is UV= TT*. A common
form is fr=i7costf+ Fsintf when the envelope is, as already seen,
1236. A conic has a given focus and given length and direction of
major axis : the envelope is two parabolas whose common focus is the
given focus and whose common latus rectum is in the given direction and
of twice the given lengtL
[This is obvious geometrically.]
1237. The envelope of the circles
a"+(y-p)(y-5')=o,
where p, q are connected by the equation pq'^(i{p'-q) = 0^ is the two
circles
a" + y«i2aa5=0.
[It will be found best to make the ratio p : q the parameter, so that
the equation will be
pq{^-k-y')^ay{p*^f) + a'{p'qy=^0.'\
1238. The envelope of the circles
where jp, q are connected by the equation
(p + a) (y - o) + 6* « 0,
is the two circles
(a;*6)« + y'x=a'.
1239. The ellipse 5^a:*+jE?*y'=/)y has its axes connected by the
equation a'/^ = 9^ (p " 9"') ^ prove that the envelope is the two circles
a;* + y**2aa;=0;
and, if the relation between the axes be a'jp* = g^ (m'jp* - ny ), the
envelope will be
m V + «y * 2nax = 0.
1240. The envelope of the ellipse
a:* + y* - 2 {ax co6a + 5ysina)r-cosa + ^sina]
+ (a* + 6*-c^ r-C0Ba + jsinaj = a* sin* a + 6* cos* a - c*
CONIC SECTIONS, ANALYTICAL. 207
is ibe two oonfocal eUipees
[The parameter tan a is involyed in the second degree.]
1241. The director circle of a conic and one point of the conic are
given : proye that the envelope is a conic whose major axis is a diameter
of the given circle.
[The equation may be taken to be "t'*'^^;^ + Ts=^f ^> ^ being
parameters connected by the equation
a* + 6'=c«(l-X'),
and a^ c, given. The envelope is -j 4* , « = 1 .]
1242. Through a fixed point is drawn a straight line meeting two
fixed straight lines parallel to each other in Z, if: the envelope of the
drole whose diameter is LM is a conic whose focus is and whose
transverse axis has its ends on the two fixed straight lines.
1243. A variable tangent to a given parabola meets two fixed
tangents, and another parabola is drawn touching the fixed tangents in
these points : prove that the directrix of this last envelopes a third
parabola touching straight lines, drawn at right angles to the two fixed
tangents through their common point, in the points where they are met
by the directrix of the given parabola.
1244. A variable tangent to a given parabola meets two fixed
tangents, and on the intercepted segment as diameter a circle is de-
scribed : the envelope is a conic touching the two fixed tangents in the
points where they are met by the directrix of the given parabola.
[The given parabola being ( - j + \jA = 1, the envelope is
{xia-^h cos 0)) + y (6 + a cos 0)) - a6 cos «}* = \ahoey sin' «.]
1245. Through a point P are drawn two circles each touching two
fixed equal circles which touch each other at A : prove that the angle at
which the two circles intersect at P is 2 sec~^ (n), where n is the ratio of
the radius of the circle drawn through P' to touch the two at ^ to the
radius of either. K P lie on BB^^ a common tangent to the two fixed circles,
the circle through P touching the two will make with BB^ an angle
4P^A
1246. Through each point of the straight line — + -~ =: 1 is drawn
a chord of the ellipse c^^ + 6V~a'6' bisected in the point : prove that
the envelope is the parabola whose focus is the point
X -y a*-y
208 CONIC SECTIONS, ANALYTICAL.
and directrix the straight line
lax + mhy = a* + b\
1247. An hyperbola has a focus at the centre of a given circle and
its asymptotes in given directions : prove that tangents drawn to it at
the points where it meets the given circle envelope an hyperbola to
which the given circle is the auxiliary circla
/•
[The equation of a tangent will be - = e cos + cos (0 -> a) where
- = 1 +60080,
a
or we may write the equation
-(l+ecosa)=ecostf + cos(tf-o),
involving only the parameter a, and giving the envelope
f — ecoBOj =f COB 8 J + sin* fl,
1248. A circle subtends the same given angle at each of two given
points : prove that its envelope is an hyperbola whose foci are the two
given points and whose asymptotes include an angle supplementary to
the given one.
1249. A circle has its centre on a fixed straight line and intercepts
on another straight line a segment of constant length : prove that its
envelope is an hyperbola of which the first straight Ime is the conjugate
axis and the second straight line is an asymptote.
1250. A conic is drawn having its focus at A, the vertex of a given
conic, passing through a focus S of the given conic, and having for
directrix a tangent to the given conic : prove that its envelope is a conic
having its focus at the given vertex and exoentridty r — . Also the
envelopes of the minor axis, the second latus rectum, and the second
directrix are each conies similar to the given oonic, and bearing to it
the linear ratios l^e : 2e, I ^e : e; and 1 + ifa 6 : e, the upper or lower
signs being taken according as aS' is the nearer or £Eui;her focus.
1251. A triangle is inscribed in the hyperbola xy=^<f whose centroid
is the fixed point (cm^ cmT^)*, prove that its sides envelope the ellipse
which touches the asymptotes of the hyperbola; and also touches the
hyperbola at the point (cm^ cfnr^)^ the curvatures at the point being as
4:1. Where the ellipse again meets the hyperbolai its tangents are
parallel to the asymptotes.
CONIC SECnONSy AHALmCAL. 209
1 252. The centre and directrix of an ellipse are giren : the enrelope
18 two parabolas having their common focus at the given centre.
[The ellipse may be taken — + — = 1, where J^ = c'(p — y); or
(yaj* + p/) p^c^qip — q)y involving onlj the parameter p : y.]
1 253. One extremity of the minor axis, and the directrix, of a conic
are given : the envelope is a circle with centre at the given point and
teaching the given line.
fiB* v* 2t/ a*
[The equation ^ ~i + ^ == "t^* where -^ — ii=^> ^' ™*7 ^ written
(6*£c* + ay)* = 4c*y* (a*6' - 6*), involving only the parameter of : h\ This
theorem is easUj proved geometricallj.]
1254. Find the envelope of the circle (a5 - A)* + y" = r*, when A^ r
are connected by the equation A' = 2 (r* 4* o^, and a is given.
1255. A circle rolls with internal contact upon a circle of half the
radius : prove that the envelope of any chord of the roUing circle is a
circle which reduces to a point when Uie chord is a diameter.
1256. A parabola rolls op an equal parabola, similar points being
always in contact : prove that the envelope pf any cftraight line perpen-
dicular to the axis of the moving parabola is a drde.
[Also obvious geometrically.]
1257. A parabola has a given focus and intercepts on a given
straight line a segment subtending a constant angle (2a) at the focus :
prove that the envelope of its directrix is an ellipse having the given
point for focus, the given straight line for minor axis, and excentricity
cos a.
1258. The envelope of the straight line
^-^-T— ^ = ocos^,
cos ^ sm ^
where 0, ^ are parameters connected by the equation
, ^ Asin0 a
is the parabola
{hx-^-ky^ay^Wexy.
[By combining the two equations we may obtain the equation in the
form
Aoj + Ay — a* — kkx ~-rhy=^ 0.]
A
1259. The envelope of the conic
a{a-kb) 6(Aa-6)
is the four straight lines (a; ifa y)' = a' - 6*.
w. p. \4k
210 <:jonic sections, analytical.
1260. A parabola is drawn touching a given straight line at a given
point Oy also the point on the normal at O, chords through which subtend
a right angle at 0, is given : the envelope is a circle in which the two
given points are ends of a diameter, and each parabola touches the
envelope at the point opposite to in the parabola.
[The equation of the parabola may be taken J\-\x-¥j\+ky = J ax,]
1261. Through each point P of a given circle is drawn a straight
line PQ of given length and direction (a given vector), and a circle is
described on FQ as diameter : prove that the enveloj)e of the common
chord of the two circles is a parabola. The envelope of the circle is
obviously two circles.
[If 05' + y* = a' be the given circle, 2h the given length, axis of y the
given direction, the envelope is a' + (y — A)' = ( a — y J .]
1262. Two points ai*e taken on a given ellipse such that the normals
intersect in a point lying on a fixed normal : prove that the envelope of
the chord joining the two points is a parabola whose directrix passes
through the centre and whose focus is the foot of the perpendicular from
the centre on the tangent which is perpendicular to the given normal.
[If a be the excentric angle of the foot of the given noimal, the
equation of one of the chords will be
X V .
o-cosa + a, sina = l,
^ a
where p, q are connected by the equation pq+p + q = 0.]
1263. A chord FQ is drawn through a fixed point (X, Y) to the
ellipse -^ + ^ = 1, the normals at P, Q meet in and from are drawn
OP', OQ^ also normals : prove that the envelope of P'$' is the parabola
whose focus is the foot of the perpendicular from the origin on the line
xX yT
-g- + ^ + 1=0, and directrix the straight line a'xT-k- h*yX= 0.
1 264. Three points are taken on a given ellipse so that their centroid
is a fixed point, the straight lines joiuing them two and two will touch
a fixed conic.
[Refer the ellipse to conjugate diameters so that the fixed point is
(-5- a, Oj, then taking the three points (a cos a, 6 sin a), &c., we may
X t/ 1
take the chord joining two to be ;?- + 9-'^ = - , where
(»»•- l)(p» + ^) - 2mp + 1=0.
The envelope is
fix \' V , ,
CONIC SECTIONS, ANALYTICAL. 211
1265. A triangle insoribed in the ellipse a'y* + 6 V = a*6* haa its
centroid at the point ( "1 , s-) : prove that its sides touch the conic
t a' a' 6' j (. "6' a* 6* j
= *^,(2xy-xY-yX)'.
[Tlie poles of the sides lie on the conic
1 266. Three ])ointM ai-e taken on a given ellipse such that the centre
of i)erpendicular8 of tlie triangle is a fixed point : the envelope of tho
chords will be a fixeil conic whose asymptotes are j>erpeudicular to the
tangents from the fixed point to the given conia
[If (A, Y) be the given point, a, )9, y excentric angles of the comers
of one of the triangles, then (1026)
(a' + 6')cosa-(a"-6')cos(a + )9 + y) = 2aX-(a* + 5')(co8)9 + co8y),
(a* + b') sin a - (a' - b') sin (a + jS + y) = 25 F- (a* + 6') (sin P + sin y);
square and add, and we obtain the relation
(a'A" + 6' r ') {p* + q') - aY - by - 2 (a* + 6') (paX + (76 F ) + {a' + ^)' = 0,
OS V
connecting the parameters in the equation ^>- + ^j=l of one of the Rides.
The equation of the enveloi>e is
+ a:'(A''-a')+y'(F'-6') = 2ayXF.]
1267. The envelope of a chord of the conic -, + f« = 1 which sub-
tends a right angle at the point (A', F) is
{{X - Xy + (y - F)'} {a* + 6« - J« - F*) = aV (^ + ^- 1)* ;
and thence that if F be the point (X, F), i^' the second focus of this
envelope, FF is divided by either axis of the given ellipse into segments
in the ratio o* + 6* : a' - 6*, and that the major axis of the envelope
bears to the minimum chord of the director circle through F the ratio
2ab : a* + b'.
1268. A parabola has its focus at the focus of a given conic and
touches the conic : prove that its directrix and the tangent at its vertex
both envelope circles, the former one of radius equal to the major axis
and with its centre at the second focus of the given conic, the latter the
auxiliaiy circle of the given conic.
212 CONIC SECTIONS, ANALYTICAL.
1269. Each diameter of a given parabola meets a fixed straight line,
and from their common point is drawn a straight line making a given
angle with the tangent corresponding to the diameter : the envelope is a
parabola, degenerating when the given angle is equal to the angle which
the fixed straight line makes with the axis.
1270. From the point where a diameter of a given conic meets a
fixed straight line is drawn a straight line inclined at a given angle to
the conjugate diameter : the envelope is a parabola. If the constant
angle be made with the first diameter, the envelope is another parabola.
[The equations of the conic and straight line being
and t the tangent of the given angle, the envelopes are respectively
{pa(j/-tx) + qb{x + ti/) + t(a*-b')Y=^iab{p{x + ty)-a}{q{i/-tx)--b},
and {bp (x •¥ty)-aq{y- te)}' + ^ah {q (a: + <y) - 6<} {p (y - to) + a/} = 0.]
1271. A fixed point A is taken on a given circle, and a chord of the
circle PQ is such that PQ ^e{AP^AQ): prove that the envelope of PQ
is a circle of radius a (1 - e*) touching the given circle at A, a negative
value of the radius meaning external contact.
1272. A fixed point A is taken within a given circle, and a chord of
the circle PQ is such that PQ=^e{AP + AQ): prove that the envelope of
PQ is a circle coaxial with the given circle and the point A and whose
radius is y(a'-c'c*)(l -e*), where a is the radius of the given circle and
€ the distance of A from its centre.
1273. One given circle U lies within another F, and PQ is a chord
of V touching Uy S the interior point circle coaxial with Usud F, PSP a
chord of F : prove that the envelope of QP is a third coaxial circle
such that the tangents drawn from any point of it to C^, F are in the
ratio a'-c^ : a', where a is the radius of U and c the distance between
their centres.
1274. A circle passes through two fixed points A, B, and a tangent
is drawn to it at the second point where it meets a fixed straight line
through A : the envelope of this tangent is a parabola, whose focus is Bj
whose directrix passes through A, and whose axis makes with BA an
angle double that which the fixed straight line makes with BA.
[If the two points be (* a, 0), and y cos a = (a; — a) sin a the given
straight line, the envelope is
{x + ay + y*«= {(a- a) cos 2a + y sin 2a}'.]
1275. Through any point on a fixed tangent to a given parabola
is drawn a straight line OPTF meeting the parabola in P, P' and a
given straight line in 7*, and OT is a mean proportional between
OP, OF : prove that PF either passes through a fixed point or en-
relopea a /Ttarabola.
CONIC SECTIONS, ANALYTICAU 213
1276. The envelope of the polar of the origin with respect to any
circle circumscribing a maximum triangle inscribed in the ellipse
[If be the fourth point in which the circumscribing circle meets
the ellipse, the equation of the polar will be
--costf + rsintf = 2-= — ^^A
1277. A chord of a given conic is drawn through a given point,
another chord is drawn conjugate to the former and equally inclined to
a given direction : prove that the envelope of this latter chord is a
parabola.
1278. A triangle is self -conjugate to the circle (a? - c)* + y* = 5*, and
two of its sides touch the circle a:* + ^= a* : prove that if the equation
of the third side be j9 (x— c) -^qy — ^^p^q will be connected by the
equation
;>'(a« + 6') + g'(a' + 6'-c*) + 2/xj + c'-.2a» = 0;
and find the Cartesian equation of the envelope.
1 279. A triangle ABC is inscribed in the circle a;*-!- ^ = a', and A is
the pole of BC with respect to the circle (« — c)' +y*=»6*: prove that
A By AC envelope the conic
( 2x^cy 2y"
2a' + 26"-c« a' + fi'-c*"
1280. From a fixed point are drawn tangents OP, OP' to one of
a series of conies whose foci are given points Sy JS' : prove that (1) the
envelope of the normals at /*, P' is the same as the envelope of PP',
(2) the circle OFF' will pass through another fixed pointy (3) the conic
OPFSS' will pass through another fixed point.
1281. A chord of a parabola is drawn through a fixed point and on
it as diameter a circle is described : prove that the envelope of the polar
of the vertex with respect to this circle is a conic which degenerates
when the fixed point is on the tangent at the vertex.
[This conic will be a circle for the point ( ^ o > ^j > ^^^ ^ rectangular
hyperbola when the point lies on the parabola y* = 4a (2a; — a).]
1282. The centre of a given circle is G and a diameter is AB^
chords APy PB are drawn and perpendiculars let fall on these chorda
from a fixed point : prove that the envelope of the straight line join-
ing the feet of these perpendiculars is a oonic whose directrices are AB
and a parallel through 0, whose excentricity is CP : COy and whose
focus corresponding to the directrix through lies on CO,
214 CX)NIC SECTIONS, ANALYTICAL.
1283. From the centres A, B oi two given circles are drawn radii
AF, BQ whose dii*ections inchide a constant angle 2a : prove that the
envelope of PQ is a conic whose excentricity is
J a* + 6* - 2ab cos 2a '
where a, h are the radii and c the distance AB.
[The conic is always an ellipse when one circle lies witliin the other,
and always an hyperbola when each lies entirely without the other;
when the circles intersect, the conic is an ellipse if 2 a be greater than
the angle subtended hy AB at a common point, reduces to the two com-
mon points when 2a has that critical value, and is an hyperbola for any
smaller numerical value. When 2a = or ^, the envelope degenerates to
a point. For different values of o, the foci of the envelope lie on a fixed
circle of radius - ,— , , and whose centre divides AB externally in the
ratio a' : 2>'.]
1284. Two conjugate chords ABy CD of a conic are taken, P is any
point on the conic, PA, PB meet CD in a, 6, is another fixed point,
and Oa, Ob meet PB, PA in Q, i? : prove that QR envelopes a conic
which degenerates if lie on ^^ or on the conic.
1285. The point circles coaxial with two given equal circles are
S, S\ a straight line parallel to SS' meets the circles in H, W so that
Sff^S'ff'y and with foci ZT, H' is described a conic passing through
Sy S* : prove that its directrices are fixed and that its envelope is a conic
having S, S* for f ocL
1286. Two conies Uy V osculate in and PP^ is the remaining
common tangent, PQ, PQ are drawn tangents to T, C7' respectively :
prove that PP\ QQ' and the tangent at meet in a point, and that, if
from any point on PP' be drawn other tangents to U, V, the stiuight
line joining their points of contact envelopes a conic touching both
curves at and touching (7, V again in Q\ Q respectively.
1287. Normals PQ, PQ' are drawn to the parabola i^==i{ix from a
point P on the curve : prove that the envelope of the circle PQQ' is the
curve
y* (a; + J j = ic" (2a - a).
which is the pedal of the parabola y" = a(aj-2a) with respect to the
origin A,
[The chord QQ* always passes through a fixed point C (- 2a, 0), and
if S be the focus of the given parabola, C, S are single foci and A a double
focus of the envelope : for a point P on the loop, CP = AP ■\- 2SP, and
for a point on the sinuous branch, CP = 2SP ^ APy so that AP may be
regarded as changing sign in vanishing.]
CONIC SECTIONS, ANALYTICAL. 215
VIII. Artal Coordinates,
[In this system tlie position of a point P with i*e8pect to three
fixeu points Ay B, C not in one straight line is determined by the values
of the ratios of the three triangles PBC^ PC A, PAB to the triangle
ABC J any one of them PBC being esteemed positive or negative accord-
ing as P and A are on the same or on opposite sides of BC, These ratios
being denoted by a:, y, z will always satisfy the equation a;4-y + «=l,
A point is completely determined by the ratios of its areal co-ordinates
{X \ Y : Z) or by two equations, as ^ = my ^nz. It is sometimes
convenient to use trilinear co-onlinates, x^ y, z being then the distances
of the point from the sides of the triangle of reference ABC and con-
nected by the equation cu; + 6y + cz == 2Ky where a, 6, c are the sides and
K the area of the triangle ABC, A point would obviously be equally
well determined by x, y, z being any fixed multiples of its areal or
trilinear co-ordinates, a relation of the form Ax + By + C7« = 1 always
existing. In the questions under this head areal co-ordinates will
generally be taken for granted.
The general equation of a straight line is |>a: + yy + r« = 0, and
Pt q, r are proportional to the perpendiculars from A^ By C on the
straight line, sign of course being always regarded. When Py q, r are
the actual peq)endiculars, px -k- qy + rz in the peqiendicular distance from
the line of the point whose areal co-ordinates are x^ y, z.
The condition tliat the straight lines p^x-hq^y4- rj»=0, p^x-^qjy + r^ =0
shall be pai*allel is
Px^ S'i. ^ =0,
I 1, 1, 1
and that they may be at right angles is
/}j9,sin'-4 + ... + ... = (^^r, + j'/j) sin ^ sin C cos ^ + ... + ...
If /?, = S', = ^j, or if /?, = 7, = r,, both these equations are true. The
straight line x-\-y-¥z^Oy t)ie line cU infinity y may then be regarded
as both parallel and perpendicular to every finite straight line, the fact
being that the direction of the line at infinity is really indeterminate.
In questions relating to four points it is convenient to take the
}>oints to be (iT, ^Yy^ Z)y or given by the equations /a: = * my = * m\
and similarly to take the equations of four given straight lines to be
px^qy^rz-^O,
The general equation of a conic is
216 CONIC SEcnoNS, analytical.
and the polar of anj point (X : T : Z) is
„du ^du ^du ^
ax df^ dz
or (the same thing)
dU dU dU ^
''dl''ydY''^dZ-^'
The special forms of this equation most useful are
(1) drcumsmbing the triangle of reference (a, 6, c = 0)
fyte-¥gzx^-kxy=0\
(2) inscribed in tiie triahgle ci reference
(fc)* + (wy)* + (na;)* = ;
(3) touching the sides AB, AC at the points B, C
but when this form is used it is often better to take such a multiple of
the ratio ^FBC : liABC for a; as to reduce the equation to the form
a? = yzi
(4) to whidi the triangle is self-conjugate (/ g^ A=0)
here again it is often convenient to use suxih multiples of the triangle
ratios as to give us the equation
aj* + y* + »* = t).
As a gehdrtd rule, when metiical iresults are wanted, it will be found
simpler to keep to the true areal or ttilineto coK)rdinate8.
The form (4) is probably the most generally useful. We may denote
any point on such a conic by a fi&n^e variable, as with the excentric
angle in the case, of a conic referred to its axes, which is indeed a
particular case of this form. Thus any pibint on the conic a:* + y* + 2* =
may be represented by the equations
^ _ y — -^ .
cosfl""Binfl"" '
and we may call this the point $.
The equation of the tangent at is
a:cos0 + ysin0=ts;,
that of the chord through 0, ^ is
a;oos^ (0 + ^) -fysin^ (0 + ^) discos ^ (0-^) ;
CONIC SECTIONS, ANALYTICAL. • 217
and the intersection of the tangents at 0, ^ is
X y (z
C08^(^ + ^) Bin^(6l + ^) cos^(0-^)'
The equations of any two conies may be taken to be
«• V* «*
their common points being : = -^— = -. . The multipliers of the
areal co-ordinates will not all be real when the triangle of reference is
real, and when the conies have two real and two impossible common
points the triangle will be imaginaiy.
Any point on the conic 7?^yz may be denoted by the co-or-
dinates (X : 1 : X') and called the point K The tangent at this point
is
X*y-2Aa; + a; = 0,
and the chord through X^ fi is
X/iy — (X + /i) a; + « = 0.
Any point on the conic fyz + gzx + Ikxy = may be taken to be
X cos* _y sin' ^ z
the tangent at the point being
?cos*fl + ^8in*fl + | = 0;
and any point on the conic (fe)» + (w»y)» + (n«)» » to be
2a!; my
cos*e sin^fl '^'
with the corresponding tangent — ^a + . % k ^''^^ ; but these equa-
tions are not often required.]
1288. The equations of the straight lines each bisecting two of the
sides of the triangle formed by joining the feet of the perpendiculars of
the triangle ABC are
a5=ycot* J9 + «cot*C, &c.,
and the perpendiculars from (x^ y, ^ on tiiem are
.2i?sin*i?sin'C(ycot*J9 + «cofC-aj), &c
1289. The sides of the triangle of reference are bisected in the
points uip B^^ C^; the triangle A^Bfi^ is treated in the same way
and so on n times : prove that the equation of BJS^ is
218 . CONIC SECTIONS, ANALYTICAL.
1 290. The equation of the straight line passing through the centres
of the inscribed and cii^cumsciibed circles is
^ /™ j> \^^n^ , y
sin
-7 (cos B - cos C) + . - „ (cos G — cos A)-^ -r—r, (cOS A - COS -5) = :
A^ ' sin-D^ ' sinC '
and the point sin A {m + n cos il) : sin B(m + n cosB) : sin (7 (w + ncos C7)
lies on this straight line for all values of m : n.
1291. If a;, y, « be perpendiculars from any point on three straight
lines which meet in a i>oint and make with each other angles A, By C\
the equation laf + my* + wz* = will i-epresent two straight lines which
will be real, coincident^ or imaginary, according as
7nn sin' A -{-rd sin' B + Im sin* C
is negative, zero, or positive.
1 292. The perpendiculars from A, B,C on a straight line are p, q, r,
and the areal co-ordinates of any point on the line are x, y, zi pix>ve
that /Kc + 9y + rz = 0, and the perpendicular distance of any point (a;, y, z)
from the line va px ■\- qy + rz.
1293. The perpendiculars from A^ By C on the straight line joining
the centres of the inscribed aud circumscribed circles are p, q, r : prove
that
« _ 2ig'(l - cos^^) (1 - cos^) (1 - cosC)
^ ~ 3-2 cos ^ - 2 cos 5 - 2 cos C '
and two similar equations for q, r.
1294. The straight lines bisecting the external angles at ^, ^, C
meet the opposite sides in A'y B, C", and p, q, r are the per|>endiculai's
from A, By C upon the stiuight line A'B'C \ pi-ove tliat
. , . r> . ^ 2/? sin A sin B sin C
psmA=:qsmB = r8mC
^3 - 2 cos -4 - 2 cos -tf-- 2 cos C
1295. Within a triangle ABC are taken two points 0, (7; AG, BG,
CG meet the opposite sides in ^', B\ C, and the points of intersection
oiaA, B'C'\ G'B, C'A'; GO, A'F are respectively Z>, E, F: prove that
A'Dy BE, C'F will meet in a point which remains the same if 0, 0' be
interchanged in the construction.
[If (a?, : y^ : z) and {x^ : y, : z^ be the points 0, G' the point is
determined by the equations
X y ^ __ _z ,
1296. The perpendiculars /?, q, r are let fall from A, B^C on any
tangent (1) to the inscribed circle, (2) to the circumscribed circle, (3) to
the ninc-jK)int«* circle, and (4) to the polar circle : prove that
CONIC SECTIONS, ANALYTICAL. 219
(1) psin A i-ysin^+rsin C=2R8inA Bin^sin C ;
(2) psin2A +q sin 25 + r ain 2C = 4/? sin -4 sin -5 sin C ;
(3) /> sin ^ cos (5 - C) + ... + ... = 2/? sin -4 sin -5 sin C;
... ^ . ^ n X •^ 2/? sin ^ sin 5 sin C
^- cos A cos Ji cos C
and also that in (4) p' tan A -hq' tan 5 + r* tan (7=0.
[The general relation for the tangent to any circle, whoso centre is
(z^ : y^ : Zj) and radius p, is
;w^i + (72^1 + »*^, = (^1 + y, + -,) p.]
1 297. The feet of the perpendiculars let fall from {x^ : y, : z^ on
the sides of the triangle of reference are A\ B\ C \ prove that straight
lines drawn through AyB^C perpendicular to B'C\ CA\ A'B^ respectively
meet in the point
1 298. A triangle LMN' has its angular points on the sides of the
triangle ABC, and AL^ BM, CN meet in a point (x^ • y, • s;,) ; a straight
line px-\- qy + rz-0 \a drawn meeting the sides of LMN in three points
which are joined to the corresponding angular points of ABC \ prove
that the joining lines meet the sides of ABC in points lying on the
straight line
X V 25 ^
+ : — —. — + 1 ^- = 0.
((/+r)a;, (r+p)y, {p + q)z,
1299. Tlie two points at which the esciibed circles of the triangle
of reference subtend equal angles lie on the straight line
(b-c)x cot A + {c — a) y cot 5 + (a - 6) « cot C = 0.
1300. Four straight lines form a quadrilateral, and from the middle
points of the sides of the triangle formed by three of them perpendiculars
are let fall on the straight line which bisects the diagonals : prove that
these perpendiculars are invensely proportional to the perpendiculars
from the angular points of the triangle on the fourth straight line.
[The three being taken to form the triangle of reference, and the
fourth being px + qy-^rz^O, the equation of the bisector of the diagonals
will be found to be
2 {qrx + rpy +pqz) = {qr-hrp -^pq) (« + y + z).]
1301. A straight line meets the sides of the triangle ABC in
A\ B", C'y the straight line joining A to the point (BB', CC) meets BC
in a, and 6, e are similarly determined : prove that if any point be
taken the straight lines joining a, 6, c to the intersections of OA, OB^
OC with A'FC will pass through a point 0^] and that 00' will pass
through a ^^oint whose position is independent of 0,
220 CONIC SECTIONS, ANALYTICAL.
\1£ px-^qy-^-rz^O be the line A'FC\ and («j : y^ : z^ the point 0,
the Btrsoght Ime OO is
passing through the point ^s^^=rs.]
1302. The two points whose distances from A^ B, C are as BO, CA^
AB respectively both lie on the straight line joining the centroid G and
the centre of perpendiculars L of the triangle.
[The two points are given by the equations
S-h^z^c^y _ S-c'x-a'z __ S-a'y-h'x
a* b' ? '
where S = a'yz + b'zx + c'xy.]
1303. The distances of L from A, B, C are as cos A : cos B : cos (7:
prove that the other point F whose distances from A, By C are as
cos A : cos B : cos C also lies on the straight line GL and is reciprocal
to L with respect to the circumscribed circle. Also
AW BU" GL*" OL" l-8cos^cos^cos(7'
where is the centre of the circle ABC.
1304. Each of the straight lines
X sin* -4 + y sin' J? + « sin* C = 0,
X cos* -4 + y cos' J? + » cos' C = 0,
is perpendicular to the straight line joining the centroid and the centre
of perpendiculars.
1305. The equation of the straight line bisecting the diagonals of
the quadrilateral, whose four sides are px^qy^rz^O^ is
and that of the radical axis of the three circles whose diameters are the
diagonals is
(^-r')(6*« + c'y) + ... + ... =0.
1306. One of the sides of a quadrilateral passes through the centre
of one of the four circles which touch the diagonals : prove that each of
the other three passes through one of the other three centres, and that
the circles whose diameters are the diagonals touch each other in a point
lying on the circle circumscribing the triangle formed by the diagonals,
ihe common tangent being normal to the circumscribed circle.
[Generally the three circles intersect in real points if each of the
four sides has two of the four centres on each side of it. K the four
sides hQpx^qy^TZ = Oy the two common points are such that
V " Q ^ r '^
CONIC SECnONS, ANALYTICAL. 221
1307. The equation of a circle passing through B and C and whose
segment on £C (on the same side as A) contains an angle 0, is
1308. The locus of the radical centre of three circular arcs on
BCj CAf ABy respectively, containing angles A-^O^ B + O^ C-^0, for
different values of ^ is the straight line
^sin(jB-C) + |sin((7--4) + |sin(-4-^ = 0j
when = H » ^0 radical centre is the centre of the circumscribed circle,
when = 0, the radical centre is the point of concourse of the three
straight lines joining Ay B, C respectively to the points of intersection
of the tangents to the circumscribed circle ; and generally the radical
centre divides the distance between these two points in the ratio
cos0(l +cosuico8 J^cosCT) :sin0sinuisin^sin(7.
1309. A straight line drawn through the centre of the inscribed
circle meets the sides of the triangle ABC in a, b, c, and these points
are joined to the centres of the corresponding escribed circles : prove
that the joining lines meet two and two on the sides of the triangle ;
and, if a\ b\ c' be their points of intersection, the circles on €m\ hh'f ecf
as diameters will touch each other in one point lying on the circum-
scribed circle, their common tangent being normal to l^e circumscribed
circle.
1310. The equation of a circle which passes through the centres of
the escribed circles of the triangle of reference is
hcaf + coy* + aha? -{• {a '\' h '\' c) (ayz + hzK + cxy) = ;
and, if we change the sign of one of the three a, 6, e throughout^ we get
the equation of the circle through the centres of the inscribed circle and
two of the escribed circles.
1311. A circle meets the sides of the triangle ABC in P, P'\ Q, Qf\
i?, K respectively, and AP^ BQy CR meet in the point X ; T : Z : prove
that Ar, BQ, CR meet in the point X' : T i Z', where
XX(Y'^Z){Y'-^Z) YY'{Z + X){Z'-^X') _ ZZ' {X -¥ Y ) (X' -^ Y')
a* " 6* "" c* •
1 31 2. The lines joining the feet of the perpendiculars of the triangle
ABC meet the corresponding sides in A\ ^, C': prove that the circles
whose diameters are AA\ BB', CC will toudi each other if
sec* A + sec* B + sec* C-2sec-48ec-SsecC+7 = 0.
222 CONIC SECTIONS, ANALYTICAL.
1313. Two circles cut each other at right angles and from three
]X)ints Ay By Con one are drawn tangents whose lengths are /?, y, r to the
other : prove that
p' sin 2A + q' sin 2J? + r« sin 2C = SAABC.
1314. The two point-circles coaxial with the circumscribed circle
and the nine-points* circle of the triangle of reference are
Tnhc sin A (xcot A ^y cot J? + « cot C) (x + y + z)= a^yz + l^zx + cVy,
in being a root of the equation
w* (1-8 cos ^ cos -5 cos C) — 2m (1 - 2 cos J cos J5 cos C) + 1 = 0.
1315. The equation of a circle in which the centroid and the centre
of perpendiculars of the triangle of reference are ends of a diameter is
26c sin -4 (as cot -4 + cot -5 + « cot (7) (« + y + «) = 3 (a'yz + h*zx + (?Qcy) \
and the tangent to it from an angular point bears to the tangent from
the same point to the nine-points' circle the ratio 2 : ^Z,
131 6. The conic h? + wy* + rmf = will represent the polar circle of
the triangle of reference if
I tan A=^m tan J? = n tan (7.
1317. The necessary and sufficient conditions that the equation
h^ + my* + T\sf + 2/?y« + 2qzx + 2rxy =
may represent a circle are
wi+n— p n-^l—q l + m — r
1318. The lengths of the tangents from A, B, C ix) a certain circle
are Pf q, r : prove that the equation of the circle is
{p'x + <^y + r^z) (a; + y + «) - a'yz - b'zx - (fxy = 0,
and that the square of the tangent from {x, y, z) is the left-hand member.
1319. The nine-points' circle of the triangle of reference touches the
inscribed and escribed circles in the points /*, P^, P^, P^ : prove that
(1) the equations of the tangents at these points are
X y z ^
6-c c—a a—b
and the three equations formed from this by changing a into —a,
b into - 6, or c into — c ; (2) PP,, PgP, meet BC in the same points as
the straight lines bisecting the internal and external angles at A ;
(3) PI\y P^P^ intersect in the point
-X y z ^
5«_c« e-a^ a*-6-'
CONIC SECTIONS, ANALYTICAL. 223
and (4) the tangents at P, Pp P^, P^ all touch the maximum ellipse in-
scribed in the triangle.
1320. The straight line lx + mi/ + nz = meets the sides of the
triangle ABC in A\ B', C: prove that the circles on AA\ BB^ CO'
have the common radical axis
/ (/» - w) X Qiot A -V m {n - T) y cot jB + n (/- m) « cot C = ;
and the circles will touch each other if
(mn + n/ + Im)^ sin ^ sin j5 sin (7 = 2/mn {I sin 2ui + m sin 2iS + 9) sin 2C).
1321. A circle having its centre at the point {X \ Y \ Z) cuts at
right angles a- given circle u = : prove that its equation is
1322. A conic touches the sides of the triangle ABC in the points
a, 6, e and Aa meets the conic again in ^' : prove that the equation of
the tangent at ^' is
2A' Y Z'
where {X \ Y \ Z) \b the point of concourse of Aa, Bb, Ce,
1323. The two conias circumscribing the triangle of reference,
passing through the point (X : Y : Z)y and touching the straight line
px + qi/ + rz = will be real if
pqr(pX -hq Y + rZ)
XYZ "•
Interpret this result geometrically.
1324. Find the two points in which the straight line y = fe meets
the conic (Ix)^ + (my)^ + ()iz)^ = 0, and from the condition that one of the
points may be at infinity determine the direction of the asymptotes.
Prove -that the conic will be a rectangular hyperbola if
/V + m^b' + n'c* + 2nnbc cos -4 + 2nlca cos B + 2lmah cos C = 0.
1325. A conic touches the sides of the triangle ABC^ any point is
taken on the straight line which passes through the intersections of the
chords of contact with the corresponding sides, and the straight lines
joining this point to A, B, C respectively meet BC, CA, AB in a, b, c:
prove that corresponding sides of the triangles ABC, abc intersect in
points lying on one tangent to the conic.
1326. A conic touches the sides of the triangle A BC, and the straight
lines joining A, B, C to the points of contact meet in ; through
is drawn a straight line meeting BC, CA, AB respectively in a, 6, c,
and from a, 6, e are drawn respectively three other tangents : prove
that the intersections of these tangents two and two lie upon a fixed
conic circumscribing the triangle.
224 CONIC SECTIONS, ANALYTICAL.
1327. If 6 be the exceniaricity of a conic inscribed inihe triangle
ABC^ and x : y : z the trilinear co-ordinates of a focus,
- - xi/z (ctx +hf/ + ez) {ayz -f hzx -h cxy)
"" '"i2'(y" + a* + 2y«cos-4)(«" + a* + 2«a;co8-B)(a* + 3/* + 2a:ycoBC)*
1328. Two parabolas are inscribed in the triangle of reference such
that triangles can be inscribed in one whose sides touch the other ; and
(« : y : »), {x' : j/ : z*) axe their foci: prove that
&*(BHff-o-
1329. The equation of the axis of a parabola inscribed in the tri-
angle ABC iapx + qy-\-rz = 0: prove that
a'p Vq <?r ^
q-r r-p p-q
1330. Prove that the equation of a conic, inscribed in the triangle
ABC and having paj + ^^y + r» = for an asymptote, is
Pj{q-r)x*qj{r-p)x^rj(p'-q)x=0.
1331. The condition that the straight line px -^ qy -^ rz = may be
an asymptote of a rectangular hyperbola circumscribing the triangle
ABC IB
p (q -r)* cot A + q (r -pY cot B + r (p- qY cot C = 0.
1332. Prove that, at any point F on the minimum ellipse circum-
scribing the triangle ABC,
AP BF CF
anBFC^amCFA'^BmAFB'^ '
and cot J?PC + cot CFA + cot AFB^^cotA + cot J9 + cot (7,
the angles BFC, CFA, AFB being so measured that their sum is 360^
1333. A conic lyz + mzx-k-nxy^ is such that the normals to it at
the points A, B, C meet in a pointy prove that
fcc/(m*-n') + cam(n*-/^ + aJn(Z*-m") = j
the point of concourse of the normals must lie on the curve
re (y* — a*) (cos ^ - cos ^ cos (7) + ... + ... =0;
and the centre of the conic on the curve
6ca5 (y*- 5^) + cay («*-a5*) + abz{x*-^) = 0.
[Trilinear co-ordinates are here employed.]
1334. The conic (&)'+ (wy)*+ {nzy = is such that the normals at
its points of contact with the sides meet in a point : prove that
-(m-n) + -^(n-l) + -^(l-m) = 0,
CONIC SECTIONS, ANALYTICAL. 225
that the centre lies on the same curve as in the last question ; and that
when with a point on this curve as centre are described two conies, one
touching the sides and the other passing through the angular points, the
directions of their axes will be the same.
1335. An ellipse inscribed in a given triangle passes through the
centre of the circumscribed circle : prove that the locus of its centre is
the conic whose foci are the centres of the circumscribed and nine-points
circles and whose major axis is ^B. Also if an inscribed ellipse pass
through the centre of perpendiculars, the locus of its centre is a conic
whose centre is the centre of the nine-points circle, to which the
perpendiculars of the triangle are normals, the sum of whose axes is K
and the difference is the distance between the centre of the circumscribed
circle and the centre of perpendiculars.
1336. Find all the common points of the two conies
{lx + m^-hnz){x + y-\-z) = 2 (w + w)y«+ 2 (n+ /)«a: + 2(/+ w)a;y;
and prove that their areas are as
(mn + nl + /m\ f (m + n) (n + J) (Z + m)
1337. A conic passes through the cornel's of a triangle and through
its centroid, prove that the pole of the mid point of any side with respect
to this conic is the straight line bisecting the other two sides.
[For, when ^ + w + n = 0, the equations lyz + mzx + nxT/ = 0,
^ (y + « - «)• + w (« + a - y)* + n (oj +y -«)• = 0,
coincide.]
1338. The minimum excentricity of any conic through the four
a:* y* «■
points xft = yi = ;^a ^ given by the equation
e* (X' cotA-^Y'cotB + Z' cot C)'
1339. Two conies have a common director circle, one being in-
scribed in the triangle ^jS(7 and the triangle self-conjugate to the other :
prove that the common centre must lie on the line
X cot ii + y cot -ff + « cot C = 0.
1340. Two concentric conies are drawn one circumscribing the
triangle of reference and to the other the triangle bisecting the sides is
self-conjugate: prove that the two conies are similar and similarly
situated, and their areas in the ratio
- Sxyz : (- a: + y + ») (as -y + «) (a; + y - «),
where (a; : y : 2) is their common centre.
[Also very easily proved by orthogonal projection.]
W. P. V^
•^^^^ ^a^m^.f '
l:S^:S. Of ^ eilm^a bueribtd in a girea cireSe, tkit ku tke
kwii 4tf«etM' etfd« wbcae ee&tre is ti^ eentre of perpeiii&akza^ tbe
m^iM ^ il« dinttor m^ hmg ZBJefMAcmBoMC ud tke area of
M^idud ibii cf ilie tiegni€»U into wkich eadi pcq>endimlar is dirided
Mr ib« eettinf (4 fN^pir^liealan the s^ment next the base is kss than
tfc« OM to tha rertex* (JK denoiei the nulins ci the circle ABC,)
1344. A cofiic is inscribed in the trisn^eX0t7 with its centre at the
«!Miir« of p^rpen/licolani, and is the an^ which its axis makes with
iiw MUi0 UO ; prore that
tan 2^ cos A - cos ^ cos C
"tan ir- tan 6'"" c(mA''2coBBoMC'
also ibaii if il", //, C^ be its points of contact with the sides, the centre
fft iim^Kmdictilars of the triangle ABC will lie on the conic and the
iangsfit thoro wUl \je jiarallel to BC.
1341!f, If (a; :^ :«) be the centre of an inscribed conic, the sum of
ihs sr|uares cm its seini-axes is
itt*co§A COB BcobC+p* ;
whsm p is the distance of its centre from the centre of perpendiculars.
1346. Tlis centre of a conic is the point (x :y:z)y its excentricity
In 0f tlm radius of its director circle p, and K denotes twice the area of
tliA triantflo of reforonoo ABO : prove that (1) for a conic inscribed in
tlio triatigio AW
(«• - 2/ (g* c ot i< -f y* cot jg + g* cot C)'
!-#• "^ («4-y4-«)(-a} + y + «) (iB-y + «)(« + y-«)'
, „ «*ooti< +w*cot J5 + «*cotC7
'^ (aj + y + «)'
CONIC SECTIONS, ANALYTICAL. 227
(2) for a drcomscribed conic
(«• — 2)*_ {ci*yz+ ... + ... ^ofbccoaA — ... — ...)*
1-e* "" iC*(iB + y + «)(-« + y + «)(x-y +«)(« + y-«)'
,_ ixj/z(a*f/z-h ...—x^bccosA — ...~ ,,,)
(x + y + «)■(-« + y + «)(«- y + «) (x + y-«)'
(3) for a conic to which the triangle is. self-conjugate,
(e' - 2 )' _ (a'yz + 6*gg -»- c'a^)'
I-e" " K'xi/z{x-k-y-k-z) '
, _ o*y« + 6'«a5 -f c^xy
^ (x + y + zy
1347. Two similar conies have a common centre, one is inscribed
and the other circumscribed to the triangle of reference : prove that
their common centre lies either on the circumscribed circle or on the
circle of which the centroid and centre of perpendiculars are ends of a
diameter.
1348. The equation of an asymptote of the conic yz=Jb? is
2fjLkx — ki/ — fjL^z = 0,
where fi is given by the equation fi'+fi + A;=0; and the asjmptoteSy
for different values of k, envelope the parabola
(y - »)• + 405 (a5 + y + «) = (y + « + 2a;)* - 4y« = 0.
1349. A conic is inscribed in the triangle ABC and its centre lies
on a fixed straight line parallel to BC (y + « = kx) : prove that its asymp-
totes envelope the conic
(*-l)(x + y + «)*=16y».
1350. The radius of curvature of the conic 9? = kyz at the point B
. kEsva'C
sin A sin B '
1351. Prove that the equation re* = 4yz represents a parabola ; and
that the tangential equation of the same parabola is ^==p'.
1352. The tangents to a given conic at two fixed points A^ B meet
in C, and the tangent at any point F meets CAy CB in B'^ A' respec-
tively : find the locus of the point of intersection of AA\ BB^ ; and, if
AP, BP meet CB, CAmck^h respectively, find the envelope of ab.
[Taking the original conic to be «* = hxy, the locus and envelope are
respectively isf^hey, «*=4fccy.]
228 CONIC SECTIONS, ANALYTICAL.
1353. The tangents. to a given conic B,t A, B meet in C and a, h are
two other lixed points on the conic ; a tangent to the conic meets CA, CB
in B", A' : prove that the locns of the intersection of aB'y bA' is & conic
passing through a, b and the intersections of Ca, Bb, and of Cb, Aa,
[Taking the given conic to be «* = a^, and (x^ : y^ : z^^ (^^ : y, : z^ the
points a, 6, the locus is
Va?! yj/Va;, yj \x^ «i/ Vy, ^J ■"
1354. The tangents to a given conic at J, J? meet in C ; jP is any
other point on the conic and AP^ BP meet CBy CA in a, 6 : prove that
the triangle abP is self-conjugate to another fixed conic touching the
former at A, B,
[Taking the given conic to be «* = xy and (x^ : y, : z^) the point Pj the
fixed conic is »" = 2xy, which is equivalent to
(?L . ^y + (y. _ aV = f^ ^ y _ ^y .]
\^i «i/ v^i »i/ V^i yi «i/
1 355. Prove that the locus of the foci of the conic a* = kyz, for
different values of k, is the circular cubic
a; (^ - «•) + 2yz (y cos B-^z cos C) = 0,
trilinear co-ordinates being used.
■
1356. Three tangents touch a conic in A, B, C and form a triangle
ahc; BC, CA, AB meet a fourth tangent in a, fi, y, and Aa, Bfi, Cy
meet the conic again in A\ B\ G'\ prove that B'C\ be ; G'A\ ca ; A'Bf^ ab
intersect in points on the fourth tangent, and aA\ bB^, dC' meet in a
point. If a family of conies be inscribed in the quadrilateral forme 1 by
the four tangents, the centre of homology of the two triangles abc^ A'BCf
lies on the curve
(]pxf^(<iyf^(Tzf^%
Xy y, z and ^-k-qy-^rz being the four tangents.
1357. Three points A, B, C axe taken on a conic and the tangents
form a triangle abc, a fourth point (X : Y : Z) is taken on the conic
and Otty Ob, Oc meet the tangents at -4, ^, C in a, )8, y from which
points other tangents are drawn forming a triangle A'BG* : prove that
AA\ BBfy CC will meet in and that the axis of the two triangles
ABCy A'FG' envelopes the curve
©'^©'^(f)*--
[If the points be fixed and the conic variable, the straight lines
BG , G'A'y A'B each envelope a fixed tricusp, two of the cusps (for BG')
being B, G and the tangents OB, OGy and the third cusp, lying on OA
and having OA for tangent, being at a point 0^ such that, if ^ meet
WinD, {J{/0D}=1]
CONIC SECTIONS, ANALYTICAL. 229
1358. In the last question, if ABC and the conic be fixed and the
point vary, the axis envelopes the curve (Zar)"*-f (?«y)~^+(?u)~* = 0,
the conic being (Ix)"^ + (my)~* + {nz)"^ = 0.
1359. Prove that the general equation of a conic, with respect to
which the conic lixf + viy' + nz' = is its own polar reciprocal, is
{la^ + my* + 7tK*) (p'mn + g^til + r'lm) = 2bnn {px + $'y + rz)'.
1360. Every hyperbola is its own polar reciprocal with respect to
a parabola having double contact witli it at the ends of a chord which
touches the conjugate hyperbola.
1361. An ellipse is its own reciprocal polar with respect to a
rectangular hy|)erbola which has double contact with it at the ends of a
chord touching the hyperbola which is confocal with the ellipse and has
its asymptotes along the equal conjugate diameter of the ellipse.
1362. A parabola is its own reciprocal with respect to any rect-
angular hyperbola which has double contact with it at the ends of a
chord touching the other parabola which has the same latus rectum.
1363. An hyperbola is its own reciprocal with respect to either
circle which touches both branches of the hyperbola and intercepts on
the transverse axis a length equal to the conjugate axis.
1 364. Each of two conies U, T is its own reciprocal with re8i)ect to
the other, prove that they must have double contact and that each is its
own reciprocal with respect to any conic which has double contact with
both U and V provided the contacts are different.
1365. Through the fourth common point of the two conies
lyz + mzx + nxy = 0, I'l/z + m'zx + rixi/ = 0,
is drawn a straight line meeting the conies again in P, Q : prove that
the locus of the intersection of the tangents &t F, Q is the curve (tricusp
quartic)
{U: {ttm - m'n) yz]^ + {ninh {rW - ril) zx}^ + {iiri (Ini - Z'm) xy}^ = 0.
1366. From any point on the fourth common tangent to the two
conies o;^ + y* + z' = 0, {Ixy + (mi/Y + (nzy = are drawn two other tan-
gents to the conies : prove that the envelope of the straight line joining
their points of contact is the curve
{I {m - n) a}* + {m (n - Q y}* + {n (/ - m) s}* = 0.
1367. The sides of the triangle ABC touch a conic U; 0, 0^, 0„
are the centres of the inscribed and escribed circles of ABC^ a conic V
is described through J?, (7, 0, 0^, and one focus of 27, and a conic W
through By C, Op 0^ and the same focus of U : prove that the fourth
common point of T, IT will be the conjugate focus of U \ also that, if
the conic W be fixed, the major axis of the conic U will always pass
through a fixed point on the internal bisector of tlie angle A, and if the
conic V be fixed, through a fixed point on the external bisector of the
angle A,
230 CONIC SECTIONS, ANALYTICAL.
1368. Four conies are described with respect to each of which three
of the four straight lines px^qy^rz^O form a self -conjugate triangle
and the fourth is the polar of a fixed point {X :Y:Z): prove that all
four will have two common tangents, meeting in (X:F:Z), whose
equation is
p'X <f7 f^Z
yZ-zY zX'xZ xY-yX
= 0.
1369. The triangle ABC is self-conjugate to a given conic and on
the tangent to the conic at any point P is taken a point Q such that the
pencil Q \ABOF\ is constant : prove that the locus of Q is a quartic
iiaviog nodes &t A^ JB, C and touching the conic in four points.
[If the conic be la^ + my* + n»* = 0, and k the given anharmonic ratio,
the locus of C is ^ ~ ^ + — ^ + — ^. = 0.1
MT mtf nz "*
1370. Two given conies intersect m A, B,C, D and from any point
on AB are drawn tangents OP^ OQ to one conic, Op, Oq to the other :
prove that Pp, Qq intersect in one fixed point and Pq, Qp in another ;
that these points remain the same if A, B he interchanged with C, D ;
and that the 'six such points corresponding to all the common chords lie
on four straight lines.
[Taking the conies to be a:"+y* + 2* = 0, oaf + 6y* + cz* = 0, the six
points are a: = 0, 6 (c - a) y* = c (a — 6) «■, &c.]
1371. A triangle A'BG* is drawn similar to the triangle of reference
ABC and with its sides passing respectively through A^B^ C; another
similar triangle abc is drawn with its sides parallel to those of the former
and its angular points upon the sides BC, CA, AB respectively : prove
that the triangle ABC is a mean proportional between the triangles abCf
A'B'C'i ^^^ ^^^ ^^ straight lines A' a, B'b, C'c meet in the ];K>int
X y
8in2ul + Bin2(j5-^) + 8in2((7 + ^) 8in2J? + sin 2((7-^) + sin2(^ + ^)
_ z
'■Bin2(7 + sin2(^-tf)+sin2(jB + ^)^
where 6 is the angle between the directions of BCy EC. Prove also
that the locus of ^Kis point is a conic having an axis along the straight
line joining the centroid and the centre of perpendiculars.
[The equation of the conic is w* + r* = m;*, where
u = a;cos2ii+yoos2J9 + 2;cos2(7+4(a; + y + «)cosiioosJ9cos(7,
r =a (sin 25 - sin 2(7) + y (sin 2(7 -sin 2 J) +« (sin 2^ -sin 25),
fi7=2(a;co8 2ii +ycos2j5 + 2C08 2(7) + a; + y + «,
of which u and to are parallel to each other and perpendicular to v.]
CONIC SECTIONS, ANALYTICAL. 231
1372. Four fixed tangents are drawn to a given conic forming a
quadrilateral whose diagonals are a^', hb'y ec'; three other conies are
drawn osculating the given conic at the same point F and passing through
a, a'; b, b'; and c, & respectively : prove that the tangents at a, a', 6, 6',
c, c' all meet in one point ; that the locus of this point as F moves is the
envelope of the straight line joining it to P and is a fourth class sextic
having two cusps on each diagonal and touching the given conic at the
points of contact of the four tangents.
[If the four tangents bepx±gy*r2 = and the conic
h? + my* + W2* = 0,
the locus is
(/mna:')* + (g* nZy*)* + (flm^)^ = 0.]
1373. A conic is drawn through B^ C osculating in F the conic
{Ix)^ + (my)* + {nz)^ = :
prove that the locus of the pole of BC with respect to this conic is
the cubic
{Ix + 4my + 4nz)' = 27 Ix {my — nzy :
also if A' be the point of contact of BC and another conic be drawn also
osculating the given conic in F but passing through A, A', the tangents
Sit B, CfA, A' will meet in a point.
1374. A parabola touches the sides of the triangle J i?(7 and the
straight line B'C joining the feet of the perpendiculars from B, C on the
opposite sides : prove that its focus lies on the straight line joining A to
the intersection of BC, BC.
1375. A triangle is self-conjugate to a parabola: prove that the
straight lines each bisecting two of the sides are tangents to the parabola,
and thence that the focus lies on the nine-points' circle and the directrix
passes through the centre of perpend iculara.
1376. A triangle is self-conjugate to a parabola and the focus of the
parabola lies on the circle circumscribing the triangle : prove that the
poles of the sides of the triangle with respect to the circle lie on the
parabola.
1377. A rectangular hyperbola is inscribed in the triangle ABC i
prove that the locus of the pole of the straight line which bisects the
two sides AB^ AC \& the circle
a;'(a' + 6'+c") + (y* + 2ajy)(a« + 6'-c«) + («' + 2ja)(a«-6' + (0 = 0;
that this circle is equal to the polar circle of the triangle and its centre
is the point of the circle ABC opposite to A.
232 CONIC SECTIONS, ANALYTICAL.
1378. A conic is drawn touching the four straight lines
prove that its equation is &* + my* + 712* = 0, where I, m, n are connected
by the equation p*mn-\-^nl + r^lm = 0, and investigate the species of
this conic with respect to the position of its centre on its rectilinear
locus.
[If the middle points of the internal diagonals of the convex quad-
rangle be L, M, and that of the external diagonal be N, L, M, N' being
in order, the conic is an hyperbola when the centre lies between - oo
and Lf an ellipse from Z to if , an hyperbola from M to N^ and an
ellipse from iV to + 00 . Hence there are two true minimum excen-
tricities.]
1379. A conic is drawn touching the four straight lines
px^qy^rz==0^
prove that any two straight lines p^x + q^y + r^z = 0, pjic + qjy + r^z = 0,
will be conjugate with respect to this conic if
p' " ^ " f^ '
1 380. The straight lines p^x + q^y + r^z = 0, /? a; + 7^ + r,2; = will
be conjugate with respect to all parabolas inscribed in the triangle of
reference if
[A particular case of the last with different notation.]
1381. The two points {x^ : y : z\ {x^ : y, : 2 ), will be conjugate
with respect to any conic through tne lour points (Jc : a F : * Z), if
X^ - y« - ^- •
1382. A triangle is self-conjugate to a rectangular hyperbola : prove
that the foci of any conic inscribed in the triangle will be conjugate with
respect to the hyperbola.
[A particular case of the last]
1383. The locus of the foci of all conies touching the four straight
lines px^qy^rz = is the cubic whose equation is, if
(/, m, w) = ^ sin* A + m' sin* B-^n* sin* C - 2ww sin J? sin C cos -4
- 2nl sin C sin J cos ^ - 2/m sin ^ sin J? cos C^
(h w, n) ^ ( - /, m , n) ^ (/, - m, n) (I, m , - w)
Ix + my-^-nz —Ix + my + nz Ix—my + nz Ix + my-nz'
and this equation may be reduced to the form
{x + y + z) {I'x' cot^ + m'y* cot £ + n*z' cot C) he sm A
= (l^x + m*y + w*2j) (a'yz + h'zx + c^xy).
CONIC SECTIONS, ANALYTICAL. 233
1384. Of all the conies inscribed in a given quadrilateral there are
only two which have an axis along the straight line which is the loctis
of the centres of the conies, and the two conies will be real and the
axis the major axis when the centre of perpendiculars of the triangle
foimed by the diagonals of the quadrilateral is on the opposite sides of
the locus of centres to the three comers of the triangle.
1385. The equations determining the foci of the conic
h^ + wy* + Tw* =
are
or { I m n) ¥ \ m n I) c K n I m)
1386. One directrix of the conic h^-^-rmf + w«* = passes through
A : prove that
mn = l(m cot* C + n cot* B) ;
and that the conjugate focus lies on the straight line joining the feet of
the perpendiculars from B, C on the opposite sides.
1387. A conic is described to which the triangle ABC is self -conju-
gate and its centre lies on the straight line bisecting two of the sides of
the triangle formed by joining the feet of the perpendiculars of the tri-
angle ABC, prove that one of its foci is a fixed point.
[It is at the foot of the perpendicular from A on BC^
1388. Given a point on a conic and a triangle ABC self-conjugate
to the conic ; AO, BOy CO meet the opposite sides in three points and the
straight lines joining these two and two meet the corresponding sides in
A\ F, C'\ prove that the intersections of BE, CC \ CC\ AA'; and
AA\ BB' also lie on the conic.
1389. Any tangent to a conic meets the sides of the triangle ABC
which is self-conjugate to the conic in a, b, c; the straight line joining
A to the intersection of Bb, Cc meets BC iu A', and if, C" are similarly
determined: prove that B'C\ C'A\ A!B are also tangents to the
conic.
1390. Two conies U^ V have double contact and from a point on
the chord of contact are drawn tangents OP, OQ ; Op, Oq ; another conic
W is drawn through p, q touching OP, OQ : prove that the tangent to
W at any point where it meets U will touch Y.
1391. A conic passes through four given points: prove Uiat the
locus of tangents drawn to it from a given point is in general a cubic,
which degenerates into a conic if the point be in the same straight line
with two of the former and in that case the locus passes through the
other two points and the tangents to it at them pass through the fifth
point.
234 CONIC SECTIONS, ANALYTICAL.
1392. A conic is inscribed in a given quadrilateral and tangents
are drawn to it from a given point : prove that the locus of their points
of contact is a cubic passing through the ends of the diagonals of the
quadrilateral, through the given point, and through any point where the
straight line joining the given point to the intersection of two diagonals
meets the third : there is a node at the given point the tangents at
which form a harmonic pencil with the straight lines to the ends of any
diagonaL
[The node is a crunode when the given point lies within the convex
quadrangle or in any of the portions of space vertically opposite any
angle of the convex quadrangle.]
1393. Prove that, if pX'\-qy-\-rz = Ohe the equation of the axis of
a parabola inscribed in the triangle ABC, or the asymptote of a rect-
angular hyperbola to which the triangle is self-conjugate,
a'p Vq (?r ^
q-r r—p p-q
1394. A parabola is inscribed in the triangle ABC and S is its focus
(a point on the circle ABC), the axis meets the circle ABC again in 0:
prove that, if with centre a rectangular hyperbola be described to
which the triangle is self-conjugate, one of its asymptotes will coincide
with OS.
1395. The conies passing through two given points and touching
three given straight lines are either all four real or all four impossible.
[If the three given straight lines form the triangle of reference and
(a?, : j/i : z^) («, : y^ : z^) be the two given points, the conies will be
(Ix)^ + (my)i + {nz)^ = 0,
where
I m n
in which ambiguities an odd number of negative signs must be taken.
If the points of contact with BC he A^y A^, ^,, A^ these can always be
taken so that
BA^.BA^ : CA^ . CA, = BA^.BA^ : CA^.CA^,]
1396. The locus of the foci of a rectangular hyperbola, to which
the triangle ABC is self-conjugate, is the tricyclic sextic
a«
x{U''X{X'hy + z)bccoBA\
where l/= a'yz + b*zx -f <?xy.
I ... t" ... ^ V/,
CONIC SECTIONS, ANALYTICAL. 235
1397. A triangle circumscribeg the conic a" + y* + «" = and two of
its angular points lie on the conic h? + m^ + nz' = : prove that the
locus of the third angular point is the conic
^ .M ^t
i+ /7.^ ^x. -O;
(- / + w + 7i)* (/ — w + n)* (/ -h w - n)
that this will coincide with the second if ^ + m* + n' = ; and that the
three conies have always four common tangents.
1398. The angular points of a triangle lie on the conic
hi' + my* + 7MJ* =
and two of its sides touch the conic flB* + y* + «* = 0; prove that the enve-
lope of the third side is the conic
/(-^+w + n)*«" + m(;-m + n)*y* + n(Z + m-n)*«*=Oj
that this will coincide with the second if ^^ + m^ + n^ = 0, and that the
three conies have always four common points.
1399. A triangle is self-conjugate to the conic a? -¥ if •¥ 7? — and
two of its angular points lie on the conic la^ + m^ + ns^ == ; prove that
the locus of the third angular point is the conic
(w» + w) a" + (n + Q y + (^ + m) «• = ;
that this will coincide with the second if Z + m + n=0; and that the
three have always four common points. Also prove that the straight
line joining the two angular points will touch the conic
^ y* «• ^
1400. A triangle is self -conjugate to the conic W + my* + n«* ==
and two of its sides touch the conic a:'4-y' + «*«=0; prove that the enve-
lope of the third side is the conic
that this will coincide with the second if / + m-f-n = 0; and that the
three have always four common tangents. Also prove that the locus of
the intersection of the two sides is tibe conic
Z (m -»■ n) «• + m (n + Z ) y* + n (/ + m) «• = 0.
236 CONIC SECTIONS, ANALYTICAL.
TX. Anharmonic Ratio, nomographic Pencils and Ranges, Invo-
lution.
[The anharmonic ratio of four points Ay B, C, D in one straight line,
denoted hj [ABCD]^ means the ratio -^ : ^^, or ^ p^ ; the order
of the letters marking the direction of measurement of the segments
and segments measured in opposite directions being affected with
opposite signs. So, if -4, ^, C, i> be any four points in a plane and P
any other point in the same plane, P {ABCD] denotes ~ — 'J~pn~' — oX»/»»
the same rules being observed as to direction of measurement and sign
for the angles in this expression as for the segments in the other.
Either of these ratios is called harmonic when its value is - 1 ; in
which case AD \& the harmonic mean between AB and AC, and DA is
the harmonic mean between DB and DC. The anharmonic ratio of
four points or four straight lines can never be equal to 1 ; as that value
leads immediately to the result AD . BC = or sin APD sin BPC =
making two of the points or two of the lines coincident.
A series of points on a straight line is called a range, and a series of
straight lines through a point is called a pencil, the straight line or
point being the axis or vertex of the range or pencil respectively. If
two ranges ahcd . . . , a'h'c'd' ... be so connected that each point a of the
first determines one point a' of the second and each point a' of the
second determines one point a of the first, the ranges are homographic.
8o also two pencils, or a range and a pencil, may be homographic ; and
in all such cases the anharmonic ratio of any range or pencil is equal
to the anharmonic ratio of the corresponding range or pencil in any
homographic system.
If four fixed points A, B, C, Dhe taken on any conic and P be any
other point on the same conic, P [ABCD] is constant for all positions of
P and is harmonic when BC, AD are conjugates with respect to the
conic. Also, if the tangent at P meet the tangents bX A, B,Cy D in. the
points a, 6, c, J, the range {ahcd) is constant and equal to the former
pencil
A range of points on any straight line is homographic with the pencil
formed by their polars with respect to any conic.
If the equations of four straight lines can be put in the form
u = /ijV, u=fi^v, u = fi^y u=fi^v, the anharmonic ratio of the pencil
formed by them or of the range in which any straight line meets
them is
A very great number of loci and envelopes can be determined imme-
diately from the following theorems : (1) The locus of the intersection of
corresponding rays of two homographic pencils is a conic passing through
the vertices ot the pencils (0, 0') and the tangents at 0, C>' are the
CONIC SECTIONS, ANALYTICAL. 237
rays corresponding to O'O, 00* respectively: (2) The envelope of a
straight line which joins corresponding points of two homographic ranges
is a conic touching the axes of the two ranges in the points which cor-
respond to the common point of the axes.
A series of pairs of points on a straight line is said to be in involu-
tion when there exist two fixed points {f^ f) on the line such that, a, a'
being any pair, {aff'a!)=- 1. The points / /' are called the foci or
double points of the range, since when a is at y, a* will also be at f.
The middle point {C) oiff is caUed the centre and Ga . Ca' = Gf\ The
foci may be either both real or both impossible, but the centre is always
real ; and when two corresponding points are on the same side of the
centre the foci are real. Similarly a series of pairs of straight lines, or
rays, drawn from a point is in involution when there exist two fixed
rays forming with any pair of corresponding rays a harmonic pencil in
which the two fixed rays are conjugate. This pair of fixed straight
lines is called the focal lines or double rays.
Any straight line is divided in involution by the six straight lines
joining the points of a quadrangle, and any two corresponding points of
the involution will lie on a conic round the quadrangle.
The pencil formed by joining any point to the six points of intersec-
tion of the sides of a quadrilateral is in involution and any pair of cor-
responding rays touch a conic inscribed in the quadrilateral.
The locus of the intersection of two tangents to a given conic drawn
from corresponding points of an involution is the conic which passes
through the double points of the involution and through the points of
contact of tangents to the given conic drawn from the double points.
The envelope of a chord of a given conic whose ends lie on
corresponding rays of a pencil in involution is a conic touching the
double rays of the involution and also touching the tangents drawn to
the given conic at the points where the double rays meet it.
These two theorems will be found to include as particular cases many
well-known loci and envelopes.
It may be mentioned that a large proportion of the questions which
are given under this head might equally well have appeared in the next
division : Reciprocal Polars and Projection.]
1401. Two fixed straight lines meet m A] B^ Cy D are three fixed
points on another straight line through A ; any straight line through D
meets the two former straight lines in 6, c and Bbf Cc meet in P, Be, Ch
in Q : prove that the loci of P, Q are straight lines through A which
make with the two former a pencil whose ratio is {{ABCD\y.
•
1402. On a straight line are taken points 0, A, B, C, A', ff, C
such that
{OABC] = {OAB^C} = [OA'BC] = [OA'FC] ;
prove that each = {OA'FC), and that the ranges {OBCA'}, {00 AB], and
{OABC'\ will each be harmonic.
238 CONIC SECTIONS, ANALYTICAL.
1403. Two fixed straight lines intersect in a point on the side BC
of a triangle ABC ; any point F being taken on. AO the straight lines
FB, PC meet the two fixed straight lines in jB , jB^, (7,, (7, respec-
tively : prove that Bfi^ and B/)^ pass each througn a fixed point on BC.
1404. From a fixed point are let fall perpendiculars on conjugate
rays of a pencil in involution : prove that the straight line tlurough
the feet of these perpendiciilars passes through a fixed point.
1405. Two conjugate points a, a! of a range in involution being
joined to a fixed point 0^ straight lines drawn through a, a! at right
angles to aO^ a'O meet in a point which lies on a fixed straight line.
1406. Chords are drawn through a fixed point of a conic equally
inclined to a given direction ; prove that the straight line joining their
extremities passes through a fixed point.
1407. Through a given point are drawn chords PF, QQf of a
given conic so as both to touch a confocal conic : prove that the points
of intersection of FQ^ F'Q\ and of /^^, PQ are fixed.
1408. A circle is described having for ends of a diameter two conju-
gate points of a pencil in involution : prove that this circle will be
cut orthogonallj by any circle through the two double points of the
range.
1409. Two triangles are formed each by two tangents to a conic and
their chord of contact; prove that their angular points lie on one
conic.
1410. Four points A^ B^ C^ D being taken on a conic, any straight
line througli D meets the conic again in ly and the sides of the triangle
ABC vcL A\ Bj C : prove that the range {A'B>C'D'\ is equal to the pencil
{ABCD] at any point on the conic.
1411. The sides of a triangle ABC touch a conic in the points
A'y By C and the tangent at any point meets the sides of the two
triangles in a, 6, c, a', h\ c' respectively : prove that {Oo^} = {Oa'h'c').
1412. Four chords of a conic are drawn through a point, and two
other conies are drawn through the point, one passing through four
extremities of the four chords and the other through the other four
extremities : prove that these conies will touch each other at the point
of concourse.
[Also very easily proved by projection.]
1413. Through a given point is drawn any straight line meeting
a given conic in <?, Q', and a point F is taken on this line such that the
range [OQQ'F] is constant : prove that the locus of jP is an arc of a
conic having double contact with the given conic.
1414. Given two points Ay B oi a, given conic; the envelope of a
chord FQ such that the pencil {AFQB} at any point of the conic has a
^ven value is a conic touching the given conic at A, B,
CONIC SECTIONS, ANALYTICAL. 239
1415. Through a fixed point is drawn any straight line meeting two
fixed straight lines ia Q, Ji respectively; E, F are two other fixed
points : prove that the locus of the point of intersection of QE, EF Ib
a conic passing through E^ F, and the common point of the two fixed
straight lines.
1416. Three fixed points A, B, C being taken on a given conic,
two other points F, F are taken on the conic such that the pencils
[PABC]^ {FABG] are equal at any point on the conic : prove that P-P,
CA, and the tangent at B, meet in a point.
1417. Six fixed points A, B,C, A', Bf^ C are taken on a given conic
such that, at any point on the conic, [RABC] = {BA'B'C] ; and P, F are
two other points on the conic such that, at any point on the conic,
{PABC]^{FA'FC}: prove that FF, AA\ BB, OCT all intersect in
one point.
Of course the she points subtend a pencil in involution at any point
le conic, and conjugate rays pass through F^ FJ\
of tL
1418. A conic passes through two given points A, A\ and touches
a given conic at a given point ; prove that their other common
chord will pass through a fixed point B on AA\ and that if the straight
line through A, A' meet the given conic in (7, C and the tangent at
in F, the points A, A^ ; B, B' ^ Cy C will be in involution.
1419. Two chords A By CD of a conic being conjugate, the angle
ACB is a right angle, and any chord DF through D meets AB m Q ;
prove that the angle FCQ is bisected by CA or CB,
1420. Three fixed points Ay B, C being taken on a conic, and F
being any other point on the, conic, through F is drawn a straight line
meeting the sides of the triangle ABC in points a, b, c such that {Po&c}
has a given value : prove that the straight line passes through a fixed
point on the conic such that the pencil {OABC} at any point of the
conic has the same given value.
1421. Prove that the two points in which a given straight line meets
any conic through four given points are conjugate with respect to the
conic which is the locus of the pole of the given straight line with
respect to the system of conies.
1422. Three fixed tangents to a given conic form a triangle ABC,
and on the tangent at any point F is taken a point such that the
pencil {FABC} has a given value : prove that the locus of is a
straight line which touches the conic.
1423. Two conies circumscribe a triangle ABC, any straight line
through A meets them again in F, Q : prove that the tangents at F, Q
divide BC in a constant anharmonic ratio.
1424. Conies are described touching four given straight lines, of
which two meet in A, and the other two in ^ ; on the two meeting
240 CONIC SECTIONS, ANALYTICAL.
in A are taken two fixed points (7, 2>, and the tangents di*awn from them
to one of the conies meet in P : prove that the locus of P is a straight
line through B which forms with BC, BD and one of the tangents
through B a pencil equal to that formed by BA^ BCy BD and the other
tangent through B.
1425. The diagonals of a given quadrilateral are AA\ BB, CG\
and on them are taken points a, a' ; 6, 5' ; c, c', so that each diagonal is
divided harmonically: prove that if a, 6, c be collinear, so also will
o', h\ c\ and their common point will be the point whei-e either of
them is touched by a conic insciibed in the quadrilateral.
[This is also a good example of the use of Projection.]
1426. Two fixed tangents OA, OB are drawn to a given conic and a
fixed point C taken on ^^ ; through C is drawn a straight line meeting
the fixed tangents in ^', ^ : prove that the remaining tangents from
A\ B^ intersect in a point whose locus is a fixed straight line through 0.
1427. Two fixed points A^ B are taken on a given conic and a
fixed straight line drawn conjugate to AB', any point P being taken
on this last straight line chords APQ^ BPQf are drawn ; prove that QQ^
passes through a fixed point on AB.
1428. A conic is inscribed in a triangle ABC, the polar of A meets
BO in a, and aP is drawn to touch the conic ; prove that if from any
point Q on aP another tangent be drawn, this tangent and QA will
form with QBj QO a harmonic pencil.
1429. Two chords AO, BO of a conic are conjugate, any chord
OP meets the sides of the triangle ABO in a, 6, c : prove that the range
{abcP\ is harmonic.
1430. Two fixed points A^ B are taken on a given conic, P is any
other point on the conic : prove that the envelope of the straight line
joining the points where PA^ PB meet two fixed tangents to the conic is
a conic which touches At A, B the straight lines joining these points to
the points of contact of the corresponding fixed tangents, and which
also touches the two fixed tangents.
1431. One diagonal of a quadrilateral circumscribing a conic la AA* :
prove that another conic can be described touching two of the sides of the
quadiilateral in ^, ^' and passing through the points of contact of the
other two.
1432. On the normal to an ellipse at a point P are taken two points
0, & such that the rectangle PO . PO* is equal to that under the focal
distances of P, and from these points tangents are drawn to the ellipse :
prove that their points of intersection lie on the circle whose diameter
is QQ', where Q, Q' are the points in which the tangent at P meets the
director circle.
CONIC SECTIONS, ANALYTICAL. 241
1 433. A range of points in involution lie on a fixed Btraight line
and a homographic system on another fixed straight line ; a, a' are con-
jugate points of the former and A, A' the corresponding points on the
latter : prove that the locus of the intersection of ctAf a! A* or of aA\
cdA is a straight line.
1434. A pencil in involution has a point for its vertex, and a
homographic pencil is drawn from another point 0\ oon*esponding rays
of the two intersect in P and the conjugate rajs in P : prove that PF
passes through a fixed point
1435. In any conic the tangent at A meets the tangents at (7, ^ in
\ c which are joined to a point by straight lines meeting BC in b\ c' :
prove that AC, cb' intersect on the polar of 0, as also AB^ he.
1436. The triangle ABC is self-conjugate to a given conic U, a
conic V is inscribed in the triangle and its points of contact are
A\E,C\ prove that, if BC touch U, so also will CA', A'ff, and the
straight line in which lie the points {BG, SC% (CA^ C'A'), and
(AB, A'B\
1437. A variable tangent to a conic meets two fixed tangents in
P, Q ; A,B are two fixed points : prove that the locus of the intersection
of AP, BQ is a conic, passing through A, B, and the intersections of
(OAy Bh) and {OB, Aa) ; Oa, Oh being the fixed tangents.
1438. Parallel tangents are drawn to a given conic and the point
where one meets a given tangent is joined to the point where the other
meets another given tangent : prove that the envelope of the joining
line is a conic to which the two given tangents are asymptotes.
1439. Through a fixed point of an hyperbola is drawn a straight
line parallel to an asymptote, and on it are taken two points P, B such
that the rectangle OP . OF is constant ; the locus of the intersection of
tangents drawn from P, F ia two fixed straight lines passing through
the common point of the tangent at and the asjrmptote, and forming
with them an harmonic pencil.
1440. Four fixed points A, B, C, D are taken on a given conic ;
through D is drawn any straight line meeting the conic again in P and
the sides of the triangle ABC in A\ B, (j\ prove that the range
{PA'BG\ is constant
1441. The tangent to a parabola at any point P meets two fixed
tangents CA , CB in a, 5, the diameters through the points of contact
^, i^ in a\ h\ and the chord of contact AR'uic', prove that
Pa. Pel \ Ph.PV^ac' : he'.
1442. A tangent to an hyperbola at P meets the asymptotes in a, 6,
the tangent at a point Qme, and the straight lines drawn through Q
parallel to the asymptotes in a\ h': prove that
Pa! '. PV=^ca ihe.
W. P. 16
242 CONIC SECTIONS, ANALYTICAL.
1443. The anharmonic ratio of the pencil subtended hj the four
points whose exoentric angles are a,, a,, a,, a^ at any point of an
ellipse is
sin I (g^ - tt,) sin ^ (a, - a J
8inJ(a,-03)8ini(a,-oJ*
1444. Tangents are drawn to a conic at four points A, B, C, />, and
form a quadrilateral whose diagonals are aa\ hh\ cc', the tangents at
A^ B, C forming the triangle ahc, and being met by the tangent at Z> in
a', h\ c'\ the middle points of the diagonals are A\ By C and the centre
of the conic is : prove that the range \A'B'C'0\ is equal to the pencil
\ABGD\ at any point of the conic.
1445. A conic is drawn through four given points A^ B, (7, 2);
BGy AD meet in A') CA, BD in ^; AB, CD in C] and is the centre of
the conic : prove that the pencil {ABCD] on the conic is equal to the
pencil {A'BCO) on the conic which is the locus of 0.
1446. The anharmonic ratio of the four common points of the two
conies
at any point on the former is one of the three
a-h b—c e-a
a-c
y
h-a' c-b'
or the reciprocal of one of them, according to the order of taking the four
points; also these are the values of the range formed on any tangent to
the second conic by their four common tangents.
1447. Two fixed tangents are drawn to a given conic intersecting
each other in and a fixed straight line in L, M; from any point on
LM are drawn two tangents to the conic meeting the two fixed tangents
in Ay B] A'y B^, respectively: prove that a conic drawn to touch the
two fixed tangents at points where they are met by LM, and touching
one of the straight lines AB^^ A'B, will also touch the other.
1448. A quadrilateral circumscribes a conic and AA\ BB' are two
of its diagonals ; any point F being taken on the conic, BPy BP and
the tangent at P meet AA' in the points i\ <, |? respectively : prove
that
V : jy = il<.^<': A'i.Ai,
[Also easily proved by projecting J, A' into foci]
'1449. Four tangents TP, TQ, TF, TQf are drawn to a parabola :
prove that the conic TPQTFQf will be a circle if Tr' be bisected by
the focus.
[A parabola can be drawn with its focus at T' touching PQ and the
normals at /*, Q) and another with its focus at T touching FQf and the
normals at F^ Q' ; and the axis of the given parabola will be the tangent
at the vertex of either of these.]
CONIC SECTIONS, ANALYTICAJU 243
1450. Four tangents TP, TQ, TF, TQ are drawn to a given
ellipse : prove that the conic TPQVFii will be a circle when CT, CT
being equally inclined to the major axis and T^ T' on the same side of
the minor axis, CT . GT'=GS*y where C is the centre and S a focus of
the given ellipse.
[A parabola can be drawn with T' for focus and GT for directrix
touching PQ and the normals at P, ^ ; and another parabola with T for
focus and GT' for directrix will touch FQf and the normals at 7^, C;
and these parabolas are the same for a series of oonics confocal with the
given ellipse.]
1451. The locus of the intersection of tangents to the ellipse
flKB* + bf/'-h 2hxi/ = 1 drawn parallel to conjugate diameters of the ellipse
aV + b'y' + 2h'xy = 1 is
{ab - h') (a V + by + 2h'xi/) = ab' + a'b - 2hh\
1452. Through each point of the conic (mo^ + by* -|» 2hxt/ = 1 is drawn
a pencil in involution whose, double rays are parallel to the co-ordinate
axes : prove that the chord cut off by a pair of conjugate rays passeH
through a fixed }X)int whose locus is ihe conic
oic* + 6y* + 2/iay = ^ .
1453. Two conjugate rays of a pencil in involution meet the oonio
u = aaf + bt^-k-c +2/t/ + 2gx -^ 2hxy =
in the points P, F; Q, Q', the double rays of the pencil being the
axes of coordinates : prove that the conic enveloped by PQ^ PQ\ FQ^
FQ'iB
i(/g- ch)xy={fy + gx + c)'.
\li fg = ch, the double rays are conjugate with respect to the conic ti,
and the chords pass through the two fixed points where the double rays
meet the polar of the vertex : if c = 0, the vertex is on the curve and
the chord determined by conjugate rays passes through the point
H- -01
1454. Four fixed tangents to a conic form a quadrilateral of which
AA\ BE are two diagonals, any other tangent meets AA' vaP and the
range \APFA'\ is harmonic : prove that the locus of the intersection of
JIF^ or of FF^ with the last tangent is a conic passing through AA! and
touching the given conic where BE meets it.
[Taking the given conic to be a:* = ^ and the straight line AA' to be
px + qy +rz = Of the locus is
p*(«»-yaj)=(5^y-r2)«,
degenerating to the straight line^ + rs; = ^ when />' = 4^; that is,
when A A' is a tangent to the given conic]
244 CONIC SECTIONS, ANALYTICAL.
1455. Three fixed tangents are drawn to a conic and their points of
intersection joined to a focus ; any other tangent meets these six lines
in an involution such that the distance between the double points
subtends a right angle at the focus. Also the locus of the double points
for different positions of the last-named tangent is the curve
-=ecosfl + cos(a + )3 + y-3tf),
where c = r(I 4- 0cos0) is the equation of the given conic, and a, /9, y
are the values of $ at the points of contact of the fixed tangents.
X. Reciprocal Pcla/ra and Projections,
[If there be a system of points, and straight lines, lying in the same
plane and we take the polars of the points and the poles of the straight
fines with respect to any conic in that plane, we obtain a system of
straight lines and points reciprocal to the former ; so that to a series of
points lying on any curve in the first syutem correspond a series of
straight lines touching a certain other curve in the second system, and
ffice versd: and, in particular, to any number of points lying on a
Straight line or a conic, correspond a number of straight lines passing
through a point or touching a conic. Thus from any general theorem
of position may be deduced a reciprocal theorem. It is in nearly all
cases advisable to take a circle for the auxiUary conic with respect to
which the system is reciprocated; the point (p) corresponding to any
proposed straight line being then found by drawing through 0, the
centre of the circle, OP perpendicular to the proposed straight line and
taking on OP a point p such that OP. Op = ^f k being the radius of the
circle ; and similarly l^e straight line through p at right angles U) Op i&
the straight line corresponding to the point P.
To draw the figure reciprocal to a triangle ABC, with respect to a
circle whose centre is or more shortly vnth respect to the point 0, draw
Oa perpendicular to BC and on it take any point a ; through a, draw
straight lines perpendicular to OC, CAy meeting in b ; and through b,
draw straight lines perpendicular to OA, AB meeting in c ; then the
points a, (, e will be the poles of the sides of the triangle ABC and the
straight lines bCy ca^ ab the polars of the points A, B, 0, with respect to
some circle with centre 0. Now suppose we want to find the point
corresponding to the perpendicular from A on BC ; it must lie on be and
on the straight line through at right angles to Oa since Oa is parallel
to the straight line whose reciprocal is required; it is therefore
determined. Hence to the theorem that the three perpendiculars of a
triangle meet in a point corresponds the following : if through any point
(0) in the plane of a triangle (abc) be drawn straight lines at right
angles to Oa, Ob, Oc to meet the respectively opposite sides, the three
points ao determined will lie on one straight Ime, or be coUinear.
CONIC SECTIONS, ANALYTICAL. 245
So from the theorem that the bisectors of the angles meet in a point
we get the following : the straight lines drawn through bisecting the
external angles (or one external and two internal angles) between 06, Oe\
Oc^ Oa; Day Ob, respectivelj, will meet the opposite sides in three
collinear points.
If a circle with centre A and radius R be reciprocated with respect
to Oy the reciprocal curve is a conic whose focus is 0, major axis along
OAj excentricity OA -r- if, and latus rectum 2^ -r E or 2 -r /? if we take
the radius of the auxiliary circle to be unity. The centre A is reciprocated
into the directrix. Focal properties of conies are thus deduced from
theorems relating to the circle. For instance, if be a point on the
circle and OFy OQ chords at right angles, FQ will pass through the
centre. Reciprocating with respect to 0, to the circle corresponds a
parabola and to the points F^ Q two tangents to the parabola at right
angles to each other; perpendicular tangents to a parabola therefore
intersect on the directrix*
Again, to find the condition that two conies which have one focus
common shoidd be such that triangles can be inscribed in one whose
sides touch the other. Take two circles which have this property, and
let Ry rhQ their radii, h the distance between their centres ; then
^ = R^^2Rr.
Keciprocate the system with respect to a point at distances
Xy y from the centres, and let a be the angle between these distances.
Then a will be the angle between the axes of the two oonicSi and, if
2cp 2c, be the latera recta, 6,, e, the excentricities,
^»"r' ^«"5' ^•'^r' *•"!' 8' = «' + y"-2ayco8a = i2**2i?r,
, 1 2 «/ 6« ^«/. •
whence — r* — =-*i + -^, -2-^*ooea,
c c c c c c c
or c* * 2CjC, = e/c* + «/c/ - 2e^e/:^c^ cos a ;
the required relation.
If a system of oonfocal conies be reciprocated with respect to
one of the foci, the reciprocal system will consist of circles having a
common radical ipds ; the radical axis being the reciprocal of the second
focus, and the first focus being a point-circle of the system.
The reciprocal of a conic vrith respect to any point in its
plane is another conic which is an ellipse, parabola, or hyperbola ac-
cording as the point lies within, upon, or without the conic. To the
points of contact of tangents firom ihe point correspond the asymptotes,
and to the polar of the point the centre of the redprocaL So also to
the asymptotes and centre of the original conic correspond the points of
contact and polar with respect to the redprocaL
As an example we may redprocate the elementary property
that the tangent at any point of a conic makes equal angles with the
focal distances. The theorem so obtained is that if we tdce any point
246 CONIC SECTIONS, ANALYTICAL.
in the plane of the conic there exist two fixed straight lines (recipro-
cals of the foci) snch that if a tangent to the conic at F meet them in
C Q\ OP makes equal angles witi OQ^ 0Q\ (More correctly there are
two such pairs of straight lines, one pair only being real.) If however
the point lie on the curve the original curve was a parabola ; and one
of the straight lines being the reciprocal of the point at infinity on the
parabola wiU be the tangent at 0, Another property of the focus, that any
two straight lines through it at right angles to each other are conjugate,
shews us that if on either of the two straight lines we take two points
X| L' such that LOL' is a right angle, Z, L' will be conjugate.
Since the anharmonic ratio of the pencil formed by any four
rays is equal to that of the range formed by their poles with respect to
any conic it follows that^ in any reciprocation whatever, a pencil
or range is replaced by a range or pencil having the same anharmonic
ratio.
The method of Projections enables us to make the proof of any
general theorem of position depend upon that of a more simple par-
ticular case of that theorem. Given any figure in a plane we have five
constants disposable to enable us to simplify the projected figure, three
depending on the position of the vertex and two on the direction of the
plane of Projection. It is clear that relations of tangency, of pole and
polar, and anharmonic ratio, are the same in the oiiginal and projected
figure.
As a good example of the use of this method, we will by means
of it prove the theorem that if two triangles be each self-conjugate to the
same conic their angular points lie on one conic.
Let the two triangles be ABC^ DBF, and abc, def their pro-
jections ; project the conic into a circle with it* centre at d^ then c, /
will be at infinity, and de, df a,t right angles. Draw a conic through
abcde, then since abc is self-conjugate to a circle whose centre ia d, d is
the centre of perpendiculars of the triangle abc, the conic is therefore a
rectangular hyperbola, and e being one of its points at infinity, / must
be the other. Thus ahcdef lie on one conic, and therefore ABC DEF
also lie on one conic. Again, retaining the centre at d, take any other
conic instead of a circle ; de, df will still be conjugate diameters, and
therefore if any conic pass through a, 6, c, d, its asymptotes will be
parallel to a pair of conjugate diameters of the conic whose centre is d
and to which ahc is self-conjugate. The same must therefore be the case
with respect to the four conies each having its centre at one of the four
points a, 6, c, (/, and the other three ix)int8 comers of a self-conjugate
triangle. These four conies must therefore be similar and similarly
situated. Moreover if we draw the two parabolas which can be drawn
through a, 6, c, d their axes must be parallel respectively to coincident
conjugate diameters of any one of the four conies ; that is to the asymp-
totes. But the axes of these parabolas must be parallel to the asymp-
totes of the conic which is the locus of the centres of all conies through
a, 6, c, dy since the centre is at infinity for a parabola. Hence, finally,
if we have four points in a plane, the four conies each of wliich has one
tho four points for its centre and the other three at the comers of a
CONIC SECTIONS, ANALYTICAL. 247
self -conjugate triangle are all similar and similarly situated to each
other and to the conic which is the locus of centres <^ all conies through
the four points.
(The same results might also be proved by orthogonal projection,
making d the centre of perpendiculars of the triangle abc, in which case
the five conies are all circles.)
Let Ay B he any two fixed points on a circle, oo, oo' the two
impossible circular points at infinity, F any other point on the circle ;
then P{Ax> CO '£] is constant. Hence FA, FB divide the segment
terminated by the two circular points in a constant anharmonic ratia
Hence two straight lines including a given angle may be projected into
two straight lines dividing a given segment in a constant anharmonic
rStio. In particular, if AFB be a right angle, AB passes through the
centre of the circle (the pole of 00*00'), and the ratio becomes har-
monic.
Thus, projecting properties of the director circle of a conic,
we obtain the following important theorem : the locus of the intersec-
tion of tangents to a conic which divide a given segment harmonically is
a conic passing through the ends of the segment and through the points
of contact of tangents to the conic drawn from the ends. If the straight
line on which the segment lies touch the conic, the locus degenerates to
a straight line joining the points of contact of the other tangents drawn
from the ends of the segment.
Reciprocating, we get the equally important theorem : if a chord
of a given conic be divided harmonically by the conic and by two
given straight lines its envelope will be a conic touching the two given
straight lines and also the tangents to the given conic at the points
where the given straight lines meet it ; but when the two given straight
lines intersect on the given conic the chord which is divided harmoni-
cally will pass through a fixed point, the intersection of the tangents
to the given conic at the points where the given straight lines again
meet it.
If tangents be drawn to any conic through 00 , 00 ' their four
other points of intersection are the real and impossible foci of the conia
When the conic is a parabola the line joining oo , 00 ' is a tangent, and
one of the real foci is at infinity, while the two impossible foci are the
circular points. Many focal properties, especially of the parabola, may
thus be generalized by projection. Thus since the locus of intersection
of tangents to a parabola including a constant angle is a conic having
the same focus and directrix, it follows that if a conic be inscribed in a
triangle ABO, and two tangents be drawn dividing BO in a constant
range, the locus of their point of intersection is a conic touching the
former in the points where AB^ AG touch it. Here B, are
the projections of oo , 00 ', ^ of the focus, and the directrix is the polar
of the focus.
The circular points at infinity have singular properties in rela-
tion to many other well-known curves. All epicycloids and hypocycloids
pass through them, the cardioid has cusps at them, and may be {mjected
into a tliree-cusped epicycloid.]
248 CONIC SECTIONS, ANALTTICAL.
1456. Two oonics have a common focus Sf and two common tan-
gents PF"^ QQf ; prove that the angles PSF, QSQ^ are equal or supple-
mentary.
1457. Two oonics have a common focus, and (1) equal minor axes,
(2) equal latera recta : prove that (1) the common tangents are parallel,
(2) one of their common chords passes through the common focus.
1458. The straight line drawn through the focus S at right angles
to any straight line SO will meet the polu" of on the directrix.
1459. The fixed point is taken on a given conic also any three
other points on the conic L, M, N: straight lines drawn through at
right angles to OL, OM^ ON meet MN^ NL^ LM in three points lying
on a straight line which meets the normal at in a fixed point (chords
through which subtend a right angle at 0),
1460. Given a conic and a point : prove that there are two real
straight lines such that the distance between any two points on
either, which are conjugate with respect to the conic, subtends a right
angle at 0,
1461. A fixed point is taken on a conic, and OR is the chord
normal at 0, OP, OQ any other chords : prove that a certain straight
line can be drawn through the pole of OR such that the tan-
gents at P, Q intercept on it a segment which subtends at an angle
2F0Q.
1462. On any straight line can be found two points, conjugate to a
given conic, such that the segment between them subtends a right angle
at a given point.
1463. A point being taken in the plane of a triangle ABC^
straight lines drawn through at right angles to OA^ OB, 00 meet the
respectively opposite sides in A'y B^ C : prove that any conic which
touches the sides of the triangle and the straight line A' EC* subtends a
right angle at 0.
1464. An ellipse is described about an acute-angled triangle ABC^
and one focus is the centre of perpendiculars of the triangle : prove that
its latus rectum is
^„cos^ cos^cos(7
^^ . i . jj . (}'
1465. A parabola and hyperbola have a common focus and axis,
and the parabola touches the directrix of the hyperbola; prove that
any straight line through the focus is harmonically divided by a
tangent to the parabola and the two parallel tangents to the hyper-
bolA,
CONIC SECTIONS, ANALYTICAL. 249
1466. A series of conies are described having equal latera recta, a
focus of a given conic their common focus, and tangents to the conic
their directrices : prove that the common tangents of any two intersect
on the directrix of the given conic at a point such that the line joining
it to the focus is at right angles to one of their common chords which
passes through the focus.
1467. A point S is taken within a triangle ABC such that the sides
subtend at S equal angles, and four conies are drawn with S as focus
circumscribing the triangle : prove that one of these will touch the other
three, and that the tangent to this conic at A will meet £C in a point A'
such that ASA' is a right angle.
1468. Prove that, with the centre of the circumscribed circle as
focus, three hyperbolas can be described circumscribing a given triangle
ABC; that their excentrieities are cosec ^ cosec (7, &c. ; their latera
recta 2B cot B cot C, &c, ; their directrices the straight lines joining the
middle points of the sides ; and that the fourth common point of any two
lies on the straight line joining one of the points A, By C to the mid
point of the opposite side.
1469. With a given point as focus four conies can be drawn cir-
cumscribing a given triangle, and the latus rectum of one of these will
be equal to the sum of the other three. Also if any conic (/ be drawn
touching the directrices of the four conies the polar of the given point
with respect to it will be a tangent to the conic V which has the given
point for focus, and which touches the sides of the triangle, and the
conic U will subtend a right angle at the given point.
[If /, /j, /,, ^3 be the four latera recta, and /^ + /, + /, = /, the latus
rectum of V will be 77 — , v /, ^',\ /, — t\ •]
1470. From a point F on the circle ABC are drawn PA\ FB, FC
at right angles to PA^ FB^ FC respectively to meet the corresponding
sides of the triangle ABC : prove that the straight line A'BC passes
through the centre of the circle.
1471. With a point on the circumscribed circle of a triangle ABC
as focus are described four conies circumscribing the triangle : prove
that the corresponding directrices will pass each through the centre of
one of the four circles touching the sides.
1472. A triangle is inscribed in an ellipse so that the centre of the
inscribed circle coincides with one of the foci ; prove that the radius of
the inscribed circle is . ; 2c being the latus rectum, and e the
l + i/l+d*
excentricity.
1473. A triangle is self-conjugate to an hyperbola, and one focus is
equidistant from the sides of the triangle : prove that each distance is
—. r , 2c being the latus rectum, and $ the excentricity.
ije — 2
250 CONIC SEcnoNS, analttical.
1474. Two conicf have a common focus, and triaDgles can be
inacribed in one which are self-conjngate to the other ; prove that
2e' + c/ = e'e/ + e/c,* - 2e^e^efi^ cos a ;
c,9 e^ being their latera recta, e,, e^ their excentricities ; and a the
angle l>etween their axes. Prove also that in this case triangles can be
circumscribed to the second which shaU be self-conjugate to the first
1475. A conic passes through two fixed points A^ A' and touches a
given conic at a fixed point : prove that their chord of intersection
meets AA' in a fixed jioint B; and, if the given conic meet AA' inC, C
and the tangent at meet it in if', AA', BBfy GC will be in involution.
1476. Four points being taken on a circle, four parabolas can be
drawn having a common focus, and each touching the sides of the
triangle formed by joining two and two three of the four points.
1477. Three tangents to an hyperbola are so drawn that the
centre of perpendiculars of the triangle formed bj them is at one
of the foci ; prove that the polar circle and the circumscribed circle of
the taiangle are fixed.
1478. Three tangents to a parabola form a triangle ABC^ and per-
pendiculars Xy y, z are let £bJ1 on them from the focus S\ prove that
yz sin BSC + »r sin CSA + ay sin ASB = 0,
the angles at S being measured so that their sum is 360^ Also prove
that if 2/ be the latus rectum,
sin 2A sin 2^ sin 2C _ 8 sin -4 sin j5 sin (7
1479. The minor axis of an ellipse is BB\ and B is the centre of
curvature at J^' ; a point P is taken on the circle of curvature at B', and
tangents drawn from F to the ellipse meet the tangent at ^ in Q, i^' :
prove that a conic drawn to touch QB^ Q[B with its focus at B and
directrix passing through Bf will touch the circle at P.
1480. An hyperbola is drawn osculating a given parabola at P,
passing through the focus, and having an asymptote parallel to the
axis : prove that the tangent to it at the focus and the asymptote
aforesaid intersect in the centre of curvature at P.
1481. Given a circle and a straight line not meeting it in real
points, the two point-circles S, S! have with the given circle the given
straight line for radical axis ; two conies are drawn osculating the circle
at P and having one a focus at aS' and the other a focus at S' : prove that
the corro8i>onding directrices coincide and pass through the point of
contivct of the parabola which osculates the given circle at P and touches
tlio given straight line.
CONIC SECTIONS, ANALYTICAL. 251
1482. An ellii)8e is drawn osculating a given circle at P and having
one focus at a point O of the circle ; a [mrabola is also drawn osculating
At P and touching the tangent at : prove tliat the directrix of the
ellipse is parallel to the axis of the parabola and passes through the
point of contact of the parabola with the tangent at 0.
1483. A point is taken within a circle, and with as focus is
described a paralK>Ia touching the radical axis of the circle and the
point-circle ; A OA' is a chord of the circle bisected in : prove that
tangents from Ay A' to the jiarabola touch it in points lying on the circle.
1484. A chord LL' of a given circle is bisected in and P is its
pole ; a parabola is drawn with its focus at and directrix passing
through L : prove that the tangents drawn to this [)arabola at points
where it meets the circle pass through Lor L'; and, if two such parabolas
intersect the circle in any the same |x)int, the angle between their axes
is constant.
1485. Two fixed points are taken on a given conic and joined to
any point on a given straight line : prove that the envelope of the
straight line joining the points in which these joining lines again meet
the conic is a conic having double contact with the given conic at the
points where the given straight line meets it and also touching the
straight line joining the two fixed points.
1486. Any straight line drawn through a given point meets two
fixed tangents to a given conic in two points irom which are drawn other
tangents to the given conic : the locus of the common })oiiit of these last
tangents is a conic which touches the given conip at the points of
contact of tangents from the fixed point and passes through the common
point of the fixed tangents.
1487. Four fixed points 0, A, B, C being taken, OB, CA meet in
B^y OC, AB inC\ and from a fixed jwint on OA two tangents are drawn
to any conic through 0, A, B, C: prove that the points of contact and
the points B, C, B\ C lie on a fixed conic.
1488. With the centre of perpendiculars of a triangle as focus are
described two conies, one touching the sides and the other passing
through the feet of the perpendiculars ; prove that these conies will
touch each other and that their point of contact will lie on the conio
which touches the sides of the triangle at the feet of the peq>endiculars,
1489. A conic is inscribed in a triangle and one focus lies on the
polar circle of the triangle : prove that the corresponding directrix
passes through the centre of perpendiculars.
1490. With the centre of the circumscribed circle of a triangle as
focus are described two ellipses, one touching the sides and the other
passing through the middle points of the sides : prove that they will
touch each other.
252 CONIC SECTIONS, ANALYTICAL.
1491. Four fixed straight lines form a quadrilateral whose diagonals
are AA\ BB^^ CC'i prove that the envelope of tangents drawn to any
conic inscribed in the quadrilateral at the points where it meets a fixed
straight line through il is a conic which touches BB^ CO' and the two
sides of the quadrilateral which do not pass through A \ and if BE^ CC
meet A A' in c, 6 and the fixed straight line through A in h\ c\ that
hh\ cc' are also tangents to this envelope.
1492. Five points are taken no three lying in one straight line, and
with one of the points as focus are described four conies each touching
the sides of a triangle formed by joining two and two three of the
remaining four points: prove that these four conies have a common
tangent.
[If Ay By CyDyEhQ the five points, A the one taken for focus, AP,AQ
two chords at right angles of the conic ABCDE^ then the common
tangent is the locus of the intersection of the tangents at P, Q.]
1493. Through a fixed point are drawn two straight lines
meeting a given conic in P, jP; QjQ] and a given straight line in i?, Rfy
an4 BB! subtends a right angle at another fixed point : prove that PQy
P^y FQy PQ[ all touch a certain fixed conic.
1494. Given a conic and a point in its plane 0: prove that there
exist two real points X, such that if any straight line through L meet the
polar of L mP and P' be the pole of this straight line, Pfi will subtend
a right angle at 0.
1495. Any conic drawn through four fixed points meets two fixed
straight linjes drawn through one of the points again in P, Q: prove
that the envelope of PQ is a conic touching the straight lines joining
the other three given points.
1496. Two equal circles Uy V touch at a point Sy a tangent to V
meets U in Py Qy and is its pole with respect to U: prove that the
directrices of two of the conies described with focus S and circumscribing
the triangle OPQ will touch the circle U,
1497. A conic touches the sides of a triangle ABC in a, &, c and
Aoy Bhy Cc meet in S^, three conies are drawn with S for focus osculating
the former at a, 6, c; prove that all four conies have one common
tangent which also touches the conic having one focus at S and touching
the sides of the triangle ABC.
1498. Given four straight lines, prove that two conies can be
constructed so that an assigned straight line of the four is directrix and
the other three form a self-conjugate triangle ; and that, whichever
straight line be taken for directrix, the corresponding focus is one of two
fixed points.
1499. A quadrilateral can be projected into a rhombus on any plane
parallel to one of its diagonals, and the vertex will be any point on a
certain circle in a certain parallel plane.
CONIC SECTIONS, ANALYTICAL. 253
1500. A conic inscribed in a triangle ABC touches BC in a and Aa
again meets the conic in A'\ the tangent at any point P meets the
tangent at A' in T: prove that the pencil T{ABCP\ is harmonic.
1501. A conic is inscribed in a triangle ABC and OP^ OQ are two
other tangents; anotlier conic is drawn through OPQBC and T is the
pole of BC with respect to it: prove that A [OBCT] is harmonic. Also
prove that if lie on the straight line joining A to the point of contact
of BC^ T will coincide with A.
1502. A conic is inscribed in a given triangle ABC and touches BC
in a fixed point a; 5, e are two other fixed points on BC: prove that
tangents drawn from 6, c to the conic intersect in a point lying on
a fixed straight line through A,
1503. A triangle is self-conjugate to a rectangular hyperbola £7'and
its sides touch a parabola V; a diameter of 6^ is drawn through the focus
of Vi prove that the conjugate diameter is parallel to the axis of V,
1504. Two tangents OP, OQ are drawn to a parabola; an hyperbola
drawn through 0, P, Q with one asymptote parallel to the axis of the
parabola meets the parabola again in if : prove that its other asymptote
is parallel to the tangent at i^ to the parabola.
1505. Two tangents OP, OQ are drawn to an hyperbola; another
hyperbola is drawn through 0, P, Q with asymptotes parallel to those of
the former : prove that it will pass through the centre C of the former
and that CO will be a diameter.
1506. A triangle is self-conjugate to a conic U and from any other
two points conjugate to U tangents are drawn to a conic V inscribed in
the triangle : prove that the other four points of intersection of these
tangents are two pairs of conjugate points to U,
1507. A conic drawn through four fixed points A, B, C, D meets a
fixed straight line L in P, Q : prove that the conic which touches the
straight lines AB, CD, L and the tangents at P and Q will have a fourth
fixed tangent which with L divides AB and CD harmonically.
1508. Through two fixed points 0, 0' are drawn two straight lines
which are conjugate to each other with respect to a given conic U \
prove that the locus of their common point is a conic V passing through
0, 0' and the points of contact of the tangents from 0, 0' to the given
conic. Also, if two points be taken on the polars of 0, 0' which are
conjugates with respect to U, the envelope of the straight line joining
them is a conic V which touches the polimi of 00^ and the tangents from
0, (7 to tr.
1509. From two points 0, 0' are drawn tangents OP, 0Q\ ffP\
0'Q[ to a given conic fj^ and a conic F is drawn through OPQO'P'Q' ;
a triangle is inscribed in Y^ two of whose sides touch U : prove that the
254 CONIC SECTIONS, ANALYTICAL.
third side passes through the common point of PQ, FQ, Also the
tangents to CT" at the points where the straight line 00' meets it meet Y
in the points of contact of the common tangents to U^ V.
[ F is the locus of the intersection of tangents to U which divide 00*
harmonically, and U is the envelope of straight lines divided har-
monicallj bj V and by the tangents to V at the points where 00'
meets it.]
1510. From two points 0, 0' are drawn tangents OP, OQ; (X-P,
O'Q' to a given conic U; a conic V is drawn through OPQO'PQ'y and
another conic F' touches the sides of the triangles OPQ, 0'PQ\ prove
that F, F' are polar reciprocals of each other with respect to U, Also
PQ^ FQ' and the tangents to F at 0, 0' intersect in one point.
1511. Any conic is drawn touching four fixed straight lines and
from a fixed point on one of the lines a second tangent is drawn to the
conic: prove that the locus of its point of contact is a conic circum-
scribing the triangle formed by the other three given lines.
[If the four be the sides of a triangle ABC and a straight line
meeting the sides in A', B^ (7 and the fixed point be on the last, the locus
passes through A^ B, C and through the point of concourse of Aa, Bb, Ccy
where a is the point {BB\ CC) ; also if any other straight line through
meet the sides of the triangle ABC yd. A'\ F\ C, and BB", CC"
meet in a, &c., then Aa, Bpy Cy intersect in a point on the locus.]
1512. A conic is inscribed in a given quadrilateral and from two
fixed points on one of the sides are di*awn other tangents to the conic :
prove that the locus of their common point is a conic passing through
the two given points and the points of intei'section of the other three
straight lines.
1513. Two common tangents to two conies meet in Ay the other two
in A'; from a point on A A' tangents OP, OQ, OB, OS are drawn to
the two conies, and the conic through OPQRS meets A A' again in O'
and the conies again in F, Q\ F, S': prove that O'F, O'Q, O'R, O'S'
will be the tangents to the two conies at F^ Q, F, S', and that the
conic OPQRS will pass through the other four points of intersection,
of the four common tangents.
1514. A tangent OP is drawn from a given point to a conic
inscribed in a given quadrilateral of which AA\ BF, CC are diagonals,
and a straight line drawn through P which with PO divides AA'
harmonically : prove that the envelope of this line is also the envelope
of the polar of and is a conic which touches the three diagonals.
Also, if OP, OF be the two tangents from the conic through OAA'PF
will pass through a fourth fixed point.
1515. A conic is inscribed in a given quadrilateral and from two
given points on one of the diagonals tangents are drawn : prove tliat
their points of intersection lie on a fixed conic which passes through the
ends of the other two diagonals and divides harmonically the segment
terminated by the two given points ; also if tangents be drawn to the
CONIC SECTIONS, ANALYTICAL. 255
former conic at points where the second conic meets it four of their
points of intersection will lie on a conic which passes through the points
of contact of the given quadrilateral and through the ends of the given
diagonal.
1516. A conic U is inscribed in a given quadrilateral and another
conic V is drawn through the ends of two of the diagonals : j)rove that
the tangents to C^ at the points where it meets V pass through the
points of intersection of V with the third diagonal ; and the points of
contact with V of the common tangents to CT, Tlie on the tangents to U
at the points where it meets the third diagonal.
1517. A conic is drawn through four given points : prove <^at the
envelope of the straight line joining the points where this conic again
meets two fixed straight lines through one of the points is a conic which
touches the two fixed straight lines and the straight lines joining two
and two the other three given points.
1518. Find the locus of a point such that one double ray of the
involution determined by the tangents from the point to two given
conies may pass through a fixed point ; and prove that the other double
ray will envelope a conic, which touches the diagonals of the quadrilateral
formed by the common tangents to the two given conies.
1519. Three conies U, K, W have two and two double contact, not
at the same points : prove that the chords of contact of F, W with U will
pass through the intersection of the common tangents to F, W and
form with the common tangents an harmonic penciL
1520. Two ellipses have the same (impossible) asymptotes: prove
that any ellipse which has double contact with both will touch them so
that the chords of contact will lie along conjugate diameters.
1521. A point Q is taken on the directrix of a parabola whose
focus is ^, a circle is described whose centre lies on SQ produced and
whose radius is a mean proportional between OQ^ OS : prove that the
points of contact with the circle of the common tangents lie on the
tangents drawn from Q to the parabola.
1522. Two chords OP, OQ of a given conic are at right angles,
another conic is described with a focus at and PQ has the same pole
with respect to the two conies : prove that tangents to the second oonio
at points where it meets the first pass through P or Q,
1523. Two fixed points A, B are taken on a given conic and
another fixed point in the plane : a chord PQ of the conic such that
[PABQ] is harmonic will have for its envelope a conic touching OA,
OB and the tangents to the given conic Q,t A, B, Also if PQ meet AB
in R and R be taken in PQ so that {PRR'Q] is harmonic, the locus of
i^ is a conic through A, B and having double contact with the envelope
oiPQ.
1524. Two points P, Q are taken on a given hyperbola and straight
lines drawn from P, Q each parallel to an asjrmptote meet in 0; a
256 CONIC SECTIONS, ANALYTICAL.
parabola is drawn touching the sides of the triangle OPQ : prove that
the tangents to the parabola at points where it meets the hyperbola pass
through the two points where the hyperbola is met by a straight line
through parallel to PQ,
1525. A point L is taken on the directrix of a parabola whose
focus is S, and a circle is drawn such that the radical axis of the circle
and S is Uie straight line through L at right angles to LS : prove that
the points of contact with the circle of common tangents to it and the
parabola lie on the tangents drawn to the parabola from L ; and that
the tangents to the parabola at their common points pass through
the points on the circle where the straight line through S at right angles
to SL meets it.
1526. Through a fixed point are drawn two chords PP% QQ^ of
a given conic such that the two bisectors of the angles at are fixed :
prove that the straight lines PQ, P'Q, PQ\ P^Q^ all touch a fixed conic
which degenerates when the two bisectors are conjugate with respect to
the given conic.
1527. The equation of the polar reciprocal of the e volute of the
ellipse ay + 5 V = aV with respect to the centre is
a* V (a'-hy
1528. Two fixed points 0, 0* are taken, and on the side BC of a
triangle ^^(7 is taken a point A' such that the pencil A' {A OCX B} is
harmonic; B\ C are similarly determined on the other sides : prove ^at
AA', BB*, C& meet in a point, that the four such points correspond-
ing to the four triangles formed by any four straight lines are collmear,
and that tangents drawn from any point on this line, to the conic which
touches the four straight lines and 00\ will divide 00' harmonically.
1529. Two conies touch at 0, and any straight line through O
meets them in P, Q ; prove that the tangents at P, Q intersect in a
point lying on the chord of intersection of the two conies.
1530. Four tangents a, 5, e, d are drawn to a conic, and the
straight line joining the points of contact of 6, c meets a, d m A, D \
prove that a conic drawn touching a, 6, e, (i so that J is its point
of contact with a will also have D for its point of contact with d,
1531. Two conies U, V intersect in A, B, C, 2), and the pole of
AB with respect to (7 is the pole of CD with respect to V: prove that
the pole of CD with respect to £^ is the pole of AB with respect to V.
1532. A conic is drawn through four given points : prove that its
asymptotes meet the conic which is the locus of centres of all conies
through the four points in two points at the ends of a diameter.
1533. Four points and one straight line being given, four conies
are described such that with respect to any one of them three of the
CONIC SECTIONS, ANALYTICAL. 257
points are comers of a self-conjugate triangle, and the fourth is the pole
of the given straight line : prove that these four conies will meet the
given straight line in the same two points which are points of contact
of the two conies through the four points touching the line. Also prove
that any conic through the four points will divide the segment between
the two common points harmonically.
1534. Four straight lines and a point being given, four conies are
described such that with respect to any one of them three of the straight
lines form a self-conjugate triangle and the fourth is the polar of the
given point : prove that these four conies will have two common
tangents from the given point, and these tangents are tangents to the
two conies through the given point touching the given lines. Also
prove that tangents from the given point to any conic touching
the given lines form a harmonic pencil with the two common tangents.
1535. Two fixed tangents CA^ CB are drawn to a given conic :
prove that the envelope of the straight line, joining any point in AB to
the point in which its polar meets a fixed straight line, is a conic
touching the sides of the triangle ABC and the fixed straight line.
1536. The sides of the triangle ABC are met by a transversal in
A\ B^ C \ the straight line joining A to the point (BB^ CC) meets
BC in a, and 6, c are similarly determined : four conies are drawn
touching the sides of the triangle ABC^ and meeting the transversal in
the same two points : prove that the other common chord of any two of
these conies passes through either a, 6, or c ; that these six common
chords intersect by threes in four points ; and that these four points
are the poles of the transversal witii respect to the four conies which
touch the sides of the triangle ahc and pass through the before-men-
tioned two points on the transversaL
1537. Two conies Z7, V have double contact, and from a fixed
point on their chord of contact are drawn tangents OP, OB ; OQ^ OQ' ;
another conic W is drawn through Q, Q^ to toiich OP^ OB : prove that
the tangents to TT at the points where it meets U will touch either V or
another fixed conic V which has double contact with both U and F.
The pole of QQ^ with respect to W lies either on the chord of contact of
U and F or on a fixed straight line through dividing PB^ and QQ\
harmonically to the chord of contact of U and F : if the former, the
tangents to IT at the points where it meets U touch V; if the latter,
they touch F, which touches V At Q, (/ and U at the points where the
fixed straight line before mentioned meets it.
1538. The four conies which can pass through three given points
and touch two given straight lines are drawn, and their remaining pair
of common tangents drawn to every two : prove that the six points of
intersection will lie by threes on four straight lines, and that the
diagonals of the quadrilateral formed by these four lines pass one
through each of the three given points.
1539. Two conies intersect in 0, A, B, C ; through is drawn a
straight line to meet the airves again in two points : prove that the
w. P. VI
260 CONIC SECTIONS, ANALYTICAL,
such that quadrilaterals can be circumscribed to Uy of which the ends of
two diagonals lie on U'j U' ia the locus of the intersection of tangents
to U which divide harmonically a certain chord of U' lying on the third
diagonal of any of the quadrilaterals, and U is the envelope of a chord
of U^ which is divided harmonically by two tangents to U drawn
from a point which is the pole of the ihird diagonal with respect to
either conia
Another method of investigating such invariant relations is as
follows : let U, U' he two conies such that triangles can be circum-
scribed to U whose angular points lie on U\ then generally if any-
tangent to U meet U' in P, Q, the second tangents drawn from P, Q
will intersect in a point J? on U', Hence, if we take F at one of the
oonmion points of (T, U'; Q, R must coincide ; or, if P be a common
point of If and U\ and PQ the tangent to ^ at P be a chord of U\ the
tangent to U' vA, Q will also touch U. We may therefore, by properly
choosing a triangle of reference, write
and thus A = 1, ®=2/, 0' =/*, A' = g', and the obvious invariant
relation is 0* «= 40' A. Of course a common tangent might be used in-
stead of a common point but would give us exactly the same result.
So, when U, U' are such that triangles can be circumscribed to U
which are self-conju^te to V\\i any tangent be drawn to U^ and we
take its pole with respect to U\ the two tangents drawn to V from this
pole will be conjugate with respect to TJ\ By considering this tangent
to ^ to be a common tangent we may see that by a proper choice of the
triangle of reference
Z7= y* + 2yz + 5?" + 2gzx^ U' =a? + 2fy%,
giving A = 5f*, = 0, 0' = 2/) A'=/*, or the invariant relation is
= 0. When triangles can be inscribed in U' which are self-conjugate
to Uy we get in the same way, by considering a common point,
U=-Qi?-^ 2yz, [7' = y* + 2gzx + 2hxy,
so that A=l, = 0, 0^=2^^, A' = ^, or the relation is again
= 0.]
1541. Denoting two conies by U, U\ the locus of points from
which tangents drawn to the two form a harmonic pencil by P, the
envelope of straight lines divided harmonically by the two by P', the
polar reciprocal of U with respect to U' by F, and that of C^' with respect
to C by r, the discribiinant of P being AA' (00' - AA'), and (00' - AA')*
that of P': prove that
F = 0^'-P=P'-0'^, r = 0'^-P = 0i7'-P'.
1542. Prove that, when iJand U' are circles, their centres are the
foci of P', and the excentricity of P' is
y
a' + 5* + 2ah cos a
a* + 6' - 2ab cos a '
where a, h are the radii and a the angle at which U and U' intersect.
CONIC SECnoXS, ANALYTICAL. 2G1
1543. A triangle ABC is inscribed in Z7 so that A is the pole of
BC with respect to i7' : prove that BG envelopes F, and ABy AC en-
velope F\ Also if a triangle ABC circumscribe (7, and A be the
pole of BC with respect to U\ the locus of ^1 is P, and that of B and
C is F.
1544. A triangle ABC is self-conjugate to Z7, and B^ C lie on U' ;
prove that the locus of A is ^U- A6^' = 0, the envelope of BC is F'
and that of iljB and AC is F.
1545. A triangle ABC is self-conjugate to 27, and its sides AB^
AC touch ^' : prove that the envelope of BC is
0"^-.0'i?" + AA'2r = O;
the locus oiAiAF, and that of ^, C7 is T'.
1546. A triangle is inscribed in U^ and two of its sides touch U' :
prove that the envelope of the third side is
(0^ - 40A') U + 4AA'£r' = 0.
1547. A triangle is circumscribed to 27, and two of its angular
points lie on U' : prove that the locus of the third is
(0* - 40'A)' 27+ 4A (0* - 40'A) F-^ IGA'A'tT = 0.
1548. A conic osculates U sX P and U' at F, and the tangents at
P, F meet in Q : prove that the locus of Q is
A'27«-A27'« = 0,
and the envelope of PF is
AA"tr» + A'A'CT'" + S^^'UU'F^ F".
1549. Two conies for which F and F' degenerate into two straight
lines and two points respectively (00' = AA') will have either four real
common points and no real common tangents, two real common points
and two real common tangents, or no reaJ common points and four real
common tangents. When they have four real common points, these
will be the points of contact of tangents from either of the points into
which F' degenerates, to the two conies ; and when they have four real
common tangents these are the tangellits drawn to the two conies at the
points, in which either of the straight lines into which F degenerates,
meets them.
1550. The condition 00^ = A A' is satisfied by a circle and rectangu-
lar hyperbola when one of their common chords is a diameter of tho
circle, and the other (therefore) a diameter of the hyperbola.
[When the second common chord is real, there are four real common
points and no real common tangents, and F is two impossible straight
lines, having one real common point where the two common chords
meet. When the second common chord meets the conies in unreal
points, there are two real common points and two real common tangents,
262 CONIC SECTIONS, ANALYTICAL.
F is two real straight lines forming a harmonic pencil with the two
common chords, and the two points into which F* degenerates are the
poles of the two common chords.]
1651. An hyperbola is described whose asymptotes are conjugate
diameters of a given ellipse : prove that the relation 00' = AA' is satis-
fied for the two conies : that when there are four real common points
the two points F' are two real points at oo , the poles of the common
diameters; and the two straight lines F are two impossible diameters :
when there are four real common tangents, the points of contact lie on
two diameters (the straight lines F) and the points F' are impossible.
1652. The general equation of a conic, for which the relation
00' = A A' is satisfied with the given conic h? + wiy* + nz* = 0, is
(Zlc* + my* + ns?) [^ + — + — J = 2 (px ■hqy + rz){p'x + 5^y + r'«).
1663. Prove that the equations of two conies, satisfying the rela-
tion 00' = AA', may be always reduced to the forms
a* + y*-»*=Oy a*-y" + wi5*=0 :
and reduce in this manner the two pairs of circles
(1) x* + y* = 49, JB' + y"-20a;+99 = 0;
(2) »« + y" = 16, «"+y-10aj + 16 = 0.
[(1) (6a;-49)"-24y« + 49(aj-5)' = 0, (5«- 49)" + 243^ = (a;- 5)';
(2) {(2 + t)a;-4(l4.20r+{(2-t)a;-4(l-2t)}«4.6y' = 0,
{(2 4.i)a;-4(l+2i)}"-{(2-t)a;-4(l-2i)}''+8i/ = 0.]
1664. The equations of two comes, for which the relation 00' = A A'
is satisfied, can always be put in the forms (area! co-ordinates)
and, if the two straight lines F meet the two curves in P, Q, P', Q';
Pi 9* P^9 fl'J t^® ranges {PF'Q'Q], {pp'q'q\ will be equal; and similarly the
tangents drawn to the two curves from the two points F' will form
pencils of equal ratios.
1656. If ABO be the triangle of reference in the last question, the
quadrangle formed by the points of contact of the common tangents
with either curve will have the same vertices as the quadrangle formed
by the points in which AB, AC meet U,
1656. The tangent and normal to a rectangular hyperbola at P
meet the transverse axis in T, G, and a circle is drawn with centre G and
radius GP: prove that straight lines drawn through T parallel to the
asjmptotes will pass through the points of contact of common tangents
CONIC SECTIONS, ANALYTICAL. 263
drawn to this circle and to the auxiliary circle, and the tangents drawn
to the two circles from any point on either of these strai^t lines wiU
form a harmonic pencil.
1557. The harmonic locus and envelope of the conies
7^—2pyz^ 9f=^2qyz
are respectively 7^ = 2ri/z, af = 2/ySy
where r, / are the arithmetic and geometric means between p and q.
1558. The harmonic locus and envelope of the conies
2laf+2Xxi/ + f/'=^2ax^ 2mic* + 2/u:y + y* = 2oa5,
are respectively
{^ + m-J(X- /jlY] oif + {\ + fjL)aci/-hi/''- 2ax = 0,
{l + m + KX- /*)•} as* + (X + /i) a:y + y* - 2ax « 0.
1559. Prove that when two conies have contact of the third order
the harmonic locus and envelope coincide.
[2maf -h y* = 2ax, 2n«* + y* = 2aa:, (m ^ n) af -^ y' = 2ax.]
1560. Four tangents are drawn to a circle U forming a quadri-
lateral such that the extremities of two of its diagonals lie on another
circle W: prove that if a, a be the radii, and b the distance between the
6 = a' or (6« - ay = 2a* (6* + a").
[In the former case U' is the locus of the points from which tangents
drawn to U divide harmonically the diameter of W drawn from the
centre of U, and U is the envelope of chords of -ZZ' divided harmonically
by the radical axis and by the diameter of U^ which is at right angles
to the line of centrea]
1561. Two conies will be such that quadrilaterals can be circum-
scribed to either with the ends of two diagonals on the other, if
0*= 20'A, and 0'* = 20A':
and the curves »*- y* = a*, »* + y'A2 »JSax+ 2a* = are so related.
•
1562. Prove that if a circle and rectangular hyperbola be described
80 that each passes through the centre of the other, and a parabola be
described with its focus at the centre of the hyperbola and directrix
touching the hyperbola at the centre of the circle, the three form a
harmonic system, such that, if any two be taken as U and U'^ the
covariants F, F\ F, F' all coincide with the third, and thus that an
infinite number of triangles can be inscribed in the first whose sides
touch the second and which are self-conjugate to the third, whatever be.
the order in which the three are taken.
264 CONIC SECTIONS, ANALYTICAL.
1563. A straight line is divided harmonically by two conies and
its pole with respect to either lies on the other: prove that the same
property is true for every other straight line divided harmonically; that,
for the two conies, = 0, 0^ = 0; and that the harmonic locus and
envelope coincide and form with the two a harmonic system.
1564. The conies U^, U , U^ form a harmonic system and any
triangle ABC is inscribed in U whose sides touch 6^, in a, 5, c; then
abc will be a triangle whose sides touch U^ (in A\ By C) and A'BC
will be a triangle whose sides touch U^ia A, B, C,
1565. Prove that any two conies of a harmonic system have two
real common points and two real common tangents; and, if ^, il' be
the common points, the common tangents BC, BC can be so taken that
ABy AC, A'B, A'C are the tangents to the two at ^, ^', and the third
conic of the system will touch AB, AC at B, C, and A!By A'C at By C\
1566. Prove that the equations of three conies forming a harmonic
system can be obtained in areal co-ordinates in the forms
a" = 2pyZy y' = 2qzXy «* =* 2rxt/y
where pqr + 1 = 0, the triangle of reference being either of two triangles.
[By using multiples of areal co-ordinates, the equation may be
written in the more symmetrical form
aj'+2y« = 0, y + 2«c=0, «* + 2iry = 0.]
1567. In a harmonic system
sif + 2yz^0, y' + 2«aj = 0, «» + 2a5y = 0, (1, 2, 3)
a triangle ABC is taken whose sides touch (2) in the points a, 6, c and
angular points Ay By C lie on (3^ : prove that Aoy Bb, Cc intersect in a
point lying on (3) such that if ^ be the point (^ : — X' : X), and ^, C7 be
similarly denoted by /a, v, the point of concourse will be {^ : -k* : k)
where
3 111
T+r+-+- = 0, X + /i + v=0, 4X/iv+l = 0,
and
X u V -i Xftv
4X»+1 4fi»4.1 4v*+l 3 /iv + vX+Xfi"
Also prove that be, ca, ah touch (1) in points similarly denoted by X, /x, v.
1568. Prove that the three conies whose equations are
3{r'-y* + 4(y4.2a)*A6icy = 0, Sir* - 3^^ - Say - 8a« = 0,
form a harmonic system, which may be reduced to the standard form
by either
jr=y+4a + a:, Y=y+ ia-x, Z^i(y + a),
or by X=3y + 4a + 3x, r=3y + 4a-3a?, £=4a.
CONIC SECTIONS, ANALYTICAL. 265
1569. A circle and rectangular hyperbola are such that the centre
of either lies on the other, and the angles at which they intersect (in
real points) are 0, V : prove that
(2 (cos S)^ + 1)(2 (cos &)^ + 1) = 3,
and the squares of their latera recta are as (1 + 8 008*0)1 : 8 sin' coe
(the same ratio as (1 + 8 cos*^)^ : 8 sin* V cos V).
1570. A circle and parabola are such that the focus of the parabola
lies on the circle and the directrix of the parabola passes through the
centre of the circle, and the two intersect in two real points at angles
0, & : prove that
(2 (cos tf)J + 1) (2 (cos ^)« + 1) = 3,
and that the latus rectum of the parabola is to the diameter of the
circle as
Ssin'^costf : (l4-8cos*tf)i
1571. A parabola and rectangular hyperbola are such that the
focus of the parabola is the centre of the hyperbola and the directrix of
the parabola touches the hyperbola, and they intersect in two real points
at angles 2^, W : prove that
(2 (sin fl)* + 1)(2 (sin ^* + 1) = 3,
and that the squares of their latera recta are as
8co8»fl8infl:(l + 8sin*tf)*.
THEORY OF EQUATIONS.
1572. The product of two unequal roots of the equation
aaf + baf + cx-hd^O
a-c
is 1 : prove that the third root is 7 — -^
1573. The roots of the equation a? —px + g' = 0, when real, are the
limits of the infinite continued fractions
^ ^ ^ ^ ^
p — p — p — ,,, p — p "-
Explain these results when p^ < 4^.
1574. Prove that, when the equation 7? -pa? -^-qx—r^O has two
equal roots, the third root must satisfy either of the equations
' 05(35 -^)* = 4r, (cc-jE?) (3a;+p) + 45' = 0.
1575. Find the relation between p, q, r in order that the roots of
the equation af — paf -{- qx — r = may be (1) the tangents, (2) the
cosines, (3) the sines, of the angles of a triangle.
[The results are
(l)jt? = r, (2)y-.2(7 + 2r=l, (3) ;>*-4pV + 8i>r + 4*^ = 0.]
1576. Prove that the roots of the equations
(1) aj*-5aj"+6a;-l=0; (2) aj» - 6aj« + lOaj - 4 = ;
(3) a*-7a»+15«*-10a;+l=0;
(4) ai'-llaj* + 45«*-84aJ» + 70x'-42a;+ll=0;
are
(1) 4cOS*y, 4c08*y, 4 COS* y j
(2)4cos*g, 4cos*-^, 4cos'-^;
(3) 4cos*^, 4cos*^, 4cos*y, 4cos*-^;
(4) 4 cos* ~, 4cos»Y^, ^^^*i3-
THEORY OF EQUATIONS 2G7
1577. Determine the relation between q and r necessary in order
that the equation o^ — ^x + r = may be put into the form
(ic* + nw + n)* = »* ;
and solve in this manner the equation
8aJ* - 36a; + 27 = 0.
1578. Find the condition necessary in order that the equation
aaj" + 6«*+ca; + ci=0
may be put under the form (af-^px-h qY = x* ; and solve in this manner
the equation aJ* + 3x' + 4a5 + 4 = 0.
[The condition is (f -^ ibcd -\- Sad* = ; and the proposed equation
may be written («* + 2a5 + 4)' = a?*.]
1579. Prove that, if the roots of the equation af - paif + ga - r =
are in H.P., those of the equation
{/>' (1 - «) + n* {pq - nr)} «* - {p^-^^npq + 3nV)a* + {pq - 3nr) aj-r =»0
are also in h.p.
1580. Reduce the equation a^-joa^ + g«- r = to the form
y* lAi 3y + m = by assuming a; E ay + 5 ; and solve this equation by
assuming y^z^- , Hence prove the condition for equal roots to be
4 (i>" - 3^)* = (2y - 9/>y + 27r)*.
1581. Prove that the roots of the auxiliary quadratic, used in
solving a cubic equation by Cardan's (or Tartaglia's) rule, are
(2a-/3-y)(2i3-y~a)(2y-a-i3)*37-308^y)(y-a)(a./3) .
54 '
where ol, P^y &^ ^^ roots of the cubia
1582. Prove that any cubic equation in x can be reduced to the
form (ay + 5)' = cy* by putting x^y-^z^ and the roots of the quadratic
for 2; will be
»08-Y)' + /3(Y-»)' + y(»-i8)'*7^(i8-Y)(Y-<')(»-/3)
C8-y)'+(y-»r+(«-i3)'
1583. Prove that, if the cubic (p , />», />-, P^^i 1)* = be put in
the form A (« + a)* + i? (« + /3)" = 0, o, p will be the roots of the quad-
ratic
(Pi* -/>«JPJ «" - iPjP,-P.P^ « + (p.* -PJ>^ = ;
and thence deduce the condition for equal roots.
J The true condition for two equal roots is given by making this
Iratic have equal roots; yet, if a = j3, the equation reduces to
(A + B) {x-\- a)' =» 0. The student should explain this result]
268 THEORY OF EQUATIONS.
1584. A cubic equation is solved by putting it in the form
{x •\- pY = z {x -k- qY : prove that the roots of the quadratic for z are
/a + 4a>* + yii)\" /o + ^cd + vioV , ^ . . - .,
( ^p ^ , ) , ( _^f. t' ) f where a, ^, y are the roots of the
cubic, and cd an impossible cube root of 1.
Solve the equation a:* + 9«*-33aj + 27 = in this manner.
[5(aj-l)'=4(«-2)".]
1585. Fix)ve that the equation
(«-a)(«-5)(«-c)-/V(a;-a)-/»«(a:-5)-AV(a;-c) + 2>&Aaj» = 0,
when a, b, c are all of the same sign, will have two equal roots only when
of _ hg ch
f-gh" g-hf h-fg'
[The equation may be reduced to the form
gj^ ¥ /9_
X a X P X y
where o, )8, y are the three ttj-t i ^.]
1586. The equation aj* - 4ic" + 5iB* - 3 = can be solved as follows :
(aV5«*-3)« = 16a;*, therefore (a;*-3a:»+5)« = 16, or a:*-3a:» + 5i4 = 0:
prove that the equation a;* — ^aaf + (a* + 1) a;* = a* — 1 can be solved in
the same way; solve it, and select the roots which belong to the original
equation.
1587. Prove that the equation
a^ + (a + 5 + c) 05* + 2 (6c + ca + oft - a* - 6* - c*) a; - Mbc =
has all its roots real for all real values of a, &, c^ and that the roots
are separated by the three
I he ^ ca , a&
a-6-c , b-c-a'-T 9 c-a-6 .
a c
[If these three expressions be denoted by oi, j3, y, the equation may
be written
- hV c'a^ aV T
x-a x — p x — y •*
1588. Investigate whether the general cubic equation can be
reduced by assuming it to coincide with either of the forms
(1) (2aJ» + (a + a')aj + 6 + 6y = (2«' + (a.a')a; + (6-6y;
(2) (a?* + aaj + 6 + c)* = (a;'-aa; + 6-c)'.
THEORY OF EQUATIONS. 209
1589. Froye that, if a, ^, y, 8 be the roots of the equation
05* + 5'aj* + raj + # = 0,
the roots of the equation
«^a;* + (7«(l -«)V + r (1 -«)»« + (1 -«)*=
will be jS + y + 8 + ^-^, &c.
1590. Prove that the equation x* - 2a^ + m (2x — 1) = has two real
and two impossible roots, for all real finite values of m, except when
m- 1.
[The equation may be written
(a:' - a? + 2)* = (2z + 1 ) a:* - 2 (m + 2) 05 + m + s*,
and the dexter is a square when 22* = m (m— 1).]
1591. Prove that the equation x* + 2pa^ -h 2rx -h rp = has in
general two real and two impossible roots : the only exception being
when three roots are equal.
1592. Prove that the roots of the equation
»■ - 6x* + 9x- 4 sin* a =
are all real and positive, and that the difference between the greatest
and least lies between 3 and 2 ^^3.
[/(O) is negative, /(I) positive, ^(3) negative, and /(^) positiva
The actual roots are readily found, by putting 05 = 4 sin' ^, to be
4 sin* — s— and 4 sin* ^ , and a may be supposed to lie between
and ^.]
1 593. In the equation x* -piS^ + p^ — ft^ + ^^4 = 0> prove that the
sum of two of the roots will be equal to the sum of the other two, if
^Pz "" ^PiPa "*"/'i* = ^ i and the product of two equal to the product of the
other two, i^ p'p^=p/»
1594. The roots of a biquadratic are a, j3, y, S, and it is solved by
putting it in the form
(aj* + ox + 6)* = (ca - (f)* ;
prove that the values of 2b are
/3y + a8, yo + )88, o)8 + y8;
those of ^ 2c are
j8 + y-a-8, y + a-/S-S, a + /3-y-8;
and those of ^ 2d are
Py — aS, ya - )88, afi — y8.
270 THEORY OP EQUATIONS.
1595. A biquadratic in x may be solved by putting x = my -^ n
and maViTig the equation in y reciprocal : prove that the three values of
n are
)8y — aS ya — )88 a^ — yS
)8 + y-a-8' y + a-)8-8' a + jS-y-S'
and those of m' are
(a-y)(a-8)(i8-y)(i8-S)
(a + i8-y-8)« ''^''•
1596. Prove that the equation 3a* + Sa:* - 6a;* - 24a; + r = will
have four real roots, if r<— 8> — 13; two real roots if r> — 8<19;
and no real roots, if r > 19.
1597. Prove that, if ^ be the n^ convergent to the infinite con-
tinued fraction
a—' a — a — , ,
jB^** - qjc +JE>, will be divisible by as* — aa; + 1, and conversely.
1598. Prove that, if '^ be the w*** convergent (unreduced) to the
infinite continued fraction
b_ b_b_
<t— a— a— f
as"** - qjc +p^ will be divisible by a:* - oa; + 6, and the quotient will be
1599. Prove that, if — " be the w*^ <5onvergent to the smaller root
of the equation as - oos + 5, which has real roots, the convergents to the
other root will be
1 9l 92 S'-l Pi P, Pn
1600. The n roots of the equation
(a-5)(a-c)^'' (b--c){h-ay'' (c- a) (c -6) ""^ '
ilifierent from o^ &, c are given by the equation
a' + J7,a:-' + J7,«-* + ... + jy = 0,
where J7, is the sum of the homogeneous products of powers of a, &, e of
p dimensions.
THEORY OF EQUATIONS. 271
1601. The n roots of the equation
different from a,, a,, ... a^, the roots of /{a;) = 0, are given by the
equation
or + J7,a:""* + ffjKr"+ ... + isr = 0,
where J7, is the sum of the homogeneous products of powers of a^^ a,...a^
of p dimensions.
1602. Prove that, if
(1 +05 +»* + ... +af'')* = a^-^a^x + a^+
and S^ = a^ + a^^., + a^^,„ + . . . , where r may have any of the p values
0, 1, 2, ... /> - 1, then of the;? quantities S^, S^, S^, ... S^^^^p^ 1 are
equal to each other, and differ from the |>*^ by 1.
[If n = 0(mod.;>), ^, = ^.- ... =^,., = ^,- (- 1)-;
andif w = r(modp), /S, = iSr,= ... =iy,., = /S;.,-(- 1)".]
1603. Prove that the equation os^ - ra:""' + # = will have two
equal roots if
1604. Prove that, if (x) have two roots equal to a, and the corre-
sponding partial fractions in \.; ! be -. c, + 7 r ,
^ f{x) {x-ay (x-a)
24>{a) ^_ 2 3<^-(a)/-(a)-^ («)/-- (g)
1605. The coefficients a^ a^, «,...a, con be so deteimined as to make
the expression
a, (x*'*«+l)-.a,a;(«*+l) + a^ («*•-•+ l)... + (-l)-a.ar(jr«+l)
equal to («-!)*" {(^* + 1) («* + 1) + 2na?}.
[The necessary value of a^^^ is ^ — =-^ t»^i^A
1606. Prove that^ if x^, x^,,.,x^ be determined by the n simple
equations
x^ - 2''a?, + S'^a?, « ... + (- l)-»w*'a;, = (- 1)-' (n+ 1)*',
r having successively the values 1, 2,... ti^
«-2« + 9 « _ (2n + 2)(2n+l) ig!LL?
272 THEORY OF EQUATIONa
1607. Prove the identity (for integral values of n)
2-(a;-l)--(n-l)2--«(«-l)--«(x+l)«+^^^-4^^^
if
the number of terms being n in the dexter, and — ^~- or ^ + 1 in the
sinister.
1608. Prove that the expression
is unchanged, or changed in sign only, if 4 -a; be substituted for x; and
deduce the identity
n [2 w(n-l)
, (2yt-r-f l)(2n-r)...(2n~2r.f 2)
" 7i(w- l)...(w-r + 1) ^'
[The roots of the expression are 4cos'^, 4cos'^,... where a^, a,,...
^- ^ . sin (w + 1) tf ^ T
the roots of --—n ^^-J
are «-.. *w^v» ^* . ^
sin^
1609. Prove that the roots of the equations
(1) ;^-(2n-l)<^-'*(^^^-I^><^-'
If
(2n-3)(2n-4)(2n-5) _
are resj)ectively
(1) 4co8\ ^, 4cos'^r ^,...4cos';7 r-;
^ ' 271+1 2n+l 27i+ 1 '
(2) 4sm'-, 4sm" — ,...4sm"-^ — ' .
n' n ' 2n
1610. Prove that, if a, o,,...a^ be the roots (all unequal) of /(ar),
and the coefficient of a^ in/ (a;) be 1,
THEORY OF EQUATIONS. 273
will be equal to the sum of the homogeneous prodacts of r dimensionB
of powers of the n quantities a^ a^... a^. Prove also that
- 1 1 1/11 IV
1611. Prove that the equation in x
a, a. a
-^ + — *- + ...+ •
will be an identical equation if
S(a) = 0, S(a6) = 0, S(a6«) = 0,...S(a6->) = 0;
but that these conditions are equivalent to S (a) = 0, J^ = J, = . . . = 6,.
1612. If four quantities a> ft, c, c2 be such that
^j + oc? + ca + W + oft + erf = 0,
^^^ i :>+" 0+ iL . j =0;
6c + aa ca + bd ab + cd
1111
while a + J + c + rf, and- + 7- +- + -5are real and finite, two of the four
abed
will be real and two impossible.
1613. Prove that, if all the roots of the cubic of - Zpaf + 3 ga;-r«
be real, the difference between any two roots cannot exceed 2y3( p*— g ),
and the difference between the greatest and least must exceed 3 ,y^-g.
Also, if ^ be the mean root,
^>-r ^5-^ and <^ ^, ^ .
1614. Prove that the sum of the ninth powers of the roots of the
equation a^ + 3x + 9 = 0is0.
1615. The system of equations of which the type is
a/ajj + ajx^ + ... + ajx^ = (f
is true for integral values of r from r=l tor = n + l: prove that they
are true for all values of r.
1616. Having given the two equations
coBfia+|)jCOs(n-l)a+/7,oos(n — 2)a+ ... -♦•/>. = 0,
mnna+;?jBin(n-l)a+/?,sin(w-2)a+ ...p^_^Bina = 0;
prove that
1 +;?j0osa+j9,cos2a+ ... +|>,oosno=0,
and jp,Bina+;7,8in2a-»-... +;?^smna = 0.
W. P. "^"^
274 THEOBT OF EQUATIONS.
1617. The Bom of two roots of the equation
a;*-8«'+21a;'-20aj + 5 =
is equal to 4 : explain why, on attempting to solve the equation from the
knowledge of this fact, the method fails.
1618. The equation o^ - 209a; + 56 = has two roots whose product
is 1| determine them : also determine the roots of the equation
a* - 387aj + 285 =
whose sum is 5.
1619. Prove that all the roots of the equation
(l-:t)--«»«a!(l-«r-+!l%iy^ii^^«'(l-a;)— -...=0,
the number of terms being m + 1, are all real, and that none lie beyond
the limits 0, 1 ; m, n being whole numbers and m > n.
1620. Find the sum of the ri^ powers of the roots of the equation
0^ — a;' + l=0; and form the equation whose roots are the squares of
the differences of the roots of the proposed equation.
[If /S^^ denote the simi of the r*** powers, >S'^_i = 0, ^S'^ =■ 4 cos -«- ; and
the required equation is
(«* + 4a; + 3)(a« + 2a;V3 + 4)(a^ + 7-4V3) = 0.]
1621. The sum of the r^ powers of the roots of the equation
is denoted by «^, and 4^^=* +«, + «,+ ... + «^; prove that, if S^ have a
finite limit when m is indennitely increased, that limit is
;?, + 2p, + 3;>,4-...-hny,
1+iPi +;>• + •••+/'•
1622. The roots of the equation
^"^ «> A y> 8,... : prove that
2(2a-/9-y)(2/9-y-a)(2y-a-i8)
= (n-l)(n-2)p/-3n(n-2);>ji, + 3n>,;
and determine what symmetrical functions of the differences of the roots
are equal to
(1) (n - 2) (n - 3)^/ _ 2 (« - l)(n - 3)^.^ + 2« (» - \)p^
(2) -(n-l)(n-2)(n-3);>/ + 4n(»-2)(n-3)j9>,
- 8n» (n - 3) p^, + 8n'/»«.
THEOBT OF EQUATIONS.
m
1623. The roots of the equation
«* -;?jX* +pj3if 'P^ +;>«« -ft =
exceed those of the equation
iC* - g^a* + g,«" - 7^ + y ^aj - 7, =
respectively, each by the same quantity : prove that
2ft"-5ft=27,'-57.,
^," - 15/1 J?, + 25/1, = 47,» - 157,7, + 257„
3p/ " 8/> A + 20/>, = 37/ - 87^7. + 2O7,,
^P^P^ - 3/>,p/ - 50/l,ft + 5/?,/>, + 25()p,
= 87/7. - 37,7,- - 507,7, + 57.7. + 25O7,.
1624. Prove that if the n roots of an algebraical equation be
«h A y> 8, ,..,
62(a-i8)(a-y)(a-8)«(n-3)S(2a-.i8-y)(2j8-y-a)(2y-a-i8).
1625. Prove that, if iJ. denote the sum of the homogeneous
products of r dimensions of the powers of the roots of the equation
ar+/?,ar~*+/i^"* + ...+ft = 0,
1626. Two homogeneous functions of a;, y of n dimensions are
denoted by w^, v^\ prove that the equation found by eliminating y
between the two equations u^ a a, 1;, = 6, will be a rational equation of
the n*^ degree in a;".
1 627. Prove that
(1)
— TiOy a + 6, a-^c
6 + a, - 26, 6 + c
c + a, c + 6, — 2c
= 4(6 + c)(c + a)(a-»-6);
(2)
(6 + c)", c', 6*
c', (c + a)*, a*
6', a*, (a + hY
(3) la-6-c, 2a, 2a
26, 6-c-a, 26
2c, 2c^ c-a-6
-6c
= 2(6c + ca + a6)*;
= (a+6 + c)";
(4)
6 + c
-ca
, 6, c
a,
c+ a
a, 6,
-06
a4-6
_ (6c + eg + 06)' ,
"■ (6 + c) (c + a) (a + 6) '
\$.— ^
276
(fi)
(«)
THEORY OF EQUATIONS.
a*, (c + a)*, c*
a«, &•, (a + J)'
a-n6-n^ (n + l)a, (n + l)a
(n + l)J, J-7ic-na^ (n + l)J
(n+l)c, (n + l)c, c-wa-«6
= n*(a + 6+c)"j
(7)
2a», -6(c' + a^-J'), 2c»
2«^, 26", -c(a'+y-c»)
= aJc(a' + 6* + 0'.
1628. Frore that
(1)
1, ooB08 + y), Bin'^i^Bm'^
1, 00B(y + a), Bin' i^ Bin' -y
, oos(a + /3), Bin" — 5— Bin" i-g—
s2Bm^^Bml^Bm^^{8m2fl + 28m(a + /9 + y-fl)
vL-Ji Mn 1--5 — Bin — J- /si]
-.Bm()3 + y) - Bin (y + a) -.Bm(o + i8)} ;
^ ' 1, co8(a + 0), Bin* ^-^— Bin* ^-2—
1, coB03+fl), Bin'^Bin*^
1, cos (y + fl), sin* 5^-^ sin'^-g-
= 2iin^^Bin^Bin^!~^{2sin2fl + 8in(a + /9 + y-fl)^
- sin (a + fl) -sinOS + ^-sin (y + fl)}.
1629. Prove that the determinant
I9 cobO, cos209 oo8(n-l)0
coB0« con29, cobSO, cosnd
cos 20, cos 30, cos 40, 008(71-1-1)0
cos(n-l)0, cosn0, oos(2n-2)0
and all its first 2nd, ... minors, to the n-3^, =0, if n be any integer
THSOBT OF EQUATIONS*
277
1630, Prove that the valae of the determinant
1, 1, 1, 1
1, 2, 3, n
1,3.6. ^
n(n+l)(n + 2)
1, », IV, rg
• n(n + l) 71(714-1) ...(2n-l)
18 1, and that of its firsts second, &c principal minors are
n(n + l)
' ■^*^*\a?, x, a, »4 «/'
2 '
1631. Prove that the determinant
(«-«i)', «/, <, <
«i"» «/i «.*, («-«J"
where « = ajj + a, + a:, + a^.
1632. The determinant of the (w + 1)«* order
«i» «,» «»>
a , z
« I «>
a.
«, a^ a.
«, ttj, a„
««-i» ««
(•I- 1)*
18 eqnal to (--l)~^a,"*» (1 -«;*>) (1- «/*»), (1 - «."*") ; where
^ii ^ti ••• ^« ^^^ ^^ roots of the equation
a^sB" + a^.jO*"* + +aja5 + »=0.
1633. Prove that the determinant
1, cosa, cos(a + )8), cos(a + ^ + y)
cos()3 + y + 8), 1, cosjS, cos(/9 + y)
C08(y + S), cos(y + 8 + a), 1, cosy
cosS, C08(S + a), cos (S + a -)- ^), 1
and all its first minors, will vanish when a + /3 + y + S=8 2ir.
278
THEOBY OF EQUATIONS.
1631
Prove that, if u^ denote the determinant of the n^ order,
a, 1, 0, 0,
1, a, 1, 0,
0, 1, a, 1, 0,
0, 0, 1, a, 1, ......
0, 0, 0, 1, a, 1
0, 0, 0, 0, 1, a
w^^ — aw^-f «i,., = 0; and thence (or otherwise) obtain its value in one
of the three equivalent forms
(1) a'-(n-l)a'-»->- ^"-^^<"-^) a-«- ,
(2) sin (n + 1) a -^ sin a^ where 2 cos a = a,
(3) (p"** -/"**) -5- (p-2')> where;?, ^ are the roots of a^ - ox + 1 =0.
1636. Prove that
0)
(2)
1-n, 1, 1,
1, 1-n, 1,
1, 1, 1-n,
1
1
1
1, 1, 1, 1-n
X, 1, 1, 1, 1
1, «, 1, h 1
1, 1, aJ, 1, 1
= 0, n being the order of the determinant;
(3)
(4)
1, 1, ^
«i, h 1, 1
1, «,i h 1
1, 1, i»,i 1
= (aj-.l)"->(a: + n-l);
=A-;>-.+ 2;>...--3/>..,+ ... + (-!)-» (n- 1),
where x^, «,, ... x^ are roots of the equation
1, 1, 1, ^n
jc*, a^, X f a^, X
of, ai*, x", X, 7f
Jar, OS, a?,
a;
,»-!
(-1) « a;^(«"-l)-».
THEORY OF EQIJATION&
279
1636. Prove that
CO80, 00820, CO830, cosn0
COS20, 00830, 00840, ... 008710, 0080
00830, 00840, ... 006910, 0060, OOS20
008 n0, 008 09 008 20, ,.\ oos (n - 1)
{co80-oos(n'fl)0}'-(l-oosn0)*
2(-l) • (l-oosnfl)
1637. Prove that
a, a, a, a, 1
^, ^, /S*. A 1
/. /. /, y, 1
8*1 8', 8*, 8, 1
*', «•, «•, «, 1
-6
«. «> «» ■>
P\ P". /s-, A
y*> y*> /> y»
8*, 8*, 8*, 8,
«•, «\ «•, «.
= (a-i8)(«-y)(«-8)(.-.)09-y)(^-8) a(.-)8)'.
1638. Prove that the determinants
0, 0, 0, a, J, c =0.
0, 0, z, a, 6,
0, y, 0, a, 0, c
X, 0, 0, 0, 6, c
«, y, «, 0, 0,
1639. Prove that, if i^, denote the determinant of the n^ order,
a, 1, 0, 0, 0,
a, a, 1, 0, 0,
1, a, a, 1, 0,
0, 1, a, a, 0,
0, 0, 0, 0, 1, a, a, 1
0, 0, 0, 0, 0, 1, a, a
^••f 1 * on, + au,.i <- 1^,., = ; and express the developed determinant in
the forma
^(n,:^)^ ^
280
THEOBT OF EQUATIONS.
(2) {p"" -!»"*• -P -f*' + 9r*' + 9)-^(p-q){p + 9- 2),
where p, q are the roots of the equation a?*— (a-l)(B+l=0;
ft
n\ - 1 f 1 . / 1 ^, Bm(n + l)tf + sin (n -1-2)^ 1
^^^ 2(l+cos^)r''("^^ s^-g /*
1640.
where 2 cos fl = 1 — a,
Prove that, if u^ denote the determinant of the n^ order,
a„ 1, 0, 0, 0, 0,
o., ap 1, 0, 0,
a., «.> ap 1> 0, 0,
0, a„ »,_„ a,, a,, 1,
a_
«S> »1J 1
0, 0, 0, a, ^,.„
0, 0, 0, 0, a,, a„ a„ a,
and ihat» if fl^i> ^gi ^g , o;^ be the roots of the equation
/(a;) = ajr-a,af-' + aX''- ......+(-!)'» =0,
n+r-l
«,
ii+r-l
n+r-I
a?""^ '
""z^"
Also prove that, when a, = a,«i = ... = a, = ttj ■« 1, <*, = except when
n = or 1 (mod. r), and is then equal to (- 1) •" or (-1) »• ,
1641. Prove that
-1, l-x^j a^,(l-«a)> «^f--^»-i(l
0,
-1,
1-aj.
t • . . • •
*• ••■ *ii-i (1 "*.)»
X, ... X,
0,
0,
0,
-1,
1
is equal to 1; the second row being formed by differentiating the first
with respect to a;,, the third by differentiating the second with teepect
to x^, and bo on.
DIFFERENTIAL CALCULUa
1642. Haying given
sin a; sin (a + a;) Bin (2a + x) ... 8in{(n-l)a + a;} = 2*~*sinnaE^
where n is a whole number and na = v: prove that
(1) cot«+cot(a + a:) +cot(2a + a;)+ ... +cot{(n-l)a + «}= nooinx^
(2) cot*a;+cot*(a+«)+cot*(2a + a:) + ...+ cot* {(n- 1) a + «} = n(n - 1)
+ n'oot*wai
1643. Prove that the limit of (cob xf^*^ as x tends to zero^ is c~*.
1644. Prove that the equation (1 -a!:')^-a^+ 1 =0 is satisfied
either by
f/Jl^af^coB'^x, or by y^ac"-l =slog(« + ^«*- 1).
1645. Prove that the equation
is satisfied by any one of the four functions
and therefore by the sum of the four functions each with an arbitrary
multiplier ; and account for the apparent anomaly.
1646. Prove that, if y = cot"* a,
cfy
^ = (- 1)" [n-J Binny sin- y ;
and, if y « tan'* ( , —\ ,
' ^ \l+«coso/'
g=(-l)-1^8m«(«-y)sm-(«-y).
sm a
y -« A
1^7. Twff0W€ Hut tibie hmtdam 4rkgx-j^-2s4^3
1M& rkcmdbrt,if»teapoBthre]iile9er,&e
win Ut pMiltTe Ibr aH postiTeTiliKSof a;; aad will be postiTe
tire fur mtpdiwt rahta of a; aeeotdiDg tmnm odd or
Hit. H«iriiigghreiia^^+x^-»-3f=0, pforetfcmt
tlifti
(1 +;r0g^ + (2»+ l)«g;y+K-»')0=O.
1651. AMomingtheexiMuiaoiiof 8iii(mUa~'a;) tobe
a,a5 + <i,-x+ ... + a, 7- + ... •
* 'jz jn '
prore that
1662. AjMumisg the expansion of {log (I + x)Y to be
of a^ sir
prove that
+ (n + l)a.,.»6(-l)-|»{i + i + J+... + l};
and thenoe that
1 053. Trove that, if y = «^ (log «)',
whonr«l, »^^=ln; whenr=2, af~J!-^x^^2\n;
«•♦•
DIFFEBEKTIAL CALCULUS. 283
and generally that
[It is a singular property, but easy to prove, that the sum of the
ooemcients of the sinister is equal to the limit of the product of the two
infinite series
dixT
{[2" |3 + II" •••}r "^^'"'"^ T2""*""]3 "^ •••/•l
1654. Prove that, if y = af"Mog (1 + a:), where r is a positive
integeri
and, if n > r,
g = (-ir'!^^{«.-".„a.-..'^af-.....torter^}
(l+a)" \n n-l ^ ' [2(n-2) ^ ^
^ ' n-r+lj
1 656. Prove that, if y = (1 + «)' log x,
>=(-ir'l^^(^-r'+^ar--...tor.lterm«);
and deduce the identity
(1 + a;)' - n (1 +«)'-> + 5i^y^\l + a^^^^
= g^-(n-r)ar'4- ^'''"''^^^^''"*"^^ af-'-...tor-Kl terms, (n>r).
1656. Prove that, in the expansion of (1 + x)' log (1 + a;), the co-
efficient of -. is (- 1)'"* In Ir- 1 ; and that of , is
|r^\n n-l *** r + 1/'
n, r, and n-r being all positive integers.
1 657. Prove that the expansion of ■^— — r^-^ is
^ (1 + «)•
n ^ ' \n n+ 1/ |2
+ n(n+l)(n + 2)f-+ =-+ =) jo-...
^ ' ^ '\7i n+ 1 n + 2/ |3
284 DIFFERENTIAL CALCULUa
and deduce the identity
1 n+1 (n+l)(n+2) 1 ^ ,
-+ — r- + ^ {^ ^ — o + «" tor terms
r r-1 [2 r-2
^|_n[r\n + r n + r-1 '*' n + lj*
[When n is not a whole number, the last identily should be corrected
by writing (n+ l)(n + 2) ... {n + r) for |n -f r -r |n.]
1658. Prove that in the expansion of (1 +a;) log (1 + a;), when
n is a positive integer, the coefficient of o;^ is 0.
1659. Prove that, in the equation/(a; + h) =/(«) + A/*' (* + ^^)> ^^
limiting value of 0, when h tends to zero, is ^; and that, if ^ be
constant, /{x) = A + Bx + C«*, where A, B^ C are independent of x.
Also prove that^ if be independent of x, f{x) = il + Bx + Cm'^ where
A^ Bf Cf m are independent of x, and find tlie value of 6 when /(x) has
this form*
[The value of ei.^jl^ log {^^}.]
1660. In the equation /(x-^h) =/{x) +hf' {x-k- Ok), prove that the
first three terms of the expansion of in ascending powers of h are
hr'(x) h^ r{x)r{x)^{r\x)y
^^24/"(x) "^48 {f"{x)Y
and calculate them when /(a?) = sina;.
ri + —tta Ao ' « I provided cot a; be finite.!
*-^ 24 48sin"aj''^ ■*
1661. In the equation
/(« + A) =/(a) + hf {a) + 1/" («) + ...+ 1/" (a + flA),
the limiting value of 6, when k tends to zero, is r ; and, if /{x) be
a rational algebraical expression of n + 1 dimensions in x, the value of
$ is always r . Also prove that, if /(a?) = c"", 9 is independent of 05 ;
and that the general form of /(a?) in order that d may be independent
of ar, is
A^ + A^x + A^-h ... + Ajuf" + Bt"".
1662. Prove that, in the equation
/(«)=/(0)+«/'(0) + ^/"(0) + ... +g/"(fe).
if /(«) = (1 - a)"* where m is any positive quantiiy, the limiting value
of as a: tends to 1 will be 1 - (- j ""* .
DIFFERENTIAL CALCULUS. 285
1663. Prove that, in the equation
F{x-^h)'-F{x) ^ F' {x 4- 6h)
/{x + h)^/{x) ~ f{x + eh) '
the limiting value of when h tends to zero is |; and that when
F(x) = mix and /(x) = cos a;, the value of is always }•
1664. Prove that the expansion of (vers"* «)• is
r"*"32 "^3.53 ■*'3.5.74 '*''7'
1665. Prove that
/1\ »^-l 1 1 1 X
<^> iiim"=rTr'"iT3'"'iT6-"*--'-*^'°'
1666. Prove that the limit of the fraction
^ 2 2.4 2.4.6 , ^
2+ + 0—5+ o g >r + ••• to 71 terms
o 0,0 0.0.1
, 1 1.3 m ' 7"""'
^"^ 2"" 2:1'" 27476-' •••*^^*^"^
IT
when n is infinite, is -^ ; and the limit of the ratio of the n^ term of
the numerator to the n*^ term of the denominator is r.
[This may be deduced from the equation
i(«^ ^)=2^34^3756-'--l
1667, Prove that
^/l+^/2 + ^/3+... + >>|n»<|{(n+l)«-l},
1 + J + J+... +->log(l+n)<l + logn;
lF-i+2^i + 3p-i + ...+n'->>lnf<-{(n + l)'-l},
111 1 I/i _1_\ 1/ 1 M
Bin(n + l)a.mnna
cosa + cofl2a + ... +00Bna> — ^ 1< :
a a
p being a positive quantity, and 2na in the last < ir.
286 DIFFERENTIAL CALCULUS.
1668. Prove that i£a,h,bhe eliminated hj differentiation from the
equation aa:' + 5y' + 2hxy=ly the resultant equation will coincide with'
thiat obtained by eliminating 6 from the equations
[From the former equation may be deduced
(^)
, = a6-A",
which may be interpreted to mean that the curvature varies as the cube
of the perpendicular from the origin on the tangent.]
1 669. Prove that the general term of the expansion of sin aiy in terms
of a^ when x, y are connected by the equation y=XQO&xy^\&
(-1)" «*•+• L ,,^ ,« \. ^ (2n+2)(2n + l), -,^
2 (2^10'*^^^ ■*"^^'*'*"^)'*^^ '^ ^'(n-l)-+...
to n -)- 1 terms}.
1670. Prove that, if {a + x-k- Ja? ■^2bx-^(x?Y^^ be expanded in
ascending powers of Xj the coefficients of af , as", and a;""*"* are
respectively
(n+l)(a + 6)(;rr2a+^^6)2-«, (n+l)(a+J)2-, and 2"*»- A-|y*\
1671. Find the limiting value, when x tends to seero, of
when/(a;) has the values
(l)sinaj, (2) tan a?, (3) log (1 + a:), (4) 1 + a; - V^^fo*, (5)€'-l,
n being of course a positive integer.
[When n is odd, the values are 0, 0, 1, ^ |w-f 1, |n, 0, 0; and
when n is even, (1) (1 . 3 . 5 ... TTH)", (2) (- \f [n, (3) 1, (4) J [n+J,
(5) \n, (6) (- 1)« (1 . 3 . 5 ... n- 1)», and (7) [n.]
1672. Prove that the limiting value, when x tends to zero, of
Si;"i\8l^; cos(n + l)a:j b 2" [n (• 1)» ; n being even.
DIFFERENTIAL CALCULUa 287
1673. Prove that the limiting value of n*, when x lends to' a value
a for which both u and t; vanish^ will always be 1, if the corresponding
limiting value of - be finite ; or if the limitiTig value of ^-j-'^^-j"
be finite.
1674. Prove that the Uniting value of u*, when x tends to a
critical value a for which u = 1 and t; — oo , is c^'^"', where m is the limit
of {x^a)v, Applj this to find the limits of {—-k — )**> *^d
^, when X tends to zero.
1675. Prove that the limiting values of
,- . sin (n + 1) a: + a^ sin na; + a^_, sin (n - 1) a; + ... + a^ sin «
,^. sin (2n + 1) a; + a^ sin (2n--l)« + a, _^ sin (2n - 3) a? -t- ..,-i-a^ sin a;
W 5 —r-^
when X tends to zero, and the coefficients a^^a^y ... a have such values
that both limits are finite, are (- 1)", (- 4)" respectivelj.
1676. Prove that the limiting values of
.•X cos (n -hi) a; -I- g^ cosna;-i-a^^^co8(n- l)a;-l- ...-t-g^ cosog
/o\ 008n« + a^ cos (n - l) a;4-a ,,, cos(n-2)aj+ ... +a,cosaj + a
W ^ ',
when X tends to zero, and the constants are so determined in each case
that the limit is finite, are (- 1)" =- , ^ ' respectively.
1677. Having given
«=</;(a:+y)+€V,(«+y) + ...+€-y.(«+y);
prove that
\dx dy" )\dx''dy ) '" \dx dy^^)^
1678. Having given
prove that
(;i + g-l)(;> + 7-.2)...(;> + gr-n)« = a,
where ;/^« denotes g) ^^^.
288 BIFFEBEKTIAL CALCULUS.
1679* Having given
prove that
(y'-«')^+('«^-i»)^+(«'-«y)^-o:
and having given
tt=/{M(«-«), tt(y-«), tt(«-<)},
prove that
du ,du du 'du^^
dx dy dz dt" '
1680. Prove that
'^ doT
[More generally,
1681. The co-ordinates of a point referred to axes inclined at an
angle co are (as, ^), and u is a function of the position of the point :
prove that
1 / d^u d^u 2 cPuN 1 /d*u d'u d'u i "X
sin'cri Vc^B* i^ dxdy)^ sin* co \da:* rfy" dxdy\ )
are independent of the particular axes.
[Their values in polar co-ordinates are
(f u 1 d"tt 1 rftt 1 ^ ^ 1 du d'u 1 /(Tu 1 du\^ ,
d?^?d0''^rd^' ? dj^ d^'^rdi^ d?''?\di^e''rde)'^
1682. Having given 2aj = r (c* + €-•), 2y = r (c« - €"•), prove that
d'u d'u __ (/"m 1 d'u 1 (f w
5? V ^ Wfl^y/ ^r* df^ dO' fd^df*"? \drde r d$J '
1683. Having given » + y = X, y = XF, prove that
d^u a*u du _ y. d^u y, d^u du
^d^'^^d^j^d^'^ dX'" ^IXdY^dX'
DIFFERENTIAL CALCULUa 289
1684. Having given « + y = €*+♦, x-y^ •-♦, prove that
d^u d^u _ fnf^'^ d*u\
dx?"'!^^^" \d^''d^y'
1 685. Having given c* = r^* *, «" = r*^ •, prove that
1686. Having given
2u;»"- = t7»- + w7»-, 2y "• = u?»-" + w*-, 2«*- = tt»- + v*-
I
prove that
^d<h ,ddi ^d<h .d<l> ^d<h ^d<h
aw av dw dx ^ dy dz
tt*" -.-? + ... + ... + 2t;"ti7" ^— ? + ... + ... +71^*"*"^+ ... + ...
du' dvdvo du
zz XT z ^ + ... + ... + TUB ~"i + ... T ...
dx' dx
1687. Having given ux^ = t?y" = ws^ = Mt?u7, prove that
/ d d d\* , ( d d dy^
\^Tu^''dv^''dJ)'^-\''^^'^dy^'dz)'*'>
or generally, when x^X^^ = x^X^^ = . . . = x^X^^ = ar,«, .-.«,>
(cf rf dV / ^ d ^ d y
1688. Prove that, if x^ + x^-^ ... +aj^ = X, + X,+ ... +-3r,,
and, if
that
d<f> d<f> d4> d<l> d<f> d<f>
dx^ dx^ '" dx^ dX^ dX^ '" dX^'
^A^^X^ -^ A^^ -^ ... + 2-4^^, + ...,
"''dry^^^dx^^'-'^^^-^dx^dx^^"'
1689. Prove that, if u be a function of four independent variables
*i» *•» ^a> ^4> *^d
«j = r sin ^ sin <^, a?, = r sin ^ cos ^ a:, = r cos tf sin ^, a;^ « r oos cos ^,
d^u dSi d'u d^ud^ \d1u 1^ dSi
dx^' '^ dx/ '^ di/ '^ dxy dr^ '^ r' dO' '^ f' ^' $ d<l>'
1 d'u 3du 2 .rt>,c?u
W. P. V^
290 DIFFERENTIAL CALCULUS.
1C90. Prove that, if a;, y, « be three variables connected bj one
iiz dz d z d z d z
equation only, and p, q, r, «, t denote ^ » 5^ » ^t » ^^- » -^ ^ ^^al,
dx _\ dx ^ q d*x ^ r
Th" p* dy^^p* d^^^p^*
d'x qr — ps d'x p't — 2pqit + (fr
dj/dz p^ ' 5/~ y '
1691. The distances of any point from two fixed points are r,, r,,
and a maximum or minimum value of /(r,, rj for points lying on a
given curve is c : prove that the curve /{r^, r^) = c will touch the given
curve.
1692. In the straight line bisecting the angle il of a triangle ABC
is taken a point P : prove that the difference of the angles AFB, AFC
will be a maximum when ^P is a mean proportional between AB, AC.
[A parabola can be drawn with its focus at A touching PB^ PC at
1693. Normals are drawn to an ellipse at the ends of two conjugate
diameters : prove that a maximum distance of their common point from
the centre is (a* + ft')^-r-3 ^/3aft, provided that a*>56'; and that a
minimum distance is always (a* — h^)-i-aJ2,
1694. Prove that <^ {/(«)} is always a maximum or minimum
when y (a;) is so ; but that, if a be the maximum or minimum value of
f(x)^ ^ (a) is not a maximum or myumum value of ^ {x),
1695. The least area which can be included between two parabolas,
whose axes are parallel and at a given distance a, and which cut each other
at right angles in two points, is a* ~- .
[The included area may be proved to be a* ~ 3 sin co cos* co, where w
is the inclination of the common chord to the axes.]
1696. From a point on the e volute of an ellipse are drawn the
two normals OP, OQ (not touching the evolute at 0): prove that if
a* < 26', PQ will have its minimum value when the excentric angle of
/2a* - ft*\i
the point for which is tlie centre of curvature is tan"* f-j^ A .
1697. Prove that, if m - 1, n - 1, and wi — 7i be positive, the expres-
m+l irfi
sion (cob a + m sin a?)**"* — (cos X'\'n sin a)"-* will be a maximum when
a; = - , and a minimum when aj = - + cof' (w) 1- cot"' (n). Also, if m — n
and wn - 1 be positive, (cos a; + fw sin a;)" -r (cos a; + w sin a^)** will be a
lu^xiniiim when a; = 0, and a minimum wlien x = tan"' m + tan"' w - - .
DIFFERENTIAL CALCULUS. 291
1698. Prove that, if n be an odd integer or a fraction whose nume-
rator and denominator are odd integers, the only maximum and minimum
values of sin" oj cos n.r are determined by the equation cos(n+ l)a:= 0.
Also, with the same form of n (> 1), the maximum and minimum values
of tan rM;(cot a;)" correspond to the values 0, ir, 2ir, ... of (n - 1) aj and
V, Srr, 5ir, ... of 2 (n+ l).r, the zero value giving a maximum, and any
value of X which occurs in both series being rejected.
1699. Through each point within a parabola y*= icuc it is obvioua
that at least one minimum chord can be drawn : prove that the part
from which two minimum chords and one maximum can be drawn is
divided from the part through which only one minimum can be drawn
by the curve
(x - 5a)"i + (4a; - 6?/ + ia)"^ + (4ic + 6y + 4a)~ J = ;
and that, at any point on the parabola y* = 4a (a: — a), one minimum
chord is that passing through the focus, (a, 0).
[The curve has two rectilinear a.symptotes 3a5 ± 3 J^y + 5a = 0, and a
parabolic asymptote 27// = 32a (Sx — a), which crosses the curve when
3d;= 17a, and is thence almost coincident with the inner branches.]
1700. The maximum value of the common chord of an ellipse and
its circle of curvatui*e at any point is
{(«• + h') (2a' - h') (a* - 26') + 2 (a* - aV + 6*)*}i
3 ^3 (a« - h')
1701. A chord PQ of an ellipse is normal at P and is its pole :
prove that, when PQ is a minimum, its length will be
3^3aV-=-(a' + 6')*
and Q will be the centre of curvature at P ; and when OP is a minimum
the other common tangent to the ellipse and the circle of curvature at P
will pass through 0,
[The minimum value of PQ here given will only exist when a* > 26*,
the axes being both maximum values of PQ ; when a' < 26*, one axis is
the maximum and the other the minimum value of PQy and there are no
other maximum or minimum values.]
1702. In any closed oval curve, PQ, a chord which is normal at /*,
will have its maximum or minimum values either when Q is the centre
of curvature at P, or when PQ is normal at Q as well as at P, which
must always be the case for two positions at least of PQ,
\ -f. 2a; — re* + 2 v/a; — a?* #—
1703. Prove that the expression ;j — -^- has 1 ^J2
for its maximum and minimum values corresponding to a; =» — 1 * ^2.
1704. Two fixed points A, B are taken on a given circle, and
another given circle has its centre at B and radius greater than BA \
292 DIFFERENTIAL CALCULUa
any point P being taken on the second circle, PA meets the first circle
again in Q : prove that the maximum lengths P^Q^, ^fit ^^ ^^ ^^
equally inclined to AB and each subtends a right angle at B^ and the
minimum lengths both lie on the straight line ti^ugh A at right angles
to AB : also P^^ A, B, P^ lie on a circle which is orthogonal to the first
circla
1705. The least acute angle which the tangent at any point of an
elliptic section of a cone of revolution makes with the generating line
through the point is cos~* (cos )S sec a), where 2a is the angle of the cone
and fi the angle which the plane of the section makes with the axis.
1706. Prove that three parabolas of maximum latus rectum can be
drawn circumscribing a given triangle ; and, if a, )3, y be the angles
which the axis of any one of them makes with the sides, that
cot a + cot ^S-f cot y = 0.
1707. Prove that, if aJ + y + » = 3c, /(a;)/(y)/(«) will be a maxi-
mum or minimum when a; = ^ = 2 = c, according as
/"(c)>or<{/'(c)}'-f/"'(c).
1708. The minimum value of {Ix + my + nzf -^ (yz + zx-¥ xy) is
2mn + 2rd + 2lm —l^—rn^—n*, provided this vidue be positive ; otherwise
there is neither maximum nor minimum value.
1709. Prove that the maximum value of
sinaj^asin*y + 6cos*y + oosa:^aco8"y + 6sin*y
is J a + 5 ; and the minimum value of
J a* sin* a; + 6* cos* x + Ja^ sin* y + ft* cos* y
sin (a; 7- y)
isa+&
1710. Prove that, when x^y^ z vary, subject to the single condition
xyz (yaj + «c + xy) = «+ y + 2;, the minimum value of
(1-f yg)(l -I- ga ?) (1-1- a?y)
(i.ha:')(l-hy*)"(l + 2')
is-f.
1711. Find the plane sections of greatest and least area which can
be drawn through a given point on a given paraboloid of rovolution;
proving that, if 0^, $^ be the angles which the planes of maximum and
minimum section make with the axis,
2tan^j tantf,= 3.
1712. The maximum and minimum values oif(x, y, 2), where
Xf yy z are the distances of a point from three fixed points (all in one
plane), aro to be determined from the equations
^1_ df 1 df ^ 1 df ^
sin (y, z) dx ~ sin (5, x) dy sin («, y) dz '
(y, z) denoting the angle between the distances y, z.
DIFFERENTIAL CALCULUS. 293
1713. Prove that, If A, £, C, D he comers of a tetrahedron .and F
a point the sum of whose distances from Aj £, Cy D is sl Tniniiniifn^
FA.Pa ^ PB.Pb ^ PC. Pe PD.Pd^
Aa " Bb " Cc " Dd '
a, 6, c, d being the points in which PA^ PB, PC, PD respectivelj meet
the opposite faces. Also prove that when IPA' + mPB* + nPC* + rPB'
is a minimum,
vol . PBCn ^ vol . PC DA ^ vol . PDAS ^ vol . PABC
I in n r *
1714. The distances of any variable point from the comers of a
given tetrahedron are denoted by w, Xyy^zi prove that, wheny (w, 05, y, z)
is a maximum or minimum,
1 -a»-6'-c' + 2a6c \du) 1 - a* - 6" - c" 4- 2a6 V \dx)
I /^v \—-(^l\\
- 1 _ a'' - 6» - c" + la'hc' \dy) " 1 - a" - ft'' - c' + ^a'h'c \dz) '
a, by Cf a', b\ c' denoting the cosines of the angles between the distances
(y, «), (-1 a?), («, y)y (w, ^), («*, y), («*, «) respectively.
1715. Prove that, if be the point the sum of the squares of whose
distances from n given straight lines, or planes, is a minimum, will be
the centre of mean position of the feet of the perpendiculars from on
the given straight lines or planes.
1716. A convex polygon of a given number of sides circumscribes
a given oval, without singular points : prove that, when the perimeter of
the polygon is a minimum, the point of contact of any side is the point
of contact of the circle which touches that side and the Cwo adjacent
sides produced.
1717. In the curve y* = 3aa^ - a:*, the tangent at P meets the carve
again in Q : prove that
tan ©Ox + 2 tan P0«= 0,
being the origin. Also prove that if the tangent at P be a normal at
Q, P lies on the curve
4y (3a - a?) = (2a - a;) (16a - 5a;).
1718. Prove that any tangent to the hypocycloid a;» + y» = a», which
makes an angle \ tan"* \ with the axis of a;, is also a normal to the curva
1719. The tangent to the evolute of a parabola at a point where it
meets the parabola is also a normal to the evolute.
1720. From a point on the evolute of an ellipse a*^ + 6V = aV the
two other normals to the ellipse are drawn : prove that the straight line
joining the feet of these normals will be a normal to the ellipse
(aV + 6y)(a'-67=a*6*.
2U4 DIFFERENTIAL CALCULUS.
1721. A tangent to a given ellipse at P meets the axes in two
points, through which are drawn straight lines at right angles to the
axes meeting in p : prove that the normal at ji> to the locus of p and tho
straight line joining the centre of the ellipse to the centre of curvature
at P are equally inclined to the axes.
1722. Trace the curve T-J +(f) =1 when w is an indefinitely
large integer, (1) when n is even, (2) when n is odd.
[(1) the curve is undistinguishable from the sides of the rectangle
formed by ic* = a', ^" = 6*; (2) when »■<»', y = 6; when y"<:6', a; = a; and
when Qi?>c? and y* > 6', - + ?- = 0, or the curve coincides with two sides
of the rectangle and with the part of one diagonal which is without the
rectangle.]
1723.. Trace the curve determined by the equations
.
x=^acoBO. y=a- — >;,
^ sin^
and prove that the whole curve can only be obtained by using impossible
(pure imaginary) values of 6.
a6
[The two curves (1) a; = acos0, y = a—. — 7., (2) a; = acosh^, y= . . , ,
*- ^ ' ^ sin Q ^ ' ^ sm h^'
give the same difSsreatial equation of the first order
(1 -a:') g- ay +1=0,
and starting from the same point (a, a) when = must coincide.]
1 724. Trace the curve 4 (a:* + 2^ - ^ayf = a* (a* + 2/), proving that
the area of a loop is -y^ (2 — ^3) a% and that the area included between
8a*
the loops is «— t« (2w - 3 ^).
1725. A curve is given by the equations
_ g cos {g* + (ft* - y) cos' 0} __ tsin {6' + (6'- g') sin'^}
*~ g'cos«6^ + 6'8in-0 ' ^~ a' cos' 6 + b' Bm' $ '
prove that its arc is given by the equation
ds (g'Rin'g + 6'cos«0 )t
de" g* cos* tf + 6" sin* tf ■ •
172G. Prove that the curve whose intrinsic equation is yy=aaec2d>,
dii
if X, y, ---, and ^ vanish together, has the two rectilinear asymptotes
g
1>1FFEBENTIAL CALCULUS. 295
1727. Two contiguous points P, -P on a curve being taken, PO, F'O
are drawn at right angles to the radius vector of each point : prove that
the limiting value of PO when F' moves up to P is * — .
1728. Two fixed i)0ints S^ S' being taken, a point P moves so that
the rectangle SP^ S'P is constant : prove that straight lines drawn from
6\ S' at right angles respectively to iSP, ^'P will meet the tangent at P
in points equidistant from P,
1729. In a lemniscate of Bernoulli, the tangent at any point makes
acute angles 0, ^ with the focal distances r, /: prove that
1730. In a family of lemniscates the foci S, S' are given (SS' s 2a) :
prove that in any one in which the rectangle under the focal distances
(c^) is less than a*, the curvature is a minimum at the points of contact
of tangents drawn from the centre, and that these points all lie on a
lemniscate (of Bemovdli) of which S, S' are vertices. Also, when c* > a*,
the points of inflexion lie on another such lemniscate equal to the former
but with its axis at right angles to that of the foiiner.
[In any of these curves, if p denote the perpendicular from the
centre on the tangent, 2<fpr = r* + c* — a*, and the radius of curvature is
2cV"-r(3r* + a*-c*).
The points of maximum curvature are the vertices.]
1731. In the curve
r (7/4 + n tan - j = 1 + tan ^ ,
the locus of the extremity of the polar subtangent is a cardioid.
1732. The tangent at any point P of a certain curve meets the
tangent at a fixed point in T, and the arc OP is always equal to
7h, TP\ prove that the intrinsic equation of the ciirve is
1
5 = c (sin ^)"~* ;
and that the curve is a catenary when n = ^, the evolute of a parabola
when n = |, a four-cusped hypocycloid when n = f, and a cycloid
when « = 2.
*
1733. From a fixed point are let fall perpendiculars on the tangent
and normal at any point of a curve, and the straight line joining the feet
of the perpendiculars passes through another fixed point : prove that the
curve is one of a system of conf ocal conies.
296 DIFFERENTIAL CALCULUS.
1734. A circle is drawn to touch a cardioid and pass throngli tlie
cnsp : prove that the locus of its centre is a circle. If two such circles
be drawn, and through their second common point any straight line be
drawn, the tangents to the circles at the points where this straight line
again meets them will intersect on the cardioid.
[Of course this and many properties of the cardioid are most easily
proved by inversion from the parabola.]
1735. Two circles touch the curve r* = a" cos m ^ in the points P, Q,
and touch each other in the pole S : prove that the angle PJSQ is equal
to •= , n being a positive or negative integer.
1736. The locus of the centre of a circle touching the curve
f^ = ar cos mO and passing through the pole is the curve (2r)" = a" cos nO^
where n{l —m) = m,
1737. In the curve r=asec"^, prove that, at a point of inflexion
the radius vector makes equal angles with the prime radius and the
tangent ; and that the distance of the point of inflexion from the pole
increases from a to a ^c, as n increases from to oo . If n be negative,
there is no real point of inflexion.^
1738. A perpendicular, SFy is drawn from the pole S to the tangent
to a curve at P : prove that, when there is a cusp at P, the circle of
curvature at F to the locus of Y will pass through S ; also that, when
there is a point of inflexion at T in the locus of F, the choi'd of cur-
vature at F through S will be equal to iSP.
1739. The equation of the pedal of a curve is T—f{6) : prove that
the equation found by eliminating a from the equations
r cos a =f{0 - a), r sin a =/' (0 - o),
is that of the curva
1740. Prove that for any cubic there exists one point such that the
points of contact of tangents drawn from it to the curve lie on a circle.
If the equation of the cubic be
005* + ^g^y + 3/Icy* + 6y* + Aa? + 2Hxy + -By* + ... = 0,
ft ^ n h A /?
and if — - = ~r- = 7. , there will be a straight line such that, if
9 / -^
tangents be drawn to the cubic from any point of it, the points of
contact will lie on a circle^
1741. The asymptotes to a cuspidal cubic are given : prove that the
tangent at the cusp envelopes a curve which is the orthogonal projec-
tion of a three-cusped hypocycloid, the circle inscribed in the hypocy-
cloid being projected into the locus of the cusp.
[The locus of the cusp is the maximum ellipse inscribed in the
triangle formed by the asymptotes, and any tangent to the tricusp at a
point meets this ellipse in two points P, Q bo that OQ is bisected in
yV ^^ point corresponding to is P.]
DIFFERENTIAL CALCULUS. 297
1742. The equation of a curve of the n^ order being
^*,©*^-*.(l)*^-*.(9-..-«,
<f>^ (z) has two roots /x and <l>^ (/x) = : prove that there will be two cor-
responding rectilinear asymptotes, whose equations are
(» - f^y <*>:' M + 2 (y - fu;) 4,,' (^) + 2«. 0*) = 0.
1743. Two points /*, Q describe two curves so that correRponding
arcs are equal, and the radius vector of Q is always parallel to the
tangent at P : show how to find /*'s path when Q's is given ; and in
especial prove that when Q describes a straight line P describes a
catenary, and when Q describes a cardioid, with the cusp as pole, P
describes a two-cusped epicycloid.
1744. The rectangular co-ordinates of a point on a given curve
being {x, y), the radius of curvature at the point is p, and the angle
which the tangent makes with a fixed straight line is ^ : prove that
©■*(2)'%'{'Kt)'}.
and, in general,
where
fx — X V — Y\
1745. A curve represented by the equation fi , ) ~ ^
is drawn having contact of the second order with a given curve at a
point P : prove that, if be the point (X, Y), PO will be the tangent
at to the locus of 0,
1746. A rectangular hyperbola whose axes are parallel to the co-
ordinate axes has three-point contact with a given curve at the point
(x, y) : prove that the co-ordinates (X, Y) of the centre of the hyper-
bola are given by the equations
X-X y \(IX/
dx da?
298 DIFFERENTIAL CALCULUS.
and that the central radius to the point (x, y) is the tangent at (X, Y)
to the locus of the centre. Also, when the given curve is (1) the
parabola y* = ^clx, (2) the ellipse a*i^ + 6*ac* = a*6' ; prove that the locus
of the centre of the hyperbola is
(1) 4 (a; + 2a)» = 27a/, (2) {ax)l + {hyf = {a' ^ I/)\
1747. An ellipse is described having four-point contact with a
given ellipse at P, and with one of its equal conjugate diameters passing
through F : prove that the locus of its centre is the curve
[The curve consists of four loops, and its whole area is to that of
the ellipse as
(a - hy {(a' + a6 + hy - ba'h') : 2a"6».]
1748. The equation of the conic of closest contact which can be
described at any point of a given curve, when referred to the tangent
and normal at the point as axes, is aa? + hy* + 2Jixy = 2y, where
1 \ dp 1 ^/dp\' l^p
""-p' ^""^pdi' ^"p'^9pU/ "3(^-'
and p is the radius of curvature at the point.
1749. The sum of the squares on the semi-axes of the ellipse of
five pointic contact at any point of a curve is
the product of the semi-axes is 27p'-r 9 + f ^- j - 3p ^^ , the rectangle
under the focal distances is 9p*-r -19 + (-^j - 3p r^l , and the excen-
tricity e is given by the equation
,.-2). {■S.^©--3.g}-
l-e'
1750. A chord QQ' is drawn to a curve parallel to the tangent at
P, a neighbouring point, and the straight line bisecting the external
angle between FQy FQ' meets QQ' in : prove that the limi f.ing value
of P(2is6p-T-^.
DIFFERENTIAL CALCULUS. 299
1751. In any curve a chord PQ is drawn pai*allel and indefinitely
to the tangent at a point : prove that the straight line joining
the middle point of the chord to will make with the normal at an
angle whose limiting value is tan~* ( o 7^) • Keconcilo this result with
the fact that the se^^ments into which the choi*d is divided by the
normal at are ultimately in a ratio of equality. If the chord meet the
normal in i?, and F', Q' bo resiH^ctively the mid |)oint and the foot of
the bisector of the angle QOI\ the limiting value of the third propor-
tional to RP", RQT will be |p "^f .
6 da
1752. A chord FQ of a curve is drawn always parallel to the
tangent at a point : prove tliat the radius of curvature at of the
locus of the middle point of this chord is
.«<».£)
1753. A chord PQ of a curve is drawn parallel to the tangent at
and is met in i? by the bisector of the angle POQ : prove that the radius
(Is
of curvature at of the locus of i? is 3p -7 .
dp
1754. The normal chord PQ at any point P of a conic is equal to
18p-r ( 9 + 2 -^j —^p-]\)} i^ ^ parabola, if / be the centre of curvature
at P and the pole of PQ, PQ.PI = 2P0*, and 10 is perpendicular to
ds
the focal distance SP ; and, in all conies, PO = 3p-j-, and the angles
lOPf IPC are equal, C being the centre.
1755. Prove that the curves r = aO. r=^j: ^ have five-point
' 2 + cos^ ^
contact at the pole.
1756. The centre of curvature at a point P of a parabola is 0, OQ
is drawn at right angles to OP meeting the focal distance oi P vol Qi
prove that the radius of curvature of the evolute at is equal to
1757. All the curves represented by the equation
for different values of n, touch each other at the point v^'^V^ ~7ri) >
and the radius of cur\'ature is (a* + 6*)* 'Tn{a^ by.
300 DIFFERENTIAL CALCULUS,
1758. At each point P of a curve is drawn the equiangular spiral
of closest contact (four-point), and «, o* are corre.^ponding arcs of the
curve and of the locus of the pole S of the spiral : prove that
d8^
d<r df? pa
Ht)" 7^'
and that the tangents at P, aS' to the two curves are equally inclined to
JSP, Prove that, when the curve is a cycloid, the locus of /S' is an equal
cycloid, the image of the former with respect to the base ; and when the
OS
curve is a catenary of equal strength, y = a log sec - , PS is constant
(=r a), corresponding arcs are equal, and the curvatures at corresponding
points are as 1 : 3. In the curve whose intrinsic equation is
^=a(co8n^)-, ^=n,
and the curvatures at /S^, P are as 2n - 1 : n; when 2n= 1, the locus of
S\&2l straight line.
1759. At each point of a parabola is described the rectangular
hyperbola of four-point contact : prove that the locus of its centre is an
equal parabola, the image of the former with respect to the directrix.
1760. At each point of a given closed oval are drawn the parabola
and rectangular hyperbola of four-point contact : prove that the arc
traced out by the centre of the hyperbola exceeds twice that traced by
the focus of the parabola by the arc of tiie oval, provided no parabola
have five-point contact.
1761. At each point of a given curve is drawn the curve in which
the chord of curvature through the pole bears to the radius vector the
constant ratio 2 : n + 1, having four-point contact with the given curve,
and corresponding arcs of the given curve and of the locus of the pole of
the osculating curve are 5, o- : prove that
'(»-l)V(«.l)-(*)-
Also, if iS" be the pole corresponding to a point P on the given curve,
the tangents at P, S are equally inclined to PS^
™-<»*')'*vA^e?T^-
and the locus of S for different values of n is the locus of the foci of the
conies which have four-point contact ¥rith the given curve at P.
DIFFERENTIAL CALCULUa 301
1762. In the last question, prove that, when the given curve in
such that corresponding arcs traced out by S and F are equal, the in-
trinsic equation of the given curve is either
^ / . \ll- ^ n - 1 ,
-,T = c (cos 6) or -TT = c sec ,- d> :
cUfi ^ ^' dit> w + 1 ^
that in the former case FS is constant in direction and the locus of S is
the image of the given curve with res|>ect to a straight line, and in the
latter that FS is constant in length, [?i + lc or c], and the
curvatures of the two curves at F, S will be as l+n:3 — nor
n + 1 : 3w - 1.
1763. At each point of a given cui've are di-awn the cardioid and
lemniscate of four-i)oint contact, and the arcs traced out by the cusp and
node respectively corresponding to an aix; a of the given curve are cr, t/ :
prove that 2(r + <r' = 3^?.
1764. At each point of the curve whose equation 13
y _x_ -X
2«a = fay/3 + ^ V3 ^
are drawn the rectangular hyperbola and parabola of four-point contact :
prove that the distance from the point of osculation to the centre and
focus respectively are a, ;r , and corresponding arcs of the three curves
are equaL
1765. At each point of a cardioid is drawn the lemniscate of closest
contact, the locus of its node will be an epicycloid, whose fixed circle is
that with which the cardioid is generated as an epicycloid and whose
moving circle is twice the radius.
1766. At each point of the curve r* = a" sin n$ is drawn the curve
similar to r* = a" sin - and having four-point contact with the former
curve : prove that the locus of its pole is the curve whose intrinsic equa-
tion is
l-n
cfe _ a(n+ 1) / . n<f>
ds ^ a(rn-l) / jn4\ -
cUji w(3w+l)\ 3n+l/ *
that the radius of curvature of this curve bears to the common radius
of curvature of the osculating curves at the corresponding point the
ratio (n + 1)' : w (3n + 1). Also the area traced out by the radius vector
n + 1
to the pole is -q— - a* (w + 1 tf + sin 20),
1767. At each point of an epicycloid is drawn the equiangular
spiral of closest contact : prove that the locus of the pole of this spiral
wiU be the inverse of the epicycloid with respect to its centre; and,
302 DIFFERENTIAL CALCULUS.
conTerselj, the curves for which thi» property is tme are those whose
intrinsic equations are
« = a(l -coswu^), 2* = a(c^ + c-**-2),
measoring from a cusp in each case.
1768. A curve is such that any two corresponding points of its
evolute and an involute are at a constant distance : prove that the
straight line joining the two points is also constant in direction.
1769. The reciprocal polar of the evolute of a p>arabola with respect
to the focus is a cissoid, which will be equal to the pedal with respect
to the vertex when the radius of the auxiliary circle is one-fourth of the
latus rectum.
1770. In any epicycloid or hypocycloid the radius of curvature is
proportional to the perpendicular on the tangent from the centre of the
fixed circle.
1771. The co-ordinates of a point of a curve, referred to the
tangent and normal at a . neighbouriug point as axes of co-ordi-
nates, are
y - 2p- Q?dr \i/V-^ds\ ''di^)^ -
where p, -^ t -5-? t ... are the values of the radius of curvature and its
differential coefficients at the origin, and s is the arc measured from the
origin.
1772. Prove that, if the tangents at two points P, Q meet in 0, the
limiting value of opTOQ - chord I^Q '' ^' '^^* ""^ OF -^ OQ '' 3p ds ^
1 xi X J. 1 1 2 COS A . 4 /^ dp\'\ . , . ,,
and that of ^^+ ^- ^^-^-^ « ^^.(^9 + ^| j, ^ bemg the angle
between the tangents.
1773. Prove that, in the curve whose intrinsic equation is
^^ = a (1 + m€^y\
the axes of the conic of closest contact at each point are inclined at con-
stant angles to the tangent and normal. Also, at any point in the curve
-77 = asec3<^, the rectangle under the focal distances in the conic of
closest contact is constant.
1774. Tangents t/O an ellipse are drawn intercepting a given lengtli
on n fixed straight line : prove that the locus of their common point is
DIFFERENTIAL CALCULUS. 303
a qnartic having fcmr-point contact with the ellipse at the points where
the tangents are parallel to the fixed straight line ; and trace the curve
when the fixed straight line meets the eUipse, (1) in real points, (2) in
impossible points ; the given intercept being greater than the diameter
parallel to the fixed straight line.
1775. The curvature at any point of the lemniscate of Bernoulli
varies as the diflerence of the focal distances ; and in the lemniscate in
which the rectangle under the focal distances is 2a', where 2a is the
distance between the foci, the curvatui-e varies as
1776. Prove that the three equations
2a cos <^ ^ /I ^\
y = A.^ ^^y ^^ 9 = *» y-P ^^8 9 (1 ~ cos ^),
all belong to the same curve, p being the radius of curvature at the point
{x^y)) ^ the angle which the tangent makes with the axis of x and 8 the
arc.
1777. The curve in which the radius of curvature at any point is
n times the normal cut ofi* by a fixed straight line (the base) is the locus
cos — r j rolling along that fixed
straight line ; and is also the envelope* of the base of the curve, in which
the radius of cui-vature is n-l times the normal, when the curve rolls
along the same straight line. The two rolling curves may be taken to
have always the same jwint of contact Py in which case the pole of the
former, <?, will always lie on the base of the latter at the point where it
touches its envelope ; the i*adius of curvature at Q of the roulette or
glissette will be nQP^ and the radii of curvature at P will be (n— 1) PG
in the envelope curve and [1 — j PG in the locus cun^e, PG being
drawn at right angles to the fixed base to meet the moving base in G.
[All the cur\'es involved ai^ easily found for the values of w, — 2,
-1,0,1,2.]
1778. The cui've in which the radius of curvature is always three
times the nonnal cut off by the base is an involute of a four-cusped
hypocycloid which passes through two of the cusps : if 4a be the longest
diameter of this cur\-e, 2a will be the shortest, and the curve will lie
altogether within an ellipse whose axes are 4a, 2a, the maximum
distance cut off on any normal to the ellipse being ^^ a ; and the mini-
mum normal chords in the two curves will be of lengths 1*840535 a and
1-859032 a, inclined at angles 51" 33' 39"-4 and 61" 52'28"-2 respectively
to the major axis.
1779. A point P l>eing taken on a given curve, P* is the coiTe-
sponding point on an inverse to the given curve: prove that (1) a
circle can be drawn touching the two curves at Py P', which will be its
own inverse, (2) when the diameter of this circle is always equal to the
304 DIFPERENTIAL CALCULUS.
radios of curvature at P the given curve is either an ellipse or an
epicycloid, and (3) the circle of inversion is the director circle for the
ellipse and the circle through the cusps for the epicycloid.
1780. The perpendicular from a fixed point on the tangent to a
certain curve is afsin ^- ) in + 2co8 — ^), where d> is the angle
\ n + 2/ \ w + 2/' ^ °
which the tangent makes with a fixed straight line : prove that the
radius of curvature at the point of contact is — — ^ a f sin— ^-^ )
and identify the curves when n is 1, 0, and - 1 respectively.
[If the straight b'ne from which ^ is measured be the axis of x and
the fixed point the origin, the curves are (1) the points (x tfea)* + y* = 0,
(2) the points ix?-k-{y^af = 0, and (3) the parabola y" = - 4a (as + 3a).]
1781. Prove that the equation of the first negative pedal of the
parabola y" = 4a (a + a) is 27a^* ^{a-k-x) (x - 8a)*, and that the equation
of the evolute of this curve is
■-«
(£)'^(^)'='-
[The intrinsic equation of this evolute is 5=7= m "" "o" •]
"• ^ (1 + cos ^)' 2 •*
1782. The radius of curvature p of a curve at a point whose areal
co-ordinates are (aj, y, z) is given by the equation
(d'x/dy dz\ d'y /dz dx\ d*z/dx dy\y
9 . \d?\^''Jt)'^W\di''^)'^d^\di' 'dtJ) ^
ic'p*
f^s^y^.iAt^dx^dxdyV
\ dtdt^^ dt dt'^^ di dt)
where a, ^1 ^ ^^^ ^^ sides and k is double the area of the triangle of
reference.
1783. A circle rolls on a fixed straight line, trace the curve which
is enveloped by any tangent to the circle ; proving that the whole arc
enveloped corresponding to a complete revolution of the circle is
2a[2«/3 + o)y a^d the area cut off the envelope by the fixed straight
line is a* l-j- + 4 j, a being the radius.
4a
1784. The curve r=Yn ^ ^^^^ ^ made to roll outside a parabola
(6^ - a)
of latus rectum 4a so that its pole always lies on the tangent at the
vertex, and the curvatures of the two curves at a point of contact P will
be as SP + a : SP — a, where S is the focus. Also the curve r = 2a^ can
be made to roll inside the parabola so that its pole always lies on the
axis, and its curvature bears to that of the parabola the ratio SP-ha : 2a;
so that at any point of contact the radius of curvature of the parabola
28 equal to the sum of the radii of curvature of the two rolling curves.
DIFFERENTIAL CALCULUS. SOS
1785. The curve r = &Bm— rolls within an ellipse of axes 2a, 26,
starting with its pole at the end of the major axis : prove that the pole
will remain always on the major axis and the curvatures of the two
curves when touching at F will be as 6* : 6* + SP.S'F, Similarly with
the curve r=a sin -r a^d the minor axis.
1786. The three curves of the last two questions touch at P and
Of Oj, 0^ are the centres of curvature of the ellipse and the two
roulettes : prove that
a'
{OO.O.P} = p.
1787. The curves 2r = 6 (€» -c »), 2r = a(€* +€~*) can be made to
roll on an hyperbola whose transverse and conjugate axes are 20, 2b, so
that the poles trace out these axes res|)ectively : the curvatures at any
point of contact P wUl be as a'6' : a* (6* + SF . S'F) : 6* {SF. ^P-a*),
a*
and if 0, Oj, 0, be the centres of curvature {OOfi^F} = - ti •
1788. A cardioid and cycloid whose axes are equal roll along the
same straight line so as always to touch it at the same point, their
vertices being simultaneously points of contact : prove that the cusp of
the cardioid will always lie in the base of the cycloid and will be the
point where the base touches its envelope. The curvatures of the two
curves at their point of contact will be as 3 : 1.
1789. A curve rolls along a fixed straight line : prove that the
curvature of a carried point ^ ;/~ ( )> ^here r is the distance from the
carried point to the point of contact and p the perpendicular from it on
the directrix.
1790. The curve
a CL £0 tan a ^ (-9 tan A
- = 1 + sec o sin (fl sin a), or - = 1 + ^ ,
r V /' y 2seco '
rolls on a straight line : prove that the locus of its pole is a circle.
1791. A loop of a lemniscate rolls in contact with the axis of xi
prove that the locus of the node is given by the equation
mo
and that, if p, p' be corresponding radii sA curvature of this locus and of
the lemniscate, 2pp — a*.
1792. The curve r^^a^cosmd rolls along a straight line: prove
that the radius of curvature of the path of the pole is r ( ] .
W. P. 20
306 DIFFERENTIAL CALCULUS.
1793. A plane curve roUs along a straight line: pi*oye that the
radius of curvature of the path of any point carried by the ix)Uing curve
is : — i , where r is the distance from the carried point to the
r - p sin ^
point of contact, ^ the angle which this distance makes with the
directrix, and p the radius of curvature at the point of contact.
1794. A curve is generated by a point of a circle which rolls along
a fixed curve : prove that the diameter of the circle through the gene-
rating point will envelope a curve generated as a roulette by a circle of
halt the dimensions on the same directrix.
1795. A parabola rolls along a straight line: prove that the
envelope of its directrix is a catenary.
1796. Two circles, of radii 6, a- 6, respectively, roll within a circle
of radius a, their points of contact with the fixed circle being originally
coincident, and the circles rolling in opposite directions in such a manner
^that the ^velocities of points on the circles relative to their respective
centres are equal : prove that they will always intersect in the point
which was originally the point of contact.
1797. In a hypocycloid, the radii of the rolling and fixed circles
are as n : 2n+ 1, where n is a whole number: prove that part of the
locus of the common point of two tangents at right angles to each other
is a circle.
1798. Prove that a graphical solution of the equation tan a; = a; can
be found by drawing tangents to a cycloid from a cusp ; the value of x
which satisfies the equation being the whole angle through which the
tangent has turned, the point of contact starting at the cusp.
1799. Tangents are drawn to a given cycloid inclined at a given
angle 2a (the angle through which the tangent turns in passing from one
point of contact to the other) : prove that the straight line bisecting the
external angle between them is tangent to an equal cycloid whose vertex
is at a distance 2a a tan a from the vertex of the given cycloid; and
that the straight line bisecting the internal angle is normal to another
equal cycloid whose vertex is at a distance 2a (1— a cot a) from the
vertex of the given cycloid, a being the radius of the generating circle.
1800. Find the envelopes of
(1) a;cos*tf + ysin*fl = a,
^^^ a'costf "^ft^^wStf"^'
6 being the parameter in each case.
1801. A perpendicular OF is let fall from a fixed point on any
one of a series of straight lines drawn according to some fixed law : prove
that, when OTia & maximum or minimum, Fis in general a point on the
envelope ; and that, if F be not on the envelope, the line to which OY
is the /)erpendicular is an asymptote to the envelope.
DIFFERENTIAL CALCULUa 307
1802. ¥ixkd the envelope of the BjBtem of circles
(aj-aX7 + (y-2aX)« = a«(l+X%
X being the parameter.
1803. The envelope of the directrix of a parabola whiph has four-
point contact with a given rectangular hyperbola is the curve
/a\i 2$
1804. The envelope of the directrix of a parabola having four-point
contact with a given curve is the locus of the point found by measuring
along the normal outwards a length equal to hidf the radius of curvatujre.
1805. Prove that the etivelo^ of the circle
«• + y* + a* + 6"- 2aaj cos fl -^ 26y sin fl = ^- cos tf +1 sin tf J (a*sin*fl+6«oo8*fl)
is the ellipse -5 + t,= 1, and its inverse f -5 — j-A = — + ^.
1806. The envelope of the straight line
1
X cos ^ + y sin ^ =: a (cos n^)^
is the curve whose polar equation is r*"* = a*~* cob:= .
1 —n
g
1807. On any radius vector of the curve r=asec* - is described a
n
circle ; the envelope ia the curve r = c sec^"* =• . Prove this geometri-
cally when n = 2, and when n = 3.
1808. A parabola is described touching a given circle and having
its focus at a given point on the circle : prove that the envelope of its
directrix is a cardioid.
1809. A straight line is drawn through each point of the curve
f'^^oTcoBmO at right angles to the radius vector: prove that the
envelope of such lines is the curve r^~* = a*"* cos =- d.
m— 1
1810. From the pole S is drawn ST perpendicular upon a tangent
to the curve r^= a^cos mtf, and with S as pole and T as vertex is drawn
a curve similar to 1^= a* cosntf: prove that the envelope of such curves is
t^*" =r a**" cos 0.
m-hn
1811. The n^;ative pedal of the paiabola t^^iax with respect to
the vertex is the curve 27ay*« («- 4a/.
20— ^
308 DIFFERENTIAL CALCULUS.
1812. The envelope of the straight line px-^qy-¥rz = Oj subject to
the condition
q—r r-p p-q
and, when the condition is -^ + — ^ + =0, cl b. c being the
q-r T-p p—q
sides of the triangle of reference ABC^ the envelope is
where
w = t(a + y + «)-acos|(5-C)-ycos{120' + |(C-il)}
-« cos {120" + 1(5 --4)},
and similarly for v, tr*
1813. The contact of the curve /(a?, y, a) = with its envelope will
be of the siscond order if, at the point of contact,
(kf ' dadx dy dady dx
1814. Find the envelopes of the rectangular hyperbola
a* - y* - 4 ox cos* o + iay sin* a + 3a'cos2a = 0,
and of the parabola
(x^a cos* o)* = 2ay sin* a + a* sin* a (2 + cos* a) ;
proving that the conditions for osculation are satisfied in each case.
1815. Given a focus and the length and direction of the major axis
of a conic, the envelope of the tangents at the ends of either latus
rectum is two parabolas, and that of the normals at the same points two
semi-cubical parabolas.
1816. The tangent at a point P of an ellipse meets the axes in T, tj
and a parabola is described touching the axes in jT, ^ : prove that the
envelope of this parabola is an evolute of an ellipse, and if FM, FN be
let fall perpendicular to the axes, MN will touch the parabola where it
has contact with its envelope. The curvatures of the parabola and the
envelope at the point of contact are as 2 : 3.
1817. At each point P (a cos 0, & sin 6) of an ellipse is described the
parabola of four-point contact, and ^ is its focus : prove that the point
where FS touches its envelope is (a?, y) where
X
-y 2{a'-b')
a coe' 0~ b an* $~ ef co^ e + 1^ sin* 0+ {a' -b'){ooff0- an* 6)'
DIFFERENTIAL CALCULUa 309
and| if P' be this point,
pp^ 2CP.CD'
where CP, CD are conjugate semi-diameters. Also prove that, when
a* =3 26', the envelope is the curve
'•'■-"•'■(I-?)"-
1818. At each point of a given ellipse is described another ellipse
osculating the given ellipse at P and having one focus at the centre C :
prove that its second focus will be the point P' found in the last
question, and FF* will be the tangent at P' to the locus of P'.
1819. A given finite straight line of length 2c is a focal chord
of an ellipse of given eccentricity e : prove that the envelope of the
major axis is a four-cusped hypocycloid inscribed in a circle of radius
C€\ the envelope of the minor axis is that involute of a four-cusped
hypocycloidy inscribed in a circle of radius ^ -,, which passes through
the centre and cuts the given segment at right angles ; the envelope of
the nearer latus rectum a similar involute touching the given segpnent;
the envelopes of the farther latus rectum and farther directrix are also
involutes of four-cusped hypocycloids ; and the envelope of the nearer
directrix is a circle of radims %c^
HIGHER PLANE CURVES.
1820. Prove that a cubic which passes through the angular points,
the mid points of the sides, and the centroid of a triangle, and also
through the centre of a circumscribing conic, will also pass through the
point of concourse of the straight lines each joining an angular point
to the common point of the tangents to the conic at the ends of the
opposite side.
[The equation of the cubic will be
lyz (y - «) + mzx («—«)+ naey (« — y) = 0,
and, if {X : T : Z)^ (Z' : F : iT) be the centre of the conic and the
point of concourse,
TZ'+rz=zx'-^z'x=xr'^x'T.
When the conic is a circle the cubic is the locus of a point such that,
if with it as centre be described two conies, one circumscribing the
triangle and the other touching its sides, their axes will be in the same
directions.]
1821. Two cubics are drawn through four given points A, B, C, 2),
and through the three vertices of the quadrangle ABCD : prove that, if
they touch at A^ By (7, or i>, the contact will be three-pointic.
[The equation of such a cubic majbe taken to be
/« (y*- «•) + my («*-»^ + ««(»"- y*) = 0.]
1822. Two cubics are drawn through the four points (-4, B, Gy 2>)
and the three vertices {Ey Fy G) of a quadrangle : prove that, if they
touch at Ey their remaining common point lies on FG and on the
common tangent at Ey and, if EG be the tangent at Ey FG will be a
tangent at F,
1823. Two cubics are drawn as in last question, and another
common point lies on the axis of homology of the triangles ABGy
EFG : prove that their remaining common point lies on the conic
whose centre ia D and which touches the sides of the triangle ABC.
mOHER PLANE CURVES. 811
1824. Ph)ve that an infinite number of cubioa can be drawn
through the ends of the diagonals of a given quadrilateral, and through
the three points where the straight lines joining a given point to the
intersection of two diagonals meets the third; also that the cubic which
passes through will have a node at 0.
[The equation of such a cubic will be
\xyz + (/a; + my + na) (»" + y" + 2f^ - 2 {hf + my" + naf) = 0,
where a; ^y 1^:2; = are the sides of the quadrilateral, and Ix^my^nz
the point ; and the taugents at the node will be real if lie within
the convex quadrilateral or in one of the portions of space Vertically
opposite an angle of the convex quadrilateral]
1825. A conic is drawn through four fixed points, and 0, ff are
two other fixed points which are conjugate with respect to every such
conic : prove that the locus of the intersections of tangents drawn from
Of 0' to the conic is a sextic having six nodes and two cusps.
[The cusps are at 0, O', three of the nodes are at the vertices
A, By C oi the quadrangle, and the other three are points A\ B^ C
on BGy CAy AB such that the pencils
A'{AOO'B}, F{BO(yC}, C{COO'A],
are harmonic.
1826. The evolute of the parabola y* = iax is its own polar re-
ciprocal with respect to any conic whose equation is
iy« + X(aj-2a)« = 27XV:
the cissoid a; (a:* + y*) = a^ is its own polar reciprocal with respect to
any conic whose equation is (» — a)" = 3 W - 2A*y". Also the cubic
x(f^''7^)- at^ is its own reciprocal with respect to each of the latter
family of conies.
1827. A cubic of the third class is its own reciprocal with respect
to each of a family of conies, the triangle whose sides are the tangents
to the cubic at the cusp and at the point of inflexion and the straight
line joining the cusp and inflexion is self-conjugate to any one of these
conies ; and the cubic has double contact with each of the conies, the
. chord of contact passing through the inflexion : also each of the conies
has double contact with another cubic having the same cusp and
inflexion and the same tangents at those points.
[The cubics may be taken to be a^ = ^ t/'zj and the conies are the
family
XV-3X«« + 2y«=0.]
1828. In any cuspidal cubic, A is the cusp, B the inflexion, C
the common point of the tangents eX Ay B] any straight line through
^ meets the cubic' again in P, Q : prove that the pencil A [BPQG] is
harmonic.
812 HIGHER PLANE CURVES.
1829. The three asymptotes of the cubic (areal co-ordinates)
a" (y + «) 4- y* (« + a) + «* (oj + y) =
meet in the point (1:1:1), the cubic touches at each angular point
the minimum ellipse circumscribing the triangle of reference, and its
curvature at any point of contact is to that of the ellipse as - 2 : 1.
If from any point P on this curve -4P, BP, CP be drawn to meet
the opposite sides of the triangle of reference in A\ F^ C\ the triangle
A*BG' will be equal to the triangle ABC*
1830. The area of the loop of the cubic
4 (y + «) (2 + a) (a? + y ) = (a? + y + 2)'
yg-i I
18 A'J^ ' ^j_^(l-a:-ic»)^a;,
where K is twice the area of the triangle of reference ; and the
radius of curvature of the loop at a point where the tangent is
parallel to a side is to that at the point on the side as 5 + ^5 : 2.
1831. The base BC of a triangle ABC being given, and the re-
lation tan' A = tan B tan C between its angles ; prove that the locus of
its vertex is a lemniscate whose axis bisects BC at right angles, and
whose foci are the ends of the second diagonal of a square on BC as
diagonal Investigate the nature of the singularity at B and C.
[Each is a triple point, two of the tangents being impossible.]
1832. A circle is described on a chord of a given ellipse, passing
through a fixed point on the axis, as diameter : prove that the envelope
is a bicircular quartic whose polar equation is
ma, being the distance from the centre of the fixed point.
[One focus of the envelope is always the fixed point, and the other
axial foci are at distances from the fixed point given by the
equation
m* (a* - J*) «» - 2am (r^V - r^^WJ*) 2» +
«(l-m')(l-mV-2-5mV6« + y)-2am(l-m«)6*(l-mV-J") = 0.
Hence the origin is a double focus if ni=:0, «, or !• When
9n ~ the envelope degenerates into the point circle at the centre and
the circular points ; when in^t the equation for the remaining foci is
{«s - 2a (1 — c')}* = 0, so that there are two pairs of coincident foci, and
the envelope breaks up into two circles whose vector equations are
r, : rj = c : 2*«.]
1833. Given the circumscribed and inscribed circles of a triangle,
the envelope of the polar circle is a Cartesian.
HIGHER PLANE CUHVES. 313
[The given centres being A, B and AC 9l straight line bisected in Bj
C will be the centre (or triple focus), B one of the single foci, and the
distances of the others from C are given by the equation
ca:«- 2 (a«- 3a6 + ^•)aj + c (a- 26/ = 0,
where a, ( are the radii and c the distance between the centres
(= ^a'- 2ab),]
1834. The equation of the nodal lima^on r = 2 (c cos - a) becomes,
when the origin is moved along the initial line through a space
(so that the curve is now its own inverse with respect to the pole);
and, if OPQ be any radius vector from this pole, -4, B the vertices,
arc ^(2 - arc i4P= 8a sin J ^ OP.
If She the node and any circle be drawn touching the axis in S
and meeting the curve again in P, Q, OPQ will be a straight line,
the tangents to the curve at P, Q will intersect in a point E on the
circle such that SR is parallel to the bisector of SOQ^ the locus of
R will be a cissoid, and that of R' (where RR is a diameter of the
circle) a circle.
1835. In the trisectrix r = a(2 cos tf *1), ^ is the node, RSRf a
chord to the outer loop, SpPj SqQ two chords inclined at angles of 60*
to the former {RSP^-PSQ^QSR =W], and A is the inner vertex :
prove that P, A^q^R are in one straight line and R^ p^ A^ Q ia another
straight line at right angles to the former.
1836. A circle touches a given parabola at P and passes through
the focus S, and the other two common tangents intersect in T : prove
that SP is equally inclined to ST and to the axis of the parabola, the
diameter of the circle through S blBects the angle PST^ and the locus of
T has for its equation
(y^ + 28aa; - 96a")* = 64a (3a - x) (7a - x)\
1837. The locus of the common points of circles of curvature of a
parabola drawn at the ends of a focal chord is a nodal bicircular quartio
which osculates the parabola in two points whose distance from the
directrix is equal to the latus rectum«
[The node is an acnode, and the equation of the curve when the pole
is at the node is
r = 2a (cos fl + y3+Toos^.
Two fod are at infinity, and two are the points (vertex of
parabola origin) 2a;=3a, 2ysrfi9a.]
314 HIGHER PLANE CUBVEa
1838. The envelope of the radical ans of two circles of curvatare
of the ellipse a'y* + V^ = a'6' drawn at the ends of conjugate diameters
IB the sextic (of the class 6)
haying asymptotes 2 [ - * ^ J
*3 = 0.
pThe curve has four cusps, a5*=2a*, y = 0; y* = 26*, aj = 0; four
acnodes, -| = ^ = |, and two crunodes at infinity.]
1839. A circle, is described with its centre on the arc of a given
ellipse and radiuS Ji^^ — f^^ where r is the focal distance and c a
constant : prove that its envelope is a bicircular quartic which has a
node at the nearer vertex when c=:a(l -e), and four real axial foci
when c is < a (1 — c) or > a (1 + c).
[The polar equation is, focus of ellipse being pole,
(r+^'+4««^r + ^')costf-46* = 0,
and any chord through the pole has two middle points on the auxiliary
circle of the ellipse. The dutances of the foci from the pole are given
by the equation r + — = 2 (6'ife oc), and the points of contact of the
double tangent lie on the ellipse
1840. The straight line joiniug the points of contact of parallel
timgen.ts to the cardioid r = 2a (1 — cos 9) always touches the curve
2rcos0 = a(l - 4cos'^) ; and an infinite number of triangles can be
inscribed in the cardioid whose sides touch the other curve.
[ThiB envelope is a circular cubic having a double focus at the cusp
of tne cardioid and two single foci on the prime radius at the distances
— a, 3a respectively from the cusp ; and, if r^, r^, r, be the distances of
any point on the curve from these three foci,
T^ being reckoned positive for the loop and negative for the sinuous
branch. Another form of the equation is 2r cos » « a, the origin being
at the centre of the fixed circle when the cardioid is generated as an
epicycloid.]
1841. A point P moves so that OP is always a mean proportional
between SFy UP \0^ S^ H being three fixed points in one straight line :
jMt>ve that, if lie between. S and iT, another system of three points
9", S', //'c&n be found on the same straight linQ such that O'P is always
HIGHEB PLANE CURVES. 315
a mean proportional between S^P and ff'P ; that (X will lie without
b^ir, and the ratio aS : OH' will Ue between ^2- 1 : ^2 + 1 and
^2 + 1:^2-1. .
and ? = ^ = ( ^^ ) . The locus of P is a circular cubic whosa
a + 6 ab \a-^oJ
real foci are S, H, S\ R\ and a vector equation is
1842. With a point on the directrix of the parabola ^=4aas as
centre is described a circle touching the parabola : prove that the locus
of the common point of the other two common tangents to the circle
and the parabola is the quartic
(/ - 2axy + 4a (a + 2a)' (4a; + 3a) = ;
also if on the normal at the point of contact of the circle and parabola
be measured outwards a distance equal to one-sixth of the ludius of
curvature, the envelope of the polar of this point with respect to the
circle is
y* + 8a(a;-2a)'«0.
1843. A circle drawn through the foci B,Coi& rectangular hyper-
bola meets the curve in F, the tangent at P to the circle meets BC in 0^
and OQ is another tangent to the circle : prove that (1) the locus of Q
is the lemniscate (Bernoulli's) whose foci are B^ C and that OP is
parallel to the bisector of the angle BQC ; (2) if OP'Q' be drawn at
right angles to OP meeting the circle in P, Q, the locus of P' will be
a circular cubic of which B, C are two foci, and the two other real foci
coincide at ^, a point dividing BC in the ratio ^2 - 1 : J2 + 1, (J? being
the nearer point to 0) ; (3) the vector equation is
(^2-l)(7P-(72 + l)ifP»2^P,
Tift
or AB.CP + AC.BP=^^.Ar} and U) the angle Q'QP exceeds the
angle PQP by a right angle.
1844. Any curve and its evolute have common foci, and touch each
other in the (impossible) points of contact of tangents drawn from the
foci
1845. Trace the curve
4ay (x + y - a - 6) + a5 (a? + y) = ;
and prove that if {x^ , y,), (a?,, yj be the ends of a chord through thp
origin
»i + «, + y, + y, = a + 6.
316 HIGHEB PLANE CUBVES.
Also prove that the area of the loop ia
BiD.'6d9
2(a-byj^
nA^O^F?
1846. A circle is described with its centre on the axis, and the
points of contact of the common tangents to it and to the fixed circle
a* + y* = a* lie on two straight lines : prove that the locus of the points
of contact on the variable circle is the two curves
(a:«-. y«- 2ay + (a;±y)* = 2a"(aj±y)«,
that these curves osculate in the points *> a, 0, and that the area Qf each
common loop is o* ( j - log 2 j ,
1847. The fixed points S, ffoxe foci of a lemniscate
{4nSP.HP=Sir),
and the points U^ F, F', V its vertices, a circle through S^ H meets the
lemniscate in i?, jR' (on the same side of SH) and Ulf, YR, VK, U'R
meet the circle again in (J, P, F^ Q^i prove that the sti^aight lines
QQ^j PF intersect SR in the same point as the tangent to the circle at
R^ and each is equally inclined to SR^ HR. Also each of the points
Qj Pf F, ^ is such that its distance from R is a mean proportional
between its distances from S and ff, as also is its distance from the cor-
responding vertex. The locus of any one of the points for difierent
circles is therefore an inverse of the lemniscate with respect to one of its
vertices, the constant of inversion being the rectangle under the focal
distances of the corresponding vertex, sign being re^irded.
[The curves are axial circular cubics, similar to each other, and any
one IS the inverse of any other with respect to one of its vertices, the
constant of inversion being always the rectangle under the distances of
the centre of inversion from S^ If. The loci of jP, F (and of Q, Q') are
images of each other, or their centre of inversion is at oo , Each of the
four curves has one vertex peculiar to itself, and is its own inverse with
respect to that vertex, the constant following the same rule as for
t wo di fferent curves, llie linear dimensions of the loci of Q, P are as
tjn- 1 : Jn+l, Each curve has S, H for two of its foci ; and, for the
locus of jP, the two other real foci (-F,, F^ divide SH in the ratios
SF^ : HF^^Jn (n- 1) + ^n (n + 1) + 1 : - V^(n- 1) + ^/i (n + 2) + 1,
HF^:SF^ = Jn{n-^\)-^Jn{n--\)-'l : Vn(n+ l)-,yn(n- 1)- 1 ;
and, if r,, r„ r„ r^ denote SP, HP, F^P, F^P,
2yn^r, = (^n-l+yn + l + 2V»*)^ + (^^-l-«/^ + l-2^n)r„
2V^-lr^=(7w-l-iyn+l + 2 ^n) r, + ( ^n - 1 + V^ + 1 - 2 Jn)r^.]
1848. Tangents inclined at a given angle a are drawn to two given
drcles, whose radii are a, b and centres at a distance c : prove that the
Jocas of their point of intersection is an epitrochoidy the fixed and rolling
HIQHEB PLANE CURVES. 817
circleB beiiur each of radiuB ^ / ; , and the distance of
the generating point from the centre of the moving circle being e.
1849. In a three-cosped hjpocjcloid 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 hjpocycloid.
1850. A tangent to a cardioid meets the curve again in P, Q:
prove that the tangents at P, Q divide the double tangent harmonicadly,
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~*+r" +Z~ =0 will represent a cardioid when
jr = a; + ty, Y= x - ly, Z= a ;
and a three-cusped hjpocycloid when
X=x-¥yljZ, Y=x-f/J3, Z=da-'2x,]
1851. Chords of a Cartesian are drawn through the triple focus:
prove that the locus of their middle points is
(r* - 6c) (r* - ca) (r* -ab) + aVc' sin" tf = 0,
a, (, 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 veloci-
ties which are to each other as m : n (m, n being integers prime to each
other and n>tn); the envelope of the joining line is an epicycloid whose
vertices lie on the given circle and the radius of whose fixed circle is
a . When - m is put for m, the points must describe the circle
w + w»
in opposite senses, and the envelope is a hypocycloid. Hence may be
deduced that the class number i&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 comers is
the describing point ; all the other comers will move in the same epicy-
cloid, 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
/> + ^, tangents are diuwn to the curve from any point on the circle
through the vertices : their points of contact will be comers 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,
313 HIGHER PLANE CURVES.
1855. The locus of the common point of two tangents to an epicy-
cloid inclined at a constant angle is an epiti-ochoid, for which the rsidiiis
of the fixed and moving circles are respectively
. a+b . a+6
a(a + 25)^"'^T25'' 5 (a + 25) ^ ^T25 "
a + 6 sin a ' a + 6 sin a '
and the distance of the generating point from the centre of the moving
circle is
. h
sm ^a
, „-. a + 26
(a + 26) -
sina
where a, b are the radii of the fixed and moving circles for the epicy-
cloid, 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 on
the axis is a nodal circular cubic which is its own inverse with respect
to the vertex A^ the constant of inversion being Che square on OA, If
Off be a straight line bisected in A, FQ a chord passing through
ff, OTf OZ perpendiculars on the tangents at P, Q, then A^ F, ^will be
oollinear and ii r . iiZ = il 0«.
SIf OA £= h and 4a be the latus rectum, the distances of the two
^ e foci from 0, the double focus, are given by the equation
of + 4aa; = 4a&, and the vector equation is, for the loop,
^ _Ji__+ htir =0,
Ja-k-b-Ja Ja + b-^Ja ^ *
r, being the distance from the internal focus. The difierence of the
arcs 8^ , «, from the node to corresponding points F, Z on the loop
and sinuous branch is determined by the equation
ds ds sin^ » ' -
INTEGEAL CALCULUS.
1857. 'The area common to two ellipees which have the same centre
and equal axes inclined at an angle a is
2ab tan 7-= — i«— . — .
(a — (r) Bin a
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 liie curve traced out by the feet of these perpendiculars is
1859. The areas of the curves
aYix-by^ia'-aTjibx-ay, iB- + y' = a-, {b>a)
A' — A
are A, A*\ prove that the limiting value of -7 , as h decreases to a»
is 6irflk
1860. The sum of the products of each element of an elliptic lamina
multiplied by its distance from the focus is \ Ma (2 + 6*), M being the
mass of the lamina, 2a the m^or axis, and e the excentricity ; and the
mean distance of all points within a prolate spheroid from one of the
foci is i a (3 + e*).
1861. Prove that the arc of the curve y^Ja^ — Vil—oo^j^
between x = 0, x- 2v(, is equal to the perimeter of an ellipse of axes
2a^ 2b : and determine the ratio of a : 6 in order that the area included
between the curve and the axis of x may be equal to the area of the
ellipse, [a : 6 = 2 : ^3.]
1862. Find the whole length of the arc enveloped by the directrix
of an ellipse rolling along a stnught line during a complete revolution;
and prove that the curve will have two cusps if the excentricity of the
ellipse exceed ^^-^ — •
320 INTEGRAL CALCULUS*
[The arc 8 is determined by tbe equation
^*_-*/l eoosijf (1— «")cco8^)
d^jf^ei ^Jl - e' sin^ " (1 - e* sin' i/f)«J '
^ being tbe angle tbrougb wbicb tbe directrix turns.]
1863. A spbere 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 tbe former : prove tbat tbe locus of tbe circular boundary of tbis
segment is a sphere.
1864. Two catenaries touch each other at the vertex, and the linear
dimensions of tbe 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 tbe arc Fp about the
directrix is equal to 2ir x area MQqm,
1866. The area of the curve r = a (cos tf + 3 8intf)*^(cos^ + 2sin^)'
included between tbe maximum and minimum radii is to the triangle
formed by the radii and chord in tbe ratio 781 : 720 nearly.
1866. Prove the results stated below, A denoting in each case the
whole area, x and y the co-ordinates of the centre of inertia of the area
on the positive side of the axis of y ;
(1) thecurve (a* + a;*) y* - 4a'y + «* = 8a*,
^ = 6,ra',« = |?log(2 + V3), y = y;
(2) thecurve y*(3a* + a;*)-4a*y + «*=0,
. II , ^ 4a4-31og3 » a ,«, ,« ,^,
(3) the curve y'(a* + a^)- Wy + {af- 2cf)* = 0,
8a
ul = wa*, 5=-{V3-log(2 + V3)}, y=aj
IT
(4) thecurve y* (a" + as') - 2maa*y + aj* = 0, (w>l),
A^^lm WW ^_^« K-l)' + 3{V^;;^1T-mlog(m + VW^^n')}
y = (wi - 1) a.
1867. Prove that the curve whose equation is
y" (a' + o*) - 2m(wi+ 1) a'y + 4«*-^ (m* + m- l)aV
+ (m + l)(4iii'-3m+l)a*-:0
consists of three loops, tbe area of one of which is equal to the sum of
the areas of tbe other two.
INTEGRAL CALCULUS. 321
1868. For a loop of the curve as*/ - 4a*y + {3a' - a^* = 0,
^=a'(2»-3V3), x = ia ^—Q^— . y= 3 2,-373*
1869. The area of a loop of the curve
y»(4a'-aO-4a'y + (a'-a^' = w a" (3 - 2 log 2).
[This curve breaks up into two hyperbolas.]
1870. In the curve
- a 3(w+l)*^/m~3m'-2f» + 2
^ ~ w 3m + 4
••«»•, m .1 acosStf a8in30 , ., .
1871. Trace the curve g= . ^^ , y= » n/i > ^^ prove that
sin Jcr sin J&
the internal area included bj the fotir branches is 2 ^3 a*.
1872. Trace the curve whose equation is a; = 2asin-; and prove
that each loop has the same area ira' and is bisected by the straight line
joining the origin to the point where the tangent is parallel to the axis
of y.
A ~ 2/
1873. The area of the loop of the curve a""" y* = vfaf' , when
n is indefinitely increased, is J^ir a' ; and the area between the curve
ft — OR
©■""'y* = wa:*" and the asymptote, when n is indefinitely increased,
flt + fiC
is 2 J2ia\
1874. The areas of (1) the loop of the curve y*" (a + ac) = ob^ (a - «),
(2) the part between the curve and the asymptote differ by
ira-
2n
1876. Prove that the whole arc of the curve 8a*y" = »■ (a* — 2aj') is
va ; and for the part included in the positive quadrant, the centre of
gravity of the area is [j^-ja » aa ) > ^^® centre of gravity of the arc is
(«— T^r— , "T— ); the centre of gravity of the volume generated by
revolution about the axis of a; is (0-79 » ^) > ^^^ ^^ ^^ ^® &i^^& ^
the surface generated is {^—rsi j .
w. p. \x
322 INTEGRAL CALCULUa
1876. Pipove that the curve jy = «" (2a- «) is rectifiable if
6« = (9ifc6^3)a'.
1877. The arc of the curve r (c^ + 1) = a (€^ - 1) measured from the
origin to a point (r, 0) is a0-r; and llie corresponding area is
1878. The whole arc of the curve
a?f + yt==a? is 5a|l + g-j^log (2 + ^3)1 .
1879. The arc of the curve
a; = a(2-.3co8tf + cos3e), y = 3a ,^2 (2tf - sin 25),
from cusp to cusp, is 14a.
1880. The curve 27 (y" - 8aa; - a*/ = 8aa? (9a + 8a;)* is rectifiable.
[We may put y = o^, 8aj= 3a (1 - f)\ and 8« = 3a^ (2 + f), measur-
ing from the cusp.]
1881. The arc of the curve
a; = a(68in0 + sin30), j^ = 3a (20 + sin 20)
measured from the cusp (at the origin) is a (12 sin + sin 30).
1882. The carves whose intrinsic equations are
(2) ^ = 2a ,|-'^f., .
^ ' C^ (l+cos<^)"
are both quintics: in (1), «' = aj' + |y*, and in (2), •■ = «' + lf y*, «, y,
8 vanishing together and ^ being measured from the axis of x \ also
the area of the loop in (2) is ^ -^ a* and its centre of gravity divides
3t
the axis in the ratio 63 : 80.
1883. The curve whose intrinsic equation is
, .4tan'f(2.tan.f)
is a quartic. [y * + 8a (a; - a) y* + — ^ =0.]
1884. In the curve y* - 6a*a:y + 3a* = 0, the arc, measured from a
.a'
point of contact of the double tangent through the origin, is ax
y
1886. The arc of the curve oa; = y' - 2a' log - - a*, measured from
2t/*
(0,a) is-^ -a;-2a.
INTEGRAL CALCULUS. 323
1886. The whole area of the curve
a; = a sin tf (15 - 5 sin" tf + 3 sin* $), y = 10a cob* d,
is -32— ««*; the arc in one quadrant is 17a, and its centre of gravity
lies outside the area.
1887. A hypocjcloid Ls generated by a circle of radius na rolling
within a circle of radius (2n + 1) a, (n integral), and an involute is
drawn passing through the cusps : prove that the area of this involute is
to that of the £xed circle as
2n (n+ 1) (8/1* + 8?i- 1) : (2n+ 1)*;
and the arc of one to the arc of the other as 4n (n + 1) : (2n + 1)*.
1888. If w^ denote I a;"* J(x - a) (6 - x) dx, and m-lhe positive,
•'a
2 (m + 3) w^^, - (2m + 3) (a + b) u^ + 2mdbu^,^ = 0.
1889. Prove that the limiting values of
(1) < Sin -Sin--- Sin — ... sin(n- 1) -> ,
\ n n 11 wj
(2) -Jsin-sm' — sin' — ...sin '(n-l)-> ,
^ I ?i n n ^ n)
when n is indefinitely increased are each equal to ^.
1890. An arithmetical, a geometrical, and an harmonical progression
have each the same number of terms, and the same first and last terms
a and I; the sums of their terms are respectively 'i* ',» ',» t^d the
continued products p^y p^fP^'- prove that, when the number of terms is
indefinitely increased,
— 1 *"g ~ > A 7 9 — a — *•
8^ I — a ° a 8^8^ icU p^
[The last of these equations is true whatever be the number of
terms.]
1891. Prove that, if w + 1 be positive, the area included by the
curve x = a cos*"** 0,y = h sin*"** and the positive co-ordinate axes is
(w + l)a6r(n+l)r(n + 2)4.r(2w + 3),
and tends to ah Jmr 4- 2*"** as n tends to infinity. Also, by considering
the arc of this curve, prove that
^^-^j ^/a*(l+a;)*- + 6*(l-aj)*"c&>2"*»-l<2"*';
and that the limit, when n is infinite, of
n2-"J ^a*(l+a;)*" + 6'(l -«)-(/« is 2(a + 5).
21—^
324 INTEGRAL CALCULUS.
1892. The lengths of two tangents to a parabola are a, h and the
included angle ci> : prove that the arc between the points of contact is
a" + 6* - a6 (1 - cos (I))
(a + 6)
a* + 6* + 2ah cos cii
aVsin'w . h Ja'-\-b'-\-2ahcoa(o+h+acoR<o
•f J log - ^.- .
(a*+6"+2a6cos(i))* <*iya*+6"+2a6cos(i)— a-6co8<u
1893. The general integral of the equation — ^ = — ^ may be
cos V cos Kp
written 8ec'04-8ec'^ + Bec'fi-2sec08ec^secfi = l ; and that of the
equation . = ._ may be written a;' + y* - 2Arcy + X* = 1.
[That is, the differential equation of all conies inscribed in the
naralleloffrain whose sides are a' = a*, y* = 6' is —, — = —r — .1
1894. The complete integral of the equation
dO d<l>
may be written in the form
where e*s=l — |: prove that this is equivalent to the ordinary form
costf cos^ + i7l - e' sin' u Bin sin <^ = cos u, where tan^ = a /toV-s — r^ •
1- #"7 2 V 6 (a' + X)
Also prove that a particular solution is
(1 -e'sin'd) (1 -e'8inV)= 1 -e*.
1895. The area of each curvilinear quadrangle formed by the four
parabolas y*= mux - w', when u has successively the values a, 6, c, c?, is
2 ,~ ,—
{a'<h<e<d), and this is equal to the area of the quadrangle included by
the common chords, when Jd-^ Ja = Jb-\- Je,
1896. In an elliptic annulus bounded by two confocal ellipses the
density at any point varies as the square root of the rectangle under the
focal distances : prove that the moment of inertia about an axis through
the centre perpendicular to the plane is
M being the mass, 2a, 26 the axes of one boundary, 2a', 26' those of the
other.
INTEGRAL CALCULUa 825
fl!* f/* ft*
1897. In an ellipBoid ^"^p+ii^ly ^® density at any point
(x, ^, 2) ifl ^ ( -g + ^ -f -J j : prove that the moment of inertia about the
is of « is if ^^Ij^ jV/'(x*)dic4.|V^(««)d^
axis
1898. The yalue of /// taken over one of the continuous
JJJ ay«
volumes bounded by the six spheres U=cuc, U=a'Xy U^hy, U^Vy^
U=^cz, U = cz, where J7'=x* + y* + 2* and <ia\ bb\ cc' are positive, is
1899. Prove that, if a, 6, c, cf be in descending order of nmgnitude,
(^ dx f* dx
A ^(^^)(i-6) (^c)"(^(£) " ^ ^ j.coN/K^(6-c)a:'}{a-rf+(6-d)a^
^{6-c+(a-c)a^}{6-c/+(a-c?)a'}
dx
=(2) r
= (4) f" ^
1900. Prove that, if a, 6, c, cif be in descending (xder of magnitude^
and m any positive quantity,
}(, \ a-d b — e )
{(a-x)(b-x)(c-x)(x-d)-' ,..
iLx)i^-J) \ \l,-x)\ c-x]^*'-^ '^'
ft {a-x)^x-a) {b-x){c-x)V
a-d b-e )
f" (g -«)—'(« -<;)—'<fe
I f («-«)(x-rf) . ^«-^)^g-c ))*'
* I o-rf * 6-c J
U (a-g)—'(c-«)— '«fe
^Trf— +"— ir^ J
326 INTEORAL CALCULUS.
and r , ^^r"^^"*"'^^"^r'^
'(a-a;)(aj~c?) (a;-6) (aj-c)^"*
(g - a;) (a; - (^) (6-a?)(c-a;) y^'
1901. Prove that the limiting values, when b increases to a, of
(2)
/*« (g - xr~ ' (x - cr^' dx
I ( {a'x){x-d) ^ {x'h){x^c) )^"^
h\ a—d b — c )
r {{a- x) (x - 5) (a; - c) (a; -- d)}''~'dx .
I r (a - a;) (a; ~ c^) (a; ~ 6) {x - c) -!"""' >
J6\ a-d b^c )
are respectively
(1) {r {m)Y ^ r {2m), (2) (a - c)""' (a - e/)""* {F .(m)}' -r T (2m).
1902. The limiting values, when b increases to a, of
n\ f"* (a-xy'''{x'-cY"dx
^' I aa-'X){x-d) ^ {x-^mx-'Tf ^'
h \ a—d b-c )
(2)
I f (a-a;)(a»-rf) ' (x-6)(«-c) ^-"
are respectively. (1) - (a-c)""*, (2) —{a--dy"'\
I
1903. Prove that, M[a>b>c>d,
a
^ / («-d)(6-c ) . /{a-b}(c-d)
{a-x)(x-d) {x-h)(x-c) ~ V (a-6)(c-d) V (a-d){6-c) '
6 a — c? 6 — c
f* db,- /(^niMzf) tan- /(iLzfO(*r£)
A •• V (a-6)(c-d)*^ V {a-b)(e-dy
1904. Having given 2a; = r («* + <"*), 2y = r (c — «~ ) : prove that
/ / rdxdf/= I r'rdrdOy
Jo Jo Jo Jo
V being a function of x, y which becomes V when their values are
Buhatituted,
INTEGRAL CALCULUa 827
1905. Having given Xa3P=7y* = ^25*«...=«y«..., prove that
V being a function of X, T, Z,,,. which becomes v when their values
are substituted, and n being the number of integrations.
1906. Having given a? + y + » = w, y + z = uv, z = uvWf prove that
-00 00 -OO -OO -1 -1
I / I Vdxdydz^l / / V u'v du dv div ;
Jo Jo Jo Jo Jo Jo
also, having given
prove that
and the corresponding theorem with n variables.
1907. Having given
iCj = r sin 0^ cos d^, x^ = r cos $^ cos ^3, a:, ^ r sin 0^ sin d, cos tf^,
a^ = r sin fl^ sin 0^ sin d^ , a^j = *• cos d, sin tf, cos tf, , a, = r cos tf ^ sin tf, sin tf,,
prove that
#00 /> 00
f / ... Vdoc, dx, dx, dx^ dx. dx^
Jo Jo 1 . 3 4 5 •
w w
= r r(^-y'r'Bm!'e^coB'0^6m0^mi0^drd$^...d$^.
Jo Jo Jo
Prove that m... dx^ dx^ ... dx^ taken orer all real values of
1908
ajj, a?,, . . . a;, for which
x^^ + x^* + ... +a;/ + 2m(arja;^ + a;^3+a;^j+...):f' 1,
is equal to
/ ^ \ i / l-m 1
Vl-W V l+m(wi-l) p/n^ A'
provided that m lies between =- and 1.
71— 1
1909. Prove that
cfe^cfa,+ ...+(i!g, w^ [n+r-1 |2r
{a' + «.' + V+...+«J™ ^"''*' |2n+2r-Hr'
n, r being positive whole numbers.
328 DTTEGBAL CALCULUS.
1910. FiOYethat jjj ...dx^dx^.,,dx^iB equal to
jmn \»(«+i)/ r(-+i)
limits of the int^pral being given by the equation
1911. Frove that
III <fa,_^,...<fa. y ^\ij.
WVi-«+<+...+0" 2rg)r^p)'
and that
v^
•+1 r p
the integral extending oyer all real values for which x*-¥x*-¥.». +2C.'^ 1.
1912. Proye that
/ ri ^ — 7i v;=o-S(-l) *sm — cos" * — ,
r having all integral values from 1 to ^ - 1 or — ^ according as n is
an even or an odd integer.
1913. Having given the equation
and that when x = ^ , y = ^— r^ ; prove that^ when a; = 2ay
a
y = -^log(2 + V3);
and generally that y= o'cos"' f ^j -t-Ja'-a? when »•<<»', and
=„Mog(i±4:i),7^
when a;>a.
[It would seem that the only form of solution holding generally is
dz
INTEGRAL CALCULUS. 329
1914. Having defined X^ X' by the equations
sma:=«-i3+|^.. ..+(-!)• »j^_j + (-l)._,
prove that
1915. Prove that the limit^ when n tends to oo , of
Jn I Bm^xclx is ij2w;
Jo
and that this is also the limit of fjn I sin" xdx, where a has any
IT J'i'*
value between and ^ excluding the former. Similarly the limit of
Jn I Bm*'*^xdx or of Jnj {cosx)''dx is Jw.
1916. Prove that the limits, when n tends to oo , of
«* I sin*" » (1 - sin x) dx,
and of n^ j sin*"a5rj-- — =-sina5-l j cfo, are ^ Jw and ^ ^ir respec-
tively.
[In both these also the only portion of the integral which affects the
result is that which arises from values of x differing very little
from ^.]
1917. The expansions of
Jn I sm*"«(fo and Jn I sin^^'ajcte,
when n is very large, are respectively
'^'V^""8ii'^T28S*'^102i;?'^''7' ^'V^"8;i"*"l28»*
105 \
330 nrrEORAL cALcxTLua
1918. Frove that
I (l+xr-'^-^jo ("IT^^- A ^'^^ '^
- fV* (2 -2a! +!«•)" „ /■v^(2-2a;• + a!*)"
and obtain in the same way the equation
rV2 /2-2^+^*\ cte _ /-yg / 2-2a;+aA (Za;
[The results are easily obtained by putting a; = tan z. The more
general theorem is
1919. Prove the following definite integrals :
(1) j|'Bm2«logcot?<fe=l, J^ ^^^L^.logLt^cte
1
n
<2) f rSl°8^-^=f2' j/tan (I + 0^)108 cot xcfe
Sir'
16 '
<^> r^1J**"=£^°e«, f iogxiog(i + j:)rfx
= 7ro(loga-l);
(1) a<l, 6<1; (2) a>l, 6<1; (3) a>l, 6>1;
Jo l+cosasina;"~ sina' Jo (1 +C08oBina;)"
(a - sin a cos a) / w\
(7) ( XBm2rxr"dx = '^^r€'"*, C s(f cos 2rx€"' dx
Jo ^ Jo
INTEGRAL CALCULUS: S31
= -(8in-c)', (c'<l);
(9) I coB2a;logcota;c/a; = -, / 8ina;logsina;ci!;c = Iog2- 1;
fi f/a; , /I - 2a:* cos 2a + a;*\ , (/, ^ a\* «^
I 1^ *°S ( 4-^.^ ) = i {(log cot ^ j - -^} ;
^^^^ j -v^log(l + smo8ma:)<£e = ^a(jr-a), o<|;
(12) r^logl±^^a^=2^ ('a<^);
^ ^ Jo X * l-2a;8ma + a;' ^ \ 2/'
ir »
(13) f^f-r^) c3?a; = 2[^a;cota;c^ = 7rlog2;
y \81^ Xy Jo
(15) /:
(16) r
" 8in'"+'x , 1.3.5...2«-1
ax = — T. — : — ■_ ^ — w :
X 2.4.6...2N
* cos" a« — cos" ft* , ,, .3.5...(n — 1)
(17) /_
/• V 3.5...^
«''(*-«) 271— (n-^l)'
A cos ax + B cos hx-hC cos cx-\- ,,,
00 X
m I
* -4 cos 005 + ^cos6aj + Ccos ca;+ ... ,
ax
h X
= log(a-^6-^c-^...)
(the two last integrals are finite only when ^ + J5 + C+ ... = 0) j
'* il, cosa,a; + ^,cosa.a5+ ... +-4 cos a a; ,
<i^) /_
of
(when finite) = ^^ irS {A a;"') j
(20) j_
" -4, sin ajaj + -4 J sin a^+ ... +-4^ sin a^a;
dx
00
ic"*'
(when finite) = ^" irS (i,«,-) j
332 INTEGRAL CALCULUS.
w /;
OT I
(when finite) = t^ 2 (^,a * log o,) ;
' A^mla,x + A^anaJe+ ... +A emax,
. ^ '^
(when finite) = ^^ 2 (J, a,-' log a,);
(23) j 5^ i? (sin* x)dx={' F (sin* «) <& ;
,_.. /■" log (1 + wain**) , , ,.
^^*^ J., a;" ^<fe=2»-(yiT;;-l), (»»<1);
(26) / -rlog(=--s rr:— t) = »(ft-«)i , (»»<1);
9 2 -.«.
- 2 (n I 1) S; ^*' •" ^^ ^*"' ^^ - ^*' "^ ^"^ ^j*"" ^^ ^''" ^^ - ^'' " ^") ;
«r V
(29) I ^ (sin 2a;) sin osc^ = / ^ (sin 2a;) cos a;(i!»
= ^ / ^ (cos 2a;) COS «(/«;
(30) Y^ (log sin «)• dx = Jf * (log cos «)• db = ||J^ + (log 2)'l ;
ir
(31) [Mogsina;logcosa;&j = j|(log2)*-|^l,
J log(l-cosa;)log(l+cosa;)cfa;=:w|(log2)*-^l;
(32) I log sin a; log tan a;(2iB s Y^ ;
XMTEGBAL CALCITLU& 333
^^^> j, sinx *'" 8 ' Jo sin* *** 16 '
p gogaeca;) '^ ir'.
Jq sin a 8 '
_(2--l)w^
(35) I — log= — s — : | = 2jrtan 'n, (»<1)
^ Jt X ° l-2n8inaj + »' ^ '
L T^°8 i_^rin^ = 'ram (m), (m<l);
(37) I — tan"* (w tan a^) = - log (1 + m),
j -^tan-» (mBina:) = I log (^1 +m' + m);
I — tan M 1 = o log (1 + m), (m < 1) :
1 6ii;
^nsin-/
n
00
•"^ In Sin-/
334 INTEGRAL CALCULUa
IT
f1 IT
(45) I log(cotx-l)c^a; = ^log2,
(46) f^iog( ;^f«^(^-°)^^: i
Jo X U + 2ncos (a; + a) + 7i*j
\1 + m cos a/
,.^. f* €""^cosa; - /Trjl+m'-^m
(^7) / 7 cte = Vo^-l i — >
Jo J^ ^ 2 1 + m'
(48) / €*«*•' cos (x + n sin «)(&;= :
Jo n
(49) r (1 - a;-)- cfaj = f ^ (1 + af)"^^ cfo; = J f " (1 + a")"" dx
Jo Jo Jo
/^0\ f* ^"^dx sin (n — m) a
yo «*" + 2a;" cos no + 1 ~ sin na
mir
w sin —
n
{m < 2n, na < tt) ;
(51) r{sinxy"dxj'(^xydx= — ,
I (sin a:)" da; I (sin a;)"" cfa; = — tan-^r-, (n<l).
Jo Jo ^ 2 ^
1920. Prove that, if n be a positive integer,
1921. Prove that
^33i . ,^» a;
^ <to> /a;€-*«»l _ "^ £ 1+4A*
\ (l + 4M-)t '
if 1 + ihk be positive.
nrrEGRAL CALCULUa 335
1 922. Trace the curve
2 r*8ina»8in*«
wJq
^
»
[When X ia between - oo and -2, y = — l;a; = -2toaj = 0, y = aj + -j;
a?
a: = toa; = 2, y = a;-.-; a;>2, y = l.]
1923. The limiting value of the infinite series
when n is 00 , is =^ .
1924. Prove that
/■• ,, , [^ f' 2«sina 2eBmB
I Jl-ecoBxdx- I J\''eQO&xdx = ~.-^^z^ = ,, — -
^0 Jo ^l+«co8a ^l-0cosp
j8 /r^ a
if o, /3 be angles < ir such that tan o = \/ fT ^^ o '
1925. Prove that
4sm^
^Ca dx C'' dx _ - _ „
Jo (l-8in^cosaj)» A (1 -sin j8cosa;)i co?^'
if /3 be any angle between - ^ and - , and cos a = tan ^ .
1 /■'
1926. Prove that, if <^ (c) = - / log (1 + c cos a?) c&,
VJq
2*(c)-*(2^)=log(l-0.
1927. By means of the identity
proTO that
1 1 1 , 1 1 'j ,
«''^(^n)"''^(^T2)"»'^--**'*~a*2a(a+l)"*'3a(a+l)(a+2)'^-*°*'
1928. Prove that, p — 9 being a positive 'whole number,
/."^•■*-l^'k'-'C-'>-*'-V<'-*)---
to^or^^ termsj,
336 INTEGRAL CALCULUS.
or
= CT^ {^■' ^'^p-p^p- 2)"' ^°« (p - 2)
+d^(p-4)-log0>-4)+ ... tolor^terms},
according as p — ^^ is even or odd.
1929. From the identity
/ (a + 8in*aj)"cosflBd!aj= I (1 + a - sin* a:)" sin a; rfa?,
or otherwise, prove that
^+3^* """275"^ [3:7 "* ^-
= (a+l)'— ^(a+l)* '+ g^g ^ («+l)'
2»n(n-l)(n-2), ,,,_.
and prove in a similar manner that
«-i 3n(n-l) ._, 3. «n(n-l)(n-2) ...
» (a + 2)- - » (a + 2)-' + ?^^^) (a + 2)-
_ 3.5n(n-lKn-2) ^^^,^.„^
1930. Prove that, c being < 1,
TT J sin"' (c sin a?) die = I -Itan"' f-= r)f ^
1931. On a straight line of length a + 5 + c are measured at random
two segments of lengths a + c^ 6 + c respectively : prove that the mean
value of the common segment is 6 -i- c — q- , a being > h.
oo
1932. A point is taken at random on a given finite straight line of
length a : prove that the mean value of the sum of the squares on the two
parts of the line is f a', and that the chance of the simi being less than
this mean value is -y» .
INTEGRAL CALCULUS. 337
1933. A tj-iangle is inscribed in a given circle whose radius is a :
prove that, if all positions of the angular points be equally probable, the
12a
mean value of the perimeter is — , and that the mean value of the
radius of the inscribed circle
isa(y-l).
1934. The perimeter (2a) of a triangle is given and all values of
the sides for which the triangle is real are equally probable : prove that
the mean value of the radius of the circfumscribed circle is five times,
. and that the mean value of the radius of an escribed circle is seven times
the mean value of the radius of tlie inscribed circle.
rmi ., , 47ra iira iira ,
The three mean values arc , ^- , ^^ . -; ^ •
*■ 105 21 15 ■*
1935. The whole perimeter (2a) and one side (c) of a triangle are
given, prove that the mean value of its area is ^cJa{a — o)\ and that
the mean value of this mean value, c being equally likely to have any
value from to a, is «^ a*.
193G. The mean value of the area of all acute-angled triangles
inscribed in a given cirale of radius a is — , and the mean value of the
a*
area of all the obtuse-angled triangles is — .
1937. The mean value of the perimeter of all acute-angled triangles
inscribed in a given circle of radius a is - -^ , snd that of the perimeter
of the obtuse-angled triangles is — - , - .
1938. The mean value of the distance from ono of rtie foci of all
3 -»- «*
points within a given prolate spheroid is a — - ' .
1939. Tlie mean value oi Jxyz where x, y, ;; are areal co-ordinates
47r
of a iwint within the tiiangle of reference is . ^_ ; and the mean value of
Jwxyzy where to^ Xy y, z are tetrahedral co-ordinates of a point within
the tetrahedron of reference is ^^ . Also the mean value of (wxyz)*'^
is 6 (r {n))* ^ r (4»).
1940. Prove that the mean value of Jx^x^^...x^ for all positive
values of a;,, aj^,... suchthat a5,-l-ap,-»-...-l-a;^=l isr(n)|rfH)[ "^^("2")'
and, more generally, that of (aj^aj, ... xjr\ r being positive, is
r (n) {V {r)Y + r (nr).
W. r. 22L
338 INTEGRAL CALCULUS.
1941. In the equation a^-^w + rsOitis known that q and r both
lie between — 1 and + 1 ; assuming all values between these limits to be
equally probable, prove that the chance that all the roots of the equation
shall be real is 2 -r 15,^3.
1942. A given finite straight line is divided at random in two
points : prove that the chance that the three parts can be sides of an
acute-angled triangle is 3 log 2 — 2.
1943. A rod is divided at random in two points, and it is an even
chance that n times the sum of the squares on the parts is less than the
square on the whole line : prove that
w(4ir + 3^3) = 12ir.
1944. On a given finite straight line are taken n points at random :
prove that the chance that one of the n + 1 segments will be greater than
half the line is (w+ 1) 2"".
1945. A straight line is divided at random by two points : prove
that the chance that the square on the middle segment shall be less
than the rectangle under the other two is (4ir - 3 J3) -i-^ J3 ; and the
chance that the square on the mean segment of the three shall be less
than the rectangle contained by the greatest and least is '41841...
1 946. A rod is divided at random in three points ; the chance that one
of the segments will be greater than half the rod is '5, and the chance that
three times the sum of the squares on the segments will be less than the
square on the whole is ir-r 6^3. Also the chance that in times the sum
of the squares on the segments will be less than (n + 1) times the square
on the whole (w > 3) is w ^ 2n\
1947. A given finite straight line is divided at random in (1) four
points, (2) n points ; the chance that (1) four times, (2) n times the sum
of the squares on the segments will be less than the square on the whole
line is
1948. A given finite straight line of length a is divided at random
in two points ; the chance that the product of the three segments will
exceed j^ a" is
oL^^y^
+ 2 cos 3x dx.
1949. The mean value of the distance between two points taken at
random within a circle of radius a is 121a-r45ir; the corresponding
mean value for a sphere is 36a -t- 35. The mean distance of a random
INTEGRAL CALCULUS. 339
point within a given sphere from a fixed pointy (1) without the sphere,
(2) within the sphere, is
(1) c^rc' (2) T^ 2^-20^-
c being the distance of the fixed point from the centre of the sphere and
a the radius of the sphere.
1950. The mean value of the distance of any point within a sphere
of radius a from a point in a concentric shell of radius 6 is
3 (g -f 6) (5a' -f 76')
20 a'-i-ab + b' '
1951. A rod is marked at random in three points; the chance tliat
n times the sum of the squares on the segments will be less than the
square on the whole is K^)* , *"• 673 l^" ^®- (^-^) }'
according as n lies between 3 and 4 or between 2 and 3.
1952. A point in space is determined by taking at random its
distances from three given points A, B, (J \ prove that the density of
distribution at any point will vary directly as the distance from the
plane and inversely as the product of the distances from j1, J3, (7.
1953. Points P, <?, R are taken at random on the sides of a triangle
ABC) the chance that the area of the triangle PQR will be greater than
(n + J) of the triangle ABC, {n being positive and < J), is
3-4^1 12n+l
^•'Si^i+^^'G-'^"^)-
1954. A rod is marked in four points at random, A bets B £50
even that no segment exceeds ^ of the whole: prove that^'s expectation
is 3«. llc^. nearly.
1955. A given finite straight line is marked at random in three
points; the chance that the square on the greatest of the four segments
will not exceed the sum of the squares on the other three is
12 log 2 -ir- 5.
1956. From each of n equal straight lines is cut off a piece at
random ; the chance that the greatest of the pieces cut off exceeds the
sum of all the others is 1 : |n — 1 ; and the chance that the square on the
greatest exceeds the sum of the squares on all the others is
1957. A rod ^^ is marked at random in P, and points Q, R are
then taken at random m APy PB respectively: prove that the chance
that the s\im of the squares on A Qy RB will exceoid the sum of those on
QPy PR is -5; but, when Q, R are first taken at random in AB and P
then taken at random in QR, the chance of the same event is
I- (3 - 2 log 2).
22—2.
340 INTEGRAL CALCULUS.
1958. Three points P, Qy R are taken at random on the perimeter
of a given semicircle (including the diameter) : prove that the mean value
of the area of the triangle PQR is
a being the radius.
1959. A rod being marked at random in two points, the chance
that twice the square on the mean segment will exceed the sum of the
squares on the greatest and least segments is *225 nearly.
I 1960. The curve p (2a - r) = a*, p being the perpendicular from the
! pole on the tangent, consists of an oval and a sinuous branch : the oval
' being a circle, and the sinuous branch the curve
>-«-»=0-3)('-.-^)-
P
1961. Trace the curve r* = -— — ^,, the prime radius passing through
a point of the curve where r= 2a: discuss the nature of this point and
prove that, if perpendiculars OF, OZ be let fall from the pole on the
tangent and normal at any point, TZ will touch a fixed circle.
1962. Find the differential equation of a curve such that the foot
of the perpendicular from a fixed point on the tangent lies on a fixed
circle : and obtain the general integral and singular solution.
[Taking the fixed circle to be a;* + y* = a*, and the fixed point (c, 0),
the differential equation is
(y-^)» = a'(l+y)-c';
which is of Chviraut s form.]
1963. Reduce the equation (a;-/>y)(a;--j = c* to Clairaut's form,
by putting o^ = X, y* = F, and deduce the general integral and singular
solution.
a^ f/* 1
[The genera] integral is -5 — r + t^-v = «» ^^^ ^^*® singular solution
is tkrctky^c]
1964. Along the normal to a curve at P is measured a constant
length PQ\ is a fixed point and the curve is such that the circle
described about OPQ has a fixed tangent at : find the differentia]
equation of the curve, the general integral, and singular solution.
[Taldng for origin, and the fixed tangent at for axis of a, the
differential equation is of ■¥ 2xyp - y* '= ci/ Jl +/>*; the general integral is
a:* + y* - 2aa: + 6* = 0, where b' (b' + c') = aV,
IKTBGRAL CALCULUa 341
and the singular solution i8fl^ + y' = «kcy. K the singular solution
be deduced from the general integral, the student should account for the
extraneous factor^ e and a?.]
1965. The ordinate and normal from a point P of a curve to the
axis of X are Pilf and PG : find a curve (1) in which PAf' varies as PG;
(2) in which the curvature varies as PM^-i-PG*, and prove that one
species of curve satisfies both conditions.
[The curve (1) is the catenary — =W€»+m"*e~«; and the curve (2)
V
2?/ ■ -?
is -ii = me + nc •, which coincides with the former when mn = 1 : or
c
2y = A cos - + ^ sin - .]
c c
1966. Prove that the equation ^ay^J-^fx) "^^^i""^ " *^®
general equation of a parabola touching the co-ordinate axes; and
ileduce (1) that, if in a series of such parabolas, the curvature has a
given value when the tangent is in a certain given direction, the locus
of the points where the tangent has this direction is an hyperbola with
asymptotes parallel to the co-ordinate axes and passing through the
origin where its tangent is in the given direction and its curvature
is four times the given curvature, (2) that if a straight line from the
origin meet one of the parabolas at right angles in the point {x^ y) the
radius of curvature at (x, y) will be
2a;y^x' + y* + 2xt/cos<D . ,
; r~/ rSin (On
(x •¥ 1/ COS Q}) (1/ + X cos <u) '
where <u is the angle between the co-ordinate axes.
1967. Find the general solution of the equation
and prove that a singular first integral is
* ■»(!)■=»•
[The general solution is jb* + y* = 2a (Xy + X'x + /i), and one general
first integral is x + y ^= « (^* + 2X^^ .]
1968. Prove that the equation
(a^ + y' - 2a?y;>)« = 4ay (1 -!>•)
can be reduced to Clairaut's form by putting ^ - y* = 2« ; and obtain the
general and singular solutions.
[(aj«-y«-X«-a7 = aVa*-X'); «»-y« = *2ay.l
^•'y^-^
342 INTEGRAL CALCULUS.
1969. Find the general and singular solutions of the equation
(a;'-y')'(l+;>») = 2(«'+l,3/')'.
[Reduce by putting o^-k-r^—Y^ 3xt/ = X : the solutions are
a^ + y*-3aa5y = a', »• + 60^3/" + 3^ = 0.]
1970. Find the general differential equation of a circle touching
the parabola j^ = iax and passing through the focus ; and deduce that
the locus of the extremities of a diameter parallel, (1) to the axis of y,
(2) to the axis of x, is
(1) 8y« {x + 2a)' = 27a (a:" + y*)*, (2) (a:" - y* + 4aa;)» = 27aa; {of + y*)*.
1971. The equation a^ ( -p) +2a6-^ + y*=0 has the singular
solution xy = ah and the general solution is formed by eliminating from
the equations
a:' = \a5€-«(l-Bintf), y* = X"»a6 €«(1 + sintf).
dy
x-y ^
1972. Solve the equation — + ^ = j , ; and examine the
"■^y^
X* V*
nature of the solution — + y^ = 1 .
a
2iab ,. I » b
[The general solution is a:*+y* = -;;-^-^lbg \ ^ J y so that
a* y*
— + T" = 1 is the particular integral corresponding to C = 0.]
1973. Find the general solution of the equation y + a; -~ = -- ; and
dx
examine the relation of the curve y* + 4a« = to the family of curves
represented by the equation.
[The general solution is (y* - 12aa:y - Xf + (y* + 4aa;)" = 0, and each
curve of Qie family has a cusp on the limiting curve y* + 4aiB = 0.]
1974. The general solution of the equation 2y = a; -3^ + -7- is
dx
(y' - 4aa;) (x' - 4Xy) + 2aAa:y = 27aV ;
and each such curve has a cusp Ijring on the curve y" = Soa:.
1975. The general solution of the equation x f ~ j - »wy / + a =
is found by eliminating p between the equations
1 \ m
(2m - 1) a; = ap"' + Xp"^'^ , (2m - 1) y = 2ap~* + - ;/«-"» ;
except when 2m := 1, when p'x -= 2a logp + X, />y = 4a log p + 2a + 2A.
INTEGRAL CALCULUS.
S43
1976. Find the orthogonal trajectory of the circles
X being the parameter,
1977. Find the orthogonal trajectory of the rectangular hyperbolas
a" - y* - 2Xx + a' = ; and prove that one solution is a conic.
[The general solution is y (3ar* + y* - Sa*) = 2ft*, which has an oval
lying within the conic 3x' + y' = 3a', if ft* < a*, an acnode when ft* = a*
and lies altogether without the conic when ft* > a*.]
1978. Prove that the orthogonal trajectory of the family
r^ = X"cos7ifl is r^ = ft"sinnd.
1979. The orthogonal trajectory of the system of ellipses
3u:*+y* + 3a* + 2Xa; = is y* = ft (x* - y* - a*).
1980. Integrate the equations :
(1
(2
(3
(4
(S
(6
{<
(8
T-5 - cot « J ■ + w sin' as = 0,
(far a*
rf*l« rf'lfc rf*w rf*w c?'m c?'u .
(te* dkf e/«* c^yc/s (/sc^ c/ax/y '
d^\i d*u d'u „ d'u ^
^ — . -f -f. 3 =
etc* rfy* cfo* dydz '
(i"w (£*w d*u
-zJ>-,-.^o.
dx^ c/y" ci»" dxdydz
1981. Having given the equaUon *^ + ^2^+*^ = ^> *^^
that, when a;»0, y=0, ;r=l, zS^^^i P«>v® *^*^ '^^^'^ aj is oo,
(£e
die*
IT
344 INTEGRAL CALCULU&
1982. Integrate the equation
-jZ [JcosyooBX-^ miysinx) = 2 cos ysecx :
and examine the nature of the solution ^ = -„ .
[The general solution is y + X =: 2 tan x J cos y. ]
1983. Integrate the equation
-^- cos a: cos (y - a;) = cos y ;
and examine if the solution y=2x-¥ 2rir - ^ is a general solution.
[The general solution is X cos a; = sin (a; - y).]
1984. Solve the equation
by patting y = zcoBX,
1985. The general solution of the equation .
u^ (2x + 1 - u^+j) = a^
u^ — x \x/
1986. A complete primitive of the equation
is w, + 1 = (a: + (?)■; and another is », + 1 = {C (- 1)* - J}* : also deduce
one of these as the indirect solution corresponding to the other.
1987. Solve the equations
(1) w,^, + 3w.-4u/ = 0,
(2) w,^, = a;(tt, + w,.,),
(4) 2 K,, - w,)* = K^, + 2w,) (w, + 2w,,.);
1988. Prove that the limiting value of
when n is an indefinitely great positive integer, is log 2.
1989. Prove that, if r be a positive integer so depending upon x
that X - nr always lies between - ^ and - , i dx = ^ log 2.
IS
INTEORAL CALCULUS. 345
1990. Solve the equations :
(4) {a^-yz)£^{i/'-zx)p^^^-xij,
(5) tan2^:*$^4.tan^^$ = tan?-:2'.
^ ^ 2 etc 2 dij 2
[(1) (x' + y« + ay-4aV = A, (2) y == (.1 cos a + J5 sin «)"•,
(3) y=ao + ajaj + a^+ ... +a a:^ + 6a:"loga;+^ , «^(iog«)*;
i_
(4) y- « = (a - y)/(y« + aa: + ay),
(5) COB (y + «) + cos (« +aj) + cos (a; + y)
=y {sin (y + «) + sin (« + a:) + sin (« + y)}.]
1991. Prove the following equation for Bernoulli's numbers :
j-nB ^*(^-^)(^"2) n(n~l)(n-2)(n-3)(n-4)
74-1
to ^~ terms, where n is an odd integer. The equation will still be
91 *4* 1
true when n is an even integer if we multiply the last term by ,
n
;j; being then the number of terms.
1992. Two equal circles have radii 2a, and the distance between
their centres is 4c, a series of circles is drawn, each touching the
previous one of the series and touching the two given circles sym-
metrically : prove that the radius of the n*^ of such a series is
c sin* a 4- sin (na + ^) sin (?» - 1 a + ^),
where a^e cos a. Deduce from this the result when c ~ a,
a-r(w + X)(n + X-l);
and the residt when e<a,
[If c = 2«^(p+|,-'), the n* radius » (ftp. . y/.)^fep".-?_ ^y ...^ .
where 66' -c*.]
346 INTBORAL CALCULUS.
1 993. When ^ (() ■= (a + 6() -r- (c + et), prove that
^(0-
b + e 6Bipyix-eain(y-t-2)a-2efcoaaBin(y4-l)a c
Bco8o6Bm{y-l)o— C8in(y+l)a— 2e(coaoBmya a'
where {6 + c)*=4(6c-(ie)cos'o; and hence prove that the condition tht
^ is a periodic function of the a;"" order is
«uc ■ iir, or iaa cob' — i- 6' + c" - 26c cos — = 0,
X X
i being an integer not a multiple of x.
Find iii'{t) when (b + e)'>i(be- ae), and discuss the special case whe
(6 + c)' = 4(6e-«).
SOLID GEOMETRY.
I. Straight Line and Plane,
1 994. The co-ordinates of four points are a-5, a-c, a-c?; 6-c,
h-d, b-a; c-d, c-a, c-b; and d-a, d-b, d-c, respectively:
prove that the straight line, joining the middle points of any two
opposite edges of the tetrahedron of which they are the angular points,
passes through the origin.
1995. Of the three acute angles which any straight line makes
with three rectangular axes, any two are together greater than the
third.
1996. The straight line joining the points (a, 6, c), (a', b\ c') will
pass through the origin if om' + bb' ^-cc' = pp ', p, p being the distances
of the points from the origin, and the axes rectangular. Obtain the
corresponding equation when the axes are inclined respectively at angles
whose cosines are /, m, n.
\aa' + bb' + cc' + (6c' + 6'c) I + {ca' + ca) m + {ah' + a'b) n
= pp yjl-t^- m' - n* + 2lmn.]
1997. From any point P are drawn PJf, PN perpendicular to the
planes of zx, zy, va the oiigin, and a, P, y, B the angles which OP
makes with the co-ordinate planes and with the plane OMN\ prove
that
cosec* 6 = cosec* a -i- cosec* p + cosec* y.
1998. The equations of a straight line are given in the foims
(1)
a -f- mz -ny _b + nx—lz _ c-^ly — mx
I m n *
a-^-mz-ny ^b-{-nx-lz _ c + ly — mx
i M " N '
(2)
obtain each in the standard form
348 SOLID OEOMETBY.
[(1)
mc — nb na — lc lb—ma
Mc-Nh _ Na-J^_ _ Lb-Ma
1099. A straight line moves parallel to a fixed plane and intersects
two fixed straight lines (not in one plane) : prove that the locus of a
l>oint which divides the intercepted segment in a given i*atio is a straight
line.
2000. Determine what straight line is represented by the equations
.^. a + mz-ny b-\-nx-lz c-hly — nix
^ ^ m-n " n — l l-m '
/t}\ ^ "*■ ^'^ — ni/_b-hnx — lz_^c + li/'- mx
^"'' mc-nb' " na' — lc' lb' — ma'
[(1) The straight line at infinity in the plane
x{m — n) + y {n-l) + z {l-m) = ;
unless Za + m6 + nc = 0, in which exceptional case the line is indeterminate,
and the locus of the equations is the plane
x{m-n) + y {n-r) + z{l-m) = a + b + c;
(2) the straight line at infiinity in the plane
X (mc'-rib') + y {na' - lc')+z{lb' - ma') = ;
unless la-\-mh'¥nc = 0, when the locus of the equations is the plane
x(mc'-nb') •¥ y {na' -Ic') •¥ zifb' -ma') = oa' + 66' + cc'.]
2001. The two straight lines
^ yz zx xy ^
•^ ' b-c c-a a — b
are inclined to each other at an angle -^ .
2002. The cosine of the angle between the two straight lines
determined by the equations
Ix +.my + nz=0, aa? + by* + c«* = 0,
^"(6 + 0) +m'(c+a) + n*(a+6)
IS
Jl*(f)-cy+ ... + ... +2mV (a-b){a-c)-\- ... + ...
2003. A straight line moves parallel to the plane y — z and inter-
sects the curves
(1) y = 0, «• = (»; (2) « = 0, y'^bx:
prove that the locus of its trace on the plane of y;s is two straight lines.
[The locus of the moving straight line i8a: = (y-s)(T — ).]
SOLID QEOMETBT. 349
2004. The direction ooeines of a number of fixed straight lines,
referred to any system of rectangular axes, are (l^, Wj, n,), (Z^, m^, w^), &c. :
prove that, if S (0 = S (w*) = 2 (u'), and S (wn) = S (w^ = 2 (/m) = 0,
when referred to one system of axes, the same equations will be true
for any other system of rectangular axes. Also prove that, if these
conditions be satisfied and a fixed plane be drawn perpendicular to each
straight line, the locus of a point which moves so that the sum of the
squares of its distances from the planes is constant will be a sphere
having a fixed centre which is the centre of inertia of equal particles
at the feet of the perpendiculars drawn from 0, and that the centre of
inertia of equal particles at the feet of the perpendiculars drawn from
any other point F lies on OP and divides OF in the ratio 2:1.
2005. A straight line always intersects at right angles the straight
line x + f/ = z = 0, and also intersects the curve y = 0, sc^ = (iz: prove that
the equation of its locus is
a;* — y* - az.
2006. The equations
ax + % + gz _ Iix + 6y -h/z _ gx -k-fy + cz
X y " z
represent in general three straight lines,* two and two at right angles to
each otlier ; but, if a-^==6- — = c -"^ , they will represent a plane
and a straight line normal to that plane.
2007. The two straight lines
x^a * y z
cos a sin a '
meet the axis of a; in 0, (X; and points P, F" are taken on the two
respectively such that
(1) OF^k.ffF; (2) OF. ffF = <^', (3) 0F-^0'F=^2ci
prove that the equation of the locus of FF is
(1) (a? + a) (y sin a + « cos a) = A; (as — a) (y sin a-z cos a) ;
(2) '^ ^^' I ^ _1-
^ ' a* c'cos'tt c*8in*a ^
(3) ^L_J!L = £(»._«.);
^ ' COS a sma a^ '
the points being taken on the same side of the plane xy,
[Denoting OP, O'F by 2X, 2/bt, the equations of FF may l>e wnttcn
05 _ y - (X - ft) COS a _« - (X + ft) sin a
a" (X + /bt)coBa "" (X- ft) sin a '
so that, when any relation is given between X, ft, the locus may be found
immeiliately.]
350 SOLID GEOMETRY.
2008. A triangle is projected orthogonally on each of three planes
mutuallj at right angles : prove that the algebraical sum of the tetra-
hedrons which have these projections for bases and a common vertex in
the plane of the triangle is equal to the tetrahedron which has the
triangle for base and the common point of the planes for vertex.
[This follows at once from the equation x cos a + y cos fi+z cos y = |>
on multiplying both members by the area of the triangle.]
«B 4/ 2>
2009. A plane is drawn through the straight line - = ^^ = - :
prove that the two other straight lines in which it meets the surface
(6-c)y2;(m«-ny) + (c - a) zx (nx - Iz) -¥ {a-h)xy{ly-mx) =
are at right angles to each other.
2010. The direction cosines of three straight lines, which are two
and two at right angles to each other, are (^j, m^, n^), (^^, m^ n ),
(^a, ^3, n^), and
amjWj + 6n,^j + d^m^ = am^n^ + hnj^ + cl^m^ = :
prove that am^n^ + bnj^ + cl^m^ = ; and j-j-j =
I'a-a ^i^a'^3 ^I'^a'^s
2011. The equations of the two straight lines bisecting the angles
between the two given by the equations
Ix + my + nz = Oy aa? + fty* + C2* = 0,
may be written
Ix + my + nz = 0, I {b -^ c)yz + m (c " a) zx -\- n (a - b) xy = 0.
2012. The straight lines bisecting the angles between the two
given by the equations
lx-¥ my ■\'nz = 0y oaf + by* -^ cn^ -k- 2/yz + 2(;zx + 2hxy = 0,
lie on the cone
af{nh-mff)+ ... + ... + yz{m^~ng + l {b'-c)]+ ... + ... = 0.
2013. The lengths of two of the straight lines joining the middle
points of opposite edges of a tetrahedron are a;, y, a> is the angle between
them, and a, a' the lengths of those edges of the tetrahedron which are
not met by either x or y : prove that
4:xy cos 0) = a* - a".
2014. The lengths of the three pairs of opposite edges of a tetrahe-
dron are a, a'; b, b'; c, c': prove that, if ^ be the acute angle between
the directions of a and a\
2aa' cos 6 - (5* + b") ~ (c* + O-
SOLID GEOMETRY. 351
2015. The locus of a straight line which moves so as always to
intersect the three fixed straight lines,
y = m(6-a), « = 7i(c — a); « = n(c-6), x = l(a-h);
x==l{a — c), y = m{h-e);
is
lyz (6 - c) + mzx (c-a) + nxy (a-b)- mnx (6 - c)*- ... - ...
= 2lmn (h — c) (c- a) (a — 6) :
and every such straight line also intersects the fixed line
a(x-al) _ h{y- brn) c (z-cn)
I m n '
2016. The straight line joining the centres of the two spheres,
which touch the faces of the tetrahedron A BCD opposite to ^, ^
respectively and the other faces produced, will intersect the edges
CBy AB (produced) in points P, Q respectively such that
CP :PD = tkAGB : tJLDB, and AQ:BQ = t^CAB : LCBL.
2017. On three straight lines meeting in a point are taken
points A^ a; B, b; C, c respectively: prove that the intersections of
the planes ABC, abc; aBC, Abe; AbC, aBc ; and A Be, ahC \ all lie on
one plane which divides each of the three segments harmonically to 0.
2018. Through any one point are drawn three straight lines each
intersecting two opposite edges of a tetrahedron ABCD ; and a,f\ b, g;
c, h are the points where these straight lines meet the edges BC, AD ;
CA, BD ; AB, CD : prove that
Ba ,€h. Dg = Bg . Ca . Dh,
Cb, Af,Dh = Ch.Ab.Df,
Ac, Bg . Df=^Af. Be . Dg,
Ab , Be , Ca =Ac , Ba . Cb.
2019. Any point is joined to the angular points of a tetrahedron
ABCD, and the joining lines meet the opposite faces in a, b, c, d:
prove that
Oa Ob Oc Od
Aa'^Bb'^Cc'^'Dd' '
regard being had to the signs of the segments. Hence prove that the
reciprocals of the radii of the eight spheres which can be drawn to touch
the faces of a tetrahedron are the eight positive values of the expression
1111
±-± — It— ±— ;
Pi P, Pz Pa
Pv P%y P^y Pa ^^g ^^ perpendiculars from the comers on the opposite
faces.
ftp* ^
•XZ3 p:^ 1 i^V
in- * f
- 7
tBnr"»i-^ r^ f.
r. I TT'.'B^r
^: _ 7K _ "-«"*_•-
.L X
Z- -^ X
.I^f
.=-—:•:»? —
eXOrl'I ▼-T^
n 1 ^
1:1". A T.:i^:
air:
T-r-.l
• • -• wr ~ •," 'a
srrr*
jL' 90 AS to
tie pnp
of <>in
rrrci Ti»*
■5 — ^ *'■'■ "■ ■^ _ " t "• ' ~ %
-A, A C 2). 1
4.- =
SOLID G£OM£TBY. 353
2024. The point is such that the sum of its distances from four
fixed points A^ BfC,I>iB the least possible : prove that any two opposite
edges of the tetrahedron A BCD subtend equal angles at ; and that, if
AOA\ BOB", COC\ BOB' be drawn to meet the faces, the harmonic
mean between AO, OA' will be one half the harmonic mean between
AO, BO, CO, BO.
2025. The equation of a cone of revolution which can be drawn to
touch the coordinate planes is
{Ix)^ + (my)* + (nz)* = 0,
the ratios I \m \n being given by the equations
m* + 71* + 2mn cos a _ w' + Z* + Inl cos j3 __ Z* + m* + 2lm cos y
sin* a ~ sin*/3 " sin'y '
where a, )8, y are the angles of inclination of the coordinate axes.
[For the solutions of these equations, see question 456.]
2026. The equations of the axes of the four cones of revolution
which can be drawn to touch the co-ordinate planes are
sm a sm p sin y
2027. The inscribed sphere of a tetrahedron ABCB touches the
faces in A', B, C", B' : prove that AA\ BF, CC\ BU will meet in a
point, if
a a h B c y
cos ^ cos ^ = cos^ cos ^ = cos^ cos ^ ;
where a, a; h, P; c, y are pairs of dihedral angles at opposite edges.
[For a sphere touching the face A and also the faces B, C, B pro-
duced; w — ay ir-P, TT-y must be written for a, fi, y; and for the
sphere in the compartment vertically opposite the dihedral angle BC^
ir-byW-Cyir-p, ir-y must be written for 6, c, j8, y.]
2028. There can in general be drawn two quadric cones con-
taining a given conic and three given points not in the plane of the conic*
[The general equation of a conicoid satisfying the conditions may be
written
where X is the only undetermined quantity.]
2029. The equations of the axes of the four cones of revolution
which contain the co-ordinate axes are
XV z
, a b c
where =r = a-rr = tt i
cos a ± 1 cos p^l cos y * 1
an odd number of negative signs being taken in the ambiguities, and
a, p, y being the angles of inclination of the co-ordinate axes.
w. P. 23
354 SOLID QEOMETBT.
2030. A point is taken such that the three straight lines drawn
through it, each intersecting two opposite edges of the tetrahedix>n ABCD^
are two and two at right angles, and a, /3, y, 8 denote the perpendicu-
lars let fall from on the faces of the tetrs^hedron: prove that
, <&c.
and that
1
1
A
2cos^Z>
1 1
2cosBC
a8
he
Oh.Oe'
ad
' Oa. Od'
&c.
n. Linear Trana/ormaiians. General Equation of the Second Degree.
[The following simple method of obtaining the conditions for a
surmoe of revolution is worthy of notice.
When the expression aot? + 6y* + C2* + %fgz + 2gzx + 2hxy is trans-
formed into AX* ■hBY'+ C^\ we obtain the coefficients A, B, C from
the equivalence of the conditions that
X (a:" + y* + »■) - aa^ - ...
and X(X"+P + ^)-^X'-^r«-C^«
may break up into (real or impossible) linear factors : which is the case
whenX = ^, X or C,
But, should two of the three coincide as J9 = C, then when X = ^
the corresponding factors coincide, or either expression must be a
complete square. The conditions that the former expression may be a
complete square when X = ^ give us
{B - a)/= - gh, <fec.,
J 9 ^
provided^, ^, & be all finite.
Should we have /= 0, then gh must = 0; suppose then /and <7 = 0,
then -5 = c, and we must have (c-o)a:*+ (<5-6)y'-2^ay a square^
whence
A«=(c-a)(c-6).
In the case of oblique axes, inclined two and two at angles a, fi, y,
we must have
X(a;" + y*+«* + 2^coSa+ 22sc cos j8 + 2ay cos y) - oo^ - ... -2^— ...
a complete square.
SOLID GEOMETBT. 355
It follows that the three equations
(X-o) (Xcos a-/) = (Xco8j8-^) (XcoBy-A),
(X-5)(Xco8j8-^) =(Xcosy-A) (Xcosa-/),
(X — c) (X cos y - A) = (X cos a -/) (Xcos j8 - g\
must be simultaneously true ; and the two necessary conditions may be
found by eliminating ^.]
2031. Determine the nature of the curve traced by the point
a; = aco8(tf+^j, y = acostf, « = acosftf-^j.
[A circle of radius ^/I^O
2032. In two systems of rectangular co-ordinate axes, tf , 0^, 0^ are
the angles made by the axes of x\ y\ ^ with the axis of z^ ana ^^ ^^ ^^
the angles which the planes of zoi^ zj/j vd make wiUi that of 9X\
prove that
tan'g,^ ,^^it'"ti ^^ = Q>
» 008 (<^, - <^,) cos (f^, - <^3)
with two similar equations.
2033. By direct transformation of co-ordinates, prove that the
equation
represents an ellipsoid of revolution whose polar axis is one half of its
equatoreal^ and the equations of whose polar axis are x^y^z.
2034. Prove that the surface whose equation, referred to axes
inclined each to each at an angle of ^,is ys + «B+afy + a* = 0, is cut by
the plane x-\-y-¥% = ^ in a circle whose radius is a and by the plane
x-^y + z= 12a in a circle whose radius is 7a.
2035. A quadric cone is described which touches the co-ordinate
planes (rectangular) : prove that an infinite number of systems of three
planes, two and two at right angles, can be drawn to touch it; and that,
if (/j, m,, Tij), {l^ m^y rij), (^3, m^ wj be the direction cosines of any such
system, the equation of the cone will be
(W^)^ + {m^fnjrnj/^ + (n^njnji)^ = 0.
2036. A point is taken and also the common point of its polar
planes with respect to three given spheres: prove that the sphere in
which these two points are ends of a diameter will contain a fixed circle
and will cut each of the given spheres orthogonally.
356
SOLID GEOMETRY.
2037. In the expression
oaf + hi^ + csf -h 2/yz + 2gzx + 2hxy + 2Ax + 2By + 2Cz -^ D^
prove that
and
A*{b + c)-h...-h...-/BC'-gCA-hAB,
A' {he -/*) + ... + ... + 2BC (sih-a/)-h ... + ...
are inyariants for all systems of rectangular co-ordinates having the
same origin.
2038. Prove also that the coefficients in the following equation in X
X + a, Xco8y + ^, Xcosj8 + ^, -4-0;
Xcosy + A, X + 5, Xcosa+yj B
Xcosj8+^, Xcosa+^ X + c, C
A, B, C, B
€Ly P,y being the angles between the co-ordinate axes, are invariants for
all systems of co-ordinates having the same origin.
2039. Assuming the formulsB for transforming from a system of
co-ordinate axes inclined at angles a, j9, y to another inclined at angles
«'> iS', y to be
prove that
1 = ^^» + 1; + 1^ + 2^3 cos a+ 2^^ cos j84- 21}^ cos y,
-with similar equations in m and n ; and that
cos a' = miWj + m,n, + mJ^^^r (m^w, + m^^ cos a 4- ... + ...,
with similar equations in ti, /, and /, m.
2040. Prove that, if oo* + 5/ + c«* become AX* + BY* + CZ* by any
transformation of coordinates, the positive and negative coefficients will
be in like number in the two expressions.
2041. The homogeneous equation
aa* -J- 5y" + c^* + 2fyz + 2gzx -»- 2hxy
will represent a cone of revolution if
gh J-V hf V-f
6-0 g
=
c — a
[These are of course equivalent to
SOLID GEOMETRY. 367
2042. The surface whose equation^ referred to axes inclined at
angles a, j3, y, is oaf + 6^* + c«* = 1, will be one of revolution, if
a COB a _ hcosfi _ ccosy
cos a — cos p cos y cos j8 — cos y cos a cos y — cos a cos p '
and the corresponding conditions for the surface ayz ■{■ hzx -¥ cxy = 1 are
a _ 5 _ c ^
1 ± cos a ~ 1 * cos p \^ cos y '
one, or three, of the ambiguities being taken negative.
2043. The equation
aa* + fty* + C2" + %fyz + 2gzx + 2 Aay + 2^ a: + 2 By + 2 C« + 2) =
will in general represent a paraboloid of revolution, if
and a cylinder of revolution if, in addition to these conditions,
Jgh + Bh/+ C/g = 0.
2044. The equation of a given hyperboloid of one sheet whose
equation referred to its axes is
can be obtained in the form x'-hj^ — z' = (r in an infinite number of
ways, provided that a' — c^y b' — c^ are not both negative ; and the new
axes of X and y lie on the cone
and the new axis of z on the cone
^ (&• - «0 + ^ (a'-C) - J («• + 6') + 2 («' + y' + aO= 0.
2045. The equation of a given hyperboloid may be obtained in the
form
ayz + bzx + cxy = 1
in an infinite number of ways; and, if a, j8, y be the angles between the
co-ordinate axes in any such case, the expression
ahc
1 - cos" a — cos* P — cos* y + 2 cos a cos )3 cos y
will be constant.
2046. Prove that the only conoid of the second degree is a hyper-
bolic paraboloid ; and that it will be a right conoid if the two principal
sections be equal parabolas.
[The equation of a conoid must be reducible to the form «=/(-)>
and this will be of the second degree only when
358 SOLID GEOMETRY.
2047. A cone is described having a plane section of a given sphere
for base and vertex at a point on t^e sphere; the subeontrary sectiona
are parallel to the tangent plane at 0.
2048. A cone whose' vertex is the origin and base a plane section
of the surface aa? + &^ + c«' = l isa cone of revolution : prove that the
plane of the base must touch one of the cylinders
(6-o)y"+(c-a)«' = l, (c-5)«« + (a-6)a:» = l, (a-c)a»+(6-c)y»= 1.
2049. A cone is described whose base is a given conic and one of
whose axes passes through a fixed point in the plane of the conic : prove
that the locus of the vertex is a circle.
2050. The locus of the feet of the perpendiculars let fall from a
fixed point on the tangent planes to the cone aaf-¥ (y'+ c;2^ = is a plane
curve : prove that it must be a circle, and that the point must He on
one of file three systems of straight lines
05=0, 6(c-a)y* = c(a-6)«*, &c.
[One only of the three systems is real.]
2051. Prove also that, when the point lies on one of these straight
lines, the plane of the circle is perpendicular to the other ; and that a
plane section of the cone perpendicular to one of the straight lines will
have a focus where it meets that straight line, and the excentricity will
1 .
be equal to ''J{b'-a){c-a),
2052. A plane cuts the cone ayz + bzx + cxy = in two straight lines
.at right angles to each other : prove that the normal to the plane at the
origin also lies on the cone.
2053. The centre of the surfiace
a(a»+2y«) + 6(/+2«») + c(«" + 2a?y)-2ila;-2%-2C«+l =
is (X, r, Z) : prove that
and that the surface will be a cylinder whose principal sections ar^
rectangular hyperbolas, ifa + 6 + c = 0, A+B + C = 0,
[In this case the axis of the cylinder will be
Aa Bh Cc
* a' + 6'+c""*^ a' + ft' + c'""* ^rri«lV-J
2054. The radius r of the central circular sections of the sur&ce
Q/yz + hzx + cxy = 1 is given by the equation
abci^-^{a* + 6' + c*) r* = 4;
and the direction cosines (^ : m : n) of the sections by the equations
SOLID GEOMETKY. 359
2055. The semi-axes of a central section of the surfiaco
ayz + hzx + coey + ahc = 0,
made by a plane whose direction cosines are ^ m, n, are given bj the
equation
r* (26cmn + ... -aV - ...) ''iahci^{anm +...) + 4a*6V= 0.
«
2056. The section of the surface y« + jkb + ojy = a* by the plane
Ix-^myA-nz =p will be a parabola if /^ + m^ + n^ = ; and that of the
surfiftce fB' + y" + «"-2y«-2«B- 2a:y=a' will be a parabola if
mn + nl + lm = 0,
2057. Prove that the section of the surface u = by the plane
Ix + my + w» = will be a rectangular hyperbola, if
? (6 + c) + 7i»* (<5 + a) + n' (a + 6) = 2mf\f+ 2nlg + 2lnih ;
and a parabola, if
^{hc-/*)+ ... + ... +27im(^A-q/*) + ... + ... = 0;
and explain why this last equation becomes identical when
gh = a/, hf=hg^ fg^cJh.
[The surface when these conditions are satisfied is a paitibolic
cylinder, and every plane section will obviously be a parabpla, reckoning
two parallel straight lines as a limiting case.]
2058. Prove that, when hg = A/'and ch -fg^ the eqi;ation
t*=aa:* + 6y* + c»'+ %fyz + 2gzx + 2hxy + 2Ax + 2J?y -f 2Cz + D =
represents in general a paraboloid, the direction cosines of whose axis are
as (0 : g : —h).
2059. Prove that the tangent lines drawn from the ori^ to the
surface t^ = lie on the cone
I)U''{Ax'hBy+Cz+J)y = 0;
and investigate the condition that the surface u may be a cone from the
consideration that this locus will then become two planes.
2060. The generators drawn through the point (X, F, Z) of the
surface ayz + bzx + cxy + abc = will be at right angles, if
2061. The generators of the surface t^ = drawn through a point
(X, r, ^) wiU be at right angles, if
fdu\'fd'u d'rr\ dUdUd'U
\dx) \dY' ^ dZ')'^ - "** - "^dYdZ dfdZ
+ ... T •••
360 SOLID GEOMETRY.
2062. Normals are drawn to a conlcoid at points lying, along a
generator : prove that they will generate a hyperbolic paraboloid wlioae
principal sections are equal parabolas.
[It is obvions that the surfiBM^ generated is a right conoid.]
2063. The axes of the two surfaces
^a?" + ^y* + C«* - (oa; + 6y + cjs)" = €*,
are coincident in direction.
2064. The two conicoids
aa;* + 6y" + c«' + 2/y«+ 25^20? + 2Aay = l, .(ix' + J5/ + C«'= 1,
have one, and in general only one, system of conjugate diameters coin-
cident in direction j but, if
K-?)4('-r)4(«-4).
there will be an infinite number of such systems, the direction of one
diameter being the same in all.
[If ^ m, n be the direction cdsines of any one of such a system, we
have the equations
giving for X the cubic
(a-Xil)(6-Aj5)(c-XC)-/'(a-^)-^'(6-A^)-A'(c-X(7)+2/SrA = 0;
which may be written in the form
9^ . V . f9
1 =
af-gh^kAf^ bg-hf^XB ch-fg-XC'^
2065. Prove that eight conicoids can in general be drawn contain-
ing a given conic and touching four given planes.
2066. The equation of the polar reciprocal of the surface
CKB* + fty* + ca^ + yyz + 2gzx-^2haiy = 1
with respect to a sphere, centre (X, F, Z) and radius >t, is
A{X(a;-Z) + r(y-7) + ir(«-Z)}'=(6cV*)(«-^T+...
+ 2(srA-q/')(y-r)(«-Z)+...,
where A is the discriminant.
SOLID QEOMETRT. 361
2067. Prove that, if l^, ?,,... 2^ be constants so determined tliat the
expression
where u , t^^ ... u, are given linear functions, is the product of two
factors, the two planes corresponding to these feictors will be conjugate
to each other with respect to any conicoid which touches the seven
planes it = ; and that, when the expression is a complete square, the
corresponding plane is the eighth plane which touches every conicoid
drawn to touch the other seven.
2068. Seven points of a conicoid being given, an eighth is thereby
determined; eight points A^, A , .,, A^ being given, from every seven is
determined an eighth accordingly, giving the points B , B^,.. B^\ prove
that the relation between the A points and the B pomts is reciprocal,
and that 'the straight lines A^B^, -^fit'* ••• ^^ meet in one point.
2069. The straight line, on which lies the shortest distance between
two generators of the same system of a conicoid, meets the two in ii, ^,
and any generator of the opposite system meets them in P, Q respectively :
prove tiiat the lengths a;, y of AP^ BQ are connected by a constant rela-
tion of the form
axy -k-hx + c7/ + d=^0,
2070. Two fixed generators of one system of a conicoid are met by
two of the opposite system in the points A, B ; F, Q ; respectively, and
A, B are fix^ : prove that the lengths x, y oi AP^ BQ are connected by
a constant relation of the form
(ixy + 6a; + cy = 0.
2071. An hyperboloid of revolution is drawn containing two given
straight lines which do not intersect : prove that the locus of its axis is
a hyperbolic paraboloid, and that its centre lies on one of the generating
lines through the vertex of the paraboloid.
in. Conicoida referred to their cuces.
2072. The curve traced out on the surface %- + —=x by the
be ^
extremities of the latus rectum of any section made by a plane through
the axis of x lies on the cone y* + «" = 4a;*.
2073. The locus of the middle points of all straight lines passing
through a fixed point and terminated by two fixed planes is a hyperbolic
cylinder^ unless the fixed planes are parallels.
2074. An ellipsoid and a hyperboloid are concentric and confocal :
prove that a tangent plane to the asymptotic cone of the hyperboloid
will cut the ellipsoid in a section of constant are&
362 SOLID GEOMETRY.
2075. The loons of the centres of all plane sections of a given
conicoid drawn through a given point is a similar and similarly situated
conicoid, on which the given point and the centre of the given surface
are ends of a diameter.
2076. An ellipse and a circle have a common diameter, and on any
chord of the ellipse parallel to this as diameter is described a circle whose
plane is parallel to that of the given circle : prove that the locus of theae
circles is an ellipsoid.
2077. Of two equal circles one is fixed and the other moves parallel
to a given plane and intersects the former in two points : prove that the
loons of the moving circle is an elliptic cylinder. If instead of circles
we take any two conies of which one is fixed and the other moves
parallel to a given plane without rotation in its own plane, and always
intersects the fixed one in two points, the locus of the moving conic is a
cylinder,
fThere is no need to use co-ordinates of any kind.]
2078. A given ellipsoid is generated by the motion of a point fixed in
a certain straight line, which straight line moves so that three other points
fixed in it lie always one in each of the principal planes : prove that
there are four such systems of points ; and that, if the corresponding
four straight lines be drawn through any point on the ellipsoid, the
angle between any two is eqiial to the angle between the remaining
two.
a* V* «*
[If a;, y, « be the point on the ellipsoid -j + ?« + -«= 1, the direction
£t! t/ <S
cosines of the four straight lines will be - , "^ ^ , *>^ - .]
2079. Prove that, when a straight line moves so that three fixed
points in it always lie in three rectangular planes, the normals drawn at
different points of the straight line to the ellipsoids which are traced out
by those points will in any one position of the straight line all lie on an
hyperboloid.
[when 2^ m, n are the direction cosines of the straight line, the locus
of the normals is
g(6-c) + ...+(6-c)(6 + c-2a)|+...+2(6-c)(a-6)(a-c) = 0,
20^ 26, 2c being the axes of the ellipsoid.
2080. From a fixed point on an ellipsoid are let fall perpendicu*
lars, (1) on any three conjugate diameters, (2) on any three conjugate
diametral planes of the ellipsoid : prove that in each case the plane
nassing through the feet of the perpendiculars passes through a fixed
jnt^ and that this point in (2) lies on the normal to the ellipsoid at O.
BOLID QEOMETBT, 363
[If (X, Tf Z) be the point 0, the fixed point in (1) is given by
X _ y _ _«_ _ 1
a*X" 6T^" c'^" a'+ 6'+ c"'
and in (2) by
x-X y-7 z-Z 1 ,
-^f 7" — ^ \ 1 l.'J
a' 6« c- a''^6''^c'
2081. At each point of a generating line of a conicoid is drawn a
straight line in the tangent plane at right angles to the generator : prove
that the locus of such straight lines is a hyperbolic paraboloid whose
principal sections are equal parabolas.
2082. The three acute angles made by any system of equal oonjugate
diameters of an ellipsoid will be always together equal to two right
angles, if
2 (6" + c"-2a') (c" + a"-25«) (a" + 6" - 2c') + 27a'6V = ;
2a, 25, 2c being the axe& Deduce the condition that an infinite number
of systems of three generators can be found on the cone
ula' + ^y» + C7«' = 0,
such that the sum of the acute angles in any such system is equal to two
right angles.
[The condition is found by eliminating X from the equations
ul(^-X)-» + ^(5-X)-» + (7(C7-X)-' = 0, X'=2ii^a;
and, if il, ^, (7 be roots of the equation
«"- 3/?,»" + 3p^-p,= 0,
the result is ^," + 12p,p,/?, +i>i*Pa=' ^^ft*-]
2083. The locus of the axes of sections of the surface
aa:* + 6y" + c«"= 1,
2! t/ 2f
made by planes containing the straight line ^ = ^ » - , is the cubic
cone
(6 - c) y» (w? - ny) + (c-a) «a;(wa?-&) + (a- 6) «y (^- ww;) =0.
a^ t/* 2*
2084. Two generators of the hyperboloid -? + ^-t»1 drawn
through a point intersect the principal elliptic section in points
Fy F at the ends of conjugate diameters : prove that
S6^ SOLID GEOMETRT.
20S5. The generators of a given conicoid are orthogonally projected
upon a plane perpendicular to one of the generators : prove that their
projections all pass through a fixed point.
2086. The orthogonal projections of the generators of the conicoid
cut? + fty" + c»* = 1 on the plane Ix + my + w« = in general envelope a
conic which degenerates if a^ + 6m' + cn' = 0; and which is siTnilar to
a" V* «*
the section of the reciprocal surface — + ~ + — = m* by the plane.
2087. Frpm different points of the straight line -j = ^ = -, asymp-
. a" t/* «*
totic straight lines are drawn to the hyperboloid -^ + |g - -^ = 1 : prove
that they will lie on the two planes
2088. The asymptotes of sections of the conicoid aoc^ + Jy* + <»■ = 1
made by planes parallel to ^ + mi/ + n« = lie on the two planes
{nc+m'ca-hn^db) {oaf + hi^ ■\' cs^) = abc (& + my + 7Mf)*.
2089. The locus of points from which rectilinear asymptotes at
right angles to each other can be drawn to the conicoid aa"+ 6y*+ cs8*= 1
is the cone
a" (5 + c) V + 6* (c + a) y + c* (a + 5) «" = 0.
2090. The locus of the asymptotes drawn from a point (X, F, JS) to
the system of confocal conicoids
V y" «• 1
J A = 1
a' + A 6' + X c* + X
is the cone
(a:-X)(y-(0 (y D(c«-a') (^-^(a«-6') _Q
Yz'Zy ■*■ Zx-'Xz Xy^Yx
2091. A plane which contains two parallel generators of a given
conicoid must pass through the centre, and touch the asymptotic cone.
2092. A sphere is described having for a great circle a plane section
of a given conicoid : prove that the plane in which it again meets the
conicoid intersects the plane of the former circle in a straight line which
lies in one of two fixed planes.
[With the usual notation, the two planes are
1 7^*^*5 V6*^=0.]'
SOLID GEOMETRY. 365
2093. In the hyperboloid -j + ^ ., =1, (a>6), the spheres, of
which one series of circular sections of the hyperboloid are great drclesi
-will have a common radical plane.
[If the sections be parallel to f/Ja* -b" + z J a* + 6* = 0, the common
radical plane will be y J a* - 6* - « J a* + 1* = 0. ]
2094. Two generators of the paraboloid X = ^* *^ drawn
through the point (X, 0, Z) : prove that the angle between them is
cos
2095. The perpendiculars let fall from the vertex of a hyperbolic
paraboloid on the generators lie on two quadric cones whose circular
sections are parallel to the principal parabolic sections of the paraboloid.
a:* V* 2«
[The equation of the paraboloid being -i — t, = — , those of the two
cones are aj* + y* + 2«* «fc a:y (r- + - j = 0.]
2096. Through A, A' the ends of the real principal axis of an
hyperboloid of one sheet are drawn two generators of the same system,
and any generator of the opposite system meets them in P, P* re-
spectively: prove that the rectangle contained by AP^ A'P* is constant.
a;* V* «*
[If the equation of the hyperboloid be-5+^-3=l, and AA* = 2a,
the constant rectangle is equal to 5* + c*.]
2097. The least distance between two generators of the same
system in an hyperboloid of revolution of one sheet cannot exceed the
diameter of the principal circular section.
2098. The equation of the cone generated by straight lines drawn
flc* V* »■
through the origin parallel to normals to the ellipsoid -i+f3+3 = l at
Cb if
a* V* «■
points where it is met by the confocal -= — r- + rr—r + -= — r = I is
aV by (?:^ ^
a'-X 6«-X c«-X "•
2099. The points on a given conicoid, the normals at which inter-
sect the normal at a given point, lie on a quadric cone whose vertex is
the given point.
[With the usual notation, (X, F, Z) being the given point, the
equation of the cone is
X(6'-c')^ r(c'-a') ^ ^(a'-6')
x-X y-Y z-Z ■•'
366 SOLID GEOMETBT.
2100. Normals are drawn to a cental coniooid at the ends of tliree
conjugate diameters: prove that their orthogonal projections on the plane
through the three endis will meet in a point.
«■ V* «■
2101. The six normals drawn to the ellipsoid ~b + ti + ^ = 1 drawn
fix>m the point {x^ y^ z^ all lie on the cone
and the normals drawn from the same point to any conf ocal will also lie
on the same cona
2102. The normals at the ends of a chord of a given conicoid inter-
sect each other: prove that the chord will be normal to some one
conf ocal conicoid.
2103. The six normals drawn to the conicoid aa:' + ^ + cs'= 1,
from any point on one of the lines
a (6 - c) aj = «fc 6 (c - a) y = sfc c (a - 6) «,
will lie on a cone of revolution.
sc" V* «*
2104. The normals to the ellipsoid -2+p+-3 = lat points on the
plane 2- + m~ + n-s=l all intersect one straight line : prove that normals
at all points lying on the plane ^+)^+ — +1 = also intersect the
same straight line; and that the necessary condition is
(mV-P)(6"-O' + K^-»^*)(«'-»T+(^'»"-^")(»""5y=.0.
Also prove that, when { = m = n=l, the normals all Intersect the
straight line
aa;(i'-c^ = 6y(c*-a*) = (»(a«-6').
2106. The normals to the paraboloid ~ + — = 2a:, at points on the
plane px-hqy + rz = lf will all intersect one straight line if
2106. Prove that a tangent plane to the cone r— + j- — «0 will
meet the paraboloid j- + — = 2as in points the normals at which all inter-
sect the same straight line; and the surface generated by this straight
line has for its equation
2(ft-c){«(8y«-cO-6c(j^-«')}*-(6y'-cO(*y' + <»^'-
SOLID QEOKETRY. 367
2107. A section of^the ooniooid aa^+bt^+es^— 1 is made by a plane
parallel to the axis of z, and the trace of the plane on osy is normal to
the ellipse
prove that the normals to the ooniooid at points in this plane all
intersect one straight line.
2108. Through a fixed point {x^ y^ z^ are drawn straight lines
each of which is an axis of some plane section of the coniooid
aa5* + 6/ + ca' = l:
proTO that the locus of these lines is the cone
2109. In a fixed plane are drawn straight lines each of which is an
axis of some plane section of a given conicoid : prove that the envelope
of these lines is a parabola.
21 10. Straight lines are drawn in a given direction, and the tangent
planes drawn through each straight line to a given conicoid are at right
angles to each other: prove that the locus of such straight lines is a
cylinder of revolution or a plane.
. . a:* y" z"
[With a central conicoid -a + t7 + -« = li the locus is
*■ a b c
a^ + f^'hs^-'{lx + my + nzy = a*+b*-h(f-p\
where ly m, n are the direction cosines of the given direction, and p the
central distance of a tangent plane perpendicular to the given direction.
With the paraboloid ^ + — = aj, the locus is
m {ly - mx) + n (& - wa) = 6 (Z* + w*) + c (^ + ?»•). ]
21 Ih A cone is described having for base the section of the
conicoid oaf + by* + cs' = 1 made by the plane Ix + my + nz = Of and inter-
sects the conicoid in a second plane perpendicidar to the former : prove
that the vertex must lie on the surface
(Z* + m" + n*) (oo* + ^ + ca;* - 1 ) = 2 (& + my + w«) {cUx + bmy + enz).
2112. The cone described with vertex (X, F, Z) and base the curve
determined by the equations oa" + 5y" + C8"=l, lx + my '^nz=Pf will
meet the conicoid again in the plane
(aX"+6r»+c^*-l)(to+my+n«-;>) = 2{lX-^mT+nZ-p)(axX-¥byT+€zZ-l).
2113. A chord AB of a conicoid is drawn normal at A and the
central plane conjugate to AB meets the tangent plane at ul in a straight
line, through which is drawn a plane intersecting the conicoid in a
conic U : prove that the cone whose vertex is A and base the conic U
has for its axes the normals at ul to the conicoid and to the two con-
focals through A,
368 SOLID GEOMETRY^
2114. Through the vertex of an enyeloping cone of a given ooniooid
€Lx'-{-bf/'-^a^=l is drawn a similar concentric and similarly sitiiated
conicoid : prove that this conicoid will meet the cone in a plane curve
which will touch the given conicoid if the vertex lie on the conicoid
005* + 6y" + cz* = 4.
2115. A tangent plane is drawn to an ellipsoid and another plane
drawn parallel to it so that the centre of the ellipsoid divides the
distance between them in the ratio 1:4: prove that, if a cone be drawn
enveloping the ellipsoid and have its vertex on the latter plane, the ca.
of the volume cut off this cone by the former plane will be a fixed point.
[The equation of the ellipsoid referred to conjugate diameters being
yf \f ^
"i '**7«'*';?~^» aj + a = the tangent plane, x = ia the parcel plane,
the c. G. is r J , 0, j .]
2116. Straight lines are drawn through the point (a;^, y^i «o) ^udi
that their conjugates with respect to the paraboloid ^ + — =2a: are
perpendicular to them respectively : prove that the locus of these straight
lines is the cone
y-^o «-«o a^-^o
and that their conjugates envelope the parabola
!^. + ^. = « + a,,, (_l|^.)*+(^o)4 + (6_c,i = 0.
2117. A straight line is perpendicular to its conjugate with respect
to a certain conicoid : prove that it is also perpendicular to its conjugate
with respect to any conicoid confocal with the former.
2118. Any generator of the surface y" + «" — jc* = m will be perpen-
dicular to its conjugate with respect to the surface
oo* + 5y* + 02* + 2/yz + 2gzx + 2hxi/ = 1,
if he —/' = ca — gi' = aib — h' and af=gh.
2119. An hyperboloid of one sheet and an ellipsoid are concentric
and every generator of the hyperboloid is perpendicular to its conjugate
with respect to the ellipsoid : prove that their eqiiations, referred to
rectangular axes, may be obtained in the forms
and that the locus of the conjugate straight lines is
a* yz _ 1
(5T7)' " 26^ ~ J^ •
E[f 25 = 2c = m' this locus is the hyperboloid itself, the ellipsoid being
* ere.]
SOLID OBOMETRY. 369
2120. In the two conicoids
eight generators of the first are respectiyelj perpendicular to their con-
jugates with respect to the second.
2121. A fixed point being taken, P is any point such that the
polar planes of 0, P with respect to a given conicoid are perpendicular
to each other : prove that the locus of P is the plane bisectmg chords
which are perpendicular to the polar plane of 0,
2122. A hyperbolic paraboloid whose principal sections are equal
is drawn through two given straight lines not in one plane : prove that
the locus of its vertex is a straight line.
2123. Prove that, when two conicoids have in common two generators
of one system, they have also common two generators of the opposite
system.
2124. Two given straight lines not in one plane are generators of
a conicoid : prove that the polar plane of any given point with respect
to the conicoid passes through a fixed point.
2125. Two conicoids touch each other in three points : prove that
they either touch in an infinite number of points or have four common
generators.
2126. Generators of the same system of the hyperboloid
as* + y* — m V = a*
are drawn at the ends of a chord of the principal circle which subtends
a given angle 2a at the centre : prove that the locus of the straight line
which intersects both at right angles is the hyperboloid of revolution
« + y 1 r- = » cos* a ( — — 5 — J- ) .
m cos a VI + w» cos a/
2127. A cone is described with vertex (X, F, Z) and base the
curve
S = oiB* + 6y* + c2'=l, ^ + gy + r2J = l:
prove that the equation of the plane in which the cone again meets the
conicoid /S^ = 1 is
2 {aXx + hYy + cZz - 1) {pX + qY+ rZ- 1)
= (aZ« + 5r« + c^"- 1) {px-\-qy-\-rz- 1).
[The cone will intersect the conicoid in two planes at right angles to
each other if
(/>«+5' + r')(aZ' + 57*+cZ*-l)=2(apX+6^F+crZ)(pX+^r+rZ-l);
and in two parallel planes if — = — = — , that is if the vertex lie on
the diameter conjugate to either plane section.]
W. P. 24
370 SOLID GEOMETRY.
2128. Tangent planes are drawn to a series of confocal coniooids
parallel to a given plane : prove that the locus of the points of contact
is a rectangular hyperbola which intersects both focal curves.
[The equations of the locus will be, with the usual notation,
Ix + my + TMf = -^— = — / = 1 .1
y » z ^ [ ^ _y I
m n n I I m
. ic* V* »"
2129. Two circular sections of the ellipsoid -»+?«+ ^ = 1 are such
that the sphere on which both lie is of constant radius mb : prove that
the locus of the centre of this sphere is the hyperbola
of «*
2130. A sphere of radius r has real double contact with the
cc* v* «*
ellipsoid -j + t5 + i = ^> ^^^ ^^^ altogether within the ellipsoid : prove
that the locus of its centre is the ellipse -| — j + z^ — 5 = 1 - — , z = 0;
Of ^ C "^ c c
and, if there be real double contact and the sphere lie altogether
without the ellipsoid, the locus of the centre is the ellipse
a-0 ar — c c
rin the first case, r must lie between — , t- ; in the second, r must
*- a
a* a*
lie between j- > —] and, in both cases, only a part of the ellipse can be
traced out by the centre.]
2131. In an hyperboloid of revolution in which the excentricity of
the genei*ating hyperbola is J^^ a cube can be placed with one diagonal
along the axis of the hyperboloid and six edges lying along generators of
the hyperboloid.
2132. A cone whose vertex is meets a conicoid in two plane
sections A^B; two other conicoids are described touching the former along
A^ B respectively and passing through : prove that these two conicoids
will touch at 0, and will have a common plane section in the polar plane
of with respect to the first conicoid.
2133. The axes of sections of the conicoid — + ^ + — = 1 made bv
a b c ^
planes parallel to ^ + my + ns; = lie on the two planes
— (5-c) + ...+ ... + 2aZ*(-- r-l — +••• + •••
a ^ ' \c bj mn
K \b cj ) \mn nl Im/
SOLID GEOMETRY. 371
2134. Two points are taken in the surface of a polished hollow
ellipsoidal shell and a ray proceeding from one after one reflexion passes
through the other ; prove that the number of possible points of incidence
is in general 8 ; but if the two points be ends of a diameter the number
is 4, and these four points are the ends of two diameters which lie on a
quadric cone containing the axes of the ellipsoid and of the central sec-
tion perpendicular to the given diameter.
rv. Tetrahedral Co-ordincUea.
2135. A plane meets the edges of a tetrahedron in six points and
six other points are taken, one on each edge, so that each edge is divided
harmonically : prove that the six planes, each passing through one of
these six latter points and the edge opposite to it, will meet in a
point.
2136. The opposite edges of a tetrahedron ABCD are, two and two,
at right angles : prove that the three shortest distances between opposite
edges meet in the point
x{AB' -¥ AC* + AD^ '-k)^y {BC^ -¥ BD^ + BA^ --k)^ ... ^ ,..,
k being the sum of the squares on any pair of opposite edges.
2137. Prove that any conicoid which touches seven of the planes
«fc fo; * my * tw; + ru? =
will touch the eighth ; and that its centre will lie on the plane
Vx + m'y + n'z + t^w = 0.
Prove that this plane bisects the part of each edge of the tetrahedron
of reference which is intercepted by the given planes.
2138. Determine the condition that the straight line - = ?^=-
"^ p q r
may touch the conicoid
lyz + mzx + Tixy + Vacw + m'yv) + rCzvo = ;
and thence prove that the equation of the tangent plane At the point
(0, 0, 0, 1) is
Vx + m'y + n'« = 0.
2139. The general equation of a conicoid touching the faces of the
tetrahedron of reference may be written
Iqroi? + mrpy^ + npq^ + Imnu^ + {lp-mq- nr) {Ixw +pyz)
■h{mq''nr-lp) {myto -h qzx) + {nr - Ip ~ mq) {nzw-hrocy) = 0.
Prove that this will be a ruled surface if
Pp' + wiV + w*^ > ^ninqr + 2nlrp + 21mpq ;
and that, when lp = mq = 7vr, the straight lines joining the points of
contact each to the opposite comer of the tetrahedron will meet in a
point.
24—2
372 SOLID GEOMETRY.
2140. A hyperbolic paraboloid is drawn containing the sides AB^
BC, CD, DA of a quadrangle not in one plane : prove that, if P be any
point on this paraboloid,
voL PBCD : vol. P ABC =yoI PC DA : vol. PDAB :
and that, if any tangent plane to the paraboloid meet AB^ CD in P, Q
respectively,
APiBP^DQiCQ,
2141. The locus of the centres of all conicoids which have in
common four generators, two of each system, is a straight line.
2142. Perpendiculars are let fall from the point (ar, y, z, to) on the
faces of the tetrahedron of reference, and the feet of these perpendicidarB
lie in one plane : prove that
A* B" C D' ^
— + — + — + — = 0,
X y z w
Ay B, C, D being the areas of the faces of the tetrahedron.
2143. The volume of the ellipsoid which has its centre at the point
{X : T \ Z \ TT), and to which the tetrahedron of reference is self-
conjugate, is ^irV JXYZW -^ {X ^ Y-^Z-k- W)\ where F is the volume
of itkQ tetrahedron.
2144. A tetrahedron is self-conjugate with respect to a given sphere :
prove that each edge is perpendicular to the direction of the opposite
edge, and that all the plane angles at one of the solid angles are obtuse.
2145. The opposite edges of a tetrahedron are two and two at right
angles to each other, and in each face is described a circle of which the
centroid and the centre of perpendiculars of that face are ends of a
diameter : ]>rove that the four circles so described lie on one sphere ;
and that this sphere (4), the circumscribed sphere (1), the polar sphere
or sphere to which the tetrahedron is self -con jugate (2), the sphere
bisecting the edges (3), and the sphere of which the centroid and the
centre of per]:>endicular8 of the tetrahedron ai*e ends of a diameter (5),
have all a common radical plane. Taking B, p to represent the radii
of the circumscribed and polar spheres and S the distance between their
centres, 8' = 72' + 3p* ; and the distances from the common radical plane
of the centres of these five spheres are
m^±l (2)-^-^ i3)^:il u)^::^ (5)^^"'
the radii of the five axe
(1) H, (2) p, (3) yW:^, (4) |, (5) |;
and the centres of the spheres (3), (4), (5) divide the distance between
the centres of (1) and (2) in the respective ratios 1:1, 2:1, 3:1.
2146. A tetrahedron is such that a sphere can be drawn touching
its six edges : prove that any two of the four tangent cones drawn to
this sphere from the corners of the tetrahedron have a common tangent
plane and a common plane section ; and that the planes of the coipmon
sectiona will all six meet in a point.
SOUD OEOMETRY. 373
2147. A tetrahedron is such that the straight lines joining its
angular points to the points of contact of the inscribed sphere with the
respectively opposite faces meet in a point : prove that, at any point of
contact, the edges of the tetrahedron which bound the corresponding
face subtend equal angles.
2148. The tangent planes &t Ay B^ C, i>, to the sphere circumscrib-
ing the tetrahedron ABCD, form a tetraJiedron abed: prove that Aa,
Bb, Cc, Dd will meet in a point if
BC.AD = CA.BD = AB.CD.
2149. Each edge of a tetrahedron is equal to the opposite edge :
- rt >
where c^ b, c are the edges bounding any one face.
2150. A conicoid circumscribes a tetrahedron A BCD and the
tangent planes at A, B^ C, D form the tetrahedron abed : prove that, if
Aa^ Bb intersect, (7c, Dd will also intersect.
2151. Four points are taken on a conicoid and the straight line
joining one of the points to the pole of the plane containing the other
three passes through the centre : prove that the tangent plane at that
point is parallel to the plane of the other three.
2152. The equation of a conicoid being
fn,nyz + vlzx + Imxy + Itxw + mryw + nrzv) = ;
prove that it cannot be a ruled surface, and that it will be an elliptic
j>araboloid if
2153. The surface
lyz + max + wxy + Tqcud + m'yw + tIzmo =
will be a cylinder, if
W {m+n''l) + mm' {n'\-l-'m) + nvl (/ + m - ») = 2/m7»,
and V: (w' + »'-Q + ww'(n' + /- w') + nn' (Z + m' -n')=^2lm'n.
[The relations
Zr (w + n' - Z') + mm' (n' + Z' - w) + nn' (Z' + w - n) = 2rmn\
ir (w' -hn-i)-^ mm' (n + T - w') + nn (^ + w' - n) = 2ZWn,
will of course also be satisfied, the system being equivalent to the two-
fold relation
{U')i + (mm')* + (nn')* = 0,
/m'n\i /nT\l /rm\i ,
(^) * {la) ^ (W = ^'
the first of which is the single condition for the surface to be a cone.
See question (141).]
374
SOLID GEOMETRY.
2154. The rectangles under the segments of chords of a certain
sphere drawn through the four points A^ B, Cy D (not in one plane),
are l^ m, n, r, and the radius of the sphere is p : prove that
1
1
1
1
1
1,
I.
1,
1,
1
0,
AB',
AC,
Ajy,
i+p*
BA\
0,
BC,
BD\
m + p*
CA\
CB\
0,
ciy.
n + p*
da;
Dff,
DC,
0,
r + p'
= 0.
l+p'y m + p*, n + p*, r + p\
2155. The enveloping developable of the two conicoids
will meet the planes of the faces of the tetrahedron of reference in the
oonics
nn'z*
_ ^ JXq^ mm! if .... « _ a (r
' W — Vt lar^ — m'r nr^-nr~ *
2156. The perpendiculars p, q, r, a let fall from the comers of a
finite tetrahedron on a moving plane are connected by the equation
Ap*+B^+C7^ + D^ + 2Fqr-\- 2Grp+2Hpq + 2i^>« + 26^'^«+ 2//V« = 0;
prove that the envelope is in general a conicoid, which degenerates to a
plane curve if
= 0.
A
H,
G,
F'
H,
B,
F,
G'
0.
F,
0,
W
F',
0',
ti\
D
V. Focal Cwrvea : Reciprocal Polars,
2157. The equations of the focal lines of the cone a^ + bzx + cxy =
are
(cy + bzy _ (az + ex)' __ (bx + ay)'
y* + «* s^ + af x' + y* '
2158. A parallelogram of minimum area is circumscribed about the
focal ellipse of a given ellipsoid, and from its angular points taken in
order are let fall perpendiculars p^, p^, p^, p^ on any tangent plane to the
ellipsoid : prove that
2c being the length of that axis of the ellipsoid which is normal to the
plane of the focal ellipse.
SOLID GEOMETBT. 375
2159. The perpendiculars from the ends of two conjugate diameters
of the focal ellipse on any tangent plane to the ellipsoid are m^^ vr^ w^ vr^
and the perpendicular from the centre is p : prove that
2160. With any two points of the focal ellipse as foci can be
described a prolate spheroid touching an ellipsoid along a plane curve,
and the contact will be real when the common point of the tangents to
the focal at the two foci lies without the ellipsoid*
[The plane of contact is the polar with respect to tlie ellipsoid of this
common point]
2161. Four straight lines can )>e drawn in a given direction so as
to intersect both focal curves of an ellipsoid, and they will lie on a
cylinder of revolution whose radius is J a* -p' ; a being the semi major
axis and p the perpendicular from the centre on a tangent plane normal
to the given direction.
2162. The cones whose common vertex is (X, Y, Z) and whose
a:* t/ «■
bases are the real focal curves of the ellipsoid — , + n + ~» = ^ being
denoted by U^ and U^ whose discriminants are respectively
Z' Y
«
(a* - c*) (i« - c*) ' (a* - 6') (c* - 6') '
the cone \U^-k- U^=0 will be a cone of revolution if
Jr'(6'-c0 Y'(c'-a') (b'-c')(c'-a'){a'-b') _
TTa""" X ^ («-*)+ a'-e'-\(a'-b') ""•
2163. With a given point as vertex is described a cone of revolu-
tion whose base is a plane section .of a given conicoid : prove that the
plane of this section will envelope a fixed cone whose vertex lies on one
of the axes of the enveloping cone drawn from the given point to the
given conicoid.
2164. This straight line joining the points of contact of a common
tangent plane to the two conicoids
aa:'+6y« + c««=l, (o-X)a:« + (6-X)/+ (c-X)««= 1,
subtends a right angle at the centre.
2165. Through a given point can in general be drawn two straight
lines either of which is a focal line of any cone having its vertex on the
straight line and enveloping a given conicoid : and, if two such cones
be drawn with their vertices one on each straight line, a prolate conicoid
of revolution can be inscribed in them having its focus at the given
point.
2166. A point is taken on the umbilical focal conic of a conicoid:
prove that there exist two points L such that, if any plane A be drawn
through L and a be its pole, Oa will be normal to Uie plane through
containing the intersection of A with the polar of L,
376 SOLID GEOMETRY.
2167. With a given point as vertex there can in general be drawn
one tetrahedron self-conjugate to a given conicoid and such that the
edges meeting in the point are two and two at right angles ; but when
the given point lies on a focal curve the number of such tetrahedrons is
infinite.
2168. A tetrahedron circumscribes a prolate ellipsoid of revolution
whose foci are S, S', so that the focal distance (from S) of each angular
point is normal to the opposite face : prove that the diameter of the
sphere circumscribing the tetrahedron is three times the major axis of
the ellipsoid, and that the centroid of the tetrahedron and the centre
of the circumscribed sphere divide SS' in the ratios 1:3, 3 : - 1
respectively.
2169. The vertical angles of two principal sections of a quadric
cone are a, p : prove that the ratio of the axes of any section normal to
a focaJ line is cos a : cos p,
2170. A sphere is described with centre (X, 0, Z) intersecting the
of y* i^
ellipsoid -j + ?-, + -5 = 1 in two circles : prove that the points of contact
with the sphere of common tangents to the sphere and ellipsoid lie on
the two planes
-{
_ f X(g-z) __ z(x-x) y
2171. The circumscribing developable of two conicoids, which have
not common plane sections, will in general contain four plane conies,
which are double lines on the developable.
2172. In a given tetrahedron are inscribed a series of closed
surfaces each similar to a given closed surface without singular points :
prove that the one of maximum volume will be such that the normals at
the points of contact will be generators of the same system of an
hyperboloid.
2173. Two conicoids having for their equations U=Oy U'=^0, the
discriminant of \U+U' is X*A+ X'0 + X**4.X0' + A': prove that the
condition that hexahedra can be described whose six faces touch U and
whose eight comers lie upon U' is
0* - 40« *A + 800'A» - 1 6 A'A' = 0,
and the condition that hexahedra can be described whose twelve edges
are tangent lines to U and whose eight comers lie upon U ' is
20* - 90'*A + 2700'A" - 81 A' A' = 0.
SOLID GEOMETRY. 377
VI. General Functional and Differential Equatians.
2174. A sarface is generated bj a straight line which always inter-
sests the two fixed straight lines
aj = a, y = mz; x = — a, y = - mz :
prove that the equation of the surface generated is of the form
maz —
naz — xy - fmxz — ai/\
2175. The general functional equation of surfaces generated by a
straight line which intersects the axis of z and the circle z = 0, x' + t/' =0*^
is
;»• + / = {« + ./(!)}';
and the general differential equation is
(^ + !/') (P^ + q!/-^)=a* (px + qi/Y-
2176. The general functional equation of surfaces generated hj a
straight line which always intersects the axis of « is
and the differential equation is
raf + 2$xi/ + ty* = 0.
2177. The differential equation of a family of surfaces, such that
the perpendicular from the origin on the normal always lies in the plane
of a?y, is
z (/?' ^^) ■\-px + qi/ = 0,
2178. The differential equation of a family of surfaces, generated
by a straight line which is always parallel to the plane of xy and whose
intercept between the planes of i/z^ zxia always equal to a, is
(px + qi/Y (/>• + q') = a'py.
2179. The general differential equation of surfaces, generated by a
straight line, (1) always parallel to the plane Ix + my + n? = 0, (2) always
X u z
intersecting the straight line -r = — = — , is
(1) (m + nqY r - 2 (m + nq) {I •hnp)8-^{l + np)' < = 0,
(2) (fy - mxY {q^r - 2pq8 +pU) + 2 (ty - mv) (nx - h) {qr - jm)
+ 2 (/y - 7nx) {ny - mz) (qs -pt)
+ (/«;- IzY r^2{nx-h) {ny - mn) 8 + {ny- mzY t = 0.
378 SOLID GEOMETRY.
VIL Envelopes,
2180. The envelope of the plane lx+mi/-hnz = a; I, m, n being
parameters connected hy the equations
is the cylinder
{y-zy + {z-xy + {x- y)' = 3a'.
m
2181. Find the envelope of the planes
(1) ?cos(tf-<^) + |cos(tf-<^) + -sin(tf + <^) = ain(d-<^),
X IJ z
(2) -cos(tf-^) + |(costf + cos<^) +-(sin tf + sin^) = 1;
both when 0, ^ are parameters, and when only is a parameter.
[The envelope of (1) when both 0, if> are parameters is the hyperboloid
a' 6* c" " '
and when onlj is a parameter, the plane (1) always passes througli a
fixed generator of this hyperboloid ; the envelope of (2) when 0, €f> are
parameters is the ellipsoid
2x» 2aj y «• ^
a" a c
when alone is a parameter the envelope is a cone whose vertex is the
point (-a, 6 cos ^ c sin ^).]
2182. The envelope of the plane
= a
sin cos <f> sin ^ sin ^ cos
is the surface
2183. The envelope of all paraboloids to which a given tetrahedron
is self-conjugate is the planes each of which bisects three edges of the
tetrahedron.
[More generally, if a conicoid be drawn touching a given plane and
such that a given tetrahedron is self-conjugate to it, there will be seven
other fiixed planes which it always touches, the equations of the eight
planes referred to the given tetrahedron being
^px^qy^rz'hw = 0,]
2184. A prolate ellipsoid of revolution can be described having two
opposite umbilics of a given ellipsoid as foci and touching the given
ellipsoid along a plane curve : and this will be the envelope of one
system of spheres, each of which has a circular section of the ellipsoid
for a great circle.
SOLID OEOMETRT. 379
2185. Spheres are described on a series of parallel chords of a
given ellipsoid as diameter : prove that they will have double contact
with another ellipsoid, and that the focal ellipse of this envelope will be
the diametral section of the given ellipsoid which is conjugate to the
chords. Also, if a, 6, c be the axes of the given ellipsoid, and a, )3, y of
the envelope,
y being that axis which is perpendicular to the focal ellipse.
2186. A series of parallel plane sections of a given ellipsoid being
taken, on each as a principal section is described another ellipsoid of
given form ; the enveloi)e is an ellipsoid touching the given one along a
central section at any point of which the tangent plane is perpendicular
to the planes of the parallel sections.
2187. The enveloi)e of a sphere, intersecting a given conicoid in
two planes and passing through the centime, is a quartic which touches
the given conicoid along a sphero-conic.
VIII. Curvature,
2188. From any point of a curve equal small lengths 8 are
measured in the same sense along the curve, and along the circle of
absolute curvature at the point, respectively: prove that the distance
between the ends of these lengths is ultimately
p, o" being the radii of curvature and torsion respectively at the point.
2189. Find the radius of absolute curvature and of torsion at any
point of the curves
(1) a; = a(3«-^), f/=3a^, « = a(3^ + «");
(2) a; = 2a<'(l +<), y = a^(i + 2), « = a^(«" + 2i+ 2).
2190. The radius of absolute curvature (p) at any point of a rhumb
line is a cos 0-r- J\ — sin* cos* a, where B is the latitude, and a the angle
at which the line crosses the meridians ; and the radius of torsion is
a ,- . ,^ g . a" tan a
(1 - sin' cos a) or -^ ^ ^
sm a cos a ^ ' or - p cos a
2191. Two surfaces have complete contact of the v^ order at a
point : prove that there are w + 1 directions of normal section for which
the curves of section have contact of the w + 1*** order ; and hence prove
that two conicoids which have double contact with each other intersect
in plane curves.
380 SOLID GEOMETRY.
2192. Prove that it is in general possible to determine a paraboloid,
whose principal sections are equal parabolas, and which has a complete
contact of the second order with a given surface at a given point.
2193. A paraboloid can in general be drawn having a complete
contact of the second order \vith a given surface at a given point, and
such that all normal sections through the point have four-point
contact.
2194. A skew surface is capable of generation in two ways by the
motion of a straight line, and at any point of it the absolute magnitudes of
the principal radii of curvature are a, b : prove that the angle between
the generators which intersecir in the point is cos"' ( 7 ) .
2195. The points on the surfaces
(1) ajy«= a {t/z + ».r + xf/)f
(2) xyz = a' (a; + y + z),
(3) ar* + 3^ + «■ - Zx7/z = a",
at which the indicatrix is a rectangular hyperbola lie on the cones
(1) X* (2/ + «) + y* (« + a;) + z* {x + y) = 0,
(2) a^ + y^ + 2?" + ost/z = 0,
(3) y2 + «c + a;y = 0,
respectively ; and in (3) these points lie on the circle
aj + y + « = a, a:* + y' + »* = a*.
2196. A surface is generated by a straight line moving so as always
to intersect the two straight lines
« = |, y=«tan^; ^ = -|» y = -«<»n^;
and X, /x are the distances of the points where the generator meets these
straight lines from the points where the axis of x meets them ; prove
that the principal radii of curvature at any point on the first straight
line are given by the equation
j\
a'p* sin* a - 2ap sin a ^ (X - /x cos a) ^ a* + ^« sin* a
= (|)'(«V^-Bin.«)-.
2197. A surface is generated by the motion of a variable circle,
which always intersects the axis of x, and is parallel to the plane of y«.
At a point on the axis of x, r is the radius of the circle, and the angle
which the diameter through the point makes with the axis of z : prove
that the principal radii of curvature at this point are given by the
equation
SOLID GEOMETRY. 381
2198. A surface is generated bj a straight line which always in-
temects a given circle and the normal to the plane of the circle drawn
through its centre ; is the angle which the generator makes with this
normal, and ^ the angle which the projection of the generator on the
plane of the circle makes with a fixed radius : prove that the principal
radii of curvature at the point where the generator meets the normal are
a .- -r sin ^ (cos O'^l) ;
and that at the point where it meets the circle, the principal radii are
given hy the equation
P [-fi) + ap COB = a',
2199. A siirface is generated by a straight line, which is always
parallel to the plane of xy, and touches the cylinder a:* + y* = a' : prove
that, if p be a principal radius of curvature at the point whose co-ordi-
nat^ are (a cos ^ + r sin d, a sin ^ - r cos ^, z)
'•(^■-('^^«^-l)v/'^)■={-(S)T•
2200. A straight line moves so as always to intersect the circle
a^ + i^ — a*, « = 0, and be parallel to the plane of zx ; prove that the
measure of specific curvature at the point (a cos ^, a sin ^, 0) is
__i cos** /"^y.
a* (1 - sin* ^ sin* <^)' W</»/ '
being the angle which the generator through the point makes with the
axis of z,
2201. A circle of constant radius a moves so as to intersect the
axis of X, its plane being parallel to the plane of yz : prove that, at the
point
(as, a sin * + a sin <^ - fl, a cos <^ + a cos <l> — 0\
the measure of specific curvature of the surface generated is
2202. In a right conoid whose axis is the axis of z, prove that the
radius of curvature of any normal section at a point (r cos tf, r sin 0, z) is
{■'<^'<t))J-<%)'
Wz Tdrdz
^dG'" de dO
and deduce the equation
for the principal i*adii of cui'vature at the point.
382 SOLID GEOMETRY.
2203. A straight line moves so as always to intersect the axis of z
and make a constant angle a with it : prove that, if p be a principal
radius of curvature of the surface generated at the point whose co-
ordinates are (r sin a cos ^, r sin a sin <l>, z+r cos a),
'''■^'°($)'*''""''v''*©V'^'''*^'^*(|;)'*'$)
2204. Investigate the nature of the contact of the surfaces
xt/z = a* (x + y + z), x{y- »)* + 4a' (a + y + ») = 0,
at any point on the line x — 0, y + »=0; proving that the piincipal
radii of curvature of either surface are ^ -^^ .
±2aV
2205. Prove that, in the surface
(/ + «•) (2a; - y + «») = 4aV,
(1) the points where the indicatrix is parabolic lie on the cylinder
a:*+«' = a*; (2) the lines y = 0, 25 = 0; y = 2x, z = 0, are nodal lines,
the tangent planes at any j)oint being respectively
2;'(a"-Z') = Xy, s^{a*^X^^X*(y-2x)\
2206. An ellipsoid is described with its axes along the co-ordinate
axes and touching the fixed plane px -\- qy •{- rz = I : prove that the locus
of the centres of principal curvature at the point of contact is the surface
whose equation is
(j>x ■^-qy + rz-l) {p*yz + q^zx + n^xy) = xyz (/?' + <?* + r*)*.
2207. The direction cosines of the normal to the conicoid
7? y" z' .
- +^ + - = 1
a c
at a certain point arc l, m, n, and the angle between the geodesies
joining the point to the umbilics is ^ : prove that
{Pa (c - 5) + m'b (c + a - 26) + n^c {a -5)}*
cos* ^ =
{- Pa{b - c) + m'b{c - a)'{-n*c {a - 6)}*+ im*n^bc(a - 6) (a - c) "
STATICS.
I. Composition and Resolution of Forces,
2208. A point is taken in the plane of a triangle ABC Bud a, b, e
are the mid points of the sides : prove that the system of forces Oo, 06,
Oc is equivalent to the system OAy OB, OC,
[The result is true when is not in the plane ABCJ]
2209. Forces P, Q, R &ct along the sides of a triangle ABC and
their resultant passes through the centres of the inscribed and circum-
scribed circles: prove that
P ^ g ^ /g
cos B — cos C cos C — cos A cos A - cos B '
2210. Four points Ay B, C, Z> lie on a circle and forces act along
the chords AB, BC, CD, DA in the senses indicated by the order of the
letters, each force being inversely proportional to the chord along which
it acts : prove that the i*esultant passes through the common points of
(1) AD, BC; (2) AB, DC; (3) the tangents at B, D; (4) the tangents
at A, C,
[Of course this proves that these four points are collinear.]
2211. In a triangular lamina ABC, AD, BE, CF are the perpen-
diculars, and forces BD, CD, CF, AE, AF, BF are applied to the
lamina: prove that their resultant passes through the centre of the
circumscribed circle and through the point of concourse of the straight
lines each joining an angular point to the intersection of tangents to the
circle ABC at the ends of the opposite side.
' [The equation of the line of action of the resultant in trilinear
co-ordinates is
asin (^- C) + y sin (^--4) + « sin (i4 - ^) = 0,
which passes through the points
(cos A : cos i? : cos (7), (sin A : sin i? : sin C7).]
384 STATICS.
2212. Three equal forces act at the comers of a triangle ABCy
each perpendicular to the opposite side : prove that, if the magnitude of
each force be represented by the radius of the circle ABC^ the magnitude
of the resultant will be represented by the distance between the centres
of the inscribed and circumsciibed circles.
2213. The resultant R of any number of forces P, i*, P„... is
determined in magnitude by the equation
i2» = S(i«) + 2SP,P.cos(P,P.),
A
where P^, ij denotes the angle between the directions of P, P^
2214. The centre of the circumscribed circle of a triangle ABC is
Of and the centre of perpendiculars is L: prove that the resultant of
forces LA^ LB, LC will act along LO and be equal to 2Z0.
2215. Three parallel forces act at the points A^ B, C and are to
each other as 6 + c : c + a : a + 6, where a, 6, c are the lengths of the
sides of the triangle ABC : prove that their resultant passes through the
centre of the circle inscribed in the triangle whose comers bisect the
sides of the triangle ABC,
2216. The position of a point P such that forces acting along PA^
PBy PC, and equal to I . PA , m . PB, n . PC may. be in equilibrium is
determined by the areal co-ordinates (limin),
2217. Forces act along the sides of a triangle ABC and are pro-
portional to the sides ; AA\ BB^, CC bisect the angles of the triangle :
prove that, if the forces be turned in the same sense about the points
A'f B, C respectively, each through the angle
_,/ B-C C-A A^B\
tan M - cot — ^— cot — ^ — cot — - — j ,
there will be equilibrium.
2218. Forces in equilibrium act along the sides AD, BD, CD, BCf
CA, AB oi a, frame A BCD, prove the following construction for a force
diagram : take any one of the points {!)) as focus and inscribe a conic in
the triangle ABC; let g? be the second focus and let fall da, db, de per-
pendiculars on the sides of the triangle ABC, then abed will form the
force diagram ; that is, be will be perpendicular to AD and proportional
to the force along AD, and so for the other sides.
2219. Four points A, B, C, D are taken in a plane, perpendiculars
are drawn from D on BC, CA, AB and a circle drawn through the feet
of these perpendiculars, and another circle is drawn with centre D and
radius equal to the diameter of the former circle; other circles are
similarly determined with their centres at A, B, C. Prove that these
four circles will intersect by threes in four points a, b, c, d, and that the
diagrams ahcd, ABCD will be reciprocal force diagrams.
STATICS. 385
2220. A triangular fraode ABC is kept in eqoilibriam bj three
forces at right angles to the sides, and S is the point of oonoourse of
their lines of action, the centre of the circle ABCy SS' is a straight
line bisected in : prove that the stresses at A^ B, are perpendicidar
and proportional to S'A, S'B, SV.
2221. A number of light rigid rods are freely jointed at their
extremities so as to form a polygon, and are in equilibrium under a
system of forces perpendicular and proportional to the respective sides
of the polygon and all meeting in one point : prove that the polygon is
inscribable in a circle, and, if be the centre of the circle, S the point
of concourse of the lines of action, SS' a straight line bisected in 0, that
the stress at any angular point F of the polygon is perpendicular and
proportional to S'P.
[The points Sy S' will be foci of a conic which can be drawn to touch
the lines of action of all the stresses at the angular points, and the circle
circumscribing the polygon is the auxiliary circle of this conic. If
ABC D ,,.he the comers of the polygon, and A' B CD',,, those of the
polygon formed by the lines of action of the stresses at ^, ^, C7,..., the
diagrams SA'BC'B',,,, S'ABCD,,, will be reciprocal force diagrams.]
2222. Two systems of three forces (P, (?, R), {F, ^, R) act along
the sides of a triangle ABO i prove that the two resultants will be
parallel if
P, Q, R =0.
F, Q", R
BC, CA, AB
2223. A lamina rests in a vertical plane with one comer A against
a smooth inclined plane and another point B is attached to a fixed point
C in the plane by a fine string, is the c. g., and the distances of
AyOyC from B are all equal: prove that, when the inclination of the
inclined plane to the horizon is half the angle ABGy every position is one
of equilibrium.
2224. Perpendiculars SK^ SK* are drawn from a focus on the
asymptotes of an hyperbola, and P is a point such that the rectangle
JTP, K'P is constant: prove, from Statical considerations, that the tan-
gent to the locus of P at a point where it meets the auxiliary circle of
the hyperbola will touch the hyperbola, and that the normal will pass
through S,
2225. Forces proportional to the sides a^, a,,... of a closed polygon
act at points dividing the sides taken in order in the ratios m^ : n ,
m, : n,,... and each force makes the same angle in the same sense witn
the corresponding side : prove that there will be equilibrium if
4 cot X area of the polygon.
2226. The lines of action of a system of forces are generators of the
same system of a hyperboloid: prove that the least distance of any
generator of the opposite system from the central axis of the forces
is proportional to the cotangent of the angle between the directions of
the two straight lines.
w. P. 25
386 STATICS.
2227. A system of co-planar forces whose components are (X^, F,),
(X,, I^,)|... ftot at the points (iCj, y,), (a?,, y.)»... and are equivalent to a
single couple : prove that there will be equilibrium if each force be turned
about its point of application in the same sense through the angle 0,
where
f.n/J S(Xy-7a;)
tan & = =-7-^ — . y V .
2228. The sums of the moments of a given system of forces about
three rectangular axes are respectively L, M, N; and the sums of the
components in the directions of these axes are X, Y^ Z: prove that
LX^MY + NZ
is independent of the particular system of axes.
[It is equal to RO^ where R is the resultant force and G the mini-
miun resultant couple.]
2229. Forces P, Q, R, F, (?', R act along the edges BC, CA, ABy
DAf DBf DC of a tetrahedron respectively : prove that there will be a
single resultant if
BC AD '^ CABD^ AB CD"^'
and that the forces will be equivalent to a single couple if
AD AB CA' BD^ BC AB' CD' CA BC
2230. The necessary and sufficient conditions for the equilibrium of
-four equal forces acting at a point (not necessarily in one plane) are that
the angle between the lines of action of any two is equal to that between
the lines of action of the remaining two.
2231. Necessary and sufficient conditions of equilibriimi for a
system of forces acting on a rigid body -are that the sum of the moments
of all the forces about each edge of any one finite tetrahedron shall
severally be equal to zero.
2232. Forces acting on a rigid body are represented by the edges
of a given tetrahedron, three acting from one angular point towards the
opposite face and the other three along the sides taken in order of the
opposite face : prove that the product of the resultant force and of the
minimum resultant couple will be the same whichever angular point be
taken.
[The product will be represented on the same scale by 18 T, F being
the volume of the tetrahedron.]
2233. A portion of a curve surface of continuous curvature is .cut
off by a plane, and at a point in each element of the portion a force
proportional to the element u applied in direction of the normal : prove
that, if all the forces act inwards or all outwards, they will in the limit
have a single resultant.
STATICS. 887
2234. A BTBtem of forces acting on a rigid body is reducible to a
single'couple : prove that it is possible, by rotation about any proposed
point, to bring the body into such a position that the forces, acting at
the same points of the body in the same directions in space, shall be
in equilibrium.
2235. A given system of forces is to be reduced to a force acting
through a proposed point and a couple : prove that if the proposed point
lie on a fixed straight line and through it be drawn always the axis of
the couple, the extremity of this axis will lie on another fixed straight
line.
2236. A given system of forces is to be reduced to two, both
parallel to a fixed plane; straight lines representing these forces are
drawn from the points where their lines of action are met by a fixed
straight line which intersects both at right angles : prove that the locus
of the other extremities of these straight lines is a hyperbolic paraboloid.
2237. Prove that the central axis of two forces P, Q intersects the
shortest distance c between their lines of action, and divides it in the
ratio
Q(Q-^rcoa0) :P(P-f ©costf),
being the angle between their directions. Also prove that the moment
of the principal couple is
I JT^T^T2PQ ios^ "
2238. A given system of forces is reduced to two, one of which F
acts along a given straight line : prove that
1 _ cos d c sin
$ being the angle which the given straight line makes with the central
axis, e the shortest distance between them, B the resultant force, and
the principal couple.
2239. A given system of forces is to be reduced to two at right
angles to each other: prove that the shortest distance between theiz
lines of action cannot be less than 2G -f- H, More generally, when the
two are inclined at an angle 2a, the shortest distance cannot be less than
2G-7-Jit&H€L
2240. A given system of forces is reduced to two P, Q, and the
shortest distances of their lines of action from the central axis are x, y
respectively : prove that
2241. Two forces act along the straight lines
X'^Of y = «tana; x^-^a, y^^-zUaia:
prove that their central axis lies on the surface
a? (y*-i- «*) sin 2a B 2ai/z,
the co-ordinates being rectangular.
25—2
t
388 STATica
2242. Two forces given in magnitude act along two straight lines
not in one plane, a third force given in magnitude acts through a given
point, and the three have a single resultant: prove that the line
of action of the third force must lie on a certain cone of revolution.
[If i^ be the resultant force and O the principal couple which are
together equivalent to the two given forces, F the third force, and a the
distance of its point of application from the central axis of the two, the
semi-vertical angle of the cone is
COS-* ( ^^ \ •
from which the conditions necessary for the possibility are obvious.]
2243. Forces X, F, 2^ act along the three straight lines
y = 6, « = -c; «=c, a5 = -a; x^a^ yr:z^hi
respectively : prove that they will have a single resultant if
a7Z+6ZX+cXr=0;
and that the equations of the line of action will be any two of the three
Y'' Z^ X"^' Z X^ Y"^' X'Y'^Z'^'
II. Centre of Gravity {or Inertia),
2244. A rectangular board of weight W is supported in a horizontal
position by vertical strings at three of its angular points; a weight 5W
being placed on the board, the tensions of the strings become W, 2 TF,
3W: prove that the weight must be at one of the angular points of a
hexagon whose opposite sides are equal and parallel, and whose area ia to
that of the board as 3 : 25.
2245. Particles are placed at the comers of a tetrahedron respect-
ively proportional to the opposite faces : prove that their centre of
gravity is at the centre of the sphere inscribed in the tetrahedron.
2246. A uniform wire is bent into the form of three sides of a
polygon AB, BC, CD, and the centre of gravity of the whole wire is at
the intersection of AC^ BD : prove that, if j& be the common point of
ABy DC produced,
EB : BG : GE^Aff : JBC" : CD*.
2247. A thin uniform wire is bent into the form of a triangle ABC^
and particles of weights P, Q, R are placed at the angular points : prove
that, if the centre of gravity of the particles coincide with that of the
wire,
P :Q '.R^h'\-c\c-^aia'\-h,
2248. The straight lines, each joining an angular point of the
triangle ABG to the common point of the tangents to the circle ABG at
the ends of the opposite side, all meet in : prove that, if perpendiculars
STATICS. 389
be let fall from on the sides, will be the centroid of the triangle
formed by joining the feet of these perpendiculars.
2249. Prove that a point can always be found within a tetra-
hedron A BCD such that, if Oa, 06, Oc, Od be perpendiculars from on
the respective faces, will be the centroid of the tetrahedron abed j and
that the distances of £rom the faces will be respectively proportional
to the faces.
[The point 0, for either the triangle or the tetrahedron, is the point
for which the sum of the squares of the distances from the sides or faces
is the least possibla]
2250. Two uniform similar rods AB^ BC, rigidly united at B and
suspended freely from A^ rest inclined at angles a, )3 to the vertical :
prove that
AB_ /
BC~\
l+?i^-l.
sma
2251. Two uniform rods AB, BC are freely jointed at B and
moveable about A^ which is fixed ; find at what point in BC a smooth
prop should be applied so as to enable the rods to rest in one straight
line inclined at a given angle to the horizon.
[K the weights of the rods be IF, W, the point required must divide
BCm the ratio W : W-^ W\]
2252. Four weights are placed at four fixed points in space, the
sum of two of the weights being given and also the sum of the other
two : prove that their centre of gravity lies on a fixed plane, and within
a certain parallelogram in that plane.
2253. A polygon is such that the angles a^, a,, a,, ... which its sides
make with any fixed straight line satisfy the equations
2 (cos 2a) = 0, 2 (sin 2o) = :
prove that if be the poiut which is the centre of gravity of equal
particles placed at the feet of the perpendiculars from on the sides,
then the centre of gravity of equal particles, placed at the feet of
the perpendiculars from any other point P, will bisect OF,
[Such a polygon has the property that the locus of a pointy which
moves so that the sum of the squares on its distances from the sides is
constant, is a circle.]
2254. The limiting position of the centre of gravity of the area
included between the area of a quadrant of an ellipse bounded by the
axes and the corresponding quadrant of the auxiliary circle, when the
ellipse approaches the circle as its limit, will be a point whose distance
from the major axis is twice its distance from the minor.
2255. A curve is divided symmetrically by the axis of x and is
such that the centre of gravity of the area included between the ordinates
a;=0, a; = A,isata distance ^ = h from the origin : prove that the
equation of the curve is
390 STATICS.
2256. The circle is the only curve in which the centre of gravity
of the area included between any two radii drawn from a fixed point and
the curve lies on the straight line bisecting the angle between the
radii.
2257. Obtain the differential equation of a curve such that the
centre of gravity of any arc measured from a fixed point lies on the
straight Ime bisecting the angle between the radii drawn to the ends of
the arc ; and prove that the curve is a lemniscate of Bernoulli, with its
radii drawn from the node, or a circle.
[The equation is r . / ^ -^(-ja) = «*> t^© general solution of which
is r* s a' sin 2 (0 + a), and a singular solution is r =: a.]
IIL Equilibrium of Smooth Bodies.
2258. A rectangular board is supported with its plane vertical by
two smooth pegs and rests with one diagonal parallel to the straight line
joining the pegs : prove that the other diagonal will be vertical
2259. A rectangular board whose sides are a, 5, is supported with
its plane vertical on two smooth pegs in the same horizontal line at a
distieuice e : prove that the angle 6 made by the side a with the vertical
when in equilibrium is given by the equation
2c cos 20 = h cos — a sin 0.
2260. A tmiform rod, of length c, rests with one end on a smooth
elliptic arc whose major axis is horizontal and with the other on a smooth
vertical plane at a distance h from the centre of the ellipse : prove that,
if be the angle which the rod makes with the horizon and 2a, 26 the
axes of the ellipse,
26 tan = a tan ^, where acos^ + A = ecos0;
and explain the result when a = 26 = e, A = 0.
2261. A rod of length a, whose centre of gravity is at a distance 6
from its lower extremity, rests in neutral eqiulibrium with the upper
extremity on a fixed vertical plane and the lower extremity on an
elliptic arc whose axes are 2a, 26 : prove that the moments about the
centre of the ellipse of the three forces which keep the rod in equilibrium
are in the constant ratios - a : - 6 : a + 6.
2262. A lamina in the form of a rhombus made up of two equila-
teral triangles rests with its plane vertical between two smooth pegs in
the same horizontal plane at a distance equal to a quarter of the longer
diagonal : prove that either a side or a diagonal of the rhombus must
be vertical, and that the stable position is that in which a diagonal is
vertical
2263. A straight uniform rod has smooth small rings attached to
its extremities, one of which slides on a fixed vertical straight wire and
the other on a fixed wire in the form of a parabolic arc whose axis
coincides with the straight wire and whose latus rectum is twice the
^ingth of the rod : prove that in the position of equilibrium (stable
STATICS. 391
when the vertex is upwards), the rod will be inclined at an angle of
30^ to the horizon. Which is the position of stable equilibrium when
the vertex is downwards ]
2264. An elliptic lamina of axes 2a, 25, rests with its plane
vertical on two smooth pegs in the same horizontal line at a distance e :
prove that, when c<b ^2 or > a J 2, the only positions of equilibrium
are when one axis is vertical ; and that, when e> & ^2 and < a J2^ the
positions in which an axis is vertical are both stable and there are
positions of unstable equilibrium in which the pegs are ends of conjugate
diameters.
2265. A rectangular lamina rests in a vertical plane with one
comer against a smooth vertical wall and an opposite side against a
smooth peg : the position of equilibrium is given by the equation
c yja^ + 6* = 26 (6 sin tf - a cos fl) + sin tf (6 cos tf + a sin fl)';
where 2a, 26 are the sides (the latter in contact with the peg), the
angle which the diagonal through the point of contact makes with the
vertical, and c the ^stance of the peg from the wall
2266. Two similar uniform straight rods of lengths 2a, 25, rigidly
united at their ends at an angle cz, rest over two smooth pegs in the same
horizontal plane : prove that the angle which the rod 2a makes with the
vertical is given by the equation
c (a + 6) sin (20 -a) =:a* sin a sin O — b* sin a sin (a — fl),
e being the distance between the pegs.
2267. A uniform lamina in the form of a parallelogram rests with
two adjacent sides on two smooth pegs in the same horizontal plane at a
distance c, 2h is the length of the diagonal through the intersection of
the two sides, a, P, 6 the angles which this diagonal makes with the sides
and with the vertical : prove that
A sin tf sin (a + j8) = c sin ()3 - o + 2^).
2268. A uniform triangular lamina ABC^ rough enough to prevent
sliding, is attached to a fixed point by three fine strings OA^ OB^ OC^
and on the lamina is placed a weight w : prove that the tensions of the
strings are as OA(W-\- Zxw) : OB (W-^ 3f/w) : OC(fr+ 3zw), where W
is the weight of the lamina and x^ y, z the areal co-ordinates, measured
on the triangle ABC^ of the point where w is placed. Also prove thai
the least possible value of w ior which the tensions can be equal is
^^'^^\Wb'^og'"oa)'
where OA is the longest string.
2269. A lamina in the form of an isosceles triangle rests with its
plane vertical and its two equal sides each in contact with a smooth
peg, the pegs being in the same horizontal plane : prove that the axis
of the triangle makes mth the vertical the angle or cos- (*^") ;
h being the length of the axis, a the vertical angle, and e the distance
between the pegs.
392 STATICS.
2270. A unifonn rod AB of* length 2a is freely moveable about A;
a smooth ring of weight P slides on the rod and has attached to it a fine
string which passes over a pulley at a height a yertically above A and
supports a weight Q hanging freely : find the position of equilibrium of
the system ; and prove that, if in this position the rod and string are
equaUy inclined to the vertical,
2Q{Qb-'Way^PWab.
2271. A portion of a parabolic lamina, cut off by a focal chord
inclined at an angle a to the axis, rests with its chord horizontal on two
smooth pegs in the same horizontal line at a distance c : prove that the
latus rectum of the parabola is c J 5 sin' a, that the distance between
the pegs is -^ X length of the bounding chord, and that the centre of
gravity of the lamina bisects the distance between the mid points of the
bounding chord and of the straight line joining the pegs.
2272. A portion of a parabolic lamina cut off by a focal chord
inclined at an angle a to the axis rests on two smooth pegs at a distance 6,
with its chord c parallel to the distance between the p^ and inclined at
an angle P to the vertical : prove that
56' 3 cos (2a -f j8) -f 17 cos ff
c» "" 3cos(2a + /9) + cos/5~'
2273. A small smooth heavy ring is capable of sliding on a fine
elliptic wire whose major axis is vertical ; two strings attached to the
ring pass through small smooth lings at the foci and sustain given
weights : prove that, if there be equUibrium in any position in which
the whole string is not vertical, there will be equilibrium in every
position. Prove also that, when this is the case, the pressure on the
wire will be a maximum when the sliding ring is in the highest or
lowest positions, and a minimum when its distances from the foci are
respectively as the weights sustained.
[The maximum pressures are
Wj (1 + c) + w, (1 - e), w, (1 - c) + w^ (1 + e),
and the minimum is 2jl-e^Jw^w^; where «?,, w^ are the weights
sustained at the upper and lower foci, and e the excentricity of the
ellipse. When w^{l—e)>w{l-he), the pressure will be a maximum in
the highest position, and a mmimum in the lowest, and there will be no
other maximum or minimum pressures.]
2274. A uniform regular tetrahedron has three comers in contact
with the interior of a fixed smooth hemispherical bowl of such
magnitude that the completed sphere would circumscribe the tetra-
hedron : prove that every position is one of equilibrium ; and that^ if
P, Qf a he the pressures at the comers and W the weight of the
tetrahedron,
2(Ci? + i?P + P0 = 3(P' + e' + i2'-ir').
STATICS. 393
2275. A beavj uniform tetrahedron rests with three of its faces
against three fixed smooth pegs and the fourth face horizontal : prove
that the pressares on the pegs are as the areas of the faces respectively
in contact.
2276. A heavy uniform ellipsoid is placed on three smooth p^ in
the same horizontnJ piano so that the pegs are at the extremities of a
system of conjugate diameters: prove that there will be equilibrium,
and that the pressures on the pegs will be one to another as the areas of
the corresponding conjugate central sections.
2277. Seven equal and similar uniform rods AB, BCy CDy DE,
EF^ FGf GA are freely jointed at their extremities and rest in a vertical
plane supported by rings at A and C, which are capable of sliding on a
smooth horizontal rod: prove that, 0^ ^, ij/ being the angles which BA^
AG, GF make with the vertical,
tan ^ - 5 tan ^ = 3 tan ^.
2278. Two spheres of densities p, o- and radii a, 6, rest in a para-
boloid whose axis is vertical and touch each other at the focus : prove
that pV* = cr"6'*; also that, if IT, IT' be their weights and R, R the
pressures at the points of contact with the paraboloid,
2279. Four uniform similar rods freely jointed at their extremities
form a parallelogram, and at the middle points of the rods are small
smooth rings joined by light ngid bars. The parallelogram is suspended
freely from an angular point ; find the stresses along the bars and the
pressures of the rings on the rods, and prove that (1) if the paraUelogram
be a rectangle the stresses will be equal, (2) if a rhombus the pressures
will be equal.
IV. Friction.
2280. Find the least coefficient of friction between a given eUiptio
cylinder and a particle, in order that for all positions of the cylinder in
which the axis is horizontal, the particle may be capable of resting
vertically above the axi&
[If the axes of the transverse section be 2a, 26, the least coefficient of
friction is tan-» ''*'"• *'^
(^)i
2281. Two given weights of different material are laid on. a given
inclined plane and connected by a string in a state of tension inclined
at a given angle to the intersection of the plane with the horizon, and
the lower weight is on the point of motion : determine the coefficient of
friction of the lower weight and the magnitude and direction of the
force of friction on the upper weight.
394 STATICS.
2282. A weight w rests on a rough inclined plane (m < 1) supported
hj a string which, passing over a smooth pulley at the highest point ol
the plane, sustains a weight >fiU7 and <to hanging vertically: prove
that the angle between the two positions of the plane in which t^ is in
limiting equilibrium is 2 tan"^ /x.
2283. Two weights of similar material connected bj a fine string
rest on a rough circular arc on which the string lies : prove that the
«ngle subtended at the centre by the distance between the limiting
positions of either weight is 2 tan~* /uu
2284. A uniform rod rests with one extremity against a rough
vertical wall, the other being supported by a string of equal length
fastened to a point in the wall : prove that the least angle which the
string can make with the wall is tan~' (^/^'O*
2285. A uniform rod of weight W rests with one end against a
rough vertical plane and with the other end attached to a string which
passes over a smooth pulley vertically above the former end and
supports a weight P: find the limiting positions of equilibrium, and
prove that equilibrium will be impossible if P< IT cose, c being the
angle of friction.
2286. Two weights support each other on a rough double inclined
plane by means of a fine string passing over the vertex, and both
weights are on the point of motion : prove that, if the plane be tilted
until both weights are again on the point of motion, the angle through
which the plane will be turned is twice the angle of friction.
2287. A uniform heavy rod rests, with one extremity against a
rough vertical wall, supported by a smooth horizontal bar parallel to
the wall, and the angles between the rod and wall in the limiting
positions of equilibrium are a, p: prove that the coefficient of friction is
sin" p - sin" a
sin* a cos a + sin* fi cos /3 *
2288. A heavy uniform rod of weight W rests inclined at an angle
to the vertical in contact with a rough cylinder of revolution whose
axis IB horizontal and whose diameter is equal in length to the rod : the
rod is maintained in its position by a fine string in a state of tension
which passes from one end of the rod to the other round the cyclinder :
prove that the tension of the string cannot be less than
TTcos (^ + c) -f 2 sin c,
where c is the angle of friction.
2289. A square lamina has a string of length equal to that of a
side attached at one of the comers; the string is also attached to a fixed
point in a rough vertical wall, and the lamina rests with its plane
vertical and perpendicular to that of the wall: prove that, if the
coefficient of friction be 1, the angle which the string makes with the
wall lies between j and J tan"' J.
STATICS. 393
[More generally, if the lamiDa be rectangular, of sides a, b, and the
length of the string be a, and the wall be inclined at an angle a to the
horizon, the angle which the string makes with the Wall in a position
of limiting eqoilibriam is given by the equation
Bin(2d-a + )8*c)-2sin2dcos(a + )8T€)=»sin(a-)8*c);
where a = b tan p, and c is the angle of friction.]
2290. Two weights P, Q, of similar material, resting on a rough
double inclined plane, are connected by a fine string passing over tibe
common vertex, and Q is on the point of motion down the plane : prove
that the great^t weight which can be added to P without disturbing
the equilibrium is
P8in2csin(a + )8)
sin (a — c) sin (Ji-t)*
a, 13 being the angles of inclination of the planes and c the angle of
friction.
2291. A uniform rod rests with one extremity against a rough
vertical wall (3/x = 7), the other extremity being supported by a string
three times the length of the rod attached to a point in the wall : prove
that the tangent of the angle which the string makes with the wall in
the limiting position of equilibrium is /^^ or ^.
2292. A given weight resting upon a rough inclined plane is
connected with a weight F by means of a string passing over a rough
p^, P hanging freely; the angles of friction for the peg and plane are
A, X.' respectively, (A > X') : prove that the inclination of the string to
the plane in limiting equilibrium, when P is a maximum or minimum,
is X-X'.
2293. A weight W is supported on a rough inclined plane of
inclination a by a force P, whose line of action makes an angle i with the
plane and whose component in the plane makes an angle fi with the
line of greatest inclination in the plane : prove that equilibrium will be
impossible if
fi' (1 + cos 2a cos 2i - sin 2asin 2i cos)3) > 2 sin'asin'jS coe'c
2294. A heavy particle is attached to a point in a rough inclined
plane by a fine weightless rigid wire and rests on the plane with the
wire inclined at an angle $ to the line of greatest inclination in the
plane ; determine the limits of $, the angle of inclination of the plane
being tan~* (/i cosec p).
[The limiting values of are )3, ir— )3, and must not lie between
these limits.]
2295. Two weights A, B connected by a fine stri ng are lying on a
rough horizontal plane ; a given force /*(>/* JjJT^ and < ful + ikB) is
continually applied to ^i so as just to move A and B rery slowly in the
plane : prove that A and B will describe concentric circles whose radii
areaoosec)?, aootjS, where cosjSs '1 *ar — •
396 STATICS.
y. MasHc Strmgs.
2296. A string whose extensibility varies as the distance from one
end is stretched by any force : prove that its extension is equal to that
of a string of equal length, of uniform extensibility equal to that at the
centre' of the former, when stretched by an equal force.
2297. An elastic string rests on a rough inclined plane with the
upper end fixed to the plane : prove that its extension will lie between
the limits jrr- ^ ; a being the inclination of the plane, c the angle
2X cose ' ^ . ^ ' ®
of friction, and I, X the lengths of the whole string and of a portion of it
whose weight is equal to the modulus.
2298. Two weights P, Q are connected by an elastic string without
weight which passes over two small rough pegs A, B in the same hori-
zontal line at a distance a, Q is just sustained by P, and AF = 6, BQ = e :
P and Q are then interchanged, and AQ = b\ BF = c': obtain equations
for determining the natural length of the string, its modulus, and the
coefficients of friction at A and B.
2299. A weight P just supports another weight Q by means of a
fine elastic string passing over a rough cylinder of revolution whose
axis is horizontal ; FT is the modulus and a the radius of the cylinder :
prove that the extension of the part of the string in contact with the
cylinder ia ? log (|±-^) .
2300. A heavy extensible string, uniform when unextended, hangs
symmetrically over a cylinder of revolution whose axis is horizontal, a
portion whose length in the position of rest is a - A hanging vertically
on each side : prove that the natural length of the part of the string in
contact with the cylinder is 2j2ah\og{J2-\-\)] a being the radius of
the cylinder, and 2A the length of a portion of the string whose weight
(when unextended) is equal to the modulus: also prove that the
extension of either of the vertical portions of the string is (>/a— is/^)'*
2301. An extensible string is laid on a cycloidal arc whose plane is
vertical and vertex upwards, and when stretched by its own weight is
just in contact with the whole of the cycloid, the natural length of the
string being equal to the perimeter of the generating circle : prove that
the modulus is the weight of a portion of the string whose natural
length is twice the diameter of the generating circle.
2302. A heavy elastic string whose natural length is 2Z is placed
synmietrically on the arc of a smooth cycloid whose axis is vertical and
vertex upwards, and a portion of string whose natural length is x hangs
vertically at each cusp : prove that
2^aX = (a + X)tanr-7=;
2a being the length of the axis of the cycloid, and X the natural length
of a portion of the string whose weight is equal to the modulus.
STATICS. 397
2303. A smooth right cylinder whose base is a cardioid is placed
with the axis of the cardioid yertical and vertex upwards, and a heavy
extensible string rests symmetrically upon the upper part in contact
with a portion of the cylinder whose length is twice the axis of the
cardioid, and the length of string whose weight is equal to^ the modulus
is equal to the length of the axis : prove that the natural length of the
string is to its length when resting on the cylinder as log (2 + 1^3) : ^3.
2304. An extensible string of natural length 21 just suironnda
a smooth lamina in the form of a cardioid, its free extremities being
at the cusp, and remains in equilibrium under the action of an attractive
force varying as the distance and tending to the centre of the fixed
circle (when the cardioid is described as an epicycloid) : prove that
yi|»2«cot(iyi)j
a being the radius of the fixed circle, 2kl the mass of the string, X
the modulus, and fir the acceleration of the force on unit mass at a
distance r.
VI. Catenaries, AUractuma, dx,
2305. An endless heavy chain of length 2/ is passed over a smooth
cylinder of revolution whose axis is horizontal ; c is the length of a
portion of the chain whose weight is equal to the tension at the lowest
point, and 2^ the angle between the radii drawn to the points where
the chain leaves the cylinder : prove that
tan «^ + ^.— ^ log tan ( 7 + ? J = - .
^ sm^ *=* \4 2/ c
2306. In a common catenary A is the vertex, P, Q two pointe
at which the tangents make angles ^, 2^ respectively with the horizon,
and the tangents at J, Q meet in : prove that the arc AP is equal
to the horizontal distance between and Q,
2307. Four pegs A, B, C, D axe placed at the comers of a square,
BC being vertically downwards, and an endless uniform inextensible
string passes round the four hanging in two festoons : prove that
1 l_-2
sinalogcot^ sin/Slogcot^
1 1 1/
tan^logcotg tan|logcot^
a, p being the angles which the tangents at i?, C make with the
vertical, I the length of the string, and a the length of a side of the
square.
398 STATica
2308. A heavy uniform chain rests in limiting equilibrium on
a rough circular arc whose plane is vertical, in contact with a quadrant
of the circle one end of which is the highest point of the circle : prove
that
(l./)A=2^.
2309. A heavy uniform chain rests in limiting equilibrium on
a rough cjcloidal arc whose axis is vertical and vertex upwards, one
extremity being at the vertex and the other at a cusp : prove that
(! + /*•) €"2=3.
2310. A uniform inextensible string hangs in the form of a
common catenary, the forces at any point being X, Y perpendicular
and parallel to the axis : prove that
sin^ - . +cos^T- +2Xsec^ = 0;
where ^ is the angle which the normal at the point makes with
the axis.
2311. To each point of a chain hanging under gravity only in
the form of the catenary 8 = ctan0 is appli^ a horizontal force pro-
portional to 8"* : prove that the form will be unaltered.
2312. A uniform inextensible string can rest in the form of a
two-cusped epicycloid imder the action of a constant force always
tending from the centre of the moving circle.
2313. A uniform inextensible string rests in the form of a circle
under a force which is always proportional to the square root of the
tension : prove that the force is proportional to the distance from a
fixed point on the circle and that its line of action always touches a
certain cardioid.
[More generally^ if the force vary as the rfi^ power of the tension,
its line of action will always toudi an epicycloid generated by a
circle of radius a -r 2 (2 — n) rolling on a circle of radius
a(l-n)-5-(2-»).]
2314. A uniform chain is kept in equilibrium in the form of an
ellipse by repulsive forces ^j , ^^ in the foci : prove that
where r^, r^ are the focal distances.
2315. A uniform chain is in equilibrium in the form of an
equiangular spiral and the tension is proportional to the radius vector :
prove that the force is constant and makes a constant angle with the
radius vector. When the chain is in equilibrium in the form of an
STATICS. 399
equiangular spiral under a constant force wliich at any point makes
an angle 6 with the norma!,
sin -7j = sin + cot a cos 0;
(i<p
where ^ is the angle which the tangent makes with a fixed direction.
2316. A uniform chain can rest in the form of a common catenary
under the action of a constant force if the force at any point make
with the axis an angle 6 determined by the equation
2 ^ (tan ^^i5n(9)- VBin(9 = 0;
^ being the angle which the normal at the point makes with the axis.
2317. A heavy uniform chain fastened at two points rests in the
form of a parabola under the action of two forces, one (A) parallel
to the axis and constant, and the other (F) tending from the focus:
prove that 3^=il + ^cos'^, <f> being the angle through which the
tangent has turned since leaving the vertex and B a constant
2318. Find the law of repulsive force tending from a focus under
which an endless uniform chain can be kept in equilibrium in the
form of an ellipse ; and, if there be two such forces, one in each focus
and equal at equal distances, prove that the tension at any point varies
inversely as the conjugate diameter.
2319. A uniform chain rests in the form of a cycloid whose axis
is vertical imder the action of gravity and of a certain normal force,
the tension at the vertex vanishing : prove that the tension at any
point is proportional to the vertical height above the vertex, and that
the normal force at any point bears to the force of gravity the ratio
(3cos'0-l) : 2cosd;
where is the angle which the normal makes with the vertical
2320. A heavy chain of variable density suspended from two
points hangs in the form of a curve whose intrinsic equation is 8 =y* (^),
the lowest point being origin : prove that the density at any point
will vary inversely as cos* 4^f\4>)'
2321. A string is kept in equilibrium in the form of a closed
curve by the action of a repulsive force tending from a fixed point,
and the density at each point is proportional to the tension: prove
that the force at any point is inversely proportional to the chord of
curvature through the centre of force.
2322. A uniform chain is in equilibrium under the action of
certain forces ; from a fixed point is drawn a straight line Op parallel
to the tangent at any point P of the chain and proportional to the
tension at P : prove that, (1) the tangent at p to the locus of p is
parallel to the resultant force at P, (2) the ultimate ratio of small
corresponding arcs at />, P is proportional to the residtant force at P.
400 STATICS.
2323. A uniform heavy chain rests in contact with a smooth
arc in a vertical plane of such a form that the pressure at any point
per unit of length is equal to m times the weight of a unit of length :
prove that the intrinsic equation of the curve will be
cb a
a y
d^ (m + cos if>)
that^ when m>\^ the horizontal distance between two consecutive
vertices is r , the arc between the same points ^ ,
(m'-l)* (w'-l)»
and the vertical distance between the Hues of highest and lowest points
2a-J-(m*— 1). When m = l, the curve is the first negative pedal of
a parabola from the focus.
2324. A uniform heavy string is attached to two points in the
surface of a smooth cone of revolution whose axis is vertical and rests
with every point of its length in contact with the cone : prove that
the curve of equilibrium is such that its differential equation, when
the cone is developed into a plane, is jp (r + c) = a', the vertex of the
cone being pole,
2325. A uniform chain is laid upon the arc of a smooth curve
which is the evolute of a common catenary so that a portion hangs
vertically below the cusp of a length equal to the diameter of the
catenary at the vertex : prove that the resolved vertical tension at any
point of the arc is constant, and that the resolved vertical pressure
per unit of length is equal to the weight of a unit of length of the
chain. Also, in the curve whose intrinsic equation is
« = a8in^-r^i+cos*^,
where ^ is measured from the horizontal tangent^ if a uniform chain
be bound tightly on any portion of it so that the tension at a vertex
is equal to the weight of a length a J2 of the chain, the resolved
vertical pressure per unit will be equal to the weight of a imit of length
and the resolved vertical tension at any point wUl he twice the weight
of the chain intercepted between that point and the vertex.
[The height above the directrix of the a G. of the portion of chain
included between two cusps is
-^2 log (72 + 1) + a,
and the area included between the directrix, the curve, and the tangents
at two consecutive cusps is 7ra*-f-^2.]
2326. A heavy uniform chain just rests upon a rough curve in the
form of the arc of a four-cusped hypocycloid, occupying the space
between two consecutive cusps at which the tangents are horizontal
and vertical respectively : prove that
2fi€''«=^« + 3.
STATICS. 401
2327. Find a curve such that the a o. of any arc lies in a straight
line drawn in a given direction through the intersection of the tangents
at the ends of the arc.
[It is obvious that the common catenary satisfies the condition, and
it will be found that^ when the arc is uniform, no other curve does so.
When the density is variable and the curve such that -rr<^<l^ varies
inversely as the density, the condition will be satisfied if the direction
from which ^ is measured be at right angles to the given direction.]
2328. A uniform chain rests in a vertical plane X)n a rough curve in
the form of an equiangular spiral whose constant angle between the
normal and radius vector is equal to the angle of friction, one end
being at a point where the tangent la horizontal : prove that, for
limiting equilibrium, the chain will subtend at the pole an angle equal
to twice the angle of friction. (The chain makes an obtuse angle
with the radius vector to the highest point.)
2329. A uniform wire in the form of a lemniscate of Bernoulli
attracts a particle at the node, the force varying as the distance : prove
that the attraction of any arc is the same as that of a circular arc
of the same material touching the lemniscate at its vertices and inter-
cepted between the same radii from the node. The same property
wUl hold for an equiangular spiral when the force varies inversely as
the distance, and for a rectangular hyperbola when the force varies
inversely as the cube of the distance, and generally for any curve
r^ = a" sin wtf ,
if the force vary as r^"'.
w. P. 26
DYNAMICS, ELEMENTARY.
I. Rectilinear Motion : Impulses,
2330. A ball A impinges on another ball B^ and after impact the
directions of motion of A and B make equal angles $ with the previous
direction of A: determine 0, and prove that^ when A=Bf taiiO= Je^
where e is the coefficient of restitution.
[In general j9(1 -e)co8 2^»^ + ej9.]
2331. A smooth inelastic ball, mass m, is lying on a horizontal
table in contact with a vertical wall and is struck by another ball, mass
fn\ moving in a direction normal to the wall and inclined at an angle a
to the common normal at the point of impact : prove that the angle B^
through which the direction of motion of the striking ball is turned, is
given by the equation m cot 6 cot a = m + w'.
2332. Two equal balls A^ B are lying very nearly in contact on a
smooth horizontal table ; a third equal ball impinges directly on A , the
three centres being in one straight line : prove that if c> 3 — 2 J2, the
final velocity of B will bear to the initial velocity of the stiiking ball the
ratio (1 + «)• : 4.
2333. Equal particles A^y A^^ .,, A^ are fastened at equal intervals
a on a fine string of length {n-\)a and are then laid on a horizontal
table at n consecutive angular points of a regular polygon of p sides
(p > n), each equal to a; a blow P is applied to -4, in direction A^A^i
prove that the impulsive tension of the sti'ing A^A^^^ is
, (1 -^ sin a)'-^-(l -sing)'-"
^^""^ "^ (l+sina)--(l-sina)- '
where pa is equal to 2ir.
2334. A circle has a vertical diameter AB^ and two particles fall
down two chords AP, PB respectively, starting simultaneously from
A, P: p.rove that the least distance between them during the motion is
equal to the distance of P from AB.
DTNAMICS, ELEICEHTABT. 40S
2335. A number of heavy particlee start at once from the vertex of
an obliqne circular cone whose base is horizontal and fall down generat-
ing lines of the cone : prove that at anj subsequent instant they will all
lie in a subcontrary section.
2336. The locus of a point P such that the times of fiJling down
PAj PB to two fixed points J, B may be equal is a rectangular
hyperbola in which AB is a diameter and the normals at A^ B are
vertical.
2337. The locus of a point P such that the time of falling down
PA to a fixed point A is equal to the time of falling vertically from A
to a fixed straight line is one branch of an hyperbola in which one
asymptote is vertical and the other perpendicular to the fixed straight
line.
[The other branch of the hyperbola is the locus of a point P such
that the time down AP \a equal to the time from the straight line verti-
cally to P,]
2338. A parabola is placed with its axis vertical and vertex down-
wards : prove that the time of falling down any chord to the vertex ia
equal to the time of falling vertically through a space equal to the
parallel focal chord.
2339. An ellipse is placed with its major axis vertical : prove that
the time of descent down any chord to the lower vertex, or from the
higher vertex, is proportional to the length of the parallel diameter.
2340. The radii of two circles in one vertical plane, whose centres
are at the same height, are a, h and the distance between tibeir centres is o
(which is greater than a + 6) : prove that the shortest time of descent
from one circle to the other down a straight line is *, / — ; — - .
^^ V ^ a + 6
2341. The radii of two circles in one vertical plane are a, 6, the
distance between their centres c, and the inclination of this distance to
the vertical is a : prove that, when c>{a-^ 6), the time of shortest descent
down a straight line from one circle to the other is equal to the time of
falling vertically through a space r-^ '— : and, when a > (6 + c\
the shortest time from the outer to the inner is ^ / — ^7 — i
^V ^a — 6 + ccosa
and from the inner to the outer ^ / — ^ — ; — '- , a beinir the ande
V ^a-6-ccosa ^ ^
which the line of centres makes with the vertical measured upwards from
the centre of the outer circle. Also prove that^ when c cos a > (a + 6),
there will be a maximum time of descent from one circle to the other
down a straight line, and this time will be «. / "^ . — '-rr ,
° V ^ccoea-(a+6)
26—2
404 DYNAMICS, ELEMENTARY.
[In the first caae, the length of the line of shortest descent is
Vc" + (a + 6)* + 2 (a+ 6) ccosa
and the angle which it makes with the vertical is
, a + 5 +c cosa
cos"*
^c*+ (a+ 6)*+ 2 (a + 6)c cos a
and similarly in the other cases.]
2342. A parabola is placed with its axis horizontal : prove that the
length of the straight line of shortest descent from the cui*ve to the
focus is one third of the latus rectum.
2343. A parabola is placed with its plane vertical and its axis
inclined at an angle 3a to the vertical : prove that the straight line of
shortest descent from the curve to the focus is inclined at an angle a to
the vertical
2344. An ellipse is placed with its major axis vertical : prove that
the straight line of quickest descent from the curve to the lower focus
(or from the higher focus to the curve) is equal in length to the latus
rectum, provided the excentricity exceed \.
2345. Two straight lines OP, OQ from a given point to a given
circle in the same vertical plane are such that the times of falling down
them are equal : prove that PQ passes through a fixed point.
2346. Two circles A^ Bin the same vertical plane are such that the
centre of il is the lowest point of B ; through each point F on B are
drawn two straight lines to A such that the times down them are equal
to the time from F to the centre of A : prove that the chord of A
joining the ends of these lines will touch a fixed cii*cle concentric
with A,
2347. There are two given circles in one vertical plane and from
each point of one are drawn the two straight lines of given time of
descent (t) to the other : prove that the chord joining the ends of these
lines envelopes a conic, whose focus is vertically below the centre of the
former circle at a depth ^gt*,
2348. Two weights IT, W move on two inclined planes, and are
connected by a fine string passing over the common vertex, the whole
motion being in one plane : prove that the centre of gravity of the
weights describes a straight line with uniform acceleration equal to
' WBinB-Wsma , ;
9 (]f7ip7y 7Tr«+ir" + 2ir>F'cos(a + /8);
where a, fi are the inclinations of the plane&
DTKAMICS, ELEBIENTABT. 405
2349. When there is equilibrium in the single moveable pulley,
the weight is suddenly doubled and the power is halyed : prove that, in
the ensuing motion, the tensions of the strings are the same aa in
equilibrium.
2350. In the system of pullies in which each hangs by a separate
string, P just supports W : prove that, if P be removed and another
weight Q be substituted, the centre of gravity of Q and W will descend
with uniform acceleration
2351. In any machine without friction and inertia a weight P,
supports a weight IT, both hanging by vertical strings ; these weights
are removed and weights P', W\ respectively substituted : prove that, if
in the subsequent motion F and W always move vertically, their centre
of gravity will descend with acceleration
(WF-W'Py
2352. Two weights each of 1 lb. support each other by means of a
fine string passing over a moveable pulley to which is attached another
string passing over another pulley and supporting a weight of 2 lbs. ; to
this pulley is similarly attached another string supporting a weight of
4 lbs., and so on, the last string passing over a fixed pulley and support*
ing a weight of 2* lbs. : prove that, if the r^ weight, reckoning from the
top, be gently raised through a space of 2'' — 1 inches, all the other weights
will each fall one inch ; and, ^ the r^ weight be in any way gradually
brought to rest, all the weights will come to rest at the same instant.
(The pulleys are of insensible mass.)
2353. A fine uniform string of length 2a is in equilibrium, passing
over a small smooth pulley, and is just displaced : prove that the velodly
of the string when just leaving the pulley is ,Jag*
2354. A large number of equal particles are fastened at unequal
intervals to a fine string and then collected into a heap at the edge of a
smooth horizontal table with the extreme one just hanging over the edge ;
the intervals are such that the times between successive particles b^ng
carried over the edge are equal : prove that, if c, be the length of string
between the n*** and w + 1* particle, and v^ the velocity just after the
n + 1^ particle has been carried over.
Deduce the law of density of a string collected into a heap at
the edge of the table with the end just over the edge, in order that
equal masses may always pass over in equal times.
[The density must vary inversely as the square root of the distance
from the end.]
406 DYNAMICS, ELEMENTARY.
2355. A large number of equal particles are attached at equal
intervals to a string and the whole is heaped up close to the edge of a
smooth horizontal table with the extreme particle just over the edge :
prove that^ if v^ denote the velocity just before the w + 1*** particle is set
in motion,
- ._^(!L!ll)_(2n+2)
^- ~ 3 n *
where a denotes the length between two consecutive particles. Calculate
the dissipated energy, and prove that, when a m indefinitely diminished,
the end of the string, in the limit, descends with uniform accelei*ation \g,
[The whole energy dissipated, just before the w+P** particle is
.set in motion, is ^aw (n*— 1), where w is the weight of each particle.]
2356. A large number of equal particles are attached at equal
intervals a to a fine string which passes through a very short fine tube in
the form of a semicircle, and initially there are 2r particles on one side
of the tube, the highest being at the tube, and r particles on the other
side, the lowest being in contact with a horizontal table where the
remaining particles are gathered together in a heap : prove that, if v^
denote the velocity just before the n^ additional particle is set in motion,
''•="3"r"'(/i + 3r-l)V'
and deduce the corresponding result for a uniform chain hanging over a
small pulley.
II. Parabolic Motion.
2357. A heavy particle is projected from a given point -4 in a
given direction : determine its velocity in order that it may pass through
another given point B,
[If the polar co-ordinates of B referred to ^ be (a, o), and P be the
angle which the given direction makes with the horizontal initial line,
the space due to the velocity of projection will be
a cos* a -r 4 cos )8 sin (/3 - a) .]
2358. A particle moving under gravity passes through two given
points : prove that the locus of the focus of its path is an hyperbola
whose foci are the two given points.
2359. The distances of three points in the path of a projectile from
the point of projection are r„ r,, r,, and the angular elevations of the
three points above the point of projection are a^, a,, a,: prove that
r^ oos'ttj sin (a, - aj + r,eoe* a,sin (a,-Oj) + r, cos* a, sin (a^ - oj = 0.
2360. A number of heavy particles are projected from the same
point at the same instant: prove that their lines of instantaneous motion
at any subsequent instant will meet in a point, and that this poiiit will
ascend with uniform acceleration g»
DTNA3nCS, ELEMENTABT. 407
2361. A number of heavy particles are projected in a vertical plane
from one point at the same instant with equal velocities : prove that at
any subsequent instant they will all lie on a circle whose centre descends
with acceleration g and whose radius increases uniformly with the time.
Also if, instead of having equal velocities, the velocity of any particle
whose angle of projection is ^ be that due to a height a sin'^, the particles
will at any subsequent instant all lie on a circle.
2362. Two points Ay B in the path of a projectile are such that the
direction of motion at J? is parallel to the bisector of the angle between
the direction of motion at A and the direction of gravity : prove that the
time from A i/o B in equal to that in which the velocity at A would be
generated in a particle falling from rest under gravity.
2363. A number of particles are projected from the same point
with velocities such that their components in a given xlirection are all
equal : prove that the locus of the foci of their parabolic paths is another
parabola whose focus is the point of projection, semi latus rectum the
space due to the given component velocity, and the direction of whose
axis makes with the vertical an angle which is bisected by the given
direction.
2364. A particle is projected from a given point so as just to
pass over a vei-tical wall whose height is 6 and distance from the point
of projection a : prove that, when the area of the parabolic path
described before reaching the horizontal plane through the point of
projection is a maximum, the range is f a and the height of the vertex
of the path J 6.
2365. A particle is projected from a point at the foot of one of two
parallel vertical smooth walls so as after thi'ee reflexions at the walls to
i-etum to the point of projection, and the last impact is dir^t: prove
that e* +6^ + c= 1, and that the vertical heights of the three points of
impact above the point of projection are as e* : 1 — c* ; 1.
2366. A heavy particle, mass m, is projected from a point A bo ha
after a time ^ to be at a point B: prove that the action in passing
{A H* a^t* \
+ Y^ >, and is a minimum when the focus of the
path lies in AB,
2367. In the parabolic path of a projectile, AB is a focal chord :
prove that the time from A to B \& always equal to the time of falling
vertically from rest through a space equal to AB ; and that the action in
passing from AtoB\a also equal to the action in falling vertically from
rest through a space equal U) AB,
2368. A heavy particle is projected from a point in a horizontal
plane in such a manner that at its highest point it impinges directly on
a vertical plane from which it rebounds, and after another rebound
from the horizontal plane returns to the point of projection : prove that
the coefficient of restitution is \,
[The equation for e is 20* + e - 1 = ; the student should account for
the root - 1.]
408 PYNAMICS, ELEMENTARY.
2369. A heavy particle, for which 6 = 1, falls down a chord from
the highest point of a vertical circle, and after reflexion at the arc
describes a parabolic path passing through the lowest poiut : prove that
the inclination of the chord to the vertical is J cos"* (- — j^ ) • If the
particle fall from the centre down a radius and after reflexion pass
through the lowest point, the inclination to the vertical will be cos" i*
•
2370. A particle is projected from a given point with given velo-
city up an inclined plane of given inclination so as after leaving the
pliuie to describe a parabola : prove that the loci of the focus and vertex
of the parabola for diflerent lengths of the plane are both straight lines.
2371. A particle, for which e = l, is projected from the middle
point of the base of a vertical square towards one of the angles, and
after being reflected at the sides containing that angle falls to the
opposite angle : prove that the space due to the velocity of projection
bears to the length of a side of the square the ratio 45 : 32.
[More generally, when the particle is projected from the same point
at an angle a to the horizon, the space due to the velocity of projection
must be to the length of a side as 9:16 cos a (3 sin a - 4 cos a) ; and
3 tan a mast lie between 4 and 9.]
2372. A particle («= 1) is projected with a given velocity from a
given point in one of two planes equally inclined to the horizon and
intersecting in a horizontal line, and after reflexion at the other plane
returns to its starting point and is again reflected on the original path ;
determine the direction of projection and prove that the inclination of
each plane must be 45^ Also, if the planes be not equally inclined to
the horizon, prove that they must be at right angles and that the incli-
nation of projection to the horizon ($) is given by the equation
• d
cos {0 + 2a) COS tf + T-siii a cos* o = 0,
where h is the space due to the velocity of projection, a the distance from
the line of intersection, and a the inclination of the plane from which
the particle starts.
[This equation has two roots , $^ and the times of flight in the two
paths will be as cos ($^ + 2a) : cos (O^ + 2a).]
2373. A particle being let fall on a fixed inclined plane bounds on
to another fixed inclined plane, the line of intersection being horizontal,
and the time between the planes is given : prove that the locus of the
point from which the particle is let fall is in general a parabolic cylinder,
but will be a plane if tan a tan (a + j8) = c, where a, /3 are the angles of
inclination of the planes.
2374. A heavy particle projected at an angle a to an inclined plane
whose inclination to the vertical is i, rebounds from the plane : prove
that, if 2tana=(l — e)tant, the successive parabolic paths will be
similar arcs of parabolas, and will all touch two fixed straight lines, one
of which is normal to the plane and the other inclined to it at an angle
tan"* (-T~ **^ *) •
DYNAMICS, ELEMENTARY. 409
2375. A particle projected from a point in an inclined plane at the
r^ impact strikes the plane normally and at the n^ impact is at the point
of projection : prove tiiat c" — 2^^ + 1 = 0.
2376. A particle is projected from a given point in a horizontal plane
at an angle a to the horizon, and after one rebound at a vertical plane
returns to the point of projection : prove that the point of impact must
lie on the straight line
y (1 + c) = a; tan a,
Xj y being measured horizontally and vertically from the point of pro-
jection. When the velocity of projection and not the direction is given,
the locus of the point of impact is the ellipse
a:' + y*(l +«)* = 4eAy,
where h is the space due to the velocity of projection.
2377. A particle is projected from a given point with given velocity
so as, after one reflexion at an inclined plane passing through the point,
to return to the point of projection : prove that the locus of the point of
impact is also the ellipse
a*+(l +e)*y'=4eAy,
with the notation of the last question.
2378. A heavy particle is projected from a point in a plane whose
inclination to the horizon is 30*^ in a vertical plane perpendicular to.
the inclined plane : prove that, if all directions of projection in that
vertical plane are equally probable, the chance of the range on the
inclined plane being at least one-third of the greatest possible range
is *5.
2379. A particle is projected from a point midway between two
smooth parallel vertical walls, and after one impact at each wall returns
to the point of projection: prove that the heights of the points of
impact above the point of projection will be as 0(2e+l) : 2 + e, their
depths below the highest point reached by the particle as
(1+2C-C*)* : (l-2<?-c*)';
and that this highest point lies in a fixed vertical straight line whose
distance from the point of projection is the less of the two lengths
a being the distance between the walls. Also, if the three parabolic
paths be completed, each will meet the horizontal plane through the
point of projection in fixed points.
III. Motion <m a smooth Curve under the action of Gravity.
2380. A heavy pai*ticle is projected up a smooth parabolic arc
whose axis is vertical and vertex upwards with a velocity due to the
depth below the tangent at the vertex: prove that, whatever be the
length of the arc, the parabola described by the particle after leaving
the arc, will pass through a fixed point.
410 DYNAMICS, ELEMENTARY.
2381. A heavy particle falls down a smooth curve in a vertical
plane of such a form that the resultant force on the particle in every
position is equal to its weight : prove that the radius of curvature at
any point is twice the intercept of the normal cut off by the horizontal
line of zero velocity.
2382. A heavy particle is projected so as to move on a smooth
parabolic arc whose axis is vertical and vertex upwards : prove that
the pressure on the curve is always proportional to the curvature.
2383. A heavy particle is projected from the vertex of a smooth
parabolic arc whose axis is vertical and vertex downwards with a
velocity due to a height h, and after passing the extremity of the
arc proceeds to describe an equal parabola freely : prove that, if c be
the vertical height of the extremity of the arc, the latus rectum is
4 (A- 2c).
2384. A parabola is placed with its axis horizontal and plane
vertical and a heavy smooth particle is projected from the vertex so
as to move on the concave side of the arc : prove that the vertical
height attained before leaving the arc is two-thirds of the greatest
height attained ; and that, if 2d be the angle described about the focus
before leaving the curve
A = a(tan'tf+3tand),
and the latus rectum of the free path will be 4a tan' 0; h being the
space due to the initial velocity and 4a the latus rectum of the
parabolic ara
2385. Two heavy particles, connected by a fine string passing
through a small fixed ring, describe horizontal circles in equal times :
prove that the circles must lie in the same horizontal plane.
2386. A heavy particle P is attached by two strings to fixed
points Ay B in the same horizontal plane and is projected so as just
to describe a vertical circle; the string PB is cut when i' is in its
lowest position, and P then proceeds to describe a horizontal circle :
prove that 3 cos 2PAB = 2 ; and that, in order that the tension of the
string PA may be unaltered, the angle APB must be a right angle.
2387. Two given weights are attached at given points of a fine
string which is attached to a fixed point, and the system revolves
with uniform angular velocity about the vertical through the fixed
point in a state of relative equilibrium : prove the equations
tan d' = — (a sin d + a' sin 0') = tan $ + j — sin 0" :
where a, a' are the lengths of the upper and lower strings, m, m' the
masses of the particles, $, & the angles. which the strings make with
the veiiiical, and 12 the common angular velocity.
2388. A heavy particle is projected so as to move on a smooth
cii^cular arc whose plane is vertical and afterwards to describe a parabola
freely : prove that the locus of the focus of the parabolic path is an
DYNAMICS, ELEMENTARY. 411
epicycloid formed by a circle of radius a rolling on a circle of radius 2a ;
4a being the radius of the given circle.
2389. A cycloidal arc is placed with its axis vertical and vertex
upwards and a heavy particle is projected from the cusp up the concave
side of the cur\'e wiUi the velocity due to a height A ; prove that the
latus rectum of the }>arabola described after leaving the arc is A* -r 4a,
where a is the radius of the generating circle ; also that the locus of
the focus of the parabola is the cycloid which is enveloped by that
diameter of the generating circle which passes through the generatiiig
point.
2390. In a certain curve the vertical ordinate of any point bears
to the vertical chord of curvature at that point the constant ratio 1 : m,
and a particle is projected from the point where the tangent is vertical
along the curve with any velocity : prove that the vertical height
attained before leaving the curve bears to the space due to the velocity
of projection the constant ratio 4:4 + rn.
2391. A smooth heavy particle is projected from the lowest point
of a vertical circular arc with a velocity due to a space equal in length
to the diameter 2a, and the length of the arc is such that the range
of the particle on the horizontal plane through the point of projection
is the greatest possible : prove that this range is equal to a ^9 + 6^3.
NEWTON.
2332. Two triangles CAB^ cAh have a common angle A and the
sum of the sides containing that angle is the same in each; BC^ be
intersect in D : prove that in the limit when b moves up to B^
CD : DB^AB : AC.
2393. Two equal parabolas have the same axis and the focus of
the outer is the vertex of the inner one, MPp^ NQq are common
ordinates : prove that the area of the surface generated by the revo-
lution of the arc FQ about the axis bears to the area MpqN a con-
stant ratio.
2394. Common ordinates from the major axis are drawn to two
eUipses which have a common minor axis and the outer of which
touches the directrices of the inner : prove that the area of the surface
generated bj the intercepted arc of the inner ellipse revolving about
tiie major axis will bear a constant ratio to the corresponding intercepted
area of the outer.
[In general if PM be the ordinate and PO the normal to any given
curve at P both terminated by the same fixed straight line, and MP
be produced to p so that Mp = PQ in length, the area of the surface
generated by an elementaiy arc PP* will bear the constant ratio 2v : 1
to the corresponding area MppM'.'l
2395. A diameter J^ of a circle being taken, P is a point on the
circle near to A and the tangent at P meets BA produced in T : prove
that ultimately the difference of BA^ BP bears to AT the ratio 1 : 2.
2396. The tangent to a curve at a point B meets the normal at a
point A in T ) (7 is the centre of curvature at A and a point on AC\
]>rove that, in the limit when B moves up to A^ the difference of OA and
OB bears to ^T the ratio OC : OA.
2397. In an arc PQ of continued curvature ^ is a point at which
the tangent is parallel to PQ : prove that the ultimate ratio PE : BQ
when PQ is diminished indefinitely is one of equality.
2398. The tangents at the ends of an arc PQ of continued cur-
vature meet in : prove that the ultimate ratio of
OP + OQ-BToPQ : arc PQ " chord PQ,
as PQ is indefinitely diminished, is 2 : 1.
NEWTON. 413
2399. Three oontigaoas points being taken on a curve, the tangents
form a triangle and the normals a similar triangle : prove that the
ultimate ratios of these triangles when the points tend to coincidence
at P is 1 : (-fj ) p being the radius of curvature at F and « the
arc to F from some fixed point of the curve.
2400. A point is taken in the plane of a given closed oval,
F is any point on the curve, and QFQ" a straight line drawn in a
given direction so that QF = FQ' and that each bears a constant
ratio n : 1 to OF: prove that, as F moves round the curve, Qf Q'
will trace out two closed loops the sum of whose areas is double the
area of the given ovaL
[When is within the oval, the loops will intersect if n>l, and
touch if w = 1 ; when is without the curve, the loops will intersect
if n be less than a certain value (always < 1) which depends on the
position of 0.]
2401. Two contiguous points 0, & are taken on the outer of two
confocal ellipses and tangents OP, OQ^ (fF^ CfQ drawn to the inner,
F* coinciding with F when (f moves up to : prove that in the limit
FF xQq^OF'.Oqf.
2402. At a point P of a curve is drawn the circle of curvature, and
small arcs PQ, Fq are taken such that the tangents at Q, q are parallel :
prove that Qq generally varies as P§*, but, if P be a point of maximum
or minimum curvature, Qq will vary as F^\ also that the angle which
Qq makes with the tangent at P is, in the former case two-thirds and in
the latter three-fourths of the angle which the tangent at Q or ^ makes
with that at P.
2403. Three equal particles A^B^C move on the arc of a given
circle in such a way that their cenj;re of gravity remains fixed : prove
that, in any position, their velocities are as sin 2^ : sin ^B : sin 20.
2404. The velocities at three points of a central orbit are in-
versely as the sides of the triangle formed by the tangents at these
points : prove that the centre of force is the point of concourse of the
straight lines joining each an angular point of this triangle to the
common point of the tangents to its circumscribed circle at ^e ends of
its opposite side.
2405. A parabola is described imder a force in the focus S^^ and
along the focal distance SF is measured a given length SQ\ QR drawn
parallel to the normal at P meets the axis in B, : prove that the velocity
at P bears to the velocity at the vertex the ratio QB : ^SQ.
2406. Prove that the equation 2F' = P.PFis true when a body
is moving in a resisting medium, F being the extraneous force and FY
the chord of curvature in the direction of P.
414 NEWTON.
2407. Two points P, Q move as follows ; P describes an ellipse
under acceleration to the centre, and Q describes relatively to -P an
ellipse of which P is the centre under acceleration to P, and the
periodic times in these ellipses are equal : prove that the absolute path
of (? is an ellipse concentric with the path of P,
2408. Two bodies are describing concentric ellipses under a centre
of force in the common centre : prove that the relative orbit of either
with respect to the other is an ellipse, and examine under what circum-
stances it can be a circle.
[The bodies must be at apses simultaneously, and either the sums of
the axes of their two paths equal, or the differences.]
2409. In a central orbit the velocity of the foot of the perpendicular
from the centre of force on the tangent varies inversely as the length of
the chord of curvature through the centre of force.
2410. Different points describe different circles uniformly, the accele-
ration in each varying as the radius of the circle : prove that the periodic
times will be equsd.
[Kinematic similarity.]
2411. A particle describes an hyperbola under a force tending to a
focus : prove that the rate at which areas are described by the central
radius vector is inversely proportional to the length of that radius.
2412. A rectangular hyperbola is described by a point under
acceleration parallel to one of the asymptotes : prove that at a point P
the acceleration is 2U*, MP-^CM*^ MP being drawn, in direction of the
acceleration, from the other asymptote, C the centre, and U the constant
component velocity parallel to the other asymptote.
2413. A point describes a cycloid under acceleration tending
always to the centre of the generating circle : prove that the acceleration
is constant and that the velocity varies as the radius of curvature at the
point.
2414. A particle constrained to move on an equiangular spiral is
attracted to the pole by a force proportional to the distance : prove that,
in whatever position the particle be placed at starting (at rest), the
time of describing a given angle about the centre of force will be the
same.
[This follows at once from properties of similar figures.]
2415. An endless string, on which runs a small smooth bead,
encloses a fixed elliptic lamina whose perimeter is less than the length of
the string ; the bead is projected so as to keep the string in a state of
tension : prove that it will move with constant velocity, and that the
tension of the string will vary inversely as the rectangle under the focal
distances.
2416. A small smooth bead runs on an endless thread enclosing a
lamina in the form of an oval curve, and the bead is projected so as to
NEWTON. 416
describe a curve of continuous curvature in the plane of the lamina under
no forces but the tensions of the thread : prove that the tension will
vary inversely as the harmonic mean between the lengths of the two
parts of the string not in contact with the lamina ; and apply this result
to prove that the chord of curvature of an ellipse at a point P in a
given direction is twice the harmonic mean between the tangents from
P to the confocal which touches a straight line drawn through P in the
given direction ; any tangent which is drawn from P outwards being
reckoned negative.
2417. A paral)ola is described with constant velocity under the
action of two equal forces one of which tends to the focus : prove that
either force varies inversely as the focal distance.
2418. A particle is describing an ellipse about a centre of force
^r"*j at a certain point ft receives a small increment Aft and the
excentricity is unaltered : pro^e that the point is an extremity of the
minor axis and that the major axis 2a is diminished by - A/a.
2419. A particle is describing an ellipse about a centre of force
Hr~* and at a certain point /x receives a small increment A/i : prove the
following equations for determining the corresjionding alterations in the
major axis 2a^ the excentricity e, and the longitude of the apse m,
rAa «?A« r «Aar 1 ^
= ---A/i;
a(2a-r) 1-e'a-r sin(^-m) fi
r, 6 being polar co-ordinates (from the centre of force) of the point at
which the change takes place.
2420. In an elliptic orbit about the focus, when the particle is at a
dihtance r from the centre of force the direction of motion is suddenly
turned through a small angle A)3 : prove that the consequent alteration
in the longitude of the apse is^fl+e* — j A)8, 2a being the length of
the major axis and e the excentricity,
2421. At any point in an elliptic orbit about the focus, the velocity
V receives a small increment Av : ])rove that the consequent alterations
in the excentrity e and the longitude of the apse w are given by the
equations
A6 eAtor 2vAv
h'(2a-r) ~ abja*e'-(r^af ~ i^e(2a-r) '
2422. In an elliptic orbit about the centre the resolved part of the
velocity at any point perpendicular to one of the focal distances is
constant; and if the whole velocity be resolved into two, one per-
pendicular to each focal distance, each will vary as the rectangle under
the focal distances.
2423. A particle moves along AP a rough chord of a circle under
the action of a force to B varying as the distance and AB is a diameter;
the particle starts from rest at A and comes to rest again at P : prove
that the co-efficient of friction is \ tan PAB,
416 NEWTON.
2424. A number of particles start from the same point with the
same velocity and are acted on bj a central force varying as the
distance : prove that the ellipses described are enveloped by an ellipse
having its centre at the centre of force and a focus at the point of
projection.
2425. An ellipse is described by a particle under the action of two
forces tending to the foci and each varying inversely as the square of the
distance : prove that
2a* _ {fi^' -f ft V ) (o> -f ioy
a, b being the axes of the ellipse, and co, to the angular velocities at any
point about the focL
2426. Two fixed points of a lamina slide along two straight lines
fixed in space (in the plane of the lamina) so that the angular velocity
of the lamina is constant : prove that (1) every fixed point of the lamina
describes an ellipse imder acceleration tending to the common point of
the two fixed straight lines and proportional to the distance; (2) eveiy
straight line fixed in the lamina envelopes during its motion an involute
of a four-cusped hypocycloid ; (3) the motion of the lamina is completely
represented by supposing a circle fixed in the lamina to roll uniformly
with internal contact on a circle of double the radius fixed in space ;
(4) for a series of points in the lamina lying in one straight line the foci
of the ellipses described lie on a rectangular hyperbola.
2427. A lamina moves in its own plane so that two fixed points of
it describe straight lines with accelerations /, f : prove that the accele-
ration of the centre of instantaneous rotation is
^/«+/'--.2^'cosa-j.sina,
where a is the angle between the straight lines.
[The accelerations^/' must satisfy the equations
cos'«^^(/8ece) = cos'tf'~>(/'secflO,
/costf+/'cos^ = ca>*,
where tf, ff are the angles which the straight line joining the two points
makes with the fixed straight lines, and ca is the angular velocity of the
lamina.]
2428. Two points Ay ^ of a lamina describe the two straight lines
OXy Oy fixed in space (in the plane of the lamina), F is any other point
of the lamina, and Q^ any diameter of the circle AOB ; PQy PQ meet
the circle again in Ry B! \ prove that Oi?, OBI will be the directions of
two conjugate diameters of the locus of P.
2429. Two points fixed in a lamina move upon two straight lines
fixed in space and the velocity of one of the points is uniform : prove
that every other point in the lamina moves so that its acceleration
is constant in direction and varies inversely as the cube of the distance
from a fixed straight line.
NE>VTON. 417
[If A describe Ox with uniform velocity U and B describe Oy at
right angles to Ooc, then if P be any other point fixed in the lamina and
FAf PB meet the circle on AB in a, 6, the acceleration of P will bo
always parallel to Oa and vary inversely as the cube of the distance from
Oh', and, if PM be drawn parallel to Oa to meet Oh, the acceleration of
P will be U'.AP'-r- AB'. PM'.]
2430. A lamina moves in its own plane so that two points fixed in
the lamina describe straight lines with equal accelerations: prove that
the acceleration of the centre of instantaneous rotation is constant in
direction, and that the acceleration of any point fixed in the lamina ia
constant in direction.
2431. Two ellipses are described about a common attractive force
in their centre ; the axes of the two are coincident in direction and the
sum of the axes of one is equal to the difference of the axes of the other :
prove that, if the describing particles be at cori-esponding extremities of
the major axes at the same instant and be moving in opposite senses, the
straight line joining them will be of constant length and of uniform
angular velocity duiing the motion.
2432. A lamina moves in such a manner that two straight lines
fixed in the lamina pass through two points fixed in space: prove that
the motion of the lamina is completely represented by sup]:x)sing a circle
fixed in the lamina to roll with internal contact on a circle of half the
radius fixed in space.
2433. A lamina moves in its own i)lane with uniform angular
velocity so that two straight lines fixed in the lamina pass each through
one of two points fixed in space : prove that the acceleration of any
point fixed in the lamina is compoimded of two constant accelerations,
one tending to a fixed point, and the other in a direction which revolves
with double the angular velocity of the lamina.
2434. A triangular lamina ABC moves so that the point A lies on
a straight line he fixed in space, and the side BC passes through a point
a fixed in space, and the triangles ABC, abc are equal and similai*: prove
that the motion of the lamina is completely represented by supposing a
{>arabola fixed in the lamina to roll upon an equal parabola fixed in
space, similar points being in contact.
2435. A particle describes a parabola under a repulsive force from
the focus, varying as the distance, and another force parallel to the axis
which at the vertex is three times the former; find the law of this latter
force ; and prove that, if two particles describe the same parabola under
the action of these forces, their lines of instantaneous motion will
intersect in a point which lies on a fixed confocal parabola.
[The second force is always three times the first.]
2436. Two particles describe curves under the action of central
attractive forces, and the radius vector of either is always parallel and
proportional to the velocity of the other : prove that the curves will be
similar ellipses described about their centn^s.
w. P. 27
DYNAMICS OF A PARTICLE.
I. RectUinetvr Motion, Kinematics.
2437. A Heavy particle is attached by an extensible string to a
fixed point, from whicH the particle is allowed to fall freely; when the
particle is in its lowest position the string is of twice its natural length:
prove that the modulus is four times the weight of the particle, and find
the time during which the string is extended beyond its natural length*
[The time is 2 ^- tan"* ^2.]
2438. A particle at ^ is attached by an elastic string at its natural
length to a point A and attracted by a force varying as the distance to a
point C in BA produced, A dividing BG in the ratio 1 : 3, and the
particle just reaches the centre of force : prove that the velocity will be
greatest at a point which divides CA in the ratio 8 : 7.
2439, A particle is attracted to a fixed point by a force fi (dist.)'*,
and repelled fix)m the same point by a constant force/; the particle is
placed at a distance a from the centre, at which point the attractive force
IS four times the magnitude of the repulsive, and projected directly from
the centre with velocity V: prove that (1) the particle will move to
infinity or not according as F' > or < 2qf; (2) that^ if a;, a; + c be the
distances firom the centre of force of two positions of the particle, the
time of describing the given distance e between them will be greatest
when x{x + e) = ia\ Also, when 7^j2a^ or Zj2(if, determine the
time of describing any distance.
[When r= J2af, the time of reaching a distance x from the centra
of force is
and, when 7= 3 J^af, the time is
DYNAMICS OF A PABTICLE. 419
2440. The aooeleralions of a point describing a cwcve are resolyed
into two, along the radius vector and parallel to the prime radius: prove
that these accelerations are respectively
cottf d f,M\ cPr /dO\' , _1_ d ( ,c?d\
'V~dt\dt)'"'de^\di) ^"^'rsinedty dt)'
2441. The motion of a point is referred to two axes Ox, Oy, of
which Ox is fixed and Oy revolves about the origin: prove that the
accelerations in these directions at any time t are
^x 1 d / ,d$\ d*y cote d ( ^dOS /dBy
d^'"^^mediVdiJ' df^ y dtV dt) ^\di) ' '
where denotes the angle between the axes.
2442. A point F is taken on the tangent to a given curve at a
point Qj and is a fixed point on the curve, the arc 0Q = 8, QP^r, and
^ is the angle through which the tangent revolves as the point of
contact passes from Oto Q: prove that the accelerations of Pin direction
QF and in the direction at right angles to this, in the sense in which ^
increasesi are respectively
d^'^dt' ^\di)' rdt\dt)'^dtdr
2443. A point describes a curve of double curvature, and its polar
co-ordinates at the time t are (r, 6, ^}: prove that its accelerations
(1) along the radius vector, (2) perpendicular to the radius vector in the
plane of 6 and in the sense in which $ increases, and (3) perpendicular
to the plane of in the sense in which ^ increases^ are respectively
2444. A point describes a parabola in such a manner that its
velocity, at a distance r from the focus, is a/-^(«^-c*)i where yj e are
constant: prove that its acceleration is compounded of /parallel to the
axis and/^ along the radius vector from the focus.
2445. A point describes a semi-ellipse bounded by the minor
axis, and its velocity at a distance r from the focus is a ^ -^^ ^ ,
where 2a is the length of the major axis and / a constant aocmeration :
prove that the acceleration of the point is compounded of two, each
varying inversely as the square of the distance, one tending to the
nearer focus and the other from the farther focus.
27—2
420 DYNAMICS OF A PARTICLE.
2446. A point is describing a circle, and its velocity at an angular
distance from a fixed point on the circle varies as ^1 + cos'tf ^ sin* 0\
prove that its acceleration is compounded of two tending to fixed points
at the extremities of a diameter, each varying inversely as the fifth
power of the distance and equal at equal distances.
2447. A point describes a circle under acceleration, constant, but
not tending to the centre: prove that the point oscillates through a
quadrant and that the line of action of the acceleration always touches
a certain epicycloid.
[The radius of the fixed circle of the epicycloid is -^ and of the
moving circle ^ , a being the radius of the circle described by the point.]
2448. A parabola is described with accelerations F, A, tending to
the focus and parallel to the axis respectively : prove that
r being the focal distance.
2449. A point describes an ellipse under accelerations i^,, F^
tending to the foci, and r,, r, are the focal distances of the point : prove
that
2450. The parabola ^ — iax is described under accelerations X, T
parallel to the axes : prove that
- dY dX ^-_ ^
2451. A point describes a parabola under acceleration which makes
a constant angle a with the normal, and $ is the angle described from
the vertex about the focus in a time t : prove that
«*'-"'(f)'~(l+co8<9)';
and find the law of acceleration.
[The acceleration varies as cos* ^ €"•*»"*, which is easily expressed
as a function of the focal distance.]
2452. A point F describes a circle of radius 4a with uniform
angtdar velocity <a about the centre, and another point Q describes a
circle of radius a with angular velocity 2io about F: prove that the
acceleration of Q varies as the distance of F from a certain fixed point.
DYNAMICS OF A PARTICLE. 421
2453. The only carve which can be described under constant
acceleration in a direction making a constant angle with the normal is
an equiangular spiral.
2454. An equiangular spiral is described by a point with constant
acceleration in a direction making an angle ^ with the normal : prove that
sin ^ ->T == 2 sin ^ -f cot a cos ^,
a being the constant angle of the spiral and d the angle through which
the radius vector has turned from a given position.
2455. The parabola y* = 2ex is described by a point under accelera-
tion making a constant angle a with the axis and the velocity when the
acceleration is normal is Vi prove that, at any point (Xy y) of the
parabola, the acceleration is FV -r- (c cos a — y sin a)* ; and that^ if when
the acceleration is normal the particle is moving towards the vertex, the
time in which the direction of motion will turn through a right angle
will be c -r r sin 2a cos a.
2456. A lamina moves so that two straight lines fixed in it pass
through two points fixed in sfiace and the angular velocity is uniform:
prove that any ]>oint fixed in the lamina, whose distance from the point
of intersection of the two straight lines is twice the diameter of the circle
described by that ]>oint, will move under ticceleration whose line of action
always touches a three-cus{)ed hypocycloid.
2457. The catenary 5 = ctan^ is described under acceleration
which at any point makes an angle ^ with the normal on the side towards
the vertex : prove that the acceleration varies inversely as the cube of
the distance from the directrix.
2458. A point describes a parabola, starting from rest at the
vertex, under acceleration which makes with the tangent an angle
tan"^ (2 tan ^), where is the angle through which the tangent has
turned : prove that the acceleration varies as J^r - 3a . r"*, where r is
the focal distance and a the initial value of r.
2459. The curve whoso intrinsic equation is 5 == a tan 2^ is
described by a point under constant accelei-ation : pi'ove that the direc-
tion of the acceleration makes with the tangent an angle — 2^, where
is given by the equation
2 T^ (tan 24>J^e) = ^cot^.
2460. A point describes an epicycloid under acceleration tending to
the centre of the fixed circle : prove that the pedal of the epicycloid
with respect to the centre will also be described under acceleration tend-
ing to the same point.
422 UnAMKB CfF A FXKTICIX.
24^1. The intrinsie eqaatkm ci a carre it #s/(4) and tlie carre is
dct c ri b ed under aeedenUions X, T pialld to the iangeni and noraud
at the origin (where ^ = 0): prore that
2462. The cnnre #=/(^) is described by a point with constant
acoelenUion which is at the origin in direction of the normal : prore
that its inclination to this direction at anj other point is given by the
equatioa
(S-^) tan {^-e)f{<f) •/"(4>).
2463. A catenary is described by a point nnder aooeleration whose
rertical component is constant (/) : prove that the horizontal component
when the tangent makes an angle ^ with the horizon is
ycos ^ oosec* ^ (1 + fit coe ^ + cos* ^).
2464. A curve is described nnder constant acceleration parallel to
a straight line which revolves uniformly: prove that the curve is a
prolate, common, or curtate cycloid ; or a circle.
2465. A point describes a certain curve and initially the accelera-
tion is normal ; when the direction of motion has turned through an
angle ^ the direction of acceleration has turned through an angle 2^ in
the same sense : prove that the acceleration varies as cos ^ ^ , as does
the angular velocity of the tangent, and that the velocity varies as
cos^
2466. A parabola is described under constant acceleration and
09 ^ are the angles which the direction of the acceleration at any point
and the tangent at that point make respectively with the directrix :
prove that-
3^(tan^oo8*fl)=oos4ft
2467. A point moves under constant aooeleration which is initially
normal, and when the direction of motion has turned through an angle ^
the direction of acceleration has turned through an angle nHf> (m constant)
in the same sense : prove that the intrinsic equation of the curve
described is
-^ = c(co8m- 1 ^)'*"*;
and determine the curve when m « 1, 2, or 3.
2468. A cycloid is described under constant acceleration and 0, ^
ai*o tho angles which the directions of motion and of acceleration at any
DYNAMICS OF A PARTICLE. 423
point make with the tangent and nonnal at the vertex respectively :
prove that
sin fl = cos fl sin (^ - 2^) log |n tan ^1 - ^ J I ;
or that ^ = 2$.
2469. A point describes an ellipse under accelerations to the foci
which are, one to another at any point, inversely as the focal distances ;
find the law of either acceleration, prove that the velocity of the point
varies inversely as the conjugate diameter, and that the periodic time is
~ ( A "^ ~ ) ' "^^^^ ^ ^ ^® angular velocity about the centre at the end
of either axis.
[This path so described is also a brachystochrone between any two
points for a certain force in the centre.]
2470. The cusp of a cardioid is S and the centre of the fixed circle
(l)y which it can be generated as an epicycloid) is (7, and the cardioid is
described under accelerations -F, F' tending to Sy C respectively : prove
that
*?ii^->4f.p^h-
where r, r' are the distances from S, C, and a = SC. Also prove ihat|
if the angular velocity about the cusp be constant, F will be constant,
F' will vary as /, and at the apse 2F+ F* = 0.
2471. A point F starts from A and moves along a straight line
with uniform velocity F; a point ^ starts from B and moves always
towards F with uniform velocity v: prove that, if F>t>, the least
distance c between F and Q is
l-fm l-Hi»
a (sin a)*"* (1 - cos a)* -r- (1 + m) * (1 - m) « ,
and, if < be the time after which they are at this distance,
V* _ ^^^^^ — €l(^ — cos a) ^
1 — w'
where m = 7^ , a = AB^ and a is the angle which AB makes with the path
of P.
2472. A point F is describing a parabola whose focus is S under
acceleration always at right angles to BFy the plane in which the motion
takes place having a constant velocity parallel to the axis, and equal to
the velocity of F parallel to the axis in the parabola at the end of the
latus rectum : prove that the path of P in space is a *' curve of pursuit"
to S described with a constant velocity equal to that of S.
2473. A point describes a curve which lies on a cone of revolution
and crosses all the generating lines at a constant angle, under accelera-
tion whose direction always intersects the axis : prove that the accelera-
tion makes a constant angle with the axis and varies inversely as the
cube of the distance from the vertex.
424 DYNAMICS OF A PABTICLE.
2474. The straight lines APy BP joining a moving point P to two
fixed points A^ B have constant angular velocities 2(d, 3ct> : prove that
the acceleration of P is compounded of a constant acceleration along AP
and an acceleration varying as BP along PB,
[These accelerations are \2v?ABy and IQto'PB respectively.]
2475. A point describes a rhumb line on a sphere so that the longi-
tude increases uniformly : prove that the whole acceleration varies as
the cosine of the latitude and at any point makes with the normal an
angle equal to the latitude.
[If a be the constant angle at which the curve crosses the meridians,
and (tf be the rate at which the longitude increases, the three accelera-
tions resolved as in (2443) will be
— ysin"^, y cos 2a sin ^ cos fl, y sin 2a sin cos 0,
where y sin* a = <o* x radius of the sphere.]
2476. A point P describes a circle under acceleration tending to
a point S and varying as SP, S being a point which moves on a fixed
diameter initially passing through P: prove that, if ^ be the angle
described about the centre in a time t, fjm sin ^ = Jm ^ 1 sin (t JfJL)y
and the distance of S from the centre = — sec' : where a is the radios
m
of the circle and m constant.
2477. A point describes an arc of a circle so that its acceleration is
always proportional to the n^ power of its velocity : prove that the
direction of the acceleration of the point always touches a certain
epicycloid generated by a circle of radius a -i- 2 (3 — w) rolling on a circle
of radius a (2 - 7i) -f (3 — n) ; where a is the radius of the described
ciixjle.
II. Central Forces.
2478. Prove that the parabola ^ = 4ax can be described under a
constant force parallel to the axis of y and a force proportional to y
parallel to the axis of x ; also, under two forces 4/x (c + x), /xy parallel to
the axes of x and y respectively.
2479. A particle is acted on by a force parallel to the axis of y
whose acceleration is fty, and is initially projected with a velocity a Jfi,
parallel to the axis of a; at a point where y=^a\ prove that it will
describe a catenary.
2480. A particle is acted on by a force parallel to the axis of y
whose acceleration (always towards the axis of x) is /xy~', and, when
y = a, is projected parallel to the axis of x with velocity a / — : prove
that it will describe a cycloid.
DTNAMICS OF A PARTICLE. 425
2481. Two equal particles attract each other with a force varying
inversely as the square of the distance and are projected simultaneously
with equal velocities at right angles to the joining line : prove that,
if each velocity be equal to that in a circle at the same distance, each
particle will describe a semi-cycloid.
2482. A cardioid is described with constant angular velocity about
the cusp under a constant force to the cusp and another constant force :
jirove that the magnitude of the latter is double that of the former
and that its line of action always touches an epicycloid generated by
a circle of radius a rolling u|)on one of radius 2a ; 8a being the length of
the axis of the cardioid.
2483. The force to the origin under which the hyi)erbola
rcos2d = 2^2acos^
can be described will vary as (^a* + 7^ + 0)' -^ r*.
2484. Tlie peri)endicular SY is let fall from the origin upon the
tangent at any point P of the curve 7^ = a* sin 2^, and the locus of Z is
described xinder a force to S: prove that this force will vary as
2485. In a central orbit the resolved velocity at any point perpen-
dicular to the radius vector is equal to the velocity in a circle at that
distance : prove that the orbit is a reciprocal spiral.
2486. A particle moves under a constant repulsive force from a
fixed point, and is projected with a velocity which is to that in a circle
at the same distance under an equal attractive force as ^2 : 1 : prove
that the orbit is the curve whose equation is of the form
r* = a*sec^tf.
2487. The force to the pole under which the pedal of a given curve
r=/(jp) can be described will vary as rp"^ (2r — p -^j ; and, if the
given curve be r^sin ^d = a^, this force will be constant.
2488. An orbit desciibed under a constant force tending to a fixed
point will be the pedal of one of the curves represented by the equation
^ V = p* + bp*, where a and b are constants.
2489. A parabola is described about a centre of force in (7, the
centre of curvature at the vertex A : prove that the force at any point
F of the parabola varies as CF (AS -^ SF)~', where 5 is the focus,
2490. The force tending to the pole under which the evolute of the
curve r =/{p) cwi he described will vary inversely as
<--^-l('l)V-*''(l)'-''i-
426 DYNAMICS OF A PABTICLE.
2491. A particle P is projected from a point A at right angles
to a straight line SA and attracted to the fixed point iS^ bj a force
varying as cosec PSA: prove that the rate of describing areas about
A will be uniformly accelerated.
2492. A particle is projected at a distance a with velocity equal
to that in a circle ' at the same distance and at an angle of 45* with
the distance, and attracterl to a fixed point by a force which at a
distance r is equal to fir~* (a' + 3r^) : prove that the equation of the
path is r = a tan (7—0)9 ^^^ ^^^^ ^^ time to the centre of force is
kM)-
a
2493. A particle is attracted to a fixed point by a force which
at a distance r is equal to
fir-^ (3a" + 3oV* - r^,
and is projected from a point at a distance a from the centre with a
velocity equal to that in a circle at the same distance and in a direction
making an angle cot~' 2 with the distance : prove that the equation of
the orbit is
r* = a*tan
(i-^)'
and that the time to the centre of force is
1
4^//i
log 2.
2494. A particle is describing a circle under the action of a
constant force in the centre and tiie force is suddenly increased to
ten times its former magnitude : prove that the next apsidal distance
will be equal to one fourth the radius of the circle.
2495. A particle is describing a central orbit in such a manner
that the velocity at any point is to the velocity in a circle at that
distance as I : ^n : prove that poo f^j p being the perpendicular
from the centre of force on the tangent at a point whose distance
is r, and that the force will vary inversely as r*"**. If the force
be repulsive and the velocity at any point be to that in a circle at
that distance under an equal attractive force as I : >^n, the particle
will des(iribe a path having two asymptotes inclined at an angle
n+ 1
2496. A particle acted on by a central force ftr~*(4r-3a) is
projected at a distance a, an angle 45*^, and with a velocity which is
to the velocity from infinity as J2 : Jb\ prove that the equation
of the path is a=r(l + sin^ cos $)^ and that the time from projection
a*
to an apse is — p= (4ir - 3 ,^3).
9iy3/x
DTKAXICS OF A PABTICLK 427
2497. A portion of an epicycloid is described nnder a force
tending to the centre of the fixed circle : prove that, if a straight
line be drawn from any fixed point always parallel and proportional
to the radius of curyature in the epicycloid^ the extremity of this
line will describe a central orbit.
2498. The curve whose intrinsic equation is * = a («*♦ — €"■■•♦) is
described under a central attractive force, the describing point being
initially at an apse at a distance c = 2ma -^ (1 + m') &om the centre of
force : prove that the force varies as r (r' + c*)~*, and that the end
of a straight line drawn from a fixed point always parallel and pro-
portional to the radius of curvature in the path will also describe a
central orbit.
2499. A particle describing a parabola about a force in the focus
comes to the apse at which point the law of force changes, and the
force varies inversely as the distance until the particle next comes
to an apse when the former law is restored ; there are no instantaneous
changes in magnitude : prove that the major axis of the new elliptic
orbit will be m^a-r{m* — 1), where 4a is the latus rectum of the parabola
and m is that root of the equation x* {\ —\ogx) = 1 which lies between
2
j€ and €, and that the excentricity will be 1 5 .
2500. In an orbit described under a central force a straight line
is drawn from a fixed point perpendicular to the tangent fmd pro-
portional to the force, and this straight line describes equal areas in
equal times : prove Uiat the differential equation of the orbit is of
the form
a)'=©'*&)'.
and that the rectangular hyperbola described about the centre is a
][>articular case.
2501. A uniform chain rests under normal and tangential forces
which at any point of the chain are —Uyt per unit of length of the
chain : prove tjiat a particle whose mass is equal to that of a unit of
length of the chain can describe the same curve under the action of
normal and tangential forces 2?^, t at the same point.
2502. A centre of force varying inversely as the n*** power of
the distance moves in the circumference of a circle and a particle
describes an arc of the same circle under the action of the force : prove
that the velocity of the centre of force must bear to the velocity of
the particle the constant ratio 5 - n : 1 — n, and that, when the ac-
celeration of the force at a distance r is /ttr"*, the time of describing
a semicircle is id
Vm'
2503. A particle P is repelled from a fixed point S hj k force
varying as (distance)'* and attracts another particle Q with a force
varying as (distance)"*; initially P and Q are equidistant from S in
428 DYNAMICS OF A PABTICLE.
opposite directions, P is at rest, and the aooelerations of the two foroes
are eqo^l : prove that Q, if projected at right angles to SQ with proper
velocitj, will describe a parabola with S for focus.
2504. A particle P is repelled from two fixed points S, S' by
forces, varying each as (distance)"' and equal at equal distances, and
attracts another particle Q with a force varying as (distance)"';
initially P, Q divide SS' internally and externally in the same ratio,
P is at rest, and the accelerations of the forces on the two particles
are equal: prove that, if Q be projected at right angles to JSS' with
velocity equal to that in a circle at the same distance firom P, it will
describe an ellipse of which S, JS' are foci.
2505. A particle F is describing a parabola under the action of
gravity, S is the focus and the straight line drawn through F at right
angles to SF touches its envelope in Q : prove that the velocity of Q
varies as SF,
2506. A particle F describes a central orbit, centre of force S^
and through F is drawn a straight line at right angles to FS^ which
line touches its envelope in Q : prove that the velocity of Q varies as
1 /d'r \
and is constant when the orbit is a parabola with its focus at S.
2507. A particle acted on by an attractive central force
fir {f - a')-'
is projected from an apse at a distance na with a velocity which is
to the velocity in a circle at the same distance as Jn* - 1 : n: prove
that the path will be an arc of an epicycloid and that the time before
reaching the cusp is
IT
4^//i
a" (n"- 1)1.
2508. A particle describes an involute of a circle under a force
in the centre of the circle : prove that the force, at any point at a
distance r, will vary as r (r* — a*)"**, where a is the radius of the circle.
2509. A smooth horizontal disc revolves with angular velocity <o
about a vertical axis at which is placed a material particle acted on
by an attractive force, of acceleration equal to w*. distance, to a certain
point of the disc : prove that the path of the particle on the disc will
be a cycloid, and that its co-ordinates in space, when the disc has turned
through an angle By are aO sin $, a (sin $-6 cos ^), the former being
measured along the straight line wliich initially joins the particle to the
centre of force.
2510. An orbit is described under a force F tending to a fixed
point S and a normal force N : prove that
^i(*l)*s(^''l)-»-
DYNAMICS OP A PARTICLE. 429
2511. A particle is projected from a point at a distance a from
a fixed centre of force whose acceleration at distance r = 2/xr (r' + a*)"*,
with a velocity J ft -^ 2a', and in a direction making an angle a with
the distance : prove that the orbit is a circle whose radius is a -r sin ou
2512. A particle describes a conic under the action of a centre
of force at a point on the transverse axis : prove that the time of
passing from one extremity of the ordinate through to the other
will be _
/2 2a - sin 2a
V /I sin* a '
the acceleration of the force at any point P being
'^^^"•(^^i)
-•
where POp is a chord through ; and cos a : 1 being the ratio which
the distance of from the centre bears to the semi-major axis.
III. Constrained Motion on Curves or Surfaces: Particles joined by
Strings,
2513. A particle, mass m, is constrained to move on a curve under
the action of forces such that the particle, if projected from a certain
point of the curve with velocity v, would describe the curve freely :
prove that, when projected from that point with velocity V, the pressure
on the curve at any point will be w ( K* - v')-5-f), where p is the radius
of curvature.
2514. A particle is acted on by two forces, one parallel to a fixed
straight line and constant, the other tending from a fixed point and
varying as (distance)"*, and is constrained to move on a parabola whose
focus is the fixed point and axis parallel to the fixed line : prove that
the pressure is always proportional to the curvature, and that^ if the
velocity vanish at a point where the magnitudes of the forces are equal,
the pressure will also vanish.
2515. A particle is attracted to two fixed points by two forces,
the acceleration of either force at a distance r being fir~* (a' + r"), and
is placed at rest at a point at which the two forces are equal and the
distances of the particle from the centres of force unequal : prove that
it will proceed to oscillate in a hyperbolic arc of which the centres of
force are foci.
2516. A particle is acted on by a repulsive force tending from a
fixed point and by another force in a fixed direction ; when at a distance
r from the fixed point the accelerations of these forces are
r'V J' r'W'^ a)
respectively: prove that the particle, abandoned motionless to the
action of these forces at a point where they are equal in magnitude, will
proceed to describe a parabola with its focus at the fixed point and its
axis in the fixed direction.
430 DTNAMIOS OF A PABTICLE.
2517. A particle is placed in a smooth parabolic groove wtatk
revolves in its own plane about the focus with uniform angular velociftf
io, and the particle describes in space an equal confocal parabola under
an attractive force in the focus : prove that this force at any point k
measured b7 coV (Sr ~ 4c) -r- 4c, r being the focal distance and 2c thfi lata
rectum«
2518. A bead moves on a smooth elliptic wire and is attached to
the foci by two similar elastic strings, of equal natural lengths, whidi
remain extended throughout the motion : prove that, if projected with
proper velocity, the velocity will always vary as the conjugate* diameter.
2519. Two particles A, B are together in a smooth circular tube;
A attracts B with a force whose acceleration is o)' distance and moves
along the tube with uniform angular velocity 2(i> ; ^ is initially at rest :
prove that the angle ^ subtended by AB at thef centre after a time I
is given by the equation
logtan!^ = <„fc
2520. A force f resides at the centre of a rough circular arc and
from a point of the circle a particle is projected with a velocity F(> J of)
along the interior of the circle : prove that the normal pressure on the
curve will be diminished one half after the time
I^W f ^\ V + Jaf J'
where a denotes the radius of the circle and ft the ooeffident of
friction*
2521. A heavy particle is projected horizontally so as to move on
the interior of a smooth hollow sphere of radius a and the velocity oC
projection is J2ga : prove that, when the particle again moves hori-
zontally, its vertical depth below the highest point of the sphere is equal
to its initial distance from the lowest point.
2522. A heavy particle is attached to a fixed point by a fine
inextensible string of length a, and, when the string is horizontal and at
its full length, the particle is projected horizontally at right angles to
the string with the velocity due to a height 2a cot 2a: prove that the
greatest depth to which it will fall is a tan a.
2523. A particle slides in a vertical plane down a rough cycloidal
arc whose axis is vertical, starting from the cusp and coming to rest at
the vertex : prove that the coefficient of friction is given by the equation
[More generally, if the particle come to rest at the lowest point and
6 be the angle which the tangent at the starting point makes with the
horizon,
11^ = sin " fi cos $.]
DTNAMICS OF A PABTICLE. 431
2524. A rough wire in the form of an arc of an equiangular spiral
whose constant angle is cot'* (2/x) is placed with its plane yertical and a
heavy particle falls down it, coming to rest at the first point where the
tangent is horizontal : prove that at the starting point the tangent
makes with the horizon an angle double the angle of friction, and that
during the motion the velocity will be greatest when the angle ^ which
the tangent makes with tbe horizon is given by the equation
(2fi* - 1 ) sin ^ + 3/x cos ^ = 2fu
2525. A heavy particle falls down the arc of a four-cusped hypocy-
cloid, starting at a cusp where the tangent is vertical and coming to rest
at the next cusp : prove that, if fi be the coefficient of friction|
/iic'**(V-l) = 8fi'+3.
2526. Find the equation for the curve on which, if a smooth particle
be constrained to move under a force varying as (distance)"', the pressure
will be constant ; and prove that Bernoulli's lemniscate is a particular
case.
2527. Three equal and similar particles, repelling each other with
forces varying as the distance, are connected by equal inextensible
strings and are at rest ; one of the strings is cut : prove that the sub-
sequent angular velocity of either of the uncut strings will vary as
^\
1 - 2 cos ^
2+cob6^ '
where is the angle between them.
2528. Two heavy particles are placed on a smooth qycloidal arc
whose axis is vertical and are connected by a fine string passing along
the arc ; the distance of either particle from its position of equilibrium
measured along the arc is initially e : prove that the time of readiing to
a distance a from the position of equilibrium will be
where a is the radius of the generating circle.
2529. An elliptic wire is placed with its minor axis vertical and on
it slides a smooth ring to which are attached strings which pass through
smooth fixed rings at the foci and sustain each a particle of weight equal
to the weight of the ring : determine the velocity which the particle must
have at the highest point in order that the velocity at the lowest point
may be equal to that at the end of the major axis.
[The requii'ed velocity is that due to a height
6(l-2«')-2«"(l+20.]
432 DYNAMICS OF A PARTICLE.
2530. Two particles of masses p, q are connected by a fine inex-
tensible string which passes through a small fixed ring; p hangs
vertically and q is held so that the adjacent string is horizontal : prove
that when q is let go the initial tension of the string is pqg -r Q? + g),
and the initial radius of curvature of the path of q bears to the initial
distance of q from the ring the ratio 3 {p'+ (p+g)'}^ ''P{p-^^) {^P-^^^)'
2531. A particle in motion on the surface z = <i>{x, y) under the
action of gravity describes a curve in a horizontal plane with velocity u :
prove that, at every point of the path,
g lUJ d^ ^ dxdy dxdy'^ \dx) d^] "*" \\dx) "*" \dy) ] '
iJie axis of z being vertical.
2532. In a smooth surface of revolution whose axis is vertical a
heavy particle is projected so as to move on the surface and describe a
path which differs very little from a horizontal circle : prove that the
time of a vertical oscillation is ir ^ / — r? — *i — :^ ¥-^ t where k is the
V g{k-k-^r sm a cos' a) '
distance firom the axis, r the radius of curvature of the meridian curve,
and a the inclination of the normal to the vertical in the mean position
of. the particle.
2533. A heavy particle is projected inside a smooth paraboloid of
revolution whose vertex is its lowest point and the greatest and least
vertical heights of the particle above the vertex are A^, A,, the velocities
at these points being T^, V^\ prove that V*=^2gh^^ V*^2gh^, and that
throughout the- motion the pressure of the particle on the paraboloid will
vary as the curvature of the generating parabola.
[^The pressure = 2 IF (/*, +a) (A,+a) -f ap, where W is the weight of the
particle, 4a the latus rectum, and p the radius of curvature of the
generating parabola. Also p', the radius of absolute curvature of the
path of the particle, is given by the equation
\ p ) a{a-^zy -^ (a-^z){h^'¥h^-'zy
where z denotes the height of the particle.]
2534. A heavy particle is moving upon a given smooth surface of
revolution under the action of a force P parallel to the axis : prove that
the equation of the projection of the path on a plane perpendicular to
the axis is
where T- , j are polar coordinates measured from the trace of the axis
dz
and the equation of the surfigwje is u* -j- =/{u), the axis of the surface
being the axis of z.
DYNAMICS OF A PARTICLE. 433
2535. Two particles of masses fw, w! lying on a smooth horizontal
table are connected by an inextensible string at its full length and
passing through a small fixed ring in the table; the particles are at
distances a, a' from the ring and are projected with velocities F, F' at
right angles to the string so that the paints of the string revolve in the
same sense : prove that either particle will describe a circle uniformly if
raV^a! ^inV^a'y and that the second apsidal distances will be a', a
respectively if mV^a^^^ m' V'^a^*.
2536. Two particles ??i, m', connected by a string which passes
throxigh a small fixed ring, are held so that the string is horizontal and
the distances from the ring are a, a'\ the particles are simultaneously set
free and proceed to describe paths whose initial radii of curvature are
p, p' : prove that
mrr^ 1 11 J[
p p p p a a
2537. Two particles m, m' are connected by a string, m' lies on a
smooth horizontal table and m is held so that the part of the string
(of length a) which is not in contact with the table makes an angle a
with the horizon : prove that, when m is set free, the initial radius of
curvature of its path is
3a {m* + m' (2m + mf) cos*a}^
m' (»t + 7n') {m cos a + (2m + 3m') cos^a} *
2538. Two particles A and B are connected by a fine string; A
rests on a rough horizontal table and B hangs vertically at a distance a
below the edge of the table, A being in limiting equilibrium ; B is now
projected horizontally with a velocity V in the plane normal to the edge
of the table : prove that A will begin to move with acceleration
^P-(,i + l)a,
and that the initial radias of curvature of the path of B will be a (/i + 1),
where p. is tlie coeflScient of friction.
2539. A smooth surface of revolution is generated by the curve
x*i/ = a' revolving about the axis of y, which is vertically downwards,
and a heavy particle is projected with a velocity due to its depth below
the horizontal plane through the origin so as to move on the sur&ce :
prove that it will cross all the meridians at a constant angle.
2540. A heavy particle is projected so as to move on a smooth
curve in a vertical plane starting from a point where the tangent is
vertical ; the form of the curve is such that for any velocity of projection
the particle will abandon the curve when it is at a vertical height above
the point of projection which bears a constant ratio 2 : m + 1 to the
greatest height subsequently attained : prove that the equation of the
curve is y" = cas"*"*, where c is constant.
2541. A smooth wire in the form of a circle is made to revolve
uniformly in a horizontal plane about a point ^ in its circumference
with angular velocity id; a small ring P slides on the wire and is
initially at rest at its greatest distance c from A : prove that its distance
w. P. 28
434 DYNAMICS OF A PARTICLE.
from A after any time t will be 2<; -r (c"* + €"*»') and that the tangent to
the path in space of F bisects the angle between PA and the radius
to P.
2542. Two equal particles are connected bj a fine string and one
lies on a smooth horizontal table, the string passing through a small
fixed ring in the table to the other particle, which is vertically below the
ring ; the first particle is projected on the table at right angles to the
string with a velocity due to a height c -^ n (n + 1), where c is the distance
from the ring : prove that the next apsidal distance will be equal to
cn"^ and the velocity will then bear to the initial velocity the ratio n : \;
also that the radius of curvature of the initial path of the projected
particle is 4c -^ (n* + w + 2).
2543. A heavy particle of weight W is attached to a fixed point by
a fine extensible string of natural length a and modulus X, and the
particle is projected so as to make complete revolutions in a vertical
plane : prove that if properly set in motion the angular velocity of the
string will be uniform, provided that X be not less than six times IT,
and tiiat the equation of the curve described by the particle is
the straight line from which is measured being drawn from the fixed
point vertically upwards.
2544. Two particles whose masses are />, q are connected by a fine
inextensible string passing through a small fixed ring, and p hangs
vertically while q describes a path deviating very little from a horizontal
circle : prove that the distance of p at any time from its mean position
is -4, sin (w,< + -ffj) + -4, sin (n^t + B^) ; where n^, n, are the positive roots
of the equation in x
|a;._| (1 _ cos a)} (^--X_(i + 3C08'«)} = ®/co8«(l -cosa);
where c is the mean distance from the ring and q cos a = p,
2545. A heavy particle is projected so as to move on a rough
inclined plane, the coefficient of friction being ntana and the
inclination of the plane a : prove that the intrinsic equation of the path
will be
ds F" 1
difi ^sina cos^ (1 + sin^)**"(l -sin^)
r^>
where V is the velocity at the highest point. Also prove that, if two
points be taken at which the directions of motion make equal aiigle« ^'
with the direction at the highest point, and F^, F, be the velocities,
p„ p^ r the radii of curvature, at these points and at the highest
point,
F,F,cos'^=F«, p,p, cos" ^^^r*.
DTNAMICS OF A PARTICLE. 435
2546. In the last question, in the partictdar case when n = 1, prove
that the time of moving from one of the two points to the other is
LK'^(?4)*S4}'
^sina I ° \4 2/ cos^J
the arc described is
j^Jlogtan^+^j^c-^^c^^l'
2(/sin
the horizontal space described is
2V' sin ^ (3 - sin* ^)
3^ sin a cos'^ *
and that
1 1 _ 4 sin ^ cos ^
2547. A heavy particle moves on a smooth curve in a vertical
plane of such a form that the pressure on the curve is constant and equal
to m times the weight of the particle : prove that the intrinsic equation
of the path is
ds a
S9
d<ft {m + cos ^)
^ being measured downwards &om a fixed horizontal line ; that the
difference of the greatest and least vertical depths of the particle is
2ma -i- {m' -- ly ; the time from one vertex to the next in the same
horizontal plane is 2irm-'$- (m'~ 1)^; and the arc between these points
if
iro (1 + 2171*) •4- (m* — 1)5, Also the greatest breadth of a loop is
a |(1 + 2^*) JiHr^ - 3m' cos"* f-^. -5- m (m" - 1)*
2548. A particle is placed at rest in a rough tube {ifi = 3) which
revolves uniformly in one plane about one extremity and is acted on by
no force but the pressures of the tube : prove that the equation of the
path of the particle is
5r = a (4€W + €-»).
2549. A rectilinear tube inclined at an angle a to the vertical
revolves with uniform angular velocity ci> about a vertical axis which
intersects the tube, and a heavy particle is projected from the stationary
point of the tube with a velocity g cos a -f a> ^sina ; find the position of
the particle at any given time before it attains relative equilibrium ; and
prove that the equilibrium is unstable,
[The particle will describe a space a along the tube in the time
28—2
436 DYNAMICS OF A PARTICLE,
where a<i)' sin a = ^ cos a ; and the equation of motion is
-1-5- = o)* (»- a) sin a.]
2550. A smooth parabolic tube of latus rectum I is made to revolve
about its axisy which is vertical, with angular velocity /j , and a heavy
particle is projected up the tube : prove that the velocity of the particle
is constant and that the greatest height to which the particle rises in
the tube is double that due to the velocity of projection.
2551. A smooth parabolic tube revolves with uniform angular
velocity about its axis, which is vertical, and a heavy particle is placed
within the tube very near the lowest point; find the least angular velocity
which the tube can have in order that the particle may lise ; and prove
that, if it rise, its velocity will be proportional to its distance from the
axis; also that, if one position be one of relative equilibrium, every
position will be such.
2552. A curved tube is revolving uniformly about a vertical axis
in its plane and is symmetrical about that axis ; the angular velocity is
~ , where a is the radius of curvature at the vertex : prove that the
equilibrium of a p£u:ticle placed at the vertex will be stable or unstable
according as the conic of closest contact is an ellipse or hyperbola.
Vi
2553. A circular tube of radius a revolves uniformly about a
vertical diameter with angular velocity ^ — , and a particle is projected
from its lowest point with such velocity that it can just reach the highest
point : prove that the time of describing the first quadrant is
v/
/ — YT- log (7n + 2 + JnTT).
2554. A circular tube containing a smooth particle revolves about
a vertical diameter with uniform angular velocity &>, find the ix>sition of
relative equilibrium ; and prove that the particle will oscillate about this
position in a time 27r -r- o) sin a, a being the angle which the normal at
the point makes with the vertical.
2555. A heavy particle is placed in a tube in the form of a plane
curve which revolves with uniform angular velocity cd about a vertical
axis in its plane, and the particle oscillates about a position of relative
equilibrium : prove that the time of oscillation is
27r / r sin a
CD V A; -r sin a cos* a*
h being the distance from the axis, r the radius of curvatxire, and a the
inclination of the normal to the veHical, at the point of equilibrium.
DYNAMICS OF A PABTICLE. 437
2556. A Btraight tube inclined to the vertical at an angle a revolyes
with uniform angular ve.locity cd about a vertical axis whose shortest
distance from the tube is a and contains a smooth heavy particle which
is initially placed at its shortest distance from the axis : prove that the
space 8 which the particle describes aloDg the tube in a time t is given
by the equation
n cos CI
cu sm'a^ ' ^ '
2557. A heavy particle is attached to two points in the same
horizontal plane at a distance a by two extensible strings each of natural
length a, and is set free when each string is at its natural length : prove
that the radius of curvature of the initial path of the particle is
2 ^/3a -7- (ni « n),
the moduli of the strings being re>pectively m and n times the weight of
the pai-ticle.
2558. Three equal particles P, $, ^, for any two of which « = 1,
move in a smooth fine circular tube of which ABS&k vertical diiuneter ;
P starts from A^ and Q, Q' at the same instant in opposite senses from By
the velocities being such that at the first, impact all three have equal
velocities : prove that throughout the whole motion the straight line
joining any two particles is either horizontal or passes through one of
two fixed points (images of each other with respect to the circle); and
that the intervals of time between successive impacts are all equaL
2559. A point F describes the curve y = a log sec - with a velocity
which varies as the cube of the radius of curvature and has attached to
it a particle Q by means of a string of length a; when P is at the
origin, Q is at the corresponding centre of curvature and its velocity is
equal and opposite to that of P : prove that throughout the motion the
velocity of Q will be equal in magnitude to that of P, and that Q is
always the pole of the equiangular spiral of closest contact with the given
curve at P.
IV. Motion of Strings on Curves or Surfaces.
2560. A uniform heavy chain is placed on the arc of a smooth
vertical circle, its length being equal to that of a quadrant and one
extremity being at the highest point of the circle : i)rove that in the
beginning of the motion tlie resultant vertical pressure on the circle
bears to the resultant horizontal pressure the ratio v^ — 4 : 4.
2561. A string of variable density is laid on a smooth horizontal
table in the form of a curve such that the curvature is everywhere
proportional to the density and tangential impulses are applied at the
ends : prove that the equation for determining the impulsive tension T
at any point is T= A^-i-Bt^^, where ^ is the angle which the tangent
makes with a fixed direction ; and that, if the curve be an equiangular
spiral, the initial direction of motion of any point will be at right angles
to the radius vector.
438 DYNAMICS OF A PABTICLE.
2562. A number of material particles F , P,, ... of masses m,, m^ . . .
connected by inextensible strings are placed on a horizontal plane so
that the strings are sides of an unclosed polygon each of whose angles
is IT ~ a, and an impulse is applied to P^ in the direction P^P, : proTe
that
«», (r,^. cos a - r,) = m,, . (T, - T^_, cos a),
where T^ is the impulsive tension of the r^ string; and deduce the
equation
ds* fids ds p'
for the impulsive tension in the case of a fine chain. From either equar
tion deduce the result of the last question.
2563. A fine chain of variable density is placed on a smooth
horizontal table in the form of a curve in which it would hang under
the action of gravity and two impulsive tensions applied to its ends,
which are to each other in the same ratio as the tensions at the same
points in the hanging chain : prove that the whole will move without
change of form parallel to the straight line which was vertical in the
hanging chain.
2564. A heavy uniform string PQ, of which P is the lower ex-
tremity, is in motion on a smooth circular arc in a vertical plane, O
being the centre and OA the horizontal radius : prove that the tension
at any point E of the string is
-_y|8iny , ^. sin a , ^A
fl^ ^ I — ^ cos (y + *) - -^ cos (a + tf )| ,
where 0, 2a, 2y are the angles AOP, POQ, POR respectively, and W
the weight of the string.
2565. A portion of a heavy uniform string is placed on the arc
of a four-cusped hypocycloid, occupying the space between two ad-
jacent cusps, and runs off the curve at the lower cusp where the tangent
is vertical : prove that the velocity which the strmg will have when
just leaving the arc will be that due to a space of nine-tenths the
length of the string.
2566. A uniform string is placed on the arc of a smooth curve
in a vertical plane and moves under the action of gravity : prove the
equation of motion
I being the length of the string, s the arc described by any point of
it at a time t, and y^, y^ the depths of its ends below a fixed horizontal
straight line.
2567. A uniform heavy string APB is in motion on a smooth
urve in a vertical plane, and on the horizontal ordinate from a fixed
ertical line to il, P, B are taken lengths equal to the arcs measured
BTNAMICS OF A PABTICLE. 439
from a fixed point of the curve to A, P, B respectively : prove that
the ends of these lengths are the comers of a triangle whose area
is always proportional to the tension at P.
2568. A uniform heavy stiing is placed on the arc of a smooth
cycloid whose axis is vei'tical and vertex upwards : determine the
motion, and prove that, so long as the whole of the string is in contact
with the cycloid, the tension at any given point of the string is constant
throughout the motion and greatest at the middle point (measured
on the arc).
2569. A uniform heavy chain is in motion on the arc of a smooth
curve in a vertical plane and the tangent at the point of greatest
tension makes an angle ^ with the vertical : prove that the difference
between the depths of the extremities is / cos <^
2570. A uniform inextensible string is at rest in a smooth groove,
which it just fits, and a tangential impulse P is applied at one end :
prove that the normal impulse per unit of length at a distance 8 (along
the arc) from the other end is Pa-r-apy where a is the whole length
of the string and p the radius of curvature at the point considered.
2571. A straight tube of uniform bore is revolving uniformly in
a horizontal plane about a vertical axis at a distance c from the tube,
and within the tube is a smooth uniform chain of length 2a which
is initially at rest with its middle point at the distance c from the
axis of revolution: prove that the chain in a time t will describe a
space Jc (c*»'- c"**^) along the tube, and that the tension of the chain at
a point distant x from its middle point is
where m is the mass of the chain and cu the angular velocity.
2572. A circular tube of radius a revolving with uniform angular
velocity co about a vertical diameter contains a heavy imiform rigid
wire which j ust fits the tube and subtends an angle 2a at the centre :
prove that the wire will be in relative equilibrium if the radius to its
middle point make with the vertical an angle whose cosine is
g -T- am* cos a,
and that the stress along the wire is a minimum at the lowest point
of the tube (provided the wire pass through that point) and a maximum
at the |K)int whose projection on the axis bisects the distance between
the projections of the ends of the wire. Discuss which position of
equilibrium is stable, proving the equation of motion
oa -j-^ + sin a sin0 {g - acu'cos a cos 0) = 0,
where is the angle which the radius to the middle point of the wire
makes with the vertical.
440 PYNAMICS OF A PARTICLE.
[The highest position of equilibrium is always unstable ; the oblique
position is stable if it is possible, the time of a small oscillation being
— : — T^t^l - — IT-, where ao>' cos a cos j3 = or :
<i) sin )3 V sin 2a ' r- y >
and the lowest position is stable when cut? cos o,<g^ the time of a small
oscillation being
2, /-,«-. L .]
V sin a a - a<ii cos a -•
2573. A pulley is fixed above a horizontal plane; over the pulley
passes a fiue inextensible string* which has two equal uniform chains
fixed to its ends ; in the position of equilibrium a length a of each chain
is vertical, and the rest is coiled up on the table. One chain is now
drawn up through a space Tia : form the equation of motion, and prove
that the system will next come to instantaneous rest when the upper
end of the other chain is at a depth ma below its mean position, where
(l-m)€-=(l+w)€-.
AlsOy when w= 1, prove that m = '5623 nearly.
V. Resisting Media, Uodographs,
2574. A heavy particle is projected vertically upwards, the re-
sistance of the air being mass x (velocity)' -r- c ; the particle in its ascent
and descent has equal velocities at two points whose respective heights
above the point of projection are a;, y : prove that
C * + € ^^ = 2.
2575. A heavy particle moves in a medium in which the resistance
varies as the square of the velocity, v, v, u are its velocities at the two
points where its direction of motion makes angles —<!>, <f> with the
horizon and at the highest point, and p, p', r are the radii of curvature
at the same two points and the highest point respectively : prove that
1 1 2cos*6 1 1 2cos'<^
x^ v'* u^ ' p p r '
2576. A heavy particle moves in a medium whose resistance varies
as the 2n^ power of lie velocity ; v, v\ u are the velocities of the particle
when its direction of motion makes angles —<f>y <l> with the horizon and
at the highest point, and p, p', r are the radii of curvature at the same
two points and the highest point respectively : prove that
JL JL - 2 cos'" <^ i^ i__^ / cos" <t> y
2577. A small smooth bead slides on a fine wire whose plane
is vertical and the height of any point of which is a sin — , a being
the arc measured from the lowest point, in a medium whose resistance
is mass x (velocity)* -j- c, and starts from the point where Ss = irc: prove
velocity acquired in falling to the lowest point is Jag:
DYNAMICS OF A PABTICLE. 441
2578. A heavy particle slides on a smooth curve whose plane is
vertical in a medium whose resistance varies as the square of the
velocity, and in any time describes a a\iSLce which is to the space
described in the same time by a particle falling freely in vacuo as
1 : 2n: prove that the curve must be a cycloid whose vertex is its
highest point, and that the starting point of the particle must divide
the arc between two cusps in the ratio 2/i — 1 : 2n + 1.
2579. A heavy particle falls down the arc of a smooth cycloid
whose axis is vertical and vertex upwards in a medium whose resistance
is mass x (velocity)' -r 2c, its distance along the ^arc from the vertex
being initially c : prove that the time to the cusp will be
where 2a is the length of the axis.
2580. A particle is projected from a fixed point A in & medium
whose resistance is measured by 3(o x velocity and attracted by a fixed
point S by a force whose acceleration is 2o>' x distance : prove that
the particle will describe a parabola tending in the limit to come to
rest at S,
[Taking ^S^^ = a, and u, v to be the component velocities at A along
and perpendicular to SA , the equation of the path is
{(u + ttio) y — vxY = av {ixc - (u + 2a«) y},
and the length of the latus rectum is
aVw -r (l6 + CKtf + V*) .]
2581. A heavy particle moves in a circular tube whose plane
is vertical in a medium whose resistance is mass x (velocity)' -r 2c,
starting from a point in the upper semicircle where the normad makes
the angle tan~*- with the vertical: prove that the kinetic energy
at any time while the particle moves through the semicircle which
begins at this point is proportional to the distance of the particle from
the bounding diameter.
2582. A point describes a straight line under acceleration tending
to a fixed point and varying as the distance : prove that the correspond-
ing point of the hodograph will move under the same law of ac-
celeration.
2583. The curves r" = oT sin mO, r^ = o" sin n^ will be each similar
to the hodograph of the other when described about a centre of force in
the pole, provided that vm + f?n- n = 0. Prove this property geo-
metrically for both curves when w = 1, 2n = — 1.
2584. A point describes a certain curve in such a manner that
its hodograph is described as if under a central force in its pole, and
T, N are the tangential and normal accelerations of the point : prove
that
dp'
^-<-©'=
442 DYNAMICS OF A PABTICLE.
where p is the radius of curvature, 8 the arc measured from a fixed
point, and c a constant : also prove that the acceleration at any point
of the hodograph will vary as
K'4i-vS0.
and that if the intrinsic equation of the curve be
ds .
-=-:=« sec w»^,
(i<p
the equation of the hodograph will be r = a sec mO.
2585. A point describes half the arc of a cardioid, oscillating
symmetrically about the vertex, in such a way that the hodograph
is a circle with the pole in the circumference : prove that the ac-
celeration of the point describing the cardioid varies as 2r - 3a, r being
the distance from the cusp and 2a the length of the axis : also prove
that the direction of acceleration changes at double the rate of the
direction of motion.
2586. A heavy particle of weight W is moving in a medium in
which the resistance varies as the n^ power of the velocity, and F is the
resistance when the direction of motion makes an angle ^ with the
horizon : prove that
W = nF COS" <!> J sec**' ^l^difK
2587. A heavy particle is projected so as to move on a rough plane
inclined to the horizon at the angle of friction : prove that the hodo-
graph of the path is a parabola and that the intrinsic equation of the
patii is
V f3sin<^ + 2sin'<A , , /ir 4\)
where F is the velocity at the highest point and a the angle of friction.
2588. Two points P, Q describe two curves with equal velocities,
and the radius vector of Q is always parallel to the direction of motion
of P : shew how to find F'a path when ^'s path is given ; and prove that
(1) when Q describes a straight line P describes a catenary, (2) when Q
describes the circle r=a cos $, P describes a circle of radius a, (3) when
Q describes the cardioid r = a (1 + cos tf), P describes a two-cusped epi-
cycloid.
2589. A circle is described by a point in a given time under the
action of a force tending to a fixed point within the circle : prove that,
for different positions of the centre of force, the action during a whole
revolution varies inversely as the minimum chord which can be drawn
^hrough the point
DYNAMICS OF A PARTICLE. 443
[In any closed oval under a central force to a point within it the
action during a whole revolution = -3 I -, c?<^, where r is the radius
■r Jo ]}
vector from the centre of force, p the perpendicular from the centre of
force on the tangent, <f> the angle which the tangent makes with
some fixed straight line, A the area of the oval, and F the periodic
time.]
2590. A point describes a parabola under a central force in the
vertex : prove that the hodograph is a parabola whose axis is at right
angles to the axis of the described parabola.
[In general if any conic be described under any central force the
hodograph is another conic which will be a parabola when the described
conic passes through the centre of force.]
2591. A point P describes a catenary in such a manner that a
straight line di-awn from a fixed point parallel and proportional to the
velocity of P sweeps out equal areas in equal times : prove that the
direction of P's acceleration makes with the normal at P an angle
tan"* (I tan ^), where ^ is the angle through which the direction of
motion has turned in passing from the vertex.
2592. A circle is described under a constant force not tending to
the centre : prove that the hodograph is Bernoulli's lemniscate.
2593. A curve is described with constant acceleration and its
hodograph is a parabola with its pole at the focus : prove that the
intrinsic equation of the described curve is
ds .<h
-7- = a sec -k .
dif> 2
2594. A point describes a curve so that the hodograph is a circle
described with constant velocity and with the pole on its circumference :
prove that the described curve is a cycloid described as if by a heavy
particle falling from cusp to cusp.
2595. A point describes a certain curve with acceleration initially
along the normal, and the direction of acceleration changes at double the
rate of the direction of motion and in the same sense : prove that the
hodograph will be a circle with the pole on its circumference.
2596. A particle is constrained to move in an elliptic tube under
two forces to the foci, each varying inversely as the square of the distance
and equal at equal distances, and is just displaced from the position of
imstable equilibrium : prove that the hodograph is a circle with the pole
on its circumference.
[The particle will oscillate over a semi-ellipse bounded by the minor
axis, and the hodograph corresponding to this will be a complete circle
with the diameter through its pole parallel to the minor axis.]
DYNAMICS OF A RIGID BODY.
I. MojnerUs of Inertia^ Principal Axes,
2597. The density of an ellipsoid at any point is proportional to
the product of the distances of the point from the principal planes :
prove that the moments of inertia about the principal axes are
where m is the mass and a, 6, c the semi-axes.
2598. Prove the following construction for the principal axes at O,
the centroid of a triangular lamina ABC : draw the circle OBC, and in
it the chords Ob, Oc parallel to ACy AB respectively, and let Bh^ Cc
meet in Z ; then, if aa' be the diameter of the circle drawn through L^
OcLj Ood will be the directions of the principal axes at 0,
2599. Prove the following construction for the principal axes at
the centre of a lamina bounded by a parallelogram ABCD : draw the
circle OBG and in it chords 06, Oc parallel to AB^ BC^ and let BC^ he
meet in L ; then, if aa' be the diameter of this circle drawn through Ly
Oay Oa' will be the directions of the principal axes at 0,
2600. Prove that any lamina is kinetically equivalent to three
particles, each of one third the mass of the triangle, placed at the comers
of a maximum triangle inscribed in the ellipse whose equation, referred
to the principal axes at the centre of inertia, is Aoi? + Bif = '2AB, where
mA^ mB are the principal moments of inertia and m the mass.
2601. Prove that any rigid body is kinetically equivalent to three
equal uniform spheres, each of one third the mass of the body, whose
centres are comers of a maximum triangle inscribed in the ellipse
and whose common radius is J^ {A+B -C) ; the equation of the ellip-
a:* V* «*
sold of gyration being -j + ^ "*" 7* = 1> ^^^ A<B<C,
[Since A-\-B can never be less than C the radius will always be real,
but for the spheres not to intersect in any of their positions it will be
necessary that 23 (C-jB) > 20-4, which could not be satisfied by a body
of form approaching spherical. As the spheres need only be ideal for
simplification of calculation, this condition is of no importance.]
DYNAMICS OF A KIGID BODY. 445
2602. A straight line is at every point of its course a principal axis
of a given rigid body : prove that it passes through the centre of
inertia.
2603. A tetrahedron is kinetically equivalent to six particles at the
middle points of the edges, each ^ the mass of the tetrahedron, and one
at the centroid of mass ^ the mass of the tetrahedron.
2604. The principal moments of inertia of a rigid body, whose
mass is unity, at the centre of inertia are A, B, C, and a* + 6* + c* + r*
is a principal moment of inertia at the point (a, b, c), the principal
at the centre of inertia being axes of co-ordinates : prove that
a' b' -•
' =1.
A-r' B-r' C-r
2605. The locus of the points at which two principal moments of
inertia of a given rigid body are equal is the focal curves of the ellipsoid
of gyration at the centre of inertia.
2606. The locus of the points at which one of the principal axes
passes through a given point, which lies in one of the principal planes at
the centre of inertia, is a circle.
2607. The locus of the points at which one of the principal axes of
a given rigid body is in a given direction is a rectangular hyperbola with
one asymptote in the given direction.
2608. In a triangular lamina any one of the sides is a principal
axis at the point bisecting the distance between its mid-point and the
foot of the perpendicular from the opposite comer.
2609. In any uniform tetrahedron, if one edge be at any point a
principal axis so also will the opposite edge ; the necessary condition
is that the directions of the two edges shall be perpendicular ; and the
point at which an edge is a principal axis divides the distance between
the mid-point and the foot of the shortest distance between it and the
opposite edge in the ratio 1 : 2.
2610. Straight lines are drawn in the plane of a given lamina
through a given point ; the locus of the points at which they are princi-
pal axes of the lamina is a circular cubic.
2611. The locus of the straight lines drawn through a given point,
each of which is at some point of its course a principal axis of a given
rigid body, is the cone
a {B -C) yz + b {C -- A)zx -^e (A - B) osij=0,
A, B, C being the principal moments of inertia at the given pointy ^^tC
the co-ordinates of the centre of inertia and the principal axes at the
given point the axes of reference. Also prove that the locus of the points
at which these straight lines are principal axes is the curve
■^■.,..,^_ (-g-C)y» ^ (g-^)'« f^ (^-g)«y ],
^ cy~hz cLz-ex \ by — ax )'
446 DYNAMICS OF A RIGID BODY.
[The equation of the cone on which these straight lines lie retains
the same form when A, B, C denote the principal moments of inertia at
the centre of inertia, and the co-ordinate axes are parallel to the principal
axes at the centre of inertia.]
2612. The principal axes at a certain point are parallel to the
principal axes at the centre of inertia : prove that the point must lie
on one of the principal axes at the centre of inertia.
2613. The different straight lines which can be drawn through the
point (a;, y, z), each of which is at some point of its course a principal
axis of a given rigid body, will lie on a cone of revolution if
x(B-C) = y{C^A)=z{A-jB),
the principal axes at the centre of inertia being co-ordinate axes and
Ay B, C the principal moments of inertia.
II. Motion about a fixed Axis.
2614. A circular disc rolls in one plane on a fixed plane, its centre
describing a straight line with uniform acceleration y*; find the magni-
tude and position of the resultant of the impressed forces.
[The resultant is a force i^/* acting parallel to the plane at a distance
from the centre of the disc of one half the radius on the side opposite to
the plane.]
2615. A piece of uniform fine wire of given length is bent into the
form of an isosceles triangle and revolves about an axis through its
vertex perpendicular to its plane : prove that the centre of oscillation
will be at the least possible distance from the axis of revolution when
the triangle is right-angled.
2616. A heavy sphere of radius a and a heavy rod of length 2a
swing, the one about a horizontal tangent, the other about a horizontal
axis perpendicular to its length through one end, each through a right
angle to its lowest position, and the pressures on the axis in the lowest
positions are equal : prove that the weights are as 35 : 34.
2617. The centre of percussion of a triangular lamina one of whose
sides is the fixed axis bisects the straight line joining the opposite comer
with the mid-point of the side.
2618. A lamina A BCD is moveable about AB which is parallel to
CD : prove that its centre of percussion will be at the common point of
AC and BDi£AB'= 3CD\
2619. In the motion of a rigid body about a horizontal axis under
the action of gravity, prove that the pressure on the axis is reducible to
a single force at every instant of the motion only when the axis of
revolution is a principal axis at the point M which is nearest to the
centre of inertia : and, if the axis be a principal axis at another point
M and the forces be reduced to two acting at My N respectively, the
former will be equal and opposite to the weight of the body.
DYNAMICS OF A RIGID BODY. 447
2620. A rough uniform rod of length 2a is placed with a length
e{>a) projecting over the edge of a horizontal table, the rod being
initially in contact with the table and perpendicular to the edge : prove
that the rod will begin to slide over the edge when it has turned
through an angle whose tangent is -3 — ^~ ; , fi being the coefficient
of friction.
2621. A uniform beam capable of motion about one end is in
equilibrium ; find at what point a blow must be applied perpendicular
to the rod in order that the impulse on the fixed end may be - th of the
•^ n
blow.
[The distance' of the point from the fixed end must be to the length of
the rod in the ratio Jn — 1 : JSn,'\
2622. A uniform beam moveable about its middle point is in
equilibrium in a horizontal position, a particle whose mass is one-fourth
that of the beam and such that the coefficient of restitution is 1 is let
fall upon one end and is afterwards grazed by the other end of the
beam : prove that the height from which the particle is let fall bears
to the circumference of the circle described by an end of the beam
the ratio 49 (2n + 1) : 48, where n is a positive integer.
2623. A smooth uniform rod is revolving about its middle point,
which is fixed on a horizontal table, when it strikes an inelastic particle
at rest whose mass is one-sixth of its own, and the angular velocity
of the rod is immediately reduced one-ninth : find the point of impact,
and prove that, when the particle leaves the rod, the direction of
motion of the particle will make with the rod an angle of 45^
[The point of impact must bisect one of the halves of the rod, and
during the subsequent motion
gy + (2a' + r*) CO* = y aW, (2a* + r*) co = 2a«0,
where r is the distance of the particle from the centre of the rod,
and o) the angular velocity of the rod at any time, Q the angular
velocity before impact.]
2624. A smooth uniform rod is moving on a horizontal table
uniformly about one end and impinges on a particle of mass equal to
its own, the distance of the particle from the fixed end being -th of
the length of the rod : prove that the final velocity of the particle will
be to its initial velocity in the ratio
V(5w«-l)(n« + 3) : in.
(In this case also e = 0.)
448 DYNAMICS OF A RIGID BODY.
2625. A uniform rod (mass m) is moving on a horizontal table
about one end and diiving before it a smooth particle (mass p) which
starts from rest close to the axis of revolution : prove that, when
the particle is at a distance r from the axis, its direction of motion
will make with the rod the angle cot"* ^ 1 + ^^ , where m^ is the
moment of inertia of the rod about the axis of revolution.
2626. A luiiform circular disc of mass m is capable of motion in
a vertical plane about its centre and a rough particle of mass p is placed
on it close to the highest point : prove that the angle B through which
the disc will turn before the particle begins to slide is given by the
equation
where a is the radius and wi^ the moment of inertia of the disc.
2627. A uniform rod, capable of motion in a vertical plane aboat
its middle point, has attached to its ends by fine strings two particles
which hang freely; when the rod is in equilibrium inclined at an
angle a to the vertical one of the strings is cut : prove that the initial
tension of the other string is
mpg H- (w + 3/) sin* a),
and that the radius of curvature of the initial path of the particle is
^Ip sin' a -r m cos a,
m, p being the masses of the rod and of a particle, and / the len^h
of the string.
2628. A uniform rod moveable about one end is held in a hori-
zontal position, and to a point of the rod is attached a heavy particle by
means of a string : prove that the initial tension of the string ^when
the rod is allowed to fall freely is
mpga (4a - 3c) -r (47wa* + Bjjc'),
where m, p are the masses of the rod and particle, 2a the length of
the rod, and c the distance of the string from the fixed end : also prove
that the initial path of the particle referred to horizontal and vertical
axes will be the curve
ma (4a - 3c) y" + 90c*^ (ma •{■pc)x = 0,
where I denotes the length of the string.
2629. A uniform rod moveable about one end has attached to the
other end a heavy particle by a fine string ; initially the rod and string
are in one horizontal straight line without motion: prove that the
radius of curvature of the initial path of the particle will be
4a6 -^ (a + 96),
DYNAMIGS OF ▲ BIQID BODY. 449
where a, b denote the lengths of the rod and string ; and explain why
the result does not depend on the masses of the two.
2630. A uniform rod, of length 2a and mass m, capable of motion
about one end, is held in a horizontal position and on the rod slides
a small smooth ring of mass p : prove that, when the rod is set £ree,
the radius of curvature of the initial path of the ring will be
4a — oc \ ma/
where e is the initial distance of the ring from the fixed end.
2631. A uniform rod capable of motion about one end has attached
at the other end a particle by means of a fine string, and the system
is abandoned freely to the action of gravity when the rod makes an
angle a with the string which is vertical : prove that the radius of
curvature of the initial path of the particle is
9/ A +^)sin»a4.cosa(2-3sin*a);
where m, p are the masses, and I the length of the string.
2632. A uniform rod is moveable about one end on a smooth
horizontal table and to the other end is attached a pai*tiele by a fine
string; at starting the rod and string are in one straight line, the
particle is at rest, but the rod in motion : prove that when the rod
and string are next in a straight line the angular velocities of the
rod and string will be as 6 : a, or as
b{3p{a-by^ma') : a {3p (a - 5)* + ma (a - 26)},
where m, p are the masses, and a, b the lengths of the rod and string.
III. Motion in Two Dimennons.
2633. Two equal uniform rods AB^ BC, freely jointed at B and
moveable about A, start from rest in a horizontal position, BG passing
over a smooth peg whose distance from ^ is 4a sin a (where 3 sin a < 2) :
prove that, when BC leaves the peg, the angular velocity of AB is
/3£
V 2a
cos a
2a l + sin'2a'
where 2a is the length of either rod.
2634. A uniform rod of length 2a rests with its lower end at
the vertex of a smooth surface of revolution whose axis is vertical
and passes through a smooth fixed ring in the axis at a distance 6
from the vertex : the time of a small oscillation will be
where e is thQ radios of curvature at the vertex.
W. P. 29
450 DYNAMICS OF A RIGID BODY.
26S5. Two heavy particles are fixed to the ends of a fine wire in the
form of a circular arc, which rests with its plane vertical on a rough
horizontal plane, and a, P are the angles which the radii through the
particles make with the vertical: prove that the time of a small
oscillation will be
/ . a . fi
J sm jr sm ^
SI 9
cos — r-
2
2636. Two equal and similar uniform rods, freely jointed at a
common extremity, rest symmetrically over two smooth pegs in the
same horizontal plane so that each rod makes an angle a with the
vertical : prove that the time of a small oscillation will be
V Tl4.
-, , ^w cos a
27r
^r 1 4- 3 cos" a '
where 2a is the length of either rod.
2637. A lamina with its centre of inei-tia fixed is at rest, and
is struck by a blow at the point (a, h) normally to its plane : prove
that the equation of the instantaneous axis is ^aa; + j&6y = 0, the axes
of co-ordinates being the principal axes at the centre of inertia and
Ay B being the principal moments of inertia; also that, if (a, h) lie on a
certain straight line, there will be no impulse at the fixed point.
2638. A uniform heavy rod revolves uniformly about one end
in such a manner as to describe a cone of revolution: determine the
pressure on the fixed point and the relation between the angle of
the cone and the time of revolution ; and prove that, if ^, ^ be the angles
which the vertical makes with the rod and with the direction of
pressure,
4 tan ^ = 3 tan $,
2639. A fine string of length 25 is attached to two points in the
same horizontal plane at a distance 2a and carries a particle p at its
middle point; a uniform rod of length 2c and mass m has at each
end a ring through which the string passes and is let fall from a
symmetrical position in the same straight line as the two points : prove
that the rod will not reach the particle if
(a + 6 - 2c) (m + 2p)m<2 (2c - a) p\
2640. A heavy uniform chain is collected into a heap and laid
on a horizontal table and to one end is attached a fine string which,
passing over a^smooth fixed pulley vertically above the heap, is attached
to a weight equal to the weight of a length a of the chain : prove that the
length of the chain raised before the weight first comes to instantaneous
DYNAMICS OF A RIGID BODY. 451
rest is aj^, and that when the weight next comes to rest the length
of chain which is vertical is ox, where x is given by the equation
(\^7
and that x is nearly equal to -yx .
2641. A uniform rod of length c has at its ends small smoofch rings
which slide on two fixed elliptic arcs whose planes are vertical and
semi-axes are a,h) a-k-Cy 6 + c respectively, and are inclined at angles a,
^ + a to the horizon : determine the motion of the rod and the pressures
on the arcs, the rod being initially vertical.
2642. A circular disc rolls on a rough cycloidal arc whose axis is
vertical and vertex downwards, the length of the arc being such that
the curvature at either end of the arc is equal to that of the circle :
prove that, if the contact be initially at one end of the arc, the point on
the auxiliary circle of the cycloid which corresponds to the point of
contact will move with uniform velocity which is independent of the
radius of the disc ; and that the normal pressure R and the force of
friction F in any position of the disc are given by the equations
3^=IFr(5costf-2cosa), S^^TTsintf,
where W is the weight of the disc, 9 the angle which the common <normal
makes with the vertical, and a the initial value of 6.
2643. A uniform sphere rolls from rest down a given length I ci h
rough inclined plane and then traverses a smooth portion of the plane of
length ml; find the impulse which takes place when perfect rolling
again begins, and prove that the subsequent velocity is less than would
have been the case if the whole plane had be^i rough; if f7»s=120y in
the ratio 67 : 77.
[The ratio in general is 2 + J25 + 35m ; 7 Jm + 1.
2644. A straight tube A£ of small bore, containing a smooth
uniform rod of the same length, is closed at the end B and in motion
about the fixed end A with angular velocity cd : prove that, if the end B
be opened, the initial stress at a point F of the rod is equal to
MJ'AF . FB ^ 2AB,
M being the mass of the rod.
2645. The ends of a uniform heavy rod are fixed by smooth rings
to the arc of a circle which is made to revolve uniformly about a fixed
vertical diameter ; find the positions of relative equilibriimi, and prove
that any such position in which the rod is not horisontal will be
stable.
[If a be the radius of the circle, cd its angular velocity, 2a the angle
which the rod subtends at the centre, there will be no inclined positions of
29—2
452 DYNAMICS OF A RIGID BODY.
eqiiilibriom unless aa)'cosa>^: if out' cos a cob fi == g^ the time of a small
oscillation about the inclined position will be ^^/^ v ^ + i tan'a ; the
time of oscillation about the lowest position will be
2,r^l+Jtan'a-^?-
(0* COS a ;
and, when g = oco' cos a, the equation of motion will be
(1+1^ tan* o) ;^ + o)* sin tf ( 1 - cos tf ) = C]
2646. A smooth semicircular disc rests with its plane vertical and
vertex upwards on a smooth horizontal table and on it rest two equal
uniform rods, each of which passes through two smooth fixed rings in a
vertical line ; the disc is slightly displaced, and in the ensuing motion
one rod leaves the disc when the other is at the vertex : prove that
m 4 sin a- (1 + sin P)* (2 - sin )8)
jS" si?^3 '
where m, p are the masses of the disc and of either rod, a the angle
which the radius to either point of contact initially makes with the
horizon, and fi = cos"* (2 cos a).
[When the one rod leaves the disc, the pressure of the other on the
disc IB pg(l - sin* fi). ]
2647. A uniform rod moves with one end on a smooth horizontal
plane and the other end attached to a string which is fixed to a point
above the plane ; when the rod and string are in one straight line the
rod is let go : prove that the angular velocity of the string when
vertical will be ^ j(l -. j and its angular acceleration
g la ■\-l-h
a^lSj a-l+h*
a, I, h being the lengths of the rod and string and the height of the fixed
point above the plane respectively.
2648. A uniform beam rests with one end on a smooth horizontal
table and has the other attached to a fixed point by means of a string
of length I : prove that the time of a small oscillation in a vertical plane
will be
2ir
/2l
2649. A sphere rests on a rough horizontal plane with half its
weight supported by an extensible string attached to the highest point,
who»e extended length is equal to the diameter of the sphere : j>rove
that the time of small oscillations of the sphere parallel to a vertical
14a
I5g'
'ane is 2ir ^ -.
BYKAKICS OF A BIQID BODY. 453
2650. Two equal uniform rods A By BCj freely jointed at B^ are
placed on a smooth horizontal table at right angles to each other and a
blow is applied to ^ at right angles to AB : prove that tho initial
velocities of A^ C are in the ratio 8:1.
2651. Two equal uniform rods AB, BC, freely jointed at B^ are laid
on a smooth horizoDtal table so as to include an angle a and a blow is
applied at A at right angles U) AB\ determine the initial velocity of C^
and prove that it will begin to move |>arallel to il j9 if 9 cos 2a = 1.
2652. Five equal uniform rods, freely jointed at their extremities,
are laid in one straight line on a horizontal table and a blow applied at
the centre at right angles to the line : prove that, initially,
14a 5a -a 9 -3'
where v, v^, v^ are the velocities of the three rods, oij, oi, the angular
velocities of the two pairs of rods, and 2a the length of each rod.
2653. Four equal unifonn rods AB, BO, CD, DE, freely jointed at
B, C, D, are laid on a horizontal table in the form of a square and a blow
is applied at A at right angles to AB fi*om the inside of the square :
prove that the initial velocity of ^i is 79 times that of E,
2654. Two equal uniform rods AB, BO, freely jointed at B and
moveable about A, are lying on a smooth horizontal table inclined to each
other, at an angle a ; a bl(»w is applied to (7 at right angles to BC in a
direction tending to decrease the angle ABC : prove that the initial
angular velocities of AB, BC will be in the ratio cos a : 8 - 3 cos* a ;
that Q, the least value of the angle ABC during the motion is given by
the equation
8 (5 - 3 cos tf) (2 - cos* a) = (1 - cosa)* (16 - 9 cos* a) :
mm
also prove that, when a = ;r , the angular velocities of the rods when in
a straight line will have one of the ratios — 1 : 3, or 3 : — 5.
2655. A heavy uniform rod resting in stable equilibrium within a
smooth ellipsoid of revolution about its major axis, which is vertical, is
slightly displaced in a vertical plane : prove that the length of the
equivalent simple pendulum is acc(3e*+ 1) -r 6 (a-c), where 2a is the
length of the rod, "Ic the latus rectum, and t the excentricity of the gene-
i-ating ellipse.
2656. A uniform rod of length 2a rests in a horizontal position with
its ends on a smooth curve which is symmetrical about a vertical axis :
prove that the time of a small oscillation will be
^V
ar cos a (1 + 2 cos' a)
Zg{a — r sin* a) '
r being the radius of curvature of the curve and a the angle which the
normal makes with the vertical at either end of the rod.
454 DYNAMICS OF A RIGID BODY.
2657. Four equal rods of length a and mass m are freeljr jointed
BO as to form a rhombus one of whose diagonals is vertical ; the enda of
the other diagonal are joined by an extensible string at its natural
length and the system falls through a height A on to a fixed horizontal
plane : prove that, if ^ be the angle which any rod makes with the
vertical at a time t after the impact,
(l+3sm*tf)(-^) = -4-, — o ' » 4- — (cosa-cos^)
^ ^ \dt/ a 1 + 3 am* a a ^ '
(sin tf — sin a)*;
2//ia sin a
where a is the initial value of 6 and X the modulus of the string.
IV. Miscellaneotis.
2658. A square is moving freely about a diagonal with angular
velocity Q, when one of the comers not in that diagonal becomes fixed ;
determine the impulse on the fixed point, and prove that the instantaneous
angular velocity is yO.
[If V be the previous velocity of the point which becomes fixed the
impulse will be l^MV.]
2659. A uniform heavy rod of length a, freely moveable about one
end, is initially projected in a horizontal plane with angular velocity O :
prove that the equations of motion are
sin«^^ = 0, afj\ = 3gcoBe-an*cot'e;
where 0, <t> are respectively the angles which the rod makes with the
vertical (downwards from the fixed end) and which the projection of the
rod on the horizontal plane makes with its initial position : also, if the
least value of 6 be - , prove that the resolved vertical pressure on the
o
fixed point when ^ = « will be y J IF, where W is the weight of the rod.
o
[The vertical pressure on the fixed point in any position is
-77- (1 1 + 9 cos 2^ + cos $j ;
and the horizontal pressure is -^ (9 sin 25 + sin 5 j , in the vertical
plane through the rod.]
2660. A uniform heavy rod moveable about one end moves in such
a manner that the angle which it makes with the vertical never difiTers
DYNAMICS OF A RIGID BODY. 455
mucli from a : prove that the time of its small oscillations will be
/2a -
V 3(7 1 +
cos a
3^1 + 3co8*a'
where a is the length of the rod.
2661. A centre of force whose acceleration is /x (distance) is at a
point Of and from another point ^1 at a distance a are projected simul-
taneously an infinite number of particles in a direction at right angles
to OA and with velocities in arithmetical progression from Y^»Jf^ ^
~- aJfjL : prove that, when after any lapse of time all the particles
become suddenly rigidly connected together, the system will revolve with
angular velocity xtn//**
[If the limits of the velocity be n^ajii, n^ajfi, and the time elapsed
$ -r Ijfi, the common angular velocity of the rigidly connected particles
will be 3(nj + n,) V/*-r6cos'^ + 2(w/+njW, + n/)sin*ft]
2662. A uniform heavy rod is suspended by two ineztensible
strings of equal lengths attached to its ends and to two fixed points whose
distance is equal and parallel to the length of the rod; an angular
velocity about a vertical axis through its centre is suddenly communi-
cated to the rod such that it just rises to the level of the fixed points :
find the im2)ulsive couple, and prove that the tension of either string is
suddenly increased sevenfold.
2663. Two equal uniform heavy rods AB, BG, freely jointed at B^
rotate uniformly about a vertical axis through A^ which is fixed, with
angular velocity Q : prove that the angles a, p which the rods make with
the vertical are given by the equations
(8 sin a + 3 8in)3) cot a = (9 sin o + 6 sin)8) cot )3 = -f , ;
where a is the length of each rod.
2664. A perfectly rough horizontal plane is made to revolve with
uniform angular velocity about a vertical axis which meets the plana
in ; a heavy sphere is projected on the plane at a point P so that
its centre is initially in the same state of motion as if the sphere
had been placed freely on the plane at a point Q and set in motion by
the impulsive friction only : prove that the centre of the sphere will
describe uniformly a circle of radius OQ, and whose centre R is such that
OR is equal and parallel to QP,
2665. A perfectly rough plane inclined at an angle a to the horison
is made to revolve with uniform angular velocity Q about a normal and
a heavy motionless sphere is placed upon it and set in motion by the
tangential impulse : prove that the ensuing path of the centre will be a
prolate, a common, or a curtate cycloid, according as the initial point of
contact is without, upon, or within the circle whose equation is
2n* (»• + /) = 35^ sin a,
the axis of y being horizontal and the point where the axis of revolution
meets the plane the origin. Also prove that, if the initial point of
contact be the centre of this circle, the path will be a straight lina
456 DYNAMICS OF A RIGID BODY.
2666. A rough hollow cylinder of revolution whose axis is vertical
is made to revolve with uniform angular velocity O about a fixed
generator and a heavy uniform sphere is rolling on the concave sur&oe :
prove that the equation of motion is
©■-<'* ¥t»-"»*>
where ^ is the angle which the common normal to the sphere and
cylinder makes at a time t with the plane containing the fixed generator
and the axis of the cylinder, and a-^-bfa are the radii of the cylinder and
sphere respectively.
2667. A rough plane is made to revolve at a uniform rate O about
a horizontal line in itself and a sphere is set in motion u(>on it : deter-
mine the motion, and prove that, if when the plane is horizontal the
centre of the sphere is vertically above the axis of revolution and moving
parallel to it, the contact will cease when tbe plane has turned through
an angle $ given by the equation
where a is the iTidius of the sphere.
2668. A uniform heavy rod is free to move about one end in a
vertical plane which is itself constrained to revolve about a vertical axis
through the fixed end at a uniform rate O, and the greatest and least
angles which the rod makes with the vertical during the motion are
a, p : prove that
aO* (cos a + cos fi) = 3g,
where a is the length of the ix>d : also prove that, when 3^ = 2aO' cos a,
2ir
the time of a small oscillation will be jr—. — .
(2 sin a
2669. Two heavy uniform rods of lengths 2a, 26 and masses A^ B
are freely jointed at a common end and are moveable about the other
end of A, and the rods fall from a horizontal position of instantaneous
rest : prove that the radius of curvature of the initial path of the free
end of i» wiU be 2ab {A ^ Bf -^ {aA^ + b {2A +^)'}.
2670. A rigid body is in motion about its centre of inertia under
no forces, and at a certain instant, when the instantaneous axis is the
straight line whose equations are
xjA{B^C)^zJa{A-B), y = 0,
a point on the cylinder
«• (^ - i?) + *• (if - c) + « y <i:i^i^) (C + ^) = js (C - ^)
is suddenly fixed : prove that the new instantaneous axis will be perpen-
dicular to the direction of the former. (The axes of co-ordinates are,
as usual, the principal axes at the centre of inertia, and A^ B^C the
squares of the semi-axes of the principal ellipsoid of gyration.)
DYNAMICS OF A RIGID BODY. 457
2671. A number of concentric spherical shells of equal indefinitely
small thickness revolve about a common axis through the centre, each
at a uniform rate pix>portional to the n^ power of its radius ; the shells
become suddenly rigidly united : prove that the subsequent angular
velocity bears to the previous angular velocity of the outermost shell the
ratio 5 : w + 5.
2672. An infinite number of concentric spherical shells of equal
small thickness are revolving about diameters all in one plane with equal
angular velocities, and the axis of revolution ot the shell whose radius is
T
r is inclined at an angle cos"^ - to the axis of the outermost shell : prove
that, when united into a solid sphere, the axis of revolution will make
an angle tan~* -r-^ with the former axis of the outer shell.
ID
2673. Prove that any possible given, state of motion of a rigid
straight rod may be represented by a single rotation about any one of an
infinite number of axes lying in a certain plane.
2674. A free rigid body is in motion about its centre of inertia
when another point of the rigid body is suddenly fixed and the body
then assumes a state of permanent rotation about an axis through
that point : prove that the point muut lie on a certain rectangular
hyperbola.
[With the notation of (2669) the point to be fixed must satisfy the
equations
(Aui^x + Bi^jj + Ci^jc) (b^ - + C^^ ^ + T'^B-\
+ (j5-(7)(C-ui)(^-^) = 0,
where co,, cd^, cu^^ are the previous component angular velocities ; also the
new axis of revolution must be parallel to the normal to the invariable
plane of the previous motion.]
2675. A rigid body is in motion under the action of no forces and
its centre of inertia is at rest ; when the instantaneous axis is a certain
given line of the body a point rigidly connected with the body is
suddenly fixed, and the new inntantaneous axis is parallel to one of the
principal axes at the centre of inertia : prove that the point to be fixed
must lie on a certain hyperbola one asymptote of which is the given
principal axis.
2676. A free rigid body is at a certain instant in a state of rotation
about an axis through its centre of inertia when a given point of the
body becomes suddenly fixed : determine the new instantaneous axis,
and prove that there are three directions of the former instantaneous
axis f(ir which the new axis will be in the same direction ; and these
three directions are along conjugate diameters of the principal ellipsoid
of inertia.
458 DYNAMICS OF A RIGID BODY.
2677. A rigid ]x>d7 is in motion under tlie action of no forces vith
its centre of inertia at rest and the instantaneous axis is deacribing a
plane in the body : prove that, if a point in that diameter of the
principal ellipsoid of inertia which is conjugate to this plane be
suddenly fixed, the new instantaneous axis will be parallel to the
former.
2678. Two equal uniform rods AB, BC, freely jointed at i? and in
one straight line, are moving uniformly in a direction normal to their
length on a smooth horizontal table when the point A becomes
suddenly fixed : prove that the initial angular velocities of the rods will
be in the ratio 3 : — 1, that the least subsequent obtuse angle between
them will be cos~^ (—f)* &^d that when next in one straight line their
angular velocities will be as 1 : 9.
2679. Three equal uniform rods AB, BC, CD, freely jointed at B
and C, are lying in one straight line on a smooth horizontal table iT^hen
a blow is applied at their centre in a direction normal to the line of the
rods : prove that
where 6 is the angle through which the outer rods have turned in a
time t and O their initial angular velocity. Prove also that the velocity
of BC will be %^\ 1 + — ; > , and that the direction of the stress
3 I 7l+sin«^i
B.t B or C will make with BC the angle tan"* (§ tan 6),
2680. Two equal uniform rods AB, BC, freely jointed at jB, are in
motion on a smooth horizontal table and their angular velocities are
Wj, <o, when the angle between them is ^: prove that (w^ + w^) (5 — 3 cos $)
and 5 (w^* + cu^') — 6<i>ja)g cos $ are both constant throughout the motion.
2681. Three equal unifonn rods (for all of which e = 0), freely
jointed at common ends, arc laid in one straight line on a smooth
horizontal table and the two outer are set in motion about the ends of
the middle rod with equal angular velocities, (1) in the same sense,
(2) in opposite senses: prove that, (1) when the outer rods make the
greatest angle with the direction of the middle rod produced on each
side the common angular velocity of the three will be y co, and (2) that
after the impact of the two outer rods the triangle formed by the three
will move with velocity J aw, where a is the length of a rocL
2682. A uniform rod of length 2a has attached to one end a
particle by a string of length b and the rod and string placed in one
straight line on a smooth horizontal table ; the particle is then projected
at right angles to the string : prove that the greatest angle which the
string can make with the rod (produced) will be
="»-Vra0^9.
DYNAMICS OF A BIQID BODY. 459
where m, p are the masses ; also that, if after a time t the rod and string
make angles 0, ^ with their initial directions,
where ifc*=Ja*(4 + — J and V is the initial velocity of the particle.
2683. A circular disc capable of motion about a vertical axis
through its centre normal to its plane is set in motion with angular
velocity Q, and at a given point of it is placed freely a rough uniform
sphere : prove the equations of motion
dO
|r«^ + ifc*«-*«0,
cTr
df
fde\^ 2 ^ A
r, being the polar co-ordinates of the point of contact at the time ty
measured from the centre of the disc, o> the angular velocity of the diso,
h the initial value of r and ^inJ^ = 7^, where m, p are the masses of the
sphere and disc and c the radius of the disc.
[These equations are all satisfied by
2684. A circular disc lies fiat on a smooth horizontal table, on
which it can move freely, and has wound round it a fine string carrying
a particle which is projected with a velocity V from a point of the disc
in a direction normal to the perimeter of the disc : prove that
^+l = sec^, <^ = *tan^-tfj
where 0, ^ are the angles through which the string and the disc have
turned at a time t, a is the radius, and 2ik'= 3 + — , m, /> being the masses
P
of the disc and particle.
2685. Two eqiial circular discs lying flat on a smooth horizontal
table are connected by a fine string coiled round each, which is wound up
until the discs are in contact with each other and are both on the same
side of the tangent string : one of the discs has its centre fixed and can
move freely about it, and the other is projected with a velocity V at right
angles to the tangent string : prove that after a time t either disc will
DTMAMICS OF A BIQID BODT.
/i >"'■ 1
have turned through an angle »/ 1 + ^-f - 1 and the stri
tamed through an angle —■ tan"' —j^ i where a iq the rad
diec
2686. A smooth straight tube of length 2a and luaaa -.
horizontal talile, contains a particle, masti p, which just fita i
ia aet in motion by a blow at right anjfles to the tube : pro
r being the distance of the particle from the mid-poiut of tl
the tube has turned through an angle &, c the initial value
--■(-=)■
2667. A circular disc of maxa tn and diameter d cai
smooth horizontal plane about a fixed point A in its peri
fine string is wound mund it carrying a particle of mass p
ia initially projected from the diac at the other end of
through A with velocity V normal to the perimeter am
tiien at rest : prove that the aneular velocity of the strin
when the length unwound ia that which initially subten
angle B such that
8;>{f»tanS+I)co«'tf+3m = 0;
and that the angular velocity of the disc is then
J(-»««»oo.«)-..
2688. A. rough (Sphere of radius a moves on the conct
a vertical cylinder of revolution of rtkdius a + b, and the <
sphere is initially moving horizontally with a velocity V
the depth of the centre below its iuitial position after a timi
(l-cosn(), where 7b'n'=2V';
also prove that, in order that perfect rolling may be i
coefficient of fiiction must not be less than
2689. A cylinder of revolution is fixed with ite ax
and a rough sphere ia projected so as to move ia cents
cyliuder, being initinlly in it» lowest poaitinu with its ce
horizontally in a diroction which makes «n angle a with
the cylinder : prove that, for the sphere to reach the h
the iuitial velocity must not be less than
ii a + 6 being the radii of tlie npber« and cjliuder.
DTNAiaCS OF A RIGID BODY. 461
[The equations of motion are
dz
^=Fco8aco8(«^^?),
ocD = F ^^ cos a sin (<^ J^) ;
where z is tlie distance described by the centre of the sphere parallel to
the axis, ^ the angle through which the common normal has turned,
and (I) the angular velocity of the sphere about that normal, after a
time t.]
2690. A sphere, radius a, is in motion on the surface of a cylinder
of revolution of radius a + b whose axis makes an angle a with the
vertical and is initially in contact with the lowest generator, its centre
moving in a direction perpendicular to the generator with such a
velocity that the sphere just makes complete revolutions : prove the
equations of motion
7(^y = |sina(17 + 10cos^),
d<a dz d4>
^di'^di H'
^dzV
(n9\
-T-j +2aV= 10p«co8aj
% being the distance described by the centre of the sphere parallel to
the axis of the cylinder, ^ the angle through which the common normal
has turned, and co the angular velocity about the normal, after a
time t,
2691. A rough sphere of radius a rolls in a spherical bowl of
radius a + 5, the centre of the sphere being initially at the same height
as the centre of the bowl and moving horizontally with velocity F:
prove that, if ^ be the angle which the common normal makes with
the vertical, and <^ the angle through which the vertical plane con-
taining the normal has turned at the end of a time t,
and, if i?, ^, iS" be the reactions at the point of contact along the
common normal, alons; the tangent which lies in the same vertical
plane with the common normal, and at right angles to both these
directions, that
also, if cdp CO,, 01, be the angular velocities of the sphere about these
three directions,
dS
462 DYNAMICS OF A RIGID BODY.
2692. A rough sphere of radius a rolls in a spherical bowl of
radius a + 6 in a state of steady motion, the normal making an angle a
with the vertical : prove that the time of small oscillations about
this position is
'V
76 cos a
5^ (1 + 3 cos* a) '
HYDROSTATICS.
[In the questions under this head, a fliiid is supposed to be uniform,
heavy, and incompressible, unless otherwise stated : and all cones,
cylinders, paraboloids, &c are supposed to be surfaces oi reyolution
and their bases circles.]
2693. A cylinder is filled with equal volumes of n different fluids
which do not mix ; the density of the uppermost is p, of the next 2/>,
and so on, that of the lowest being np : prove that the mean pressures
on the corresponding portions of the curve surfaces are in the ratios
1" : 2* : 3* : ... : n".
2G94. A hollow cylinder containing a weight W of fluid is held
so that its axis makes an angle a with the horizon : prove that the
resultant pressure on its curve surface is IF* cos a in a direction nia.king
an angle a with the vertical
2695. Equal volumes of three fluids are mixed and the mixture
separated into three parts; to each of these parts is then added its
own volume of one of the original fluids, and the densities of the
mixtures so formed are in the ratios 3:4:5: prove that the densities
of the fluids are as 1 : 2 : 3.
2696. A thin tube in the form of an equilateral triangle is filled
with equal volumes of three fluids which do not mix and held with
its plane vertical : prove that the straight lines joining the common
surfaces of the fluids form an equilateral triangle whose sides are in
fixed directions ; and that, if the densities be in ^. P., the straight line
joining the surfaces of the fluid of mean density will be always vertical
2697. A thin tube in the form of a square is filled with equal
volumes of four fluids which do not mix, whose densities are p^ p^ p^ p^,
and held with its plane vertical; straight lines are drawn joining
adjacent points where two fluids meet so as to form another square :
prove that, if P| + p^ = p. + Pa* ^^® diagonals of this square will be vertical
and horizontal respectively ; but, if p^ = p, and p^^p ^ every position of
the fluids will be one of equilibrium.
464 HYDROSTATICS.
2698. A fine tube in the form of a regular polygon of n
filled with equal Yolumes of n different fluids wluch do not mix and
held with it.s plane vertical : prove that the sides of the polygon formed
by joining adjacent points where two fluids meet will have its sides
in fixed directions ; and, if the densities of the fluids satisfy two certain
conditions, every position will be one of equilibrium.
[These conditions may be written
Pj cos a + pj cos 2a + ... + p^ cos wa = 0,
Pj sin a + pj sin 2a + . . . + p^ sin ?mi = 0,
where wa=27r.]
2699. A circular tube of fine uniform bore is half filled with equal
volumes of four fluids which do not mix and whose densities are as
1 : 4 : 8 : 7, and held with its plane vei-tical : prove that the diameter
joining the free surfaces will make an angle tan'' 2 with the verticaL
2700. A triangular lamina ABC, right-angled at (7, is attached to a
string at A and rests with the side AC vertical and half its length
immersed in fluid : prove that the density of the fluid is to that of the
lamina as 8 : 7.
2701. A lamina in the form of an equilateral triangle, 8us]>ended
freely from an angular point, rests with one side vertical and another
side bisected by the surface of a fluid : prove that the density of the
lamina is to that of the fluid as 15 : 16.
2702. A hollow cone, filled with fluid, is suspended freelj from
a point in the rim of the base : prove that the total pressures on the
curve surface and on the base in the position of rest are in the ratio
1 + 11 sin* a : 1 2 sin* o,
where 2a is the vertical angle of the cone.
2703. A tube of small bore, in the form of an ellipse, is half filled
with equal volumes of two given fluids which do not mix : find the
inclination of its axes to the vertical in order that the free surfaces
of the fluids may be at the ends of the minor axis.
2704. A hemisphere is filled with fluid and the surface is divided
by horizontal planes into n portions, on each of which the whole
pressure is the same : prove that the depth of the r*^ of these planes
is to the radius as Jr : Jn,
2705. A hemisphere is just filled with fluid and the surface is
divided by horizontal planes into n portions, the whole pressures on
which are in a geometrical progression of ratio k : prove that the
depth of the r^ plane is to the radius as
HYDROSTATICS. 465
2706. A lamina ABCD in the form of a trapezium with parallel
sides AB, CD is immersed in fluid with the parallel sides horizontal :
prove that the depth of the centre of pressure below E^ the point of
intersection of AB, CD is
c c (3w«- 47W + 3) - U (1 - 7w*)
2(l + m) 3A(l + w)-2c(l-m) '
where h is the depth of Ey c the distance between AB, CD, and m
the ratio CD : AB; an<l that, when the centre of pressure is at E,
the depths of AB, CD will be as
3m*- 1 : 3-m".
2707. The co-ordinates of the centre of pressure of a triangular
lamina immersed in fluid are
x^ + x^ + x^ ' ar, -fx. + iKg '
where (a?,, y^), (ar^, y ), and (x^^, y^) are the co-ordinates of the middle
points of the sides or the lamina, the axis of y being the intersection
of the plane of the lamina with the surface of the fluid and the axis
of X any other straight line in the plane of the lamina.
2708. The co-ordinates of the centre of pressure of any lamina
immersed in fluid are
whei-e (a^, y,), (x^, y^, {x^, y^) are the co-ordinates of the comers of
a maximum triangle inscribed in an ellipse whose equation referred to
the principal axes of the lamina at its c. 6. is
J A, ijB being the principal radii of gyration. The axes to which
the centre of pressure is referred are as in the previous question.
2709. Prove the following construction for finding the centre
of pressure of a lamina always totally immersed in fluid which is
capable of motion in its own plane about its c. G. : find A, B the highest
and lowest positions of the centre of pressure, through A draw a straight
line parallel to that straight line of the lamina which is horizontal
when A is the centre of pressure, and another straight line similarly
determined through B] their point of intersection is the centre of
pressure.
2710. Prove that, when the ao. is fixed below the surface of s
fluid and the lamina move about the c. G. in its own plane, the centre of
pressure describes a circle in space and in the lamina the ellipse whoae
equation referred to the principal axes is
where J A, yJB are the principal radii of gyration, and c the depth
w. P. 30
466 HYDROSTATICS.
of the c.G. measured in the plane of the lamina below the surface
of the fluid : also that, of the four points in which the circle and ellipse
intersect, the centre of pressure is the lowest and the other three are
comers of a triangle whose sides touch a fixed circle with its centre
at the CO.
2711. A lamina totally immersed in fluid moves in its own plane
so that the centre of pressure is a point fixed in space : prove that the
path of the c. o. is the curve whose equation is, referred to the centre of
pressure as origin,
of {(x- ay + 2/'} -^ X {x-aJiA + B) + AB = ;
where J A, JB are the principal radii of gyration at the ao., and
a the depth (measured in the plane of the lamina) of the centre of
pressure below the surface of the fluid.
2712. A I'ectangular lamina A BCD is immersed in fluid with the
side AB in the surface of the fluid; a point P is taken in CD and the
lamina divided into two pai'ts by the straight line APi determine for
what position of P the distance between the centres of pressure of the
two parts is a maximum.
[If the sides AB^ BC be denoted by a, 6, and DP hj os, the distance
will be a maximimi when 27aj: = 4 (9a*— 26*), and since x must be
positive and less than a, there will be no maximum unless 6 : a lie
between 3 : 2 ^2 and 3 : ^2.]
2713. A lamina in the form of the sector of a circle is immersed in
fluid with the centre of the circle in the surface: prove that the co-
ordinates of its centre of pressure are
Zi a- sin a cos a cos 20 3a ^
"A oT,rrT;,r^ » -^ COS a COS ^,
o sm a sm u 4
where the axis of ^ is in the surface of the fluid, and O — a^ $-^a sre the
angles which the bounding radii make with the axis of y.
2714. A lamina, bounded by the epicycloid generated by a circle of
radius a rolling on a circle of radius 2a, is placed in fluid with the cusp
line in the surf Jice : prove that the co-ordinates of the centre of pressure
of half the part immersed are l ^l^fray ^-|« ; and those of the oentie of
pressure of the part lying outside the fixed circle are yxff'"*> T^i ^*
axis of y lying in the surface.
2715. An isosceles triangle is immersed with its axis vertical and
its base in the surface of a fluid : pn>ve that the resultant pressure on
the area intercepted between any two horizontal planes acts through the
c. G. of that portion of the volume of a sphere, described with the axis
for diameter, which is intercepted between the planes.
2716. A conical shell is placed with its vertex upwards on a hori-
zontal table and fluid is poured in through a small hole in the vertex;
the cone begins to rise when the weight of the fluid poured in is equal
to its own weight : prove that this weight bears to the weight of fluid
which would fill the cone the ratio 9-3 ^/3 : 4.
HYDROSTATICS. 467
2717. A parabolic lamina bounded by a double ordinate perpen-
dicular to the axis floats in fluid with its focus in the sui^face and its
axis inclined at an angle tan"' 2 to the vertical : prove that the density
of the fluid is eight times the density of the lamina, and that the length
of the axis bears to the latus rectum the ratio 15:4.
2718. A lamina in the form of an isosceles triangle floats in a fluid
with its plane vertical and (1) its base totally out of the fluid, (2) its
ba^^e totally immersed, and its axis in these two positions makes angles
$, <l> with the vertical : prove that
sin* a (sec* + sec* <^) = 4 cos a,
and that both positions will not be possible unless cos a > J2 — 1 ; where
a is the vertical angle.
2719. A cone with its axis vertical and vertex downwards is filled
with two fluids which do not mix and their common surface cuts off one-
fourth of the axis from the vertex : prove that, if the whole pressures of
the fluids on the curve surfaces be equal, their densities will be as 45 : 1 .
2720. A right cone just filled with fluid is attached to a fixed point
by a fine extensible string attached to the vertex, and initially the string
is of its natural length and the cone at rest : prove that the pressure of
the fluid on the base of the cone in the lowest position is six times the
weight of the fluid.
2721. A barometer stands at 29*88 inches and the thermometer ia
at tlie dew-point ; a barometer and a cup of water are placed under a
receiver from which the air is removed and the barometer then stands
at '36 of an inch : find the space that would be occupietl by a given
volume of the atmosphere if it were deprived of its vapour without
changing its pressure or temperature.
2722. In Hawksbee's air-pump, the machine is kept at rest when
the n}^ stroke is half completed; find the difference of the tensions of
the two piston rods.
2723. In Smeaton*s air-pump, during the w*** stroke, find the poai-
tion of the piston at that instant of time when the upper valve begins
to open.
2724. Tlie volumes of the receiver and barrel of an air-pump are
Ay B ; p, a are the densities of atmospheric air and of the air in the
receiver respectively, and IT the atmospheric pressure: prove that the
work done in slowly raising the piston through one stroke is
n{^-^fiog(i.£)},
gravity being neglected.
2725. A portion of a cone cut off by a plane through the axis and
two planes perpendicular to the axis is immersed in fluid in such a
manner that the axis of the cone is vertical and the vertex in the sur-
face : prove that the resultant horizontal pressure on the curve surface
passes through tbe c. o. of the body immersed.
30—2
468 HYDROSTATICS.
2726. Assuming that the temperature of the atmosphere in aaoend-
ing from the earth's surface decreases slowly hj an amount proportional
to the height ascended, prove that the equation connecting the pressure
p and the density p at any height will be of the form p = kp^^^f where m
is a small fraction.
2727. A cylinder floats in fluid with its axis inclined at an angle
tan~^ f to the vertical, its upper circidar boundary just out of the fluid
and the lower one completely immersed: prove that the length of the
axis is nine-eighths of the diameter of the generating circle.
2728. Two equal and similar rods AB, BC, fixed at an angle a at J?,
rest in a fluid of twice the specific gravity with the angle B out of the
fluid, and the axis of the system makes an angle $ with the horizon:
prove that
cos 20=2 - sec a.
2729. A uniform solid tetrahedron has each edge equal to the
opposite edge: prove that it can float partly immeraed in fluid y^ith. any
two opposite edges horizontaL
2730. A lamina in the form of a parabola bounded by a double
ordinate rests in liquid with its plane vertical, its focus in the surface of
the fluid, and its base just out of the fluid: prove that the ratio of the
densities of the solid and liquid is 1 : (1 + cos a)', where a is the angle
given by the equation 2 cos 2a = 3 (1 — cos a).
2731. A cone of density p floats with a generator vertical in a fluid
of density o-, the base being just out of the fluid: prove that, if 2a be
the vertical angle,
^ = (cos 2a)*,
and that the length of the vertical side immersed is to the length of the
axis as cos 2a : cos a.
2732. A cone is moveable about its vertex, which is fixed at a given
distance c below the surface of a liquid, and rests with its axis, h in
length, inclined at an angle $ to the vertical and its base completely out
of the fluid : prove that
cos cos" a (rh*
(cos"^-sin'a)* P«* '
2a being the vertical angle and p, a- the densities of the liquid and cone.
Also prove that this position will be stable, but that it cannot exist
unless ah* cos* a > pc*,
2733. A homogeneous solid in the form of a cone rests with its
axis vertical and its vertex at a depth c below the surface of a liquid
whose density varies as the depth : prove that the condition for stable
equilibrium is that cos a < ^ ^ , where h is the length of the axis and
2a the vertical angle. Prove also that this is the condition that no
T)Ositions of equilibrium in which the axis is not vertical can exist.
HYDROSTATICS. 469
2734. An elliptic tube half full of liquid revolves about a fixed
vertical axis in its plane with angular velocity co : prove that the angle
which the straight line joining the free surfaces of the fluid makes with
the vertical will be tan"* (--,], where p is the distance of the axis
from the centre of the ellipse.
2735. A hollow cone very nearly filled with liquid revolves uniformly
about a vertical generator : prove that the pressure on the base is
f IT i — (1+5 cos' a) tan a + 8 sin aj- ;
where W is the weight of the fluid, 2a the vertical angle, a the radius
of the base, and a> the angular velocity.
2736. A hollow cone very nearly filled with liquid revolves about a
horizontal generator with uniform angular velocity a> : prove that the
whole pressure on the base in its highest or lowest position is
^ pa*io' r 1 + - J cos a + 5 cos* a j;
where a ia the radius of the base and 2a the vertical angle.
2737. A cone the length of whose axis is h and the radius of the
base a floats in liquid with ^^j of its volume below the surface : prove
that, when the liquid revolves about the axis of the cone with angular
velocity ^y/ 1 S | ^ , the cone will float with the length ^ or | ^ of its
axis immersed ; and investigate which of the two positions is stable.
2738. A sphere of radius a floats in liquid, which is revolving with
uniform angular velocity o> about a vertical axis, with its centre at the
vertex of the free surface of the liquid : prove that
4 (/?* + 4a*) (a -pq) = a (/? + 4a^)*,'
where p<o' = 2g and 1 + ^ : 2 is the ratio of the densities of the sphere
and liquid.
2739. A hollow paraboloid whose axis is equal to the latus rectum
is placed with its axis vertical and vertex upwards and contains seven-
eighths of its volume of liquid : find the angular velocity with which this
liquid must revolve about the axis in order that its free surface may be
confocal with the paraboloid ; and prove that in this case the pressure
on the base is greater than when the liquid was at rest in the ratio
2^/2 : 2^2-1.
2740. A liquid is acted on by two central forces, each varying as
the distance from a fixed point and equal at equal distances from those
points, one attractive and one repulsive : prove that the surfaces of equal
pressure are planes.
470 HYDROSTATICS.
2741. A liquid is at rest under the action of two forces teDding to
two filled points and each varying inversely as the square of the distaDce,
one attractive and one repulsive : prove that one surface of equal
pressure is a sphere.
2742. A mass of elastic fluid is confined within a hollow sphere
and repelled from the centre of the sphere by a force /i -r- distance :
prove that the whole pressure on the sphere bears to the whole pressure
which would be exerted if no such force acted the ratio 3k + /jl : 3k;
where p = kp ia the relation between the pressure and density.
2743. A quantity of liquid not acted on by gravity just fills a
hollow sphere and is repelled from a point on the surface of the sphere
by a force equal to /x (distance) ; the liquid revolves about the diameter
turough the centre of force with uniform angular velocity w : find the
whole pressure on the sphere, and prove that, if when the angular
velocity is diminished one half the pressure is also diminished one half,
CD* = 6/x.
2744. All space being supposed filled with an elastic fluid the total
mass of which is known, which is attracted to a given point by a force
varying as the distance ; find the pressure at any point.
2745. Water is contained in a vessel having a horizontal base and
a cone is suppoi-ted partly by the water and partly by the base on which
the vertex rests : prove that, for stable equilibrium, the depth of the
fluid must be greater than h Jm cos a, vi* being the specific gravity of
the cone, h the length of its axis, and 2a the vertical angle.
2746. A solid paraboloid is divided into two parts by a plane
through the axis and the parts united by a hinge at the vertex ; the
system is placed in liquid with its axis vertical and vertex downwards
and floats without separation of the parts : prove that the ratio of the
density of the solid to that of the liquid must be greater than x^^ w^here
X is given by the equation
3/^=.7/(l-a;),
and Z, h are the lengths of the latus rectum and axis respectively.
2747. A cone is floating with its axis vertical in a fluid whose
density varies as the depth : prove that, for stable equilibrium,
cos' a < -J
7^
where 2a is the vertical angle, p the density of the cone, and cr the
density of the fluid at a depth equal to the height of the cone.
2748. A uniform rod rests in an oblique position with half its
length immersed in liquid and can turn freely about a point in its length
whose distance from the lower end is one-sixth of the length : compare
the densities of the rod and liquid, and pix)ve that the equilibrium is
stable.
HYDROSTATICS. 471
2749. A uniform rod is moveable about one end, which is iSxed below
the surface of a liquid and, when slightly displaced from its highest
position, it sinks until just immersed : prove that, when at rest in the
highest position, the pressure on the point of support was zero.
2750. Two equal uniform rods A By BC, freely jointed at B^ are
capable of motion about A, which is fixed at a given depth below the
surface of a liquid : find the position in which both rods rest partly
immersed, and prove that for such a position to be possible the density
of the rods must not exceed one-third the density of the liquid.
2751. A hemisphere, a point in the rim of whose base is attached
to a fixed point by a fine string, rests with the centre of the sphere in
the surface of the liquid and the base inclined at an angle a to the
horizon : prove that
p 1 6 (ir — a) cos a — Stt sin a
(T 2ir (8 cos a - 3 sin a) '
where p, cr are the densities of the solid and liquid.
2752. A cone is floating with its axis vertical and vertex down-
wards in fluid and - th of its axis is immersed ; a weight equal to the
weight of the cone is placed upon the base and the cone then sinks until
just totally immersed before lising : prove that
w' + 7t* + n = 7.
2753. A hollow cylinder with its axis vertical contains liquid, and
a solid in the form of an ellipsoid of revolution is allowed to sink
freely in the liquid with axis also vertical : the solid just tits into the
cylinder and sinks until just immersed before rising : prove that its
density is one-half that of the liquid.
2754. A hollow cylinder with its axis vertical contains liquid and a
solid cylinder is allowed to sink freely in it with axis also vertical : prove
that, if it sink until just immei-sed before rising, the dennities of the
solid and liquid must be in the ratio 1 : 2. Also, if the density of the
liquid initially vary as the depth, prove that the density of the solid
must be the initial density of the liquid at a depth of one-sixth of the
whole distance sunk by the solid.
2755. A hollow cylinder with vertical axis contains a quantity of
liquid and a solid of revolution (of the curve y* a a;" about the vertical
axis of x) is allowed to sink in the li({uid, starting when its vertex is in
the surface and coming to instantaneous rest when just immersed :
prove that the density of the solid must bear to the density of the liquid
the ratio 1 : 2 (n -h 2); and that, if a similar solid be allowed to sink in an
unlimited mass of liquid of half the density of the former, this solid will
also come to rest when just immersed.
2756. A cylinder whose axis is vertical contains a quantity of fluid
whose density varies as the depth and into this is allowed to sink a solid
of revolution whose base is equal to that of the cylinder, which sinks
until just immersed before rising; in the lowest position of this solid
472 HYDROSTATICS.
the density of the surrounding fluid varies as the n^ po-wer of the depth :
])rove that the weight of the solid is to the weight of the displaced fluid
as n - 1 : 3 (27» +1), whereas if the solid can rest in this p06iti<m the
i*atio must be n ~ 1 : n + 1. Also prove that the generating carve of Uw
solid will be
^ a- V h) '
where a is the radius of the base and h the height.
[If n = 2 the solid is a paraboloid, if n — 3, an ellipsoid.]
2757. A hollow cylinder with vertical axis contains a quantity of
fluid whose density vaiies as the depth and into it is allowed to sink
slowly, with vertex downwards, a solid cone the radius of whose base is
equal to the radius of the cylinder ; the cone rests when just immersed :
prove that the density of the cone is equal to the initial density of the
fluid at a depth equal to one-twelfth of the length of the axis of the
cone. If the cone bo allowed to sink freely into the fluid, starting with
its vertex at the surface and just sinking until totally immersed, the
density of the cone will be to the density of the fluid at the vertex of
the cone in its lowest position as 1 : 30.
2758. A tube of fine bore whose plane is vertical contains a
quantity of fluid which occupies a given length of the tube ; a given
heavy particle just fitting the tube is let fall through a given vertical
height : find the impulsive pressure at any point of the fluid ; and
prove that the whole kinetic energy after the impact bears to the
kinetic energy dissipated the ratio of the mass of the particle to the wnuwy
of the fluid.
[If rriy m' be the masses of the particle and fluid, V the velocity of
the particle just before impact, the impulsive pressure at a point whose
distance along the arc from the free end is 8 will be > 7 , where / is
m + ffh C
the whole length of arc occupied by the fluid]
2759. A flexible inextensible envelope when filled with fluid has
the form of a paraboloid whose axis is vertical and vertex downwards
and whose altitude is five-eighths of the latus rectum : prove that the
tension of the envelope along the meridian will be greatest at points
IT
where the tangent makes an angle -r with the vertical.
[In general, if 4a be the latus rectum and h the altitude, the tension
per unit of length at a point where the tangent makes an angle $ with
the vertical will be -^ ( "•~'a ~'^~h) » ^^®^ ^ ^ *^® specific gravity
of the fluid.]
2760. Fluid vrithout weight is contained in a thin flexible envelope
in the form of a surface of revolution and the tensions of the envelope
at any point along and perpendicular to the meridian are equal : prove
that the surface is a sphere.
HYDROSTATICS. 473
2761. A quantity of homogeneous fluid is contained between two
pai*allel planes and is in equilibrium in the form of a cylinder of
radius b under a pressure m ; that portion of the fluid which lies within
a distance a of the axis being suddenly annihilated, prove that the
initial pressure, at a point whose distance from the axis is r, is
«^logg)^logQ.
2762. A thin hollow cylinder of length A, closed at one end and
fitted with an air-tight piston, is placed mouth downwards in fluid ; the
weight of the piston is equal to that of the cylinder, the height of a
cylinder of equal weight and radius formed of the fluid is a, the height
of fluid which meaaui-es the atmospheric pressure is c, and the air
enclosed in the cylinder would just fill it at atmosphenc density : prove
that, for small vertical oscillations, the distances of the piston and of
the top of the cylinder from their respective positions of equilibrium are
of the form A sin (Xt + a)+£ sin (jU + )3), X, /x being the positive roots of
the equation
a^ ' or
and 711 = (a + c)* -r ch.
2763. A filament of liquid PQR is in motion in a fixed tulie of
small uniform bore which lies in a vertical plane with its concavity
always upwards ; on the horizontal ordinates to F, Q, H at any instant
are taken points p, q, r, whose distances from the vertical axis of abscissas
are equal to the arcs measured to P, Q, R from a fixed point of the tube :
prove that the fluid pressure at Q is always proportional to the area of
the triangle pqr,
2764. A centre of force, attracting inversely as the square of the
distance, is at the centre of a spherical cavity within an infinite mass of
liquid, the pressure in which at an infinite distance is w, and is such that
the work done by this pressure on a unit of area through a unit of
length is one half the work done by the attractive force on a pai-ticle
whose mass is that of a unit of volume of the liquid as it moves
from infinity to the initial boundary of the cavity : prove that the time
of filling up the cavity will be ira ^ — {2 - (f)'}; where a is the initial
radius of the cavity and p the density of the fluid.
GEOMETKICAL OPTICS.
2766. Three plane mirrors are placed so that their intersections are
parallel to each other and the section made by a plane perpendicular to
their intersections is an acute-angled triangle; a ray proceeding from a
certain point of this plane after one reflexion at each miiTor proceeds
on its original course : prove that the point must lie on the perimeter of
a certain triangle.
2766. In the last question a ray starting from any point after one
reflexion at each mirror proceeds in a direction parallel to its original
direction: prove that after another reflexion at each mirror it will
proceed on its original path, and that the whole length of its path
between the first and third reflexions at any mirror is constant and
equal to twice the perimeter of the triangle formed by joining the feet
of the perpendiculars.
2767. A ray of light whose direction touches a conicoid is reflected
at any confocal conicoid : prove that the reflected ray also touches the
first conicoid.
2768. In a hollow ellipsoidal shell small polished grooves are made
coinciding with one series of circular sections and a bright point jilaced
at one of the umbilics in which the series terminates: prove that the
locus of the bright points seen by an eye in the op}X)8ite umbilic is
a central section of the ellipsoid, and that the whole length of the path
of any ray by which a bright point in seen is constant.
2769. A ray proceeding from a po^nt on the circumference of a
circle is reflected n times at the circle: prove that its point of inter-
section with the consecutive ray similarly reflected is at a di&tance from
the centre equal to ^ =-^1 + 4n(n+ l)sin*^, where a is the radius
and the angle of incidence of the ray: also prove that the caustic
biuface generated by such rays is the surface of revolution generated by
an epicycloid in which the fixed circle has the radius ^ = and the
moving circle the radius ^ r .
2770. A ray of light is reflected at two plane mirrors, its direction
re incidence being parallel to the plane bisecting the angle between
nirrors and making an angle $ with their line of intersection : prove
the deviation is 2 sin~* (sin sin 2a), where 2a is the angle between
GEOMETRICAL OPTICJS. 475
the mirrors. More generally, if D^ be the deviation after r successive
reflexions,
cos \ -0,^_j = sin ^ sin {2n - 1 a — <^), sin J D^ = sin 6 sin 2/ui,
where <^ is the angle which a plane through the intersection of the
mirrors parallel to the incident ray makes with the plane bisecting the
angle between the mirrors.
2771. Two prisms of equal refracting angles are placed with one
face of each in contact and their other faces parallel and a ray passes
through the combination in a principal plane : prove that the deviation
will be from the edge of the denser prism.
2772. The radii of the bounding surfaces of a lens are r, 8, and its
thickness is (l+-j(«-r): prove that all rays incident on the lens
from a certain point will pass through without aberration but also
without deviation.
2773. Prove that a concave lens can be constructed such that the
path of every ray of a pencil proceeding from a certain point after
refraction at the first surface shall pass through the centre of the lens;
that in this case there will be no aberration at the second refraction,
and that the only efioct of the lens is to throw back the origin of light
a distance (jjL — l)t, where t is the thickness of the lens.
2774. What will be the centre of a lens whose bounding sui'faces
are confocal paraboloids on a common axis] Prove that the distance
between the focal centres of such a lens is — r {a + b), 4a, 46 being the
latera recta.
2775. The path of a ray through a medium of variable density is
an arc of a circle in the plane of xi/ : prove that the refractive index at
I /x — <i\
any point (cc, y) must be /( ], where y is an arbitrary
function and (a, b) the centre of the circle.
2776. A ray of light is propagated through a medium of variable
density in a plane which divides the medium symmetrically : prove that
the path is such that when described by a point witli velocity always
proportional to /jl, the index of refraction, the accelerations of the point
parallel to the (rectangular) axes of x and y will be proportional to
^' ^respectively.
2777. A ray is propagated through a medium of variable density
in a plane (xy) which divides the medium symmetrically: prove that
the projection of the radius of curvaturo at any point of the path of the
ray on the normal to the surface of equal density through the point is
''*»'"' *° I" -• \/©' +(!)'•
476 GEOMETRICAL OPTICS.
2778. A small pencil of parallel rays of white light, after trann-
mission in a principal plane through a prism, is received on a screen
whose plane is perpendicular to the direction of the pencil : prove that
the length of the spectrum will be proportional to
(ft, — fi^) sin I -f cos' D cos (i> + 1 — <^) cos <^';
where t is the refracting angle, ^, «^' the angles of incidence and in-
fraction at the first surface, and D the deviation, of the mean ray.
2779. Prove that, when a ray of white light is refracted through a
prism in a principal plane so that the dispersion of two given colours is
a minimum,
sin (3<^^ - 2t) 2 ^
siu <^ yx"
where ^' is the angle of refraction at the first surface and % the re-
f i-acting angle. Hence prove that minimum dispersion cannot co-exist
with minimum deviation.
2780. A transparent sphere is silvered at the back : prove that the
distance between the images of a speck within it foi-med (1) by one
direct refraction, (2) by one direct r^exion and one direct refraction, is
2/x a c (a - c) -f- (a + c — /nc) (/tc + a - 3c),
where a is the radius of the sphere and c the distance of the speck from
the centre towards the silvered sida
2781. The focal length of the object-glass of an Astronomical
Telescope is 40 inches, and the focal lengths of four convex lenses
forming an erecting eye-piece are respectively |, ^, f , \ inches, the
intervals being the first and second, and the second and thiixi being
1 inch and \ inch respectively: find the position of the eye-lens and
the magnifying power when the instrument is in adjustment; and trace
the course of a pencil from a distant object through the instrument.
[The eye-lens must be at a distance of 41*5 inches fix)m the object-
glass.]
2782. Two thin lenses of focal lengths /, ,/, are on a common axis
and separated by an interval a; the axis of an excentric pencil, before
incidence, cuts the axis of the lenses at a distance d from the first lens:
prove that, if J" be the focal length of the equivalent single lens.
11 1^ a /2 1\
2783. The focal length F o^ & single lens equivalent to a system of
three lenses of focal lengths /,, f^, f^ separated by intervals a, 6, for an
excentrical pencil parallel to the axis, is given by the equation
L-1 i 1 ^(\
1\ h_(\_ \\ ah
2784. Prove that the magnifying power of a combination of three
OSes of focal lengths /,, /,, /, on a common axis at intervals <i^ 6 will
\ independent of the position of the object, if
SPHERICAL TRIGONOMETRY AND
ASTRONOMY.
2785. In a spherical triangle ABC^ a = 6 = ^, c = o' P^o^® *^* *^o
spherical excess is cos~*^.
2786. In an equilateral spherical triangle ABC^ a, 6, c are the
middle points of the sides : prove that 2 sin -^ = tan —^ ,
2787. In an equilateral spherical triangle whose sides are each a
a A
and angles A, prove that 2 cos ^ sin -^ = 1.
2788. Each of the sides of a spherical triangle ABC is a quadrant,
and P is any point on the sphere : prove that
cos' il/* + cos'^/* + cos* (7P = 1 ,
cos -iP COB j5P cos CP + cot BPC cot CFA cot AFB = 0,
and that tan BCP tanCAPtAnABP^^l;
the angles BPC, CPA, APB being measured so that their sum is always
four right angles, and sign regarded in the third equation.
2789. Each of the sides of a spherical triangle ABC is a quadrant
and P is any other point on the sphere within the triangle; another
spherical triangle is described with sides equal to 2-4P, 2^P, Wl*
respectively : prove that the area of the latter triangle is twice that of
the former.
2790. A spherical triangle ABC is equal and similar to its polar
triangle: prove that
sec* -4 + 8ec*-5 + Bec*C+ 28ec^ sec JS sec (7 =■ 1.
2791. Solve a spherical triangle in which the side a, the sum or
the difference of the other two sides h, e, and the spherical excess J^,
are given.
478 SPRERICAL TRIGOirOlfErKT AXD 1STB090MT.
[Either of the eqiuktioiM
(a h-i-cK/ a h-e\
1 + con - =.
2 af h — c 6 + c>
«»2
1 ^
1 - cos -r =
( h-c a\f a h*-f\
2 a f h — e b + &^
006
Hiifllice to determine 6 * c when a, ^ and 5 7 c are given.]
2792. Hie suni of the sides of a spherical triangle being given,
prove that the area is greatest when the triangle is equilateral.
2793. In a spherical triangle ABC^ a + 6 + c = ir, prove that
B C
cos^+cosi5+cosC=l, cosa = tan ^ tan - ,
A BO
and that sin -5- = cos — cos ^ sin a.
2794. In a spherical triangle A +B+ C=2ir: prove that
cos A + cot - cot 7: = 0, <fec.
2 2 '
2795. In a si)herical triangle ABC, A^B-hC: prove that
sm' ^ = sm ^ + sm* ^ .
2790. The polo of the small circle circumscribing a spherical
triangle ABC is : prove that
.,6 .,c , ^a ^ , h . e BOG
sur ;r + sin* - - sm" = 2 sin ^ sm ^ cos — ^— ;
J J 2 ALL
and that, if /* be any }K)int on this circle,
, a , PA . h . PB . e , PC ^
Bin . Bin - - + Bin ^ sm -^ + sm jr sm -5- = 0,
J J L L L L
that arc of the thi'oe PA, PB, PC being reckoned negative which
oromt^M one of the sides.
2797. Pn>vo that
sin * > oi>a o sin (* - n) + cos b sin (# - 6) + cos c sin (« — r),
and Ci>s •}? < civi .^ ci>a(-i?-<^) +coa^co8(iS-J?) + co8Cco8(5^-C),
whoro IS ^ aro the sides and A^ B^ C the angles of a spherical
iriaugK\ and
2«=(i + 6 + c, ^S^A -^B-^C.
379^ Tho centre of the sphere on which lies a spherical triangle
<.4 Z^"* ia iK aiui fv>rcea act along OAy OB^ OC proportioiial to sin a» sin ^
ain e t^^yn^xiytfy i proT« that their resoltant acts through the pole of
Ui« cirol« ABi\
SPHERICAL TRTGONOMETRY AND ASTRONOMY. 479
2799. The gi'eat circle drawn through a comer of a spherical
triangle perpendicular to the opposite side divides the angle into parts
whose cosines are as the cotangents of the adjacent sides, and divides
the opposite side into parts whose sines are as the cotangents of the
adjacent augles.
2800. Prove that a spherical triangle can be equal and similar
to its polar triangle only when coincident with it, each side being a
quadrant.
2801. In a spherical triangle A ■i-a = ir: prove that
tang- |)tan(^-0 = *tang-|)=-. tang- f)tang-^).
2802. Prove the formula
a-\- h + c cos A + cos B + cos (7-1
cos
2 ^ , A , B , C
4 sm 2 sin -^ sin ^
2803. Two sides of a spherical trim^le are given in position, and
the included angle is equal to the spherical excess : prove that the
middle point of the third side is fixed.
2804. Two sides of a spherical triangle are given in position,
including an angle 2a, and the spherical excess is 2fi ; on the great
circle bisecting the given angle are taken two points S, *S", such that
cos SA = - cos S^A = tan (a — P) cot a :
prove that, if P be the middle point of the base,
sin a sin J {SP + S'P) = sin (a - ^).
2805. Two fixed points A, B are taken on a sphere, and P is
any point on a fixed small circle of which A is pole; the great
cii-cle PB meets the great circle of which A is pole in Q : prove that
the ratio cos PQ : cos BQ will be constant.
2806. Prove that, when the Sun rises in the N.E. at a place in
latitude /, the hour angle at sunrise is cot"* (- sin l),
2807. In latitude 45** the observed time of transit of a star in
the equator is unaffected by the combined errors of level and of devia
tion in a transit : prove that these errors must be very nearly equal
to each other.
2808. The ratio of the radius of the Earth's orbit to that of an
inferior planet is m : 1, and the ratio of their motions in longitude
(considered uniform) is n : 1: prove that the elongation of the planet
as seen from the Earth when the planet is stationary is
tan"* . / — I — r- .
480 SPHERICAL TRIGONOMETRY AND ASTRONOMY.
2809. Tlie mean motions in longitude of the Earth and of an inferior
planet are m, rti\ and the difference of their longitudes is ^ : proT'e
that the planet's geocentric longitude is increasing at the rate
/ 'xkt \ '\\ (w*m')* - {ni^ - (mm'Y + nfi\ cos d>
{mm y (»r + m '•) ^ ^-r — ^ —^ — *— — — ;
m* - 2 {mm'y cos <^ + w'*
and verify that the mean value of this during a synodic period is fi».
2810. The maximum value of the aberration in declination of a
given star is
20" -5 ^1- (cos S cos o) + sin S sin cd cos a)* ;
where a, 8 are the riglit ascension and declination of the star, and a> the
obliquity of the ecliptic.
2811. Prove that all stars whose aberration in right ascension
is a maximum at the same time that the aberration in declination
vanishes lie either on a quadric cone whose circular sections are parallel
to the ecliptic and equator, or on the solstitial colure.
2812. The right ascensions and declinations of two stars are a, a';
8, 8' respectively, and A is the Sun's nght ascension at a time when
the aberrations in declination of both stars vanish : prove that
. tan 8 sin a - tan 8' sin a'
tan A = - — K 7 — Ki > .
tan cos a — tan o cos a
2813. In the Heliostat, if the diurnal change of the Sun's declina-
tion be neglected, the normal to the mirror, and the intersection of
the plane of the mirror with the plane of reflexion will each tnoe
out a quadric cone whose circular sections are perpendicular to the
axis of the Earth and to the reflected ray.
2814. The latitude of a place has been determined by observation
of two zenith distances of the Sun and the time between them and
each observed distance was too great by the same small quantity As :
prove that the consequent error in the latitude is
A« cos (a + a') -r cos (a — a') ;
where 2a, 2a' are the azimuths at the times of observation.
2815. The hour angle is determined by observation of turo aeenitb
distances of a known star and the time between ; each observed zenith
distance is too great by Az : prove that the consequent error in hour
angle is
Az sin (a + a) -r- cos I cos (a — a');
where I is the latitude of the place and 2a, 2a' the azimuths of the star
at the two observations.
[See a paper by JVIr Walton, Quarterly Journal, Vol. v., page 289.1
THE END.
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13 SCIENTIFIC CATALOGUE.
Huxley (Professor)— «waVmi«/.
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PHYSICAL SCIENCE. 13
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l6 SCIENTIFIC CA TALOGUE.
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! F.R.S., and P. G. Tait, M.A. Sixth Edition. Crown 8vo. 6j.
I " The book is one which well deserves the attention of thoughtful and
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PHYSICAL SCIENCE. 19
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the British Association. Crown 8vo. 9/.
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Henry Tanner, F.C.S., Professor of Agricultural Science,
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IS,
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THE DEPTHS OF THE SEA : An Account of the General
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" Lightning " during the Summers of 1868*69 and 70, under the
sdentific direction of Dr. Carpenter, F.R.S., J. Gwyn Jeffreys,
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who wish to be pleasantly introduced to the subject^ and rightly
to appreciate the news which artvues from time to time from the
* Cnallenger,^ should not fail to seek instruction from it.**
THE VOYAGE OF THE ** CHALLENGER."— THE ATLAN-
TIC. A Preliminary account of the Exploring Voyages of H.M.S.
"Challenger," during the year 1873 "*<* ^^^ ^^^7 P*rt of 1876.
With numerous Illustrations, Coloured Maps & Charts, & Portrait
of the Author, engraved!byC. H. Jeens. 2 Vols. Medium 8vo. 42s.
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authoritative account of the general results of the expedition^ ana
B 2
20 SCIENTIFIC CATALOGUE.
Thomson— ^dnHntud:
i^af as many of the ascertained data as may he accepted Toifkcon'
fidence should speedily find their place in the general body of
scientific 1 knowledge. No one can be more competent than the
•, , r ^acfpntpllshed scicpHfic chief of the expedition to satisfy the public in
'..". I this respect, ... 77ie paper ^ prhUifi^, and especially tke numerous
illustrations^ are of the Xighest quaiity. . . . We have rarely ^ if
ever J seen more beautiful specimens of wood engramng them aHund
in this work. . . . Sir Wyvilu ThomsorCs style ts particularly
. attractive; he is easy and graceful, but vi^rous and exceedingly
,',.., happy in thf choice of language, and throughout the work4here are
.; touches which shmo that science has nc4 banished seHtimtnt from
^ his bofom."
Thudichum and Duprg.— a TREATISE ON THE
- ORIGIN, natltre; and varieties of wine.
* Being a Complete Manual of Viticaltare and CEnologyk By J. L.
ff *W. Thudichum, M.D., and August DuFRfi, Ph.D., Lectoier on
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— Wine Trade Review.
Wallace (A. R.). — Works by Alfred Russet. Wallace.
contributions to the theory of natural
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THE GEOGRAPHICAL DISTRIBUTION OF* ANIMALS,
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.. mate inductions from them. Naturalists, ought to be grateful to
. y him for having undertaken so toilsome a task. The work, indeed,
is a credit. to cul concerned — the author, the publishers, the artist —
unfortunatdy nmo no more — of the attractive illustratumt-^Uisi
but by no means least, Mr, Stanford's map-designer^
PHYSICAL SCIENCE, 21
Wallace (A. R,)^c<m/inwd.
TROPICAL NATURE I with other Essays. 8yo. lis.
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the phases of nature in them snore clearly, shortly, and impres-
sively" — Saturday Review.
Warington.— THE WEEK OF CREATION; OR, THE
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ington, Author of ** The Historic Character of the Pentateuch
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Wilson.— RELIGIO CHEMICL By the late George Wilson,
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of Edinburgh. WiUi a Vignette beautifully engraved after a
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Wilson (Daniel).— CALIBAN : a Critique on Shakespeare's
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** TTie whole volume is ^nost rich in the eloquence of thought and
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\Vright.— METALS AND THEIR CHIEF INDUSTRIAL
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turer on Chemistry in St. Mary's Hospital SchooL Extra fcap.
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22 SCIENTIFIC CATALOGUE.
SCIENCE PRIMERS FOR ELEMENTARY
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Balfour Stewart.
Introductory. By Professor Huxley, F.R.S. [Nearly ready.
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trations. New Edition. i8mo. doth. \s.
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Botany — By Sir J. D. Hooker, K.C.S.I.. C.B., F.R.S. With
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Logic — By Professor Stanley Jevons, F.R.S. New Edition.
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SCIENCE CLASS-BOOKS. 23
Elementary Science Class-books — continued,
QUESTIONS ON LOCKYER'S ELEMENTARY LESSONS
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Physiology.— LESSONS in elementary physiology.
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fessor of Natural History in the Royal School of Mines. New
Edition. Fcap. 8vo. 4;. 6d,
QUESTIONS ON HUXLEY'S PHYSIOLOGY FOR
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Botany — lessons in elementary botany. By D.
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Edition. Fcap. 8vo. 4;. 6d,
Chemistry — lessons IN elementary chemistry,
INORGANIC AND ORGANIC. By Henry E. Roscoe,
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Fcap. Svo, y, 6d,
24 SCIENTIFIC CATALOGUE^
ElemcDtary Science Class-books — ce^nued.
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KS ELEMENTARY TREATISE. By A. B.
Tfity
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QUESTIONS ON THE SAME. u. U.
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The Properties of Solid and Fluid Bodies. iSmo. y.^. Fart
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With numerous Illustrations. [In fttfaralu 11.
AfANUALS FOR STUDENTS. 25
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tions. Crown 8vo. 71. dd.
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Hooker (Dr.)— the students flora of the British
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is.dd.
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Huxley and Parke r._ELEMENTARY biology, part
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LL.D., F.R.S. , New and Revised Edition. Crown 8vo. I2J. 6d.
26 SCIENTIFIC CATALOGUE.
Manuals for Students — continued.
Oliver (Professor) first book of Indian botany.
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Parker and Bettany.—THE MORPHOLOGY OF the
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trated. Crown 8vo. lOf. (>d,
Tait — AN ELEMENTARY TREATISE ON HEAT. By Pro-
fessor Tait, F.R.S.E. Illustrated. [In th€ Press.
Thomson.-. ZOOLOGY. By Sir C. Wyville Thomson,
F.R.S. Illustrated. [In preparation,
Tylor and Lankester.-ANTHROPOLOGY. By E. B.
Tylor, M.A., F.R.S., and Professor E. Ray Lankester, M. A.,
F.R.S. Illustrated. [In preparation.
Other vo!umes of these Manuals will follow.
MENTAL AND MORAL PHILOSOPHY, ETC. 27
WORKS ON MENTAL AND MORAL
PHILOSOPHY, AND ALLIED SUBJECTS.
Aristotle — AN INTRODUCTION TO ARISTOTLE'S
RHETORIC. With Analysis, Notes, and Appendices. Bj £.
M. Cope, Trinity College, Cambridge. 8vo. 14}.
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Balfour. — a defence of philosophic doubt : being
an Essay on the Foundations of Belief. By A. J. Balfour,
M.P. 8vo. I2J.
**i*/r. Balfour's criticism is exceedingly hrilliant and suggestive, ^^^^
Pall Mall Gazette.
•* An able and refreshing contribution to oneofthe burning questions
of the age, and deserves to make its mark in the fierce battle rurw
raging between science and theology,*^ — Athenaeum.
Birks. — Works by the Rev. T. R. BiRKS, Professor of Moral Philo-
sophy, Cambridge : —
FIRST PRINCIPLES OF MORAL SCIENCE ; or, a First
Course of Lectures delivered in the University of Cambridge.
Crown 8vo. Sj. dd,
TTiis work treats of three topics all preliminary to the direct exposi'
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of moral Science, its Spiritual Geography, or rdation to other
main subjects of human thought, and its Formative Principles, or
some eleffientary truths on whuh its whole development must
depend.
MODERN UTILITARIANISM; or. The Systems ot Paley,
Bentham, and Mill, Examined and Compared. Crown 8vo. ts. 6d.
MODERN PHYSICAL FATALISM, AND THE DOCTRINE
OF EVOLUTION ; including an Examination of Herbert Spen-
cer's First Principles. Crown 8vo. 6s,
SUPERNATURAL REVELATION; or, First Principles of
Moral Theology. 8vo. 8j.
^28 SCIENTIFIC CATALOGUE.
Boole — AN INVESTIGATION OF THE LAWS OF
THOUGHT, ON WHICH ARE FOUNDED THE
MATHEMATICAL THEORIES OF LOGIC AND PRO-
B ABILITIES. By George Boole, LL.D., Professor of
Mktheihaiics in the Queen's University, -Ireland, &c Sro. i^r.
Sutler.-^LECTURES OJf TH^ HIStORY OJF A*JCIEKT
PHILOSOPHY. By W. Archer Butler, late Professor of
Moral Philosophy in ihe University of Dublin. Edited from the
. Author's MSS.j with Notes, by Wiluam Hepworth Thom?-
SON, M.A., Master of Trinity College, and Regiua Professor of
Greek in the University, of Cambridge. New and Cheaper Edition,
revised by the Editor. 8vo, I2s.
Caird.— A critical account of the philosophy
OF KANT. With an Historical Introduction. By E. Caird,
M. A., Professor of Moral Philosophy in the University of Glasgow.
8yo. i8x.
Calderwood. — works by the Rev. Henry Calderwood, M. A.,
LL.D., Professor of Moral Philosophy in the University of £ldin-
burgh : —
PHILOSOPHY OF THE INFINITE: A Treatise on Man's
Knowledge of the Infinite Being, in answer to Sir W. Hamilton
and Dr. Mansel. Cheaper Edition. 8vo. *js, 6d.
**A book of great ability .... written in a clear stle, and may
be easily understood by even those who are not versed in sucA
discussions" — British Quarterly Review.
A HANDBOOK OF MORAL PHILOSOPHY. Sixth Edition.
Crown 8vo. 6s,
*^It M, we fed convinced t the best handbook on the subject^ inidlectually
and morally f and does infinite credit to its author.^ — Standard.
**A comfact and useful work, going aver a great deal of ground
in a manner adapted to suggest and facilitate further study. , , .
His book ivill be an assistance to many students outside his awn
University of Edinburgh, — Guardian.
THE RELATIONS OF MIND AND BRAIN. 8vo. I2s,
** // should be of real service as a clear exposition and a searching
criticism of cerebral pyschology ," — Westminster Review.
** Altogether his work is probably the best combination to bt found
at present in England of exposition and criticism on the sufyWt
of physiological psychology " — The Academy^ . .
Clifford.— LECTURES AND ESSAYS. By the late Professor
W. K. Clifford, F.R.S. Edited by Leslie Stephen and
Frederick Pollock, with Introduction by F. Pollock. Two
Portraits. 2 vols. 8vo. 25J.
MENTAL AND MORAL PHILOSOPHY, ETC. 29
Clifford — continued,
** Thi Times of October 22nd says : — ^■^ Many a friend of the author
on first taking tip .these volumes and renumbering his verscUUe
genius and his keen enjoyment of all realms of intellectual activity
must have irefnbledy lest they should be found to consist of fragmen-
tary pieces of work, too disconnected to do justice to his powers oj
consecutive reading, and too txxried to have any effect as a whole.
Fortunately these fears are groundless. . , . It is not only in
subject that the various papers are closely related. There is also a
singular cotisistency of view and of method thi'oughout, , , . It
is in the social and metaphysical subjects that the richness of his
intellect shows itself most forcibly in the rarity and originality of
the ideas which he presents to us. To appreciate this variety it ts
necessary to read the book itself for it treats in some form or other
of all the subjects of deepest inttrest in this age of questioning,^
Fiske. — OUTLINES OF COSMIC PHILOSOPHY, BASED
ON THE DOCTRINE OF EVOLUTION, WITH CRITI-
CISMS ON THE POSITIVE PHILOSOPHY. By JOHH
Fiske, M.A., LL.B., formerly Lecturer en Pliilosophy at
Haivard University. 2 vols. 8vo. 251.
** The work constitutes a very ejfeciive encyelopoedia of the evolution^
ary philosophy, and is well worth the study of all who wish to see
at once the entire scope and purport of the scientific dogmatism of
the day,^* — Saturday Review.
Harper.— THE metaphysics of the school. By the
Rev. Thomas Harper (S.J.)« ^^ 5 ^o^s. 8vo.
[ Vol I. in November,
Herbert.— THE realistic assumptions of modern
SCIENCE examined. By T. M. Herbert, M.A., late
Professor of Philosophy, &c., in the Lanctshire Independent
College, Manchester. 8vo. 141.
" Mr, Herberts work appears to us am of real ability and import-
ance. The author has shown himself well trained in philosophical
literature, and possessed efhigh critical andspeculatipepowers,*'^-
Mind.
Jardine.— THE ELEMENTS OF THE psychology OF
COGNITION. By Robert Jardine, B.D., D.Sc, Principal of
the General Assembly's CoUege, Calcutta, and FcUow of the Uni-
versity of Calcutta. Crown 8va 6s. bcL
JevOnS. — ^Works by W. StanlIcy TEVOffS, LL.D., M.A., J'.R.S.,
Professor of Political Economy, University CoUege, London.
30 SCIENTIFIC CATALOGUE.
JevonS — continued.
THE PRINCIPLES OF SCIENCE. A Treatise on Logic and
Scientific Method. New and Cheaper Edition, revised. Crown
8vo. ITS. td.
**No one in future can be said to have any true knowledge of what
has been done in the way of logical and scientific method in
Enflani without having carefitUy studied Professor JevonS
book**^ 5''-ectator.
THE SUBSTITUTION OF SIMILARS, the True Principle of
Reasoning. Derived from a Modification of Aristotle's Dictum.
Fcap. 8vo. 2s. td.
ELEMENTARY LESSONS IN LOGIC, DEDUCTIVE AND
INDUCTIVE. With Questions, Examples, and Vocabulary of
Logical Terms. New Edition. Fcap. 8vo. y. 6d,
PRIMER OF LOGIC. New Edition. iSmo. is.
MaCCOlL— THE GREEK SCEPTICS, from Pyrrho to Scxtus.
An Essay which obtained the Hare Prize in the year 1868. By
Norman Maccoll, B.A., Scholar of Downing College, Cam-
bridge. Crown 8va 3^. 6d.
M'Cosh— Works by James M'Cosh, LL.D., President of Princeton
College, New Jersey, U.S.
" He certainly shows himsdf skilful in that applicaHon of logic to
psychology y in that inductive science of the human mind which is
the fine side of English philosophy. His philosophy as a whole is
worthy of attention.** — Revue de Deux Mondes.
THE METHOD OF THE DIVINE GOVERNMENT, Physical
and Moral. Tenth Edition. 8vo. I Of. 6d.
" This work is distinguished from other similar ones by its being
based upon a thorough study of physical science, and an accurate
knowledge of its present condition, and by its entering in a
deeper and more unfettered manner than its predecessors upon the dis^
cussion of the appropriate psychological, ethical, and theological ques^
tions. The author keeps aloof at once from the k priori idealism and
dreaminess of German specukUion since Schdling, and from the
' onesidedness and narrowness of the empiricism and posttrvism
which have so prevailed in England,** — Dr. Ulrici, in "2^tsdirift
fur Philosophic.*'
THE INTUITIONS OF THE MIND. A New Edition, 8vo.
doth. lor. 6</.
" The undertaking to adjust the claims of the sensational and in*
tuitional philosophies, and of the k posteriori andk priori methods^
is accomplished in this work with a great amount of success.*^ —
Westminster Review. **/ value it for its large acquaintance
with English Philosophy, which has not led him to neglect the
great German works, I admire the moderation and clearness^ as
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