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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 
Examinations  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  0 

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  0  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  0  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  0  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  0  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  0  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  0  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  0  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  0  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  0  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  0 :  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  0  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  0  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  0  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  0  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  0  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  0  will  lie  on  the  polar  of  0, 

114.  If  0A\  OB',  OCy  be  perpendiculars  from  a  point  0  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  0  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  0  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;~ftdf)  (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  =  0 ;] 

(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     0     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  +  —  =  0 

a 

can  be  solved  directly,  and  that  the  complete  cubic  aj*  +  3/mc*  +  3qx  +  r  =  0 
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 

+  -+ =  0 


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){by-^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  0  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  _ ay(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      0      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  =  0  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  =  0  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 


«.     ,      «.     ,            ,     ». 

=  0 

—  ,..      .|.  ,         -■■•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     0     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'-  ari-^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  0  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  expression  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 +  0  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  0  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~16)~(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  0  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  -  0 ;  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  0  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  0  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  0  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  0  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  0  =  c  sin  0 ; 

also  eliminate  0  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)  =  0 ; 
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  «  0 ; 
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)  +  ...  =  0 ; 
cos  2a  +  cos  2)3  +  cos  2y  =  sin  2a  +  sin  2)3  +  sin  2y  =  0  ; 
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 0 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  0  cos  0  +  6  6in'0    a  cos'  ^  +  2h  sin  ^  cos  ^  -i-  6  sin'^  ~~  ah-h* 
if  a  -f  A  (tan  0  +  tan  <^)  +  6  tan  0  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)  =  0  ; 
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  0  COS  — ^     , 

z  z 

y  cos  —^  -  sm  0  fsintp  sin       ^  , 

z  z 

r^a  cos  6  cos  ^  4*  6  sin  ^  sin  ^, 
0  =  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  0  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  0        sin  0 


=  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  =  0 ; 

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)  =  0  ; 

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  =  0  ) 

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)  -  0 

=  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  0     sin  0  ' 

^,    ^  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  0  a  0        ^ 

^  — ^        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  0  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*  0  +  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*  0  +  cos*  20  +  oob' 30- 3ooB0  ooB  20008  30 ; 

A  o  •    30  f.    .30  ,50) 

and  2  sin  -^  •!  cos*  -^  -  cos*  -^  > 

=  sin'  0  +  sin'  20  +  sin' 30  -  3  sin  0  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  sin  (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  =  0 ; 
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  0  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  +  0        ^  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  .       0  +  /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  =  0 ; 

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  ^  =  0  =  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  =  0  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  0  '^     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  0  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  0  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  0  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'(tt4-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  0  0 

assuming  that  each  angle  lies  between  0  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  0  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  0  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  ihioMgfc  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  bgtpeeii  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. 
0  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 
tuifmlur  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  0  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  ] 
f4«>4i^W»lfg«ui^«»ifc«C  =  (l~c»wJ>(l--co«^>(l~cwO.] 


M%,  If  O^  i9  |pe  the  centres  of  tlie  dmnnBcribed  ad  inacribcd 
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  0  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  0  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  0  and  cos  0  in  terms  of  $  satisfy  the  identity 

2  sin  0  cos  0  =  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  0  +  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  0  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  0  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  0  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  0  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  0  (=  <)  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  0  (1  +  cos  0),  if  0  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   .    0  +  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  0  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  0  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)  0  =  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 0 .  0  or  0,  then,  if  cos $  +  isiji0  =  z^ 

C  +  t-S^=  l  +  2s  +  3s*+ ... +714^     = 7^ Ti '- 


_  1  -(cosn^  +  t sinn0)(n+  1)  +n  (cosn4-  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  0  —  ^  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  0  +  a:*,  and  tan  6  = pj . 

'^     ^  ^     1  +  a;  cos  0 

Thus  S  =  tan"*  ( = ^ ) ,  meaning  by  this  the  angle  between  -  - 

\i  "T"  a;  cos  0/  a 

It  X  sin  0 

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 

0      0 
result  will  be  ^  if  ^  lie  between  those  limits,  or  0  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  0  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    0  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  0  and  2ir, 

(9)  C=Jlog(cosec*tf),  /S^=tan"'(cotfl), 

(10)     cos*    .  =  sin  fl  (sin  0  +  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  0       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  0  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  0  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  0     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  0  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  0  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  0  so  that  PO  is 
bisected  by  the  axis :  prove  that  any  chord  through  0  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  0  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  0 :  prove  that  any  circle 
through  0  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  0  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  0 :  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,  0  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  0  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  0  to  a  series  of  con- 
focal  parabolas :  prove  that  the  corresponding  normals  envelope  a  fixed 
parabola  whose  directrix  passes  through  0  and  is  parallel  to  the  axis  of 
the  system,  and  whose  focus  ^S"  is  such  that  OS^  is  bisected  by  S. 

681.  A  point  0  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  0  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  0  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  0  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,  0  will  be  the  point  such  that  any 
chord  through  it  subtends  a  right  angle  at  0 ;  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  0 
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  0  on  any  ellipse  are  drawn  tangents  OF,  OQ  to 
any  confocal :  prove  that  the  chord  of  curvature  at  0  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  0  on  a  given  ellipse  are  drawn  two  tangents 
OF,  OQ  to  a  given  confocal  ellipse  and  a  diameter  parallel  to  the 
tangent  at  0  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  0 
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  0  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  0 ; 
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  0  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  0  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  0  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  0  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,  0  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  0  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  0  +  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  »  0  is 

»■  +  2x1/  sec  2a  +  y*  =  0. 

884.  The  equation  ha^  -  2kc2/  +  ay*  =  0  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'  0  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  0  +  y'  {4  cos  a  -  (1  +  cos  o)'  cos'  tf}  =  0 
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^  =  0 ; 

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  =  0 :  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^-2FgcosO 

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  0  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  0  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  0  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  0  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  0  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  0  and  equidistant  from  it. 

927.  The  points  P,  P'  are  taken  on  the  diameter  through  a  fixed 
point  0  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  0  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  0  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  0  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  0  (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  0  and  its 
pole  at  P ;  a  parabola  is  drawn  with  its  focus  at  0  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  0  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 

0       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  0  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^  =  0 : 

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  0  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  0  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 
0  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*  0 
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  0       * 

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  0  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      0 

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  0 


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  0 
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  0  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  0 ;  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  0 :  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  0  with  its 
envelope,  and  the  rectangle  under  the  perpendiculars  from  0,  C  on  the 
polar  of  0  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  0  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      0 

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  0  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  0  will  be  at  right  angles  to  each  other. 

1000.  If  0  be  the  point  in  the  normal  at  P  such  that  chords  drawn 
through  0  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  0  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  0  =  —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  0  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  0  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      0 

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  0  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  =  0  :  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  0  :  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,  0  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  0  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  0  the  centre  of  the  hyperbola,  PO  is  the 
tangent  at  (^  to  the  locus  of  0  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  0  be  its  centre,  PO  will  be  the  tangent  at  0 
to  the  locus  of  0  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  0 

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  0  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  0  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  0  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      0 

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  0  +  n  cos  tf,  —  =  wsin  ^  -  w»  cos  ^ : 
.a  0 

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  dfltmuatribgd  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  0  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  0  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  0  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  0  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_^      lyY 

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  0  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  0  the  normals  at  P,  Q. 


1081.  Two  conies  have  common  foci  Sj  8*^  a  point  0  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  0  will  be 
normals  to  a  third  conf  ocal  conic  at  points  lying  on  the  polar  of  0  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  0  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  0  to  the  confocals  through  0 :  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  =  0  :  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  —  0  , 


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  0  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  0  »  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  0  at  which  0  =  a,  0  =  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  0 ;  from  0  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  0  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  0  »  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    ■—  =  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  =  0  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'  =  0 (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  =  0  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  «•  0  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  »  0  is 

1165.  The  foci  of  the  conic  u  =  0  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  =  0  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  ««=»  0  is 

1169.  The  equation  of  the  asymptotes  of  the  conic  u  =  0  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^  ^  «  "?trj '\  (/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  »  0  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  =  0  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  ^  0  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  ^      0      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»  =  0 

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)  »  0  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  =  0  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  0 ;  from  0  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  0  =  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  0 
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  0  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  0  and  also  touching  them  again,  and  the  curvature  at  0  of  this 
locus  is  three-fourths  of  Uie  curvature  of  either  of  the  former,  and  that 
the  straight  lines  joining  0  with  the  other  two  points  of  contact  form 
with  OP  and  the  tangent  at  0  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  0  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  0  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  0  is  the  curve 

<^*^-@4)-(?4)"("--'-)' 

1210.  The  axes  of  any  conic  through  four  given  points  make  with 
the  bisectors  of  the  angle  0  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  0 


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^  0  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(;>' +^  =  0 

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  0  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  0  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  0  +  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  +  0 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  0  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  0 

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  0  and  from  0  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  0  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  0  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  0  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  0  :  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  0  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  =  0  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,  0  is  another  fixed  point, 
and  Oa,  Ob  meet  PB,  PA  in  Q,  i? :  prove  that  QR  envelopes  a  conic 
which  degenerates  if  0  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  0  and  PP^  is  the  remaining 
common  tangent,  PQ,  PQ  are  drawn  tangents  to  T,  C7'  respectively : 
prove  that  PP\  QQ'  and  the  tangent  at  0  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  0  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;)*  =  0 ; 

(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*  =  0 
may  be  represented  by  the  equations 

^   _   y  —  -^  . 

cosfl""Binfl""    ' 
and  we  may  call  this  the  point  $. 

The  equation  of  the  tangent  at  0  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  =  0  may  be  taken  to  be 

X cos* 0 _y sin' 0 ^   z 

the  tangent  at  the  point  being 

?cos*fl  +  ^8in*fl  +  |  =  0; 

and  any  point  on  the  conic  (fe)»  +  (w»y)»  +  (n«)»  »  0  to  be 

2a!;  my 


cos*e     sin^fl    '^' 

with  the  corresponding  tangent  — ^a  +   .  %  k^''^^ 0 ;  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)  =  0 : 

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*  =  0  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  0  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  0  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  0  =  H »  ^0  radical  centre  is  the  centre  of  the  circumscribed  circle, 

when  0  =  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)  =  0 ; 

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  =  0  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  =  0 

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  =  0  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  =  0 ; 

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  =  0  :  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  =  0  will  be  real  if 

pqr(pX  -hqY  +  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  0 ;  through  0 
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»  =  0  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  =  0  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^  0  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")  =  0  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  =  0  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  0  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*  cot  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  0  (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  0  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  ^  0  meet 
WinD,  {J{/0D}=1] 


CONIC  SECTIONS,  ANALYTICAL.  229 

1358.  In  the  last  question,  if  ABC  and  the  conic  be  fixed  and  the 

point  0  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'  =  0  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  =  0  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„  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 
0  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  =  0  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)^  =  0 : 

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;  =  0  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  =  0  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*  =  0 


are 


or  {     I  m     n)      ¥  \    m  n       I)      c  K     n  I      m) 

1386.  One  directrix  of  the  conic  h^-^-rmf  +  w«*  =  0  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  0  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  0  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  0  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*  +  «"  =  0  and  two  of 
its  angular  points  lie  on  the  conic  h?  +  m^  +  nz'  =  0  :  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'  =  0 ;  and  that  the 
three  conies  have  always  four  common  tangents. 

1398.  The  angular  points  of  a  triangle  lie  on  the  conic 

hi'  +  my*  +  7MJ*  =  0 

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?  —  0  and 
two  of  its  angular  points  lie  on  the  conic  la^  +  m^  +  ns^  ==  0 ;  prove  that 
the  locus  of  the  third  angular  point  is  the  conic 

(w»  +  w)  a"  +  (n  +  Q  y  +  (^  +  m)  «•  =  0 ; 

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«*  ==  0 
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  =  0  or  sin  APD  sin  BPC  =  0 
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  0  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  0  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  0  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  0 ;  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 
0  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  0  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  0  such  that  the 
pencil  0  {FABC}  has  a  given  value :  prove  that  the  locus  of  0  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  0  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  0  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  0  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  0  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  0 :  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  0  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  0  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  =  0 

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  0  or  more  shortly  vnth  respect  to  the  point  0,  draw 
Oa  perpendicular  to  BC  and  on  it  take  any  point  a ;  through  a,  0  draw 
straight  lines  perpendicular  to  OC,  CAy  meeting  in  b ;  and  through  b,  0 
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  0  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  0  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  0  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. 

0  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  0  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,  0  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  0  on  the  directrix. 

1459.  The  fixed  point  0  is  taken  on  a  given  conic  also  any  three 
other  points  on  the  conic  L,  M,  N:  straight  lines  drawn  through  0  at 
right  angles  to  OL,  OM^  ON  meet  MN^  NL^  LM  in  three  points  lying 
on  a  straight  line  which  meets  the  normal  at  0  in  a  fixed  point  (chords 
through  which  subtend  a  right  angle  at  0), 

1460.  Given  a  conic  and  a  point  0  :  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  0  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  0  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  0  being  taken  in  the  plane  of  a  triangle  ABC^ 
straight  lines  drawn  through  0  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  0 :  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  0  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  0 :  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  0  is  taken  within  a  circle,  and  with  0  as  focus  is 
described  a  paralK>Ia  touching  the  radical  axis  of  the  circle  and  the 
point-circle  0 ;  A  OA'  is  a  chord  of  the  circle  bisected  in  0 :  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  0  and  P  is  its 
pole ;  a  parabola  is  drawn  with  its  focus  at  0  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  0  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  0  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  0  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  0  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 
0  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  0  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  0  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  0  and  is  a  conic  which  touches  the  three  diagonals. 
Also,  if  OP,  OF  be  the  two  tangents  from  0  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  0  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  0  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  0  in  the  plane  :  a  chord  PQ  of  the  conic  such  that 
0  [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  0  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  0  are  drawn  two  chords  PP%  QQ^  of 
a  given  conic  such  that  the  two  bisectors  of  the  angles  at  0  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  0  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  0  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  0  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,  0'  =  2/)  A'=/*,  or  the  invariant  relation  is 
0  =  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,  0^=2^^,  A'  =  ^,  or  the  relation  is  again 
0  =  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*  =  0  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^  =  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'  0  coe  0 
(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  =  0  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  =  0 ; 

(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  =  0  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*  =  0 ;  and  the  proposed  equation 
may  be  written  («*  +  2a5  +  4)'  =  a?*.] 

1579.  Prove  that,  if  the  roots  of  the  equation  af  -  paif  +  ga  -  r  =  0 
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  =  0  to  the  form 
y*  lAi  3y  +  m  =  0   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)*  =  0  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>^  =  0 ; 

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  =  0  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  =  0  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  =  0 

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  0  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 -«)*=  0 
will  be  jS  +  y  +  8  +  ^-^,  &c. 

1590.  Prove  that  the  equation  x*  -  2a^  +  m  (2x  —  1)  =  0  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  =  0  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  =  0 

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  0 
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  =  0  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:""' +  #  =  0   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  +  3ga;-r«0 
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  =  0 

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  =  0  has  two  roots  whose  product 
is  1|  determine  them :  also  determine  the  roots  of  the  equation 

a*  -  387aj  +  285  =  0 
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  =  0 
exceed  those  of  the  equation 

iC*  -  g^a*  +  g,«"  -  7^  +  y  ^aj  -  7,  =  0 
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, 0 

1,  a,   1,  0, 0 

0,   1,  a,  1,   0, 0 

0,   0,   1,  a,  1,  ......  0 


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)  0 


{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 

0,  y,  0,  a,  0,  c 

X,  0,  0,  0,  6,  c 

«,  y,  «,  0,  0,  0 

1639.  Prove  that,  if  i^,  denote  the  determinant  of  the  n^  order, 

a,  1,  0,  0,   0, 0 

a,  a,  1,  0,   0, 0 

1,  a,  a,  1,  0, 0 

0,  1,  a,  a,  0, 0 


0,   0,   0,  0,   1,  a,  a,   1 

0,  0,  0,  0,  0,   1,  a,  a 

^••f  1  *  on,  +  au,.i  <- 1^,.,  =  0 ;  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, 0 

o.,  ap   1,  0,  0,  0 0 

a.,  «.>  ap  1>  0,   0,  0 


0,  a„   »,_„    a,,  a,,  1,  0  0 


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,  <*,  =  0  except  when 
n  =  0  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  0  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  0  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  0  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  0  when  h  tends  to  zero  is  |;  and  that  when 
F(x)  =  mix  and  /(x)  =  cos  a;,  the  value  of  0  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  +  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  0  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  0  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  0  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  =  0 ; 

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  0  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  0  be  the  point  the  sum  of  the  squares  of  whose 
distances  from  n  given  straight  lines,  or  planes,  is  a  minimum,  0  will  be 
the  centre  of  mean  position  of  the  feet  of  the  perpendiculars  from  0  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, 

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 

.  0 

x=^acoBO.  y=a- — >;, 
^       sin^ 

and  prove  that  the  whole  curve  can  only  be  obtained  by  using  impossible 
(pure  imaginary)  values  of  6. 

0  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  0  =  0  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 0 {g*  +  (ft* -  y) cos' 0}         __ tsin 0 {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  0  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  0  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  0  meets  this  ellipse  in  two  points  P,  Q  bo  that  OQ  is  bisected  in 
yV  ^^  point  corresponding  to  0  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)  =  0  :  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  0  be  the  point  (X,  Y),  PO  will  be  the  tangent 
at  0  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  0 :  prove  that  the  limif.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  0 :  prove  that  the  straight  line  joining 

the  middle  point  of  the  chord  to  0  will  make  with  the  normal  at  0  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  0  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  0 :  prove  tliat  the  radius  of  curvature  at  0  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  0 

and  is  met  in  i?  by  the  bisector  of  the  angle  POQ :  prove  that  the  radius 

(Is 
of  curvature  at  0  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  0  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  0  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. 

0 

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  0  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 

0 
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)  =  0  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  0  to  the 
intersection  of  two  diagonals  meets  the  third;  also  that  the  cubic  which 
passes  through  0  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;  =  0  are  the  sides  of  the  quadrilateral,  and  Ix^my^nz 
the  point  0 ;  and  the  taugents  at  the  node  will  be  real  if  0  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)  =  0 

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  ~  0  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  0  -  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  0  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)  =  0 ; 

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)  =  0 ; 

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 
two  different  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  0  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  0  on 
the  axis  is  a  nodal  circular  cubic  which  is  its  own  inverse  with  respect 
to  the  vertex  A^  the  constant  of  inversion  being  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  0  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^'  =  0  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  0  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  0  +  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*"**  0  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 0 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  0  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  0     \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 ' 

pgogaeca;)'^     ir'. 
Jq       sin  a  8  ' 

_(2--l)w^ 

(35)    I     —  log= — s — : |  =  2jrtan  'n,  (»<1) 

^         Jt     X     °  l-2n8inaj  +  »'  ^         ' 


L    T^°8i_^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 


0  ^ 


» 


[When  X  ia  between  -  oo  and  -2,  y  =  — l;a;  =  -2toaj  =  0,  y  =  aj  +  -j; 

a? 
a:  =  0  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  0  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\  0  is  a  fixed  point  and  the  curve  is  such  that  the  circle 
described  about  OPQ  has  a  fixed  tangent  at  0 :  find  the  differentia] 
equation  of  the  curve,  the  general  integral,  and  singular  solution. 

[Taldng  0  for  origin,  and  the  fixed  tangent  at  0  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  0  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      0 


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«  =  0  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  =  0 

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'  =  0  ;  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;  =  0  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 =  0 

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,  0  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)  =  0 ; 

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')  =  0 ; 

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  0  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 


0         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)  =  0 
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^  =  0  : 
prove  that     am^n^  +  bnj^  +  cl^m^  =  0 ;  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  0  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  0  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  0  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  0 ;  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  0  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  0  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 


=  0 


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)  =  0 

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  0  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^  =  0  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  =  0  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  =  0 
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^  =  0 ;  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  =  0  by  the  plane 
Ix  +  my  +  w»  =  0  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  =  0 

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^  =  0  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  =  0  will  be  at  right  angles,  if 

2061.  The  generators  of  the  surface  t^  =  0  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  =  0 ;  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  0  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  =  0 ; 

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  0  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«  =  0  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«  =  0  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  0  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  =  0  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  =  0  the  tangent  plane,  x  =  ia  the  parcel  plane, 


the  c.  G.  is  r J ,  0,  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  0  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        0  "^  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      0 

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  0  meets  a  conicoid  in  two  plane 
sections  A^B;  two  other  conicoids  are  described  touching  the  former  along 
A^  B  respectively  and  passing  through  0 :  prove  that  these  two  conicoids 
will  touch  at  0,  and  will  have  a  common  plane  section  in  the  polar  plane 
of  0  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;  =  0  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?  =  0 
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  =  0 ; 

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)  =  0 ; 

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  =  0 
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 


0 
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 


=  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  =  0 


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  0  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  0 
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  0 
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 


-{ 


_  fX(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  0  only  is  a  parameter. 

[The  envelope  of  (1)  when  both  0,  if>  are  parameters  is  the  hyperboloid 

a'     6*      c" "   ' 

and  when  0  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      0       c 

when  0  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  0  cos  <f>     sin  ^  sin  ^     cos  0 
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*  0  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'  0  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  0  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 ;  0  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  0  =  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</»/  ' 

0  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      0      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  0  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  0  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,  0  the  centre  of  the  circle  ABCy  SS'  is  a  straight 
line  bisected  in  0 :  prove  that  the  stresses  at  A^  B,  0  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  0  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,  0  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  0  in  the  same  sense  witn 
the  corresponding  side :  prove  that  there  will  be  equilibrium  if 


4  cot  0  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), 

0  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  0 

$  being  the  angle  which  the  given  straight  line  makes  with  the  central 
axis,  e  the  shortest  distance  between  them,  B  the  resultant  force,  and  0 
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  0 :  prove  that,  if  perpendiculars 


STATICS.  389 

be  let  fall  from  0  on  the  sides,  0  will  be  the  centroid  of  the  triangle 
formed  by  joining  the  feet  of  these  perpendiculars. 

2249.  Prove  that  a  point  0  can  always  be  found  within  a  tetra- 
hedron A  BCD  such  that,  if  Oa,  06,  Oc,  Od  be  perpendiculars  from  0  on 
the  respective  faces,  0  will  be  the  centroid  of  the  tetrahedron  abed  j  and 
that  the  distances  of  0  £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)  =  0  : 

prove  that  if  0  be  the  poiut  which  is  the  centre  of  gravity  of  equal 
particles  placed  at  the  feet  of  the  perpendiculars  from  0  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  0  —  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  0  be  the  angle  which  the  rod  makes  with  the  horizon  and  2a,  26  the 
axes  of  the  ellipse, 

26 tan 0  =  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),  0  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  0  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  0  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 
0  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  0  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  0  are  )3,  ir— )3,  and  0  must  not  lie  between 
these  limits.] 

2295.  Two  weights  A,  B  connected  by  a  fine  string  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  0 :  prove  that  the  arc  AP  is  equal 
to  the  horizontal  distance  between  0  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  0  -7j  =  sin  0  +  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  0  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  0  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  0  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  =  0 ;  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  0  ,  $^  and  the  times  of  flight  in  the  two 
paths  will  be  as  cos  ($^  +  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  0  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  0 :  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  0  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  0  is  within  the  oval,  the  loops  will  intersect  if  n>l,  and 
touch  if  w  =  1 ;  when  0  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  0  :  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  0  denotes  the  angle  between  the  axes. 

2442.  A  point  F  is  taken  on  the  tangent  to  a  given  curve  at  a 
point  Qj  and  0  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  0  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  0  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  0  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  0  —  2^,  where 
0  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 
dctcribed  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  0  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  0  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'  0 :  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  0  on  the  transverse  axis :  prove  that  the  time  of 
passing  from  one  extremity  of  the  ordinate  through  0  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  0 ;  and  cos  a  :  1  being  the  ratio  which 
the  distance  of  0  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  0  "  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  0  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- ,    0  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  0  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  0  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  0  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  0 ;  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,  0  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  =  0 ; 

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*  0  +  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  0  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  0  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  0  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  0  :  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  ^  0  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  0  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. 


CAM  OR  I  DOE  :    PBIHTED  BT  C.  J.  CLAT,   M.A.   AT  TUB   UN1VKR81TT 


Bedford  Stmxxt,  Covknt  Gardkn,  London, 

December^  1879. 


Macmillan  6-  Co:s  Catalogue  of  Works 
in  Mathematics  and  Physical  Science; 
including  Pure  and  Applied  Mathe- 
matics; Physics^  Astronomy^  Geology, 
Chemistry,  Zoology,  Botany;  and  of 
Works  in  Mental  and  Moral  Philosophy 
and  Allied  Subjects. 


MATHEMATICS. 

Airy.— Works  by  Sir  G.  B.  AiRY,  K.C.B.,  Astronomer  Royal  :— 
ELEMENTARY  TREATISE  ON  PARTIAL  DIFFERENTIAL 
EQUATIONS.     Designed  for  the  Use  of  Students  in  the  Univer- 
sities.   With  Diagrams.    New  Edition.    Crown  8vo.     5x.  (W. 

ON  THE  ALGEBRAICAL  AND  NUMERICAL  THEORY  OF 
ERRORS  OF  OBSERVATIONS  AND  THE  COMBINA- 
TION OF  OBSERVATIONS.    Second  Edition.    Crown  8vo. 

UNDULATORY  THEORY  OF  OPTICS.  Designed  for  the  Use  of 
Students  in  the  University.     New  Edition.     Crown  8vo.     (is,  6d, 

ON  SOUND  AND  ATMOSPHERIC  VIBRATIONS.  With 
the  Mathematical  Elements  of  Music  Designed  for  the  Use  of 
Students  of  the  University.  Second  Edition,  revised  and  enlarged. 
Crown  8vo.    91. 

A  TREATISE  ON  MAGNETISM.  Designed  for  the  Use  of 
Students  in  the  Univeraty.    Crown  8va    91.  6d, 

Ball  (R.  S.,  A.M.).— EXPERIMENTAL  MECHANICS.  A 
Course  of  Lectures  delivered  at  the  Royal  Collie  of  Science  for 
Ireland.  By  Robert  Stawell  Ball,  A.M.,  Professor  of  Applied 
Mathematics  and  Mechanics  in  the  Royal  College  of  Science  for 
Ireland  (Science  and  Art  Department).     Royal  8va  l6r. 

**  We  have  not  met  with  any  book  of  the  sori'in  English,     It  eluci' 
dates  instrtutivdy  the  methods  of  a^  teacher  of  the  very  highest 
rank.     We  most  cordially  recommemt  it  to  all  ottr  riaders^^ — 
Mechanics'  Magazine, 
5,000.1x79.] 


SCIENTIFtC  CATALOdUE. 


Bayma.— THE  elements  of  molecular  mecha- 
nics. By  Joseph  Bayma,  S.J.,  Professor  of  Philosofpfaj, 
Sto<iyhut8i  College.    Demy  8ro.  cloth,     ioa   6d, 

B00le.---Work8    by   G.    BootB>    D.CL,    F.K.Si,    Profi»Bor   of 

Mathematics  in  the  Queem's  University,  Ireland  \ — 

A  TREATISE  ON  DIFFERENTIAL  EQUATIONS.  Thiid 
Edition.     Edited  by  I.  ToDHUNTAk.     Crown  8v6.  dotl.  t^. 

A  TREATISE  ON  DIFFERENTIAL  EQVATIONS.  Supple- 
mentary Volume.  Edited  by  I.  Todhuntsr.  CroMrn  Svo.  dolh. 
8/.  M, 

THE  CALCULUS  OF  FINITE  DIFF^RENtES.    Ciontn  8vo. 

cloth.     lor.  6^.     New  Edition  revised. 

Cheyne.  —  AN  ELEMENTARY  TREATISE  ON  THE 
PLANETARY  THEORY,  WiUi  a  Collection  of  Problems. 
By  C.  H.  H.  CHtYKE,  M.A.,  F.R.A.S.  Second  Edition.  Crown 
8vo.  cloth.    6:.  6d, 

Clifford.— THE  ELEMENTS  OF  DYNAMIC,  An  IntroducUoVi 
to  the  study  of  Motion  and  Rest  in  Solid  and  Fluid  Bodies.  By 
W.  K.  Clifford^  F.R.S.,  Professor  of  Applied  Mathfcmatits  and 
Mechanics  at  Umv/ersity  College,  London.  Part  t. — Kincfnatic. 
Crown  8vo.    7j.  d</.  ' 

Cummihg.— AN  INTRODUCTION  TO  THE  TH£OKY  OF 
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Cuthbertdon.- -EUCLIDIAN  GEOMETRY.  By  F.  Cuth- 
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School.     Extra  fcap.  8vo.     ^.  6d. 

Everett.— UNITS  AND  PHYSICAL  CONSTANTS.  By  J. 
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PhikMophy,  Queen's  CoUege,  Belfast.     Eafra  fcap.  8va     ^r,  6^. 

Pefrcrs— Wollisby  the  Rev.  W,M.  Fbrrsrs,  M.  A.,t^.R.S.»^ellow 
and  Tutor  of  Gonville  and  Caius  College,  Cambridge  :— 

AN  ELEMENTARY  TREATISE  ON  TRILINEAR  CO-ORDI- 
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SPHERICAL  HARMONICS  AND  SUBJECTS  CONNECTED 
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MA  THEM  A  TICS. 


Frost. — ^W<irks  by  Percival  Frost»  M.A.,  late  Fellow  of  St. 
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THE  FIRST  THREE  SECTIONS  OF  NEWTON'S  PRIN- 
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Problems,  prindpallj  intended  as  Examples  of  Kewton*s  Methods. 
Third  Edition.    8vo.  doth,     I2x. 

AN  ELEMENTARY  TREATISE  ON  CURVE  TRACING. 
8vo.  I2J. 

SOLID  GEOMETRY.  Being  a  New  Edition,  revised  and  enlai^ed, 
of  the  Treatise  by  Frost  and  Wolstenholmx.  Vol.  I.  S?o.  idr. 

Godfray.— Works  by  Hugh  Godpray,  M.A.,  Mathematical 
Lecturer  at  Pembroke  Collie,  Cambridge : — 

A  TREATISE  ON  ASTRONOMY,  for  the  Use  of  Colleges  and 
Schools.     Svo.  cloth,     its,  6</. 

AN  ELEMENTARY  TREATISE  ON  THE  LUNAR 
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Newton.     Second  Edition,  revised.     Crown  8vo.  cloth.    5^.  M 

Green  (George).— mathematical  papers  of  the 

LATE  GEORGE  GREEN,  Fellow  of  Gonville  and  Caiua 
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Hemming.— AN    elementary  treatise  on  the 

differential  and  integral  CALCULUS.  For  the 
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Fellow  of  St.  John's  Collie,  Cambridge.  Second  Edition,  with 
Corrections  and  Additions.     8vo.  cloth.     9i. 

Jackson.— GEOMETRICAL  CONIC  SECTIONS.  An  Ele- 
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a  2 


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Mechanical  Engineering,  Lehigh  University,  Bethlehem,  Pienn., 
U.S.A.     Crown  l$vo.    7x.  6</. 

Morgan. ^A  collection  of  problems  and  exam- 
ples IN  MATHEMATICS.  With  Answers.  By  H.  A. 
Morgan,  M.  A.,  Sadlerian  and  Mathematical  Lecturer  of  Jesus 
Collie,  Cambridge.     Crown  8vo.  cloth,     ts,  6d, 

Newton's  Principia. — ^4to.  cloth,    su.  ed. 

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Parkinson.— A  treatise  on  optics.  By  S.  Parkin- 
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Cambridge.  Third  Edition,  revised  and  enlarg^  Crown  8vo. 
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Rayleigh. — THE  THEORY  OF  SOUND.  By  Lord  Rayijhgh, 
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Reuleaux.— THE  kinematics  of  machinery.    Out- 

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Translated  and  edited  by  A.  B.  W.  Kennedy,  C.E.,  Professor  of 
Civil  and  Mechanical  Engineering,  University  CoU^e,  Loodon. 
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Routh. — Works  by  Edward  John  Routh,  M.A,  F.R.S.,  late 
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MATHEMATICS. 


STABILITY  OF  A  GIVEN  STATE  OF  MOTION,  PARTI- 
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1877.    8vo.     8x.  bd. 

Tait  and  Steele.— dynamics  of  a  particle.    With 

namerous  Examples.  By  Professor  Tait  and  Mr.  Stseuc  Fonrdi 
Edition,  revised.    Crown  8vo.     I2J-. 

Thomson.— papers  on  electrostatics  and  mag- 
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Todhunter.— Works  by  L  ToDHUNTKR,  M.A.,  F.R.S.,  of 
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"A/r.  TodkutUer  is  chi^y  kmntm  to  students  of  mathematics  as  the 
author  or  a  series  of  admirahU  mathematical  text-boohs,  which 
possess  the  rare  qualities  of  dein^  clear  in  style  and  absolutely  fret 
from  mistaheSf  typographical  or  other" — Saturday  Review. 

A  TREATISE  ON  SPHERICAL  TRIGONOMETRY.  New 
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PLANE  CO-ORDINATE  GEOMETRY,  as  applied  to  the  Straight 
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A  TREATISE  ON  THE  DIFFERENTIAL  CALCULUS. 
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A  TREATISE  ON  THE  INTEGRAL  CALCULUS  AND  ITS 
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EXAMPLES  OF  ANALYTICAL  GEOMETRY  OF  THREE 
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A  TREATISE  ON  ANALYTICAL  STATICS.  With  numerous 
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doth.     los.  6d, 

A  HISTORY  OF  THE  MATHEMATICAL  THEORY  OF 
PROBABILITY,  from  the  Time  of  Pascal  to  that  of  Laplace. 
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RESEARCHES  IN  THE  CALCULUS  OF  VARIATIONS, 
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to  which  the  Adams'  Prize  was  awarded  in  the  University  of 
Cambridge  in  187 1.    8vo.    6s, 


SCIENTIFIC  CA  TALOGUE. 


Todhun  X,^T— continued, 
A  HISTORY  OF  THE  MATHEMATICAL  THEORIES  OF 
ATTK ACTION,  and  the  Figiwe  of  the  Ewlh,  from  the  time  of 
Newton  to  that  of  Laplace.    Two  vols.    8vo.     2^ 

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Wilson  (W.  P.)-— A  TREATISE  ON  DYNAMICS,  By 
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and  Professor  of  Mathematics  in  Queen's  CoUej^e,  Belfast     8vq, 

Wolstenholmc— MATHEMATICAL  PROBLEMS,.,on  Sub- 
jects included  in  the  First  and  Second  Divisions  of  the  Schedule 
.  of  S\;rt>jects  for  the  Cantbri^ge  Mathematical  Tcipof  ^GxamiDatkm. 
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Young.— SIMPLE  TRACTICAL  WETHODS  OF  CALCU- 
LATINO  STRAINS  ON  GIRDERS,  AJRCHES,  AND 
TRUSSES.  With  a  Supplementary  Essay  on  Economy  in  suspen- 
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London,  and  'Member,  of  the  Institution /of  CiyiI  Eogifiaars.  >ftvo. 
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PHYSICAL  SCIENCE. 


PHYSICAL   SCIRNCB. 

Airy  (G.  B.).— POPULAR  ASTRONOMY.  With  Ilkstrartoni. 
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Balfour.— A  TREATISE  QN  COMPARATIVE  EMBRY^ 
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{Shm-tly, 

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y  li  is  a  book  that  cannot  be  ignare4%  Qn4  nms$  iiuvitabiy  lead  to 
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**  Abounds  in  information  of  interest  to  tke  student  of  kioUgieal 
science"-  -Daily  News. 

BUkC— ASTRONOMICAL  MYTHS.  Based  on  Flammarion't 
*' The  Heavens.'*  By  John  F.  Blake.  With  numeix)us  Illustra- 
tions.    Crown  Svow     qj. 

Blanford  (H.  P.).— RUDIMENTS  OF  PHYSICAL  GEO. 
GRAPHY  FOR  THE  USE  OF  INDIAN  SCHOOLS.  By 
H.  F.  Blanford,  F^G.S.  With  numerous  Illustrations  and 
Glossary  of  Technical  Terms  employed.  New  Edition.  Globe 
8ya     2J.  6a. 

Blanford  (W.    T.).— geology   and  zoology   of 

ABYSSINIA.    By  W.  T.  Blanford.    8vo.    21s. 

Bosanquet — an  elementary  treatise  ON  musical 

INTERVALS  AND  TEMPERAMENT.  With  an  Account  of 
an  Enharmonic  Harmonium  exhibited  in  the  Loan  Collection  of 
Scientific  Instruments,  South  Kensington,  1876  ;  also  of  an  Enhar- 
monic Organ  exhibited  to  the  Musical  Association  of  London, 
May,  1875.  By  R.  H.  Bosanquet,  Fellow  of  St.  John's  CpHege, 
Oxford.     8vo.     6j. 

Clifford.— SEEING  AND  THINKING.  By  the  late  Professor  W. 
K.  Clifford,  F.R.S.    With  Diagrams.    Crown  8vo.    jx.  6</. 

[Nature  Series, 


8  SCIENTIFIC  CATALOGUE. 


Coal :  ITS  HISTORY  AND  ITS  USES.  By  Professors  Greew, 
MiALL,  ThojCpk,  ROcker,  and  Haesball*  of  the  Yorkshire 
College,  Leeds.     With  Illustrations.    8vo.     lis,  6d, 

**  Itjumishaa  very  eampreh€nsive  treatise  on  tke  uthalesulj^afCoal 
from  the  geological^  chemical^  mechanical^  andindustrial  points  of 
view,  concluding  with  a  chapter  on  the  important  topic  huwn  as 
the  *  Coal  Question:  "—  Daily  News. 

Cooke    (Josiah    P.,  Jun.).— -first  PRINCIPLES  OF 

CHEMICAL  PHILOSOPHY.  By  Josiah  P.  Cooke,  Tun., 
Ervine  Professor  of  Chemistry  and  Mineralogy  in  Harvard  College, 
Third  Edition,  revised  and  corrected.    Crown  8vo.    I2j.  ' ' 

Cooke  (M.  C.).— HANDBOOK  OF  BRITISH  FUNGI, 
with  full  descriptions  of  all  the  Species,  and  Illustrations  of  the 
Genera.    By  M.  C.  Cooke,  M.A.    Two  vols,  crown  8yo.    24s. 

**  WUl  maintain  its  place  as  the  standard  English  book^  on  tJU 
subject  of  which  it  treats,  for  many  years  to  eome^^* — Standard. 

Crossley.— HANDBOOK  of  double  STARS,  WITH  A 
CATALOGUE  OF  1,200  DOUBLE  STARS  AND  EXTEN- 
SIVE LISTS  OF  MEASURES  FOR  THE  USE  OF  AMA- 
TEURS.  By  E.  Crossley,'F.R.A.S.,  T.  Gledhill,  F.R.A.S., 
and  J.  M.  Wilson,  F.R.A.S.     With  Illustrations.    8vo.    2ix. 

Dawkins* — Works  by  W.  Boyd  Dawkins,  F.R.S.,  &c..  Pro- 
fessor  of  Geology  at  Owens  College,  Manchester. 

CAVE-HUNTING  :  Researches  on  the  Evidence  of  Caves  respect- 
ing the  Early  Inhabitants  of  Europe.  With  Coloured  Plate  and 
Woodcuts.     8vo.     2 1  J. 

"  T^e  mass  of  information  he  has  brought  together,  ivith  the  judicious 
use  he  has  made  of  his  materials,  will  be  found  to  inzpest  his  book 
with  much  of  f tew  and  singular  valued — Saturday  Review. 

EARLY  MAN  IN  BRITAIN,  AND  HIS  PLACE  IN  THB 
TERTIARY  PERIOD.     With  lUustrations.    8vo.  [Shortiy. 

Dawson  (J.  \V.). — ACADIAN   geology.    The  Geologic 

Structure,  Organic  Remains,  and  Mineral  Resources  of  Nova 
Scotia,  New  Branswick,  and  Prince  Edward  Island.  By  John 
William  Dawson,  M.A.,  LL.D.,  F.R.S.,  F.G.S.,  Principal  and 
Vice-Chancellor  of  M'Gill  College  and  University,  Montreal,  &c 
With  a  Geological  Map  and  numerous  Illustrations.  Third  Edition, 
with  Supplement    8vo.     2IJ.     Supplement,  separately,  zr.  6«/. 

Fiske.— DARWINISM ;  AND  OTHER  ESSAYS.  By  JOHH 
FiSKE,  M.  A.,  LL.  D.,  formerly  Lecturer  on  Philosophy  in  Harvard 
University.     Crown  8vo.     *js,  6d. 


PHYSICAL  SCIENCE. 


Fleischer.— A  system  of  volumetric  analysis. 

By  Dr.  £.  Fleischer.  Translated  from  the  Second  German 
Edition  by  M.  M.  Pattison  Muir,  with  Notes  and  Additions. 
Illustrated.     Crown  8vo.     ^s,  6t/. 

Fluckiger    and  Hanbury.— pharmacographia.    A 

History  of  the  Principal  Drugs  of  Vegetble  Origin  met  with  in 
Great  Britain  and  India.  By  F.  A.  Fl&ckioer,  M.D.,  and 
D.  Hanbury,  F.R.S.    Second  Edition,  revised.    8¥0.    aiA 

Forbes.— THE  transit  of  VENUS.  Bv  George  Forbes, 
B.  A.,  Professor  of  Natural  Philosophy  in  the  Andersonian  Univer- 
sity of  Glasgow.  With  numerous  Illustrations.  Crown  8vo.  $/.  6di 

Foster  and  Balfour.— elements  of  embryology 

By  Michael  Foster,  M.D.,  F.R.S.,  andF.  M.  Balfour,  M.A.* 
Fellow  of  Trinity  CoUege,  Cambridfi;e.  With  numerous  Illustra- 
tions.    Part  I.     Crown  ova     js,  6al 

Galton. — Works  by  Francis  Galton,  F.R.S.  :— 
METEOROGRAPHICA,  or  Methods  of  Mapping  the  Weather. 
Illustrated  by  upwardsof  600  Printed  LithographicDiagrams.  4to.  9/. 

HEREDITARY  GENIUS :  An  Inquiry  into  its  Laws  and  Con- 
sequences.    Demy  8vo.     12s. 
Th€  Times  calls  U  **  a  mast  abU  and  most  inUrtsiing  book,** 

ENGLISH    MEN   OF    SCIENCE;   THEIR   NATURE   AND 
NURTURE.     8vo.    8x.  6d. 
**  The  book  is  certainly  one  of  very  great  interest  J* — Nature. 

Gamgee.— A  TEXT-BOOK,  systematic  and  PRACTICAL, 
OF  THE  PHYSIOLOGICAL  CHEMISTRY  OF  THE  ANI- 
MAL  BODY.  By  Arthur  Gamgee,  M.D.,  F.R.S.,  Professor 
of  Physiology  in  Owens  College,  Manchester.  With  Illustrations. 
8vo.  \In  the  press, 

Geikie. — Works    by     Archibald     Geikie,     LL.D.,     F.R.S., 

Murchison  Professor  of  Geology  and  Mineralogy  at  Edinburgh : — 

ELEMENTARY   LESSONS   IN   PHYSICAL  GEOGRAPHY. 

With  numerous  Illustrations.  Fcap.  8vo.    4/.  6^.  Questions,  u.  6</. 

OUTLINES  OF  FIELD  GEOLOGY.   With  Illustrations.  Crown 

8vo.     3^.  (id, 
PRIMER  OF  GEOLOGY.     Illustrated.     i8mo.     \s, 
PRIMER  OF  PHYSICAL  GEOGRAPHY.   lUustrated.  i8mo.  is. 

Gordon.— AN  elementary  book  on  heat.  By  J.  E. 
H.  Gordon,  B.A.,  Gonville  and  Caius  College,  Cambridge. 
Crown  8vo.     2J. 


K)  SCIENTIFIC  CA  TALOGUE. 


Gray.-r-STRUCTURAL  BOTANY  ON  THE  BASIS  QF 
MORPHOtOGV.  By  Professor  Asa  Gray.  With  lUiutraiions. 
IJvo.  [In  the  fu^ess. 

Guillemin.— THE  forces  OF  NATURE:  A  Popular  Intro- 

duction    to    the  Study  of   Physical  Phenomena.      By  Am^Ak 

O^JILLEMIN.      Translated  from  the  French  by  Mrs.  Kormah 

LocKYER  ;  and  Edited,  with  Addinons  and  Notes,  by  J.  Norman 

LOCKYEA,  F.R.S.   Illustrated  by  Coloured  Plates,  and  455  Wood- 

'CHtfi.  -Third  and  eheaf>er  Edition.     Royal&vo.     2U. 

•*  Translator  and  Editor  have  done  justice  to  their  task.     Thi 

text  has  ait  the  force  and  Jlow  0/  arieiitctl  t^ritin^,  eomhimttg- 

faithfiiintss  to  the  amthof^s  tneamng  wiik  furiiy  and  indeftendenee 

in  regard  4o  idiom  ;  white  the  historical  precision  and  oftmracy 

pervading  the  work  throughout^  speak  of  the  ^oatchful  editorial 

superadfioH  fohich  has  been  given  to  every  scUMic  detail,    Nfi^ki^z 

can  well  ex(ieed  the  cletpmess  and  d^cacy  of  the  illustrative  jpood* 

(uts,    A/toge^h^r,  the,  ^oork  may  be  said  to  have  no  parallel ^  fUher 

in  point  oj  fulness  or  a^raciien^  as  a  p^lar  fvamtal  of  pkysiccU 

saence,"^SsituTd3iy  Review. 

THE  APPLICATIONS  OF  PHYSICAL  FORCES.  By  A. 
Guillemin.  Translated  from  the  French  by  Mrs.  Lockyfr,  and 
Edited  with  Notes  and  Additions  by  J.  W.  Lockyer,  F.R.S. 
With  Colouced  Plates  and  numerous  lUustrations.  .Cheaper 
Edition.  Imi>erial  8vo.  cloth,  extra  gilt  36;. 
Also  in  Eighteen  Monthly  Parts,  price  ix.  each.  Part  I.  in  Nov€aiber« 
'       1878. 

**  A  book  which  we  can  heartily  recommend^  both  on  account  4^  the 
width  and  saundness  of  its  contents,  and  also  because  of  tfu  excel* 
lettce  of  its  print,  its  illustrations,  aiul  external  appearance^* — 
Westminster  Review. 

Hanlwiry.— SCIENCE  papers  :  chiefly  Pharmacological  and 
Botanical.  By  Daniel  Hanbury,  F.R.S.  Edited,  with 
Memoir,  by  J.  Ince,  F.L.S.,  and  Portrait  engraved  by  C.  H. 
J  KENS.     8vo.     141. 

•Henslow.— THE  theory  OF  EVOLUTION  OF  LIVING 

THINGS,   and  Application  of  the  Principles  of  Evolution    to 

Religion  considered  as  Illustrative  of  the  Wisdom  and  Benefi* 

cence  of  the  Almighty.     By  the    Rev.    George  Hbnslow, 

M.A.,  F.L.S.    Crown  8vo.    6;. 

Hooker.— Works  by  Sir  J.  D.  Hooker,  K.C.S.I.,  C.B., 
F.R.S.,  M.D.,    D.C.L.  :— 

THE   STUDENT'S   FLORA  OF  THE  BRITISH  ISLANDS. 
Second  Edition,  revised  and  improved.     Globe  8vo.     tor.  6<^ 
"  Certainly  the  fullest  and  most  accurate  manual  of  the  kind  that 
has  yet  appeared,    Dr,  Hooker  has  shmvn  his  characteristic  indusiry 


PHYSICAL  SCIENCE.  %i 


H  OOker — contintuJ, 

and  ahUUy  im  the  care  and  skill  which  he  has  thrwum  into  the 
charaetcrs  of  the  plants.  These  are  to  a  great  extent  arigimd^  and 
are  really  admirahle  fer  their  eamhinatian  of  clearness^  brevity, 
and cam^ftauss.*''^^9X\  MoU  Gazette. 

PRIMER  OF  BOTANY.     Wiih  Illustrations.     iSma     ix.     New 
Edition,  Teviaed  and  corrected. 

Hooker  and  BaM.— journal  of  a  tour  in  marocco 

AND  THE  GREAT  ATLAS.  By  Sir  J.  D.  Hooker,  K.C.SJ., 
C.B.,  F.R.S.,  &c.,  and  J<iHN  Ball,  F.R.S.  With  Appendices, 
including  a  Sketch  of  the  Geology  of  Marocco.  By  G.  Maw, 
F.L.S.,  F.G.S.  With  Map  and  lUastrations.  8va  2ix. 
••  This  is,  without  domht,  one  of  the  most  interesting  and  valuable 
boohs  of  travel  published  for  many  years,** — Spectator. 

Huxley  and  Martin.— a  course  of  practical  in- 
struction IN  elementary  BIOLOGY.     By  T.   H. 

HuxLKY,  LL.D.,  Sec.  R.S.,  assisted  by  H.  N.  Martin,  B.A., 

M.B.,  D.Sc,  Fellow  of  Christ's  College,  Cambridge.     Crown  Svob 

6s, 

**  This  is  the  ntost  thoroughly  vahtahle  booh  to  teachers  and  students 

of  biology  which  has  ever  appeared  in  the  English  tongue," — 

London  Quarterly  Review. 

Huxley  (Professor).— lay   sermons,   addresses, 

AND  REVIEWS.     ByT.  H.  Huxley,  LL.D.,  F.RS.    New 

and  Cheaper  Edition.    Crown  8vo.     ^s,  6d, 

Foufteen  Diuoursa  on  the  following  subjects: — (i)  On  the  Advisable* 
ness  of  Improving  JVatural  Anotoledge  :—-{2)  Emancipation — 
Blach  and  White ;— <5)  A  Liberal  Education^  and  where  to  find 
i^;--<4)  Scientific  Education '.—{l)  On  the  Edueatianal  Value  of 
the  Natural  History  Sciences:— {6)  On  the  Study  of  Zoology:^ 
(7)  On  the  Physical  Basis  of  Li/e:—{%)  The  Scientific  Aspects  of 
Positivism  :^ {'9)  On  a  Piece  of  Chalh: — (lo)  Geological  ContcfH' 
poruneity  and  Persistent  Types  of  Life  :—{ii)  Geological  Rrfomi: — 
(la)  The  Origin  of  Species :- {IT,)  Criticisms  on  the  "  Origin  ^ 
Species:''— {\^)  On  Descarte^  '' Discourse  touching  the MHl.od of 
using  On/s  Reason  rightly  and  of  seeking  Scientific  Thith.** 

ESSAYS     SELECTED     FROM      "LAY    S?;RM0NS,     AD- 
DRESSES, AND  REVIEWS."  Second  Edition.  CrowiiSvo.  is- 

CRITIQUES  AND  ADDRESSES.     8vo.     lOf.  ed. 

Contents :— I,  Administrative  Nihilism,  2,  The  School  Boards: 
what  they  can  do,  and  what  they  may  do,  x.  On  Medical  Edu* 
cation.  4.  Yeast,  5.  On  thePormation  of  Coal,  6.  On  Coral 
and  Coral  Reefs,  7.  On  the  Methods  and  Results  of  Ethnology, 
5,   Oh  sonff  rixed  fioints  in  British  Ethnology,    9.  Paleeontology 


13  SCIENTIFIC  CATALOGUE. 

Huxley  (Professor)— «waVmi«/. 

and  the  Doctrine  of  Evolution,    la  Biogenesis  and  Alnogenesis, 
.11.  Mr,  Darufitis  Critics,      12.    The  Genealogy  of  Animals, 
13.  Eishop  Berkeley  on  the  Metaphysics  of  Sensation, 

LESSONS  IN  ELEMENTARY  PHYSIOLOGY.  With  numerous 
Illustrations.     New  Edition.     Fcap.  8vo.    4/.  6d, 

" Pure  gold  throughout" — Guardian.  '*  Unquestionably  the  clearest 
and  most  complete  elementary  treatise  on  this  stAject  that  we  possess  in 
any  language/* — ^Westminster  Reriew. 

AMERICAN  ADDRESSES:  with  a  Lecture  on  the  Study  of 
Biology.    8vo.    6x.  6d, 

PHYSIOGRAPHY:  An  Introduction  to  the  Study  of  Nature,  With 
Coloured  Plates  and  numerous  Woodcuts.  New  Edition.  Crown 
8vo.     *js.  6d. 

"  It  would  be  hardly  possible  to  place  a  more  useful  or  suggestive 
book  in  the  hands  of  learners  and  teachers^  or  one  that  is  better 
calculated  to  make  physiography  a  fceuourite  subject  in  the  science 
schools,*''-^Acadtmy, 

Jellet    (John  H.,    B.D.).— a   treatise    on   the 

THEORY  OF  FRICTION.  By  John  H.  Jellet,  B.D., 
Senior  Fellow  of  Trinity  College,  Dubun ;  President  of  the  Royal 
Irish  Academy.     8vo.    8j.  6d, 

Jones. — ^Works  by  FRANCIS  Jones,  F.R.S.E.,  F.C.S.,  Chemical 
Master  in  the  Grammar  School,  Manchester. 

THE  OWENS  COLLEGE  JUNIOR  COURSE  OF  PRAC 
TICAL  CHEMISTRY.  With  Preface  by  Professor  Roscoe. 
New  Edition.    i8mo.     With  Illustrations.    2s.  6d. 

QUESTIONS  ON  CHEMISTRY.  A  Series  of  Problems  and 
Exercises  in  Inorganic  and  Organic  Chemistry.     i8mo.    3^. 

Kingsley.— GLAUCUS !  OR,  the  wonders  OF  THE 

SHORE.  By  Charles  Kingsley,  Canon  of  Westminster. 
New  Edition,  with  numerous  Coloured  Plates.    Crown  8va    6s. 

Landauer. — blowpipe    analysis.    By  j.  landauer. 

Authorised  English  Edition,  by  James  Taylor  and  W.  E.  Kay,  of 
the  Owens  College,  Manchester.  With  Illustrations.  Extra  fcap. 
8vo.    4s.  6d, 

Langdon.— THE  application  of  electricity  to 

railway  working.  By  W.  E.  Langdon,  Member  of  the 
Society  of  Telegraph  Engineers.  With  numerous  Illustrations. 
Extra  fcap.  Svo.     4J.  td, 

•*  There  is  no  officer  in  the  telegraph  service  who  will  not  profit  by 
the  study  oj  this  book," — Mining  Journal. 


PHYSICAL  SCIENCE.  13 

Lockyer  (J.  N.).— Works  by  J.  NoMfAN  Lockyer,  F.R.S.— 
ELEMENTARY  LESSONS  IN  ASTRONOMY.    With  nu- 
merous  Uiiutnttions.    New  Edition.    Fcap^  8vo.    5/.  6d, 
"  TMe  book  is  ftUlj  clear ^  sound,  and  worthy  of  attention,  not  only  as 
a  popular  exposition,  but  as  a  scientific  ^ Index,*  **  —  Athenaeum. 
THE  SPECTROSCOPE  AND  ITS  APPLICATIONS.    By  J. 
Norman  Lockyer,  F.R.S.    With  Coloured  Plate  and  numerous 
Illustrations.    Second  E^tion.    Crown  Svo.    xs.  6d, 
CONTRIBUTIONS  TO  SOLAR  PHYSICS.    By  J.   Norman 
Lockyer,  F.R.S.    I.  A  Popular  Account  of  Inquiries  into  the 
Physical  Constitution  of  the  Sun,  with  especial  reference  to  Recent 
Spectroscopic  Researches.      II.    Communications  to  the  Royftl 
Society  of  London  and  the  French  Academy  of  Sciences^  with 
Notes.    Illustrated  by  7  Coloured  Lithographic  Plates  and  175 
Woodcuts.     Royal  Svo.  cloth,  extra  gilt,  price  3IX.  6d. 
*'  The  book  may  be  taken  as  an  authentic  exposition  of  the  present 
state  of  science  in  connection  with  the  important  subject  of  spectro- 
scopic analysis,  ...  Even  the  unscientific  public  may  darioe  much 
inforfnation  from  it.** — Daily  News. 
PRIMER  OF  ASTRONOMY.     With  lUustrations.     i8mo.     I/. 

Lockyer  and  Seabroke. — star-gazing:  past  and 

PRESENT.  An  Introduction  to  Instrumental  Astronomy.  By 
J.  N.  Lockyer,  F.R.S.  Expanded  from  Shorthand  Notes  of  a 
Course  of  Royal  Institution  Lectures  with  the  assistance  of  G.  M. 
Seabroke,  F.R.A.S.  With  numerous  Illustrations.  Royal  Svo.  2ix. 
"  A  book  of  great  interest  and  utility  to  the  astronomical  student^ 
— Athenaeum. 

Lubbock. — Works  by  Sir  John  Lubbock,  M.  P., F.R.S.,  D.C.L.: 
THE    ORIGIN    AND    METAMORPHOSES    OF    INSECTS* 
With  Numerous  Illuftrations.  Second  Edition.  Crown  Svo.  Jj.  6dl 
**Asa  summary  of  the  phenomena  of  insect  metamorphoses  his  little 
book  is  of  great  value,  and  will  be  read  with  interest  and  profit 
by  all  students  of  natural  history.     The  whole  chatter  on  the 
origin  of  insects  is  most  interesting  and  valuable.     The  illustra- 
tions are  numerous  and  good.** — Westminster  Review. 
ON  BRITISH  WILD  FLOWERS  CONSIDERED  IN  RELA- 
TION  TO  INSECTS.     With  Numerous  Illustrations.     Second 
Edition.     Crown  Svo.     4;.  dd, 
SCIENTIFIC  LECTURES.    With  Illustrations.    Svo.    %s.  6d. 
Contents  t—ZJ^Tiwrj    and    Insects^ Plants    and    Insects — Thg 
Habits  of  Ants— Introduction    to    the    Study   of   Prehistoric 
Archeology,  6r*c, 

Macmillan  (Rev.  Hugh). — ^For  other  Works  by  the  same 
Author,  see  Theological  Cataloouk 
HOLIDAYS  ON  HIGH  LANDS ;  or,  Rambles  and  Incidents  in 
search  of  Alpine  Plants.    Globe  Sva  doth.    6/.   ^ 


14  SCIENTIPIC  CATALOGUE. 


MacmiUan  (Rev-  l^x^)Ay^catuimud, 

FIRST  FORMS  OF  VEGETATION.  Second  EdHkm,  corrected 
and  enlarged;  with  Coloured  Fronti^yiece  miA  tmmeroiis  Dliistra* 
tions.    Globe  8vo.    6s. 

77^  first  edUhn  of  this  book  was  published  under  the  name  oj 
^^ Footnotes  from  the  Page  of  Nature;  or.  First  Forms  of  Vegeta- 
twn,  Probahly  the  best  popular  guide  to  the  study  of  mosses, 
lichens,  and  fungi  eoer  written.  Its  pmetical  value  as  a  kelp  to 
the  student  and  colleHor  eatmot  be  exaggerated.* — Manchester 
Examiner. 

Mansfield  (C.  B.). — Work^  by  the  late  C.  B.  Mansfield  :— 

A  THEORY  OF  SALTS.  A  Treatise  on  the  Constitution  of 
Bipolar  (two-membered)  Chemical  Compounds.     Crown  8vo.   i\s. 

AERIAL  NAVIGATION.  The  Problem,  with  HinU  for  its 
Solution.  Edited  by  R.  B.  Manspisld.  With  a  Prefiace  by  J. 
M.  Ludlow.    With  Ulustrations.    Crown.  8vo.     ios.6d. 

Mayer.-^SOUND  :  a  Series  of  Simple,  Entertaining,  and  In- 
expensive Experiments  in  the  Phenomena  of  Sound,  for  the  Use  of 
Students  of  every  i^.  By  A.  M.  Mayer,  Prof^sor  of  Physics 
in  the  Stevens  Institute  of  Technology,  &c.  With  nimierous  Illus- 
trations.   Crown  8vo.     3*.  6d, 

Mayer  and  Barnard. — light,    a  Scries  of  ShnpK  Enter- 

tainbig,  and  Useful  Experiments  in  the  Phenomena  of  Light,  for 
the  use  of  Students  of  every  age.  By  A.  M.  Maybr  and  C. 
Barnard.     With  lUustrations.    Crown  Svo.    2s.  6d. 

Miall.— STUDIES  IN  COMPARATIVE  ANATOMY.  No.  i. 
The  Skull  of  the  Crocodile.  A  Manual  for  Students.  By  L.  C. 
MtALLy  Ptofessor  of  Biology  in  Yorkshire  College.  8vo.  2i.  6d. 
No.  2,  The  Anatomy  of  the  Indian  Elephant.  By  L.  C.  Miall 
and  F.  Greenwood.    With  Plates.    $s. 

Miller.— THE  ROMANCE  OF  ASTRONOMY.  By  R.  Kalley 
Miller,  M.A.,  Fellow  and  Assistant  Tutor  of  St.  Peter's  Col- 
lie, Cambridge.  Second  Edition,  revised  and  enlaiged.  Crown 
8vo.    4J.  6d, 

Mivart  (St.  George). — Works  by  St. George  Mivart.FJLS. 

&c.,  Lectnrer  in  Comparative  Anatomy  at  St.  Mary's  HosptUl:  — 

ON  THE  GENESIS  OF  SPECIES.     Second  Edition,  to  which 

notes  Irnve  been  added  in  reference  and  reply  to  Darwin's  "  Descent 

of  Man."    With  numerous  Illustrations.    Crown   8vo.     9/. 

"  In  no  work  in  the  English  languare  lias  this  great  controversy 

been  treated  at  once  with  tlie  same  broad  and  vigorous  grasp  of 

facts,  and  the  same  libaal  and  candid  temper'*— '?i^\\aAxfY.vrrs.^vf. 


PHYSICAL  SCIENCE.  i^ 


Mivart  (St.  G^OTgt)^coniinu^ 

THE  COMMON  FROG.    With  Nttmeroos  UliMrations.    Crdwn 

8vo.  3J.  6d,    (Nature  Series.) 

*Wt  is  an  able  mamgram  of  the  Frogy  and  s&mtthing  morti  It 
throws  valuable  crosst^hts  over  wide  p&ttions  of  animateH  nature. 
Would  thai  such  Tvorks  wen  more  pletttiful*^ — Qnarterly  Journal 
of  Science. 

Moselcy.— NOTES  by  A  NATURALIST  on  THE  "CHAL- 
LENGER," being  an  account  of  various  obsenratiom  made  during 
the  voyage  of  H.M.S.  "  Challenger"  round  the  world  in  the  years 
1872—76.  By  H.  N.  MosiLEV,  M.A..  F.R.S.,  Member  of  the 
Scientific  Staff  of  the  "Challenger."  With  Map,  Coloured 
Plates,  and  Woodcuts.    8vo.    2\s, 

"  This  is  cet'toinly  the  most  interesting  and  suggestive  booh,  deserifr 
five  of  a  naiuralisCs  travels,  which  has  been  published  since  Mr* 
Darwiris  ^Journal  of  kesearches  *  appearid,  now  Piore  than  forty 
years  ago.  TluU  it  is  worthy  to  be  placed  alongside  that  ddi^tful 
record  of  the  impressions,  spcculatiofts,  and  re/lections  of  a  master 
mind,  h,  we  do  not  doubt,  the  highest  praise  which  Mr,  Mosetif 
would  desire  for  his  book,  and  we  do  not  hesitate  to  say  thai  such 
praise  is  it*  desert" — Natiire. 

Muir.— PRACTICAL  CHEMISTRY  FOR  MEDICAL  STU- 
DENTS.  Specially  arranged  for  the  first  Bf.  B.  Course.  By 
M.  M.  Pattison  M01R,  F.R.S.E.     Fcap.  Svo.     is.  6r/. 

Murphy.— .HABIT  AND  INTELLIGENCE:  a  Series  of 
Essays  on  the  Laws  of  Life  and  Mind.  By  JosEl^H  John 
Murphy.  Second  Edition,  thoroughly  revised  and  mostly  re- 
written.    With  Illustrations.    Svo.     i6j. 

Nature.— A  weekly  illustrated  journal  of 

SCIENCE.      Published  every  Thursday.       Pnce  6d.     Monthly 

Parts,  2s,  and  2s,6d. ;  Half-yearly  Volumes,  ly.  Cases  for  binding 

Vols,  is,  6d, 

*'  This  able  and  laell-ediled  Journal,  which  i>osts  up  the  science  of 
tlu  day  promptly,  and  promises  to  beof  stgncU  service  to  Hud^ts 

and  savants Scarcely  any  expressions  that  wt  can  employ 

Would  exaggerate  vur  sense  ef  the  moral  and  theological  value  of 
the  ttw/^.'— British  Quarterly  Review. 

Newcomb. — popular  ASTRONOMY.     By  Simon  New- 
COMB,  LL.D.,  Professor  U.S.  Naval  Observatory.      W*hh.ii2 
Ensravings  and  Five  Maps  of  the  Stars.    Svo.     iSx. 
"  As  affording  a  ihoroUghty  reliable  foundation  for  more  advanced 
reading.  Professor  JStetucoml^s  *  Popular  Astronomy '  is  deserving 
of  strong  recommendation^ — Nature. 
Oliver— Works  byDANtSL  Olivba,  F.R.S.,  F.L.S.»  Profetiori!rf 
Botany  in  University  College^  London,  and  Keeper  of  the  Herba* 
rium  and  Library  of  the  Royal  Gardens,  Kcw  ; — 


l6  SCIENTIFIC  CA  TALOGUE. 

OViVtX— continued, 
LE3S0NS  IN  ELEMENTARY  BOTANY.    With  nearly  Two 
Hundred  Illustrations.    New  Edition.    Fcap.  8vo.    4r.  6</. 

FIRST    BOOK    OF    INDIAN    BOTANY.      With    numerous 
Illustrations.    Extra  fcap.  8vo.    6j.  6^. 

"//  contains  a  wdl^digesUd  summary  of  all  essential  knowledge 
pertaining  to  Indian  Botany^  wrought  out  in  accordance  with  the 
best  principles  of  scientific  arrangement »^ — Allen's  Indian  Mail. 

Pasteur.— STUDIES  on  fermentation.  The  Diseases 
pf  Beer;  their  Causes  and  Means  of  Preventing  them.     By  L. 

Pasteur.  A  Thmslation  of  **  Etudes  sur  la  Bi^re,"  With  Notes 
Illustrations,  &c.  By  F.  Faulkner  and  D.  C.  Robb,  B.A. 
8vo.     2 1  J. 

Pennington.— NOTES  ON  THE  BARROWS  AND  BONE 
CAVES  OF  DERBYSHIRE.  With  an  account  of  a  Descent 
into  Elden  Hole.  By  RooKB  Pennington,  B.A.,  LL.B., 
F.G.S.    8vo.    6j. 

Penrose  (P.  C.)— on  a  method  of  predicting  by 

GRAPHICAL  CONSTRUCTION,  OCCULT ATIONS  OF 
STARS  BY  THE  MOON,  AND  SOLAR  ECLIPSES  FOR 
ANY  GIVEN  PLACE.  Together  with  more  rigorous  methods 
for  the  Accurate  Calculation  of  Longitude.  By  F.  C.  Penrose, 
F.R.A.S.     With  Charts,  Tables,  &c.    4ta     I2J. 

Perry.— AN  elementary  treatise  on  steam.  Bt 

John  Perry,  B.E.,  Professor  of  Engineering,  Imperial  Collie  of 
Engineering,  Yedo.     With  numerous  Woodcuts,  Numerical  Ex- 
amples, and  Exercises.     i8mo.    4J.  dd, 
,  **Mr,  Perry  has  in  this  compact  little  volume  brought  together  an 
immense  amount  of  information^  new  told,  regaming  steam  and 
its  application f  not  the  least  of  tts  merits  being  that  it  is  suited  to 
the  capacities  alike  of  the  tyro  tn  engirteering  science  or  the  better 
grade  of  artisan" — Iron. 

Pickering.— ELEMENTS  oF  physical  manipulation. 

By  E.  C.  Pickering,  Thayer  Professor  of  Physics  in  the  Massa- 
chusetts Institute  of  Technology.  Part  L,  medium  8vo.  los,  6d, 
Part  II.,  lOf.  6d. 

*•*  PVhen  finished  *  Physical  Manipulation^  will  no  doubt  be  con- 
sidered the  best  and  mast  complete  text-book  on  the  smfy'ect  py 
which  it  treats.*'—Sz.tvLTt, 

Prestwich.- THE  past  and  future  of  geology. 

.  An  Inaugural  Lecture,  by  T.  Prestwich,  M.A.,  F.R.S.,   &c.. 
Professor  of  Geology,  Oxford.     8vo.    2s. 

RadclifFe.— PROTEUS  :  OR  UNITY  IN  NATURE.  By.  C. 
B.  Radcliffe,  M.D.,  Author  of  "Vital  Motion  as  a  mode  of 
PhysicU  Motioa     Second  Edition.     8vo.     Js.  6d, 


PHYSICAL  SCIENCE. 


R«ndu.— THE  THEORY  OF  THE  GLACIERS  OF  SAVOY. 
By  M.  LE  Chanoine  Rendu.  Translated  by  A.  Wells,  Q.C, 
late  President  of  the  Alpine  Cli^h.  To  which  are  added,  the  Original 
Memoir  and  Supplementary  Articles  by  Professors  Tait  and  Rus- 
KIN.  Edited  with  Introductory  remarks  by  George  Forbes,  B.A., 
Professor  of  Natural  Philosophy  in  the  Andersonian  University, 
Glasgow.    8va   ^s.  6d, 

ROSCQC. — Works  by  Henky  E.  Roscoe,  F.R.S.,  Professor  of 
Chemistry  in  Owens  College,  Manchester  : — 

LESSONS  IN  ELEMENTARY  CHEMISTRY,  INORGANIC 
AND  ORGANIC.  With  numerous  Illustrations  and  Cbromo- 
litho  of  the  Solar  Spectrum,  and  of  the  Alkalis  and  Alkaline 
Earths.     New  Edition.     Fcap.  8vo.    41.  6r/. 

CHEMICAL  PROBLEMS,  adapted  to  the  above  by  Professor 
Thorpe.     Fifth  Edition,  with  Key.     2s. 

**  We  unhesitatingly  pronounce  it  tht  best  of  all  our   elementary 
treatises  on  Chemistry" — Medical  Times. 

PRIMER  OF  CHEMISTRY.     Illustrated.     i8mo.     is, 

Roscoe    and    Schorlemmer. — a  treatise  on.  in- 

ORGANIC    chemistry.     With  numerous  Illustrations.     By 
Professors  Roscoe  and  Schorlemmer. 

Vol.  I.,  The  Non-metallic  Elements.    8vo.     21s. 

Vol.  II.,  Part  I.      Metals.     8vo.     i8j. 

Vol.  II.,  Part  IL     Metals.    8vo.     i8j. 

**  Regarded  as  a  treatise  on  the  Non-nietallic  Elements^  there  can  be 

no  doubt  thai  this  volume  is  incomparably  the  most  satisfactory  one 

of  which  we  are  in  possession" — ^Spectator. 

**  //  would  be  difficult  to  praise  the  work  too  highly.  All  the  merits 
ivhich  we  noticed  in  the  first  volume  are  consticuous  in  the  second, 
ne  arrangement  is  clear  and  scientific ;  the  fuels  gained  by  modern 
research^  are  fairly  represented  and  Judiciously  selected;  and  the 
style  throughout  is  singularly  lucid,** — Lancet. 

Rumford  (Count).— the  LIFE  AND  COMPLETE  WORKS 
OF  BENJAMIN  THOMPSON,  COUNT  RUMFORD.  With 
Notices  of  his  Daughter.  By  Geoeqe  Ellis.  With  Portrait. 
Five  Vols.  8vo.    4/.  i^r.  6d. 

Schorlemmer.— A  manual  of  the  chemtstry  of 

THECARBON  COMPOUNDS  ORORGANICCHEMISTRY. 
By  C.  Schorlemmer,  F.R.S.,  Lecturer  in  Oiganic  Chemistry  in 
Owens  College,  Manchester.  .  8vo.     i^r.  .       . 

*^It  appears  to  us  to  be  as  complete  a  manual  of  the  metamorphoses  of 
carbon  as  could  be  eit  present  produced,  and  it  must  prove  eminently 
'  usefM  to  the  chemical  student,** — Athenseom* 

B 


.  I/ 


i!   I 


1 8  SCIENTIFIC  CA  TALOGUE. 

i 

Shann.— AN  elementary  treatise  on  heat,  IK 

RELATION   TO    STEAM  AND    THE    STEAM  ENGINE. 
By  G.  Shann,  M.A.    With  Illustrations.     Crown  8ya    4s.  6d. 

Smith.— HISTORIA  FILICUM  :  An  Exposition  of  the  Nature, 
Number,  and  Organography  of  Ferns,  and  Review  of  the  Prin- 
ciples upon  which  Genera  are  founded,  and  the  Systems  of  Classifi- 
cation of  the  principal  Authors,  with  a  new  General  Arrangement, 
&c  By  J.  Smith,  A.L.S.,  ex-Curator  of  the  Royal  Botanic 
Garden,  Kew.  With  Thirty  Lithographic  Plates  by  W.  H.  Fitch, 
F.L.S.     Crown  8vo.     12s,  6d. 

*•  No  one  anxious  to  work  up  a  thorough  knowledge  of  ferns  can 
ajfford  to  do  without  it^ — Gardener's  Chronide. 

South  Kensington  Science  Lectures. 

Vol.  I. — Containing  Lectures  by  Captain  Abn£Y,  F.RS.,  Professor 
Stokes,  Professor  KENtrsDY,  F.  J.  Bramwell,  F.R.S.,  Pro- 
fessor G.  Forbes,  H.  C.  Sorby,  F.R.S.,  J.  T.  Bottomley, 
F.R.S.E.,  S.  H.  Vines,  B.Sc,  and  Professor  Carey  Foster. 
Crown  8vo.     6s,  [Vol.  1 1,  nearly  ready. 

Vol.  II.— Containing  Lectures  by  W.  Spottiswoode,  P.R.S.,  Prof. 
Forbes,  H.  W.  Chisholm,  Prof.  T.  F.  Pigot,  W.  Froude, 
F.R.S.,  Dr.  Siemens,  Prof.  Barrett,  Dr.  Burden-Sander- 
son, Dr.  Lauder  Brunton,  F.R.S.,  Prof.  McLeod,  Prof. 
Roscoe,  F.R.S.,  &c.     Crown  8vo.     dr. 

Spottiswoode.— POLARIZATION    OF    LIGHT.      By    W. 

Spottiswoode,  President  of  the  Royal  Society.    With  numerous 

Illustrations.    Second  Edition.    Cr.  8vo.    y,  6d,    (Nature  Series.) 

"  The  illustrations  are  exceedingly  well  adapted  to  assist  in  making 

the  text  comprehensible,^* — Athenaeum.     "^  clear ^   trustworthy 

iWAHittz/.*'— Standard.  ' 

Stewart  (B.). — Works  by  Balfour  Stewart,  F.R.S.,{Professor 
of  Natural  Philosophy  in  Owens  College,  Mandiester : — 

LESSONS   IN   ELEMENTARY  PHYSICS.      With  numerous 
Illustrations  and  Chromolithos  of  the  Spectra  of  the  Sun,  Stars^ 
and  Nebulae.     New  Edition.     Fcap.  8vo.    4J.  dd. 
The  Educational  Times  calls  this  the  beau-idial  of  a  scientific  text- 
bookf  clear,  accurate,  and  thorough.** 

PRIMER  OF  PHYSICS.  With  Illustrations.  New  Edition,  with 
Questions.     i8mo.     is, 

1  Stewart    and    Tait.— the    UNSEEN    UNIVERSE:    or, 

I  Physical  Speculations  on  a  Future  State.    By  Balfour  Stewart^ 

!  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 

religious  readers,  ,  ,  ,  It  is  a  perfectly  sober  inquiry,  on  scientific 
grounds,  into  the  possibilities  of  a  future  existetice** — Guardian* 


PHYSICAL  SCIENCE.  19 

Stone.— ELEMENTARY  LESSONS  ON  SOUND.  By  Dr. 
W.  H.  Stone,  Lecturer  on  Physics  at  St.  Thomas'  Hospital. 
With  Illustrations.     Fcap.  8vo.     3^.  6d, 

Tait— LECTURES  ON  SOME  RECENT  ADVANCES  IN 
PHYSICAL  SCIENCE.  By  P.  G.  Tait,  M.A.,  Professor  of 
Philosophy  in  the  University  of  Edinburgh.  Second  edition, 
revised  and  enlarged,  with  the  Lecture  on  Force  delivered  before 
the  British  Association.     Crown  8vo.    9/. 

Tanner.— FIRST  principles  of  agriculture.    By 

Henry  Tanner,  F.C.S.,  Professor  of  Agricultural  Science, 
University  College,  Aberystwith,  Examiner  in  the  Principles  of 
Agriculture  under  the  Government  Department  of  Science.    i8mo. 

IS, 

Taylor.— SOUND  and  music  :  A  Nan-Mathematical  Trea- 
tise on  the  Physical  Constitution  of  Musical  Sounds  and  Harmonyt 
including  the  Chief  Acoustical  Discoveries  of  Professor  Helm- 
holtxi  By  Sedley  Taylor,  M.A.,  late  Fellow  of  Trinity  CoL 
ledge,  Cambridge.    Large  crown  8vo.     &r.  6d, 

**  In  no  previtms  scientific  treatise  do  we  remember  so  exhaustive  and 
so  richly  iliustrated  a  description  of  forms  of  vibration  and  of 
wavc'moti  n  in  fluidsP^  Musical  Standard. 

Thomson. — Works  by  Sir  Wyville  Thomson,  K.C.B.,  F.R.S. 
THE  DEPTHS  OF  THE  SEA  :  An  Account  of  the  General 
Results  of  the  Dredging  Cruises  of  H.M.SS.  "Porcupine"  and 
"  Lightning  "  during  the  Summers  of  1868*69  and  70,  under  the 
sdentific  direction  of  Dr.  Carpenter,  F.R.S.,  J.  Gwyn  Jeffreys, 
F.R.S.,  and  Sir  Wyville  Thomson,  F.R.S.  With  neariy  100 
Illustrations  and  8  coloured  Maps  and  Plans.  Second  Emtion. 
Royal  8vo.  cloth,  gilt.     3IJ.  6</. 

The  Athenaeum  says:  "  The  book  is  full  of  interesting  matter^  and 
is  7vritten  by  a  master  of  the  art  of  popular  exposition.  It  is 
excellently  illustrated^  both  coloured  maps  and  woodcuts  possessing 
high  merit.  Those  who  have  already  become  interested  in  dredging 
operations  will  of  course  make  a  point  0/  reading  this  work;  those 
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. 
The  Times  says :—"  It  is  right  that  the  public  should  have  some 
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 
.'     Chemistry  at  Westminster  Hospital.     Medium  Svso.  cloth  gilt.  25J. 

**A  treatise  almost  unique  for  Us  usefulness  either  to  thCanne^grffwery 
the  vendor,  or  the  consumer  of  wine.  The  analyses  bfvHne  are 
the  most  complete  we  have  yet  seen,  exhUnting  at  agicmce  the 
constitucftt principles  of  nearly  all  the  wines  known  in  this  Country,  ** 
— Wine  Trade  Review. 

Wallace  (A.  R.). — Works  by  Alfred  Russet.  Wallace. 

contributions  to  the  theory  of  natural 

SELECTION.        A  Series   of  Essays.       New   Edition,  with 
Corrections  and  Additions.     Crown  8vo.     ts.  6d. 

The  Saturday  Review  says:  "Ne  has  combined  an  abundance  of 
fresh  afid  original  facts  with  a  liveliness  and  sagacity  of  reasoning 
.  which  are  not  often  displayed  so  effectively  on  so  small  a  scale" 

THE  GEOGRAPHICAL  DISTRIBUTION  OF*  ANIMALS, 
with  «  study  of  the  Relations  of  Living  and  Extinct  Faunas  as 
Elucidating  the  Past  Changes  of  the  Earth*s  Surface.  2  vols.  8vo. 
with  Maps,  and  numerous  Illustrations  by  Zwecker,  42J. 

The  Times  says:  ^*  Altogether  it  is  a  wonderful  and  fascinating 
story,  whatever  objections  may  be  taken  to  theories  founded  upon 
it.  Mr.  IVallace  has  not  attempted  to  add  to  its  interest  by  any 
adornments  of  style  i  lie  has  given  a  sitriple  and  clear  statement  vf 
httrinsicdlly  int&esling' facts,  and  what  he  considers  to  be  f^giti* 
..  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. 

**  Nowhere  amid  the  many  desoHptt^ns  of  the  tropics  that  have  been 
given  is  to  be  found  a  summary  of  the  fast  history  and  acfttal 
phenomena  of  the  tropics  which  gives  thai  which  is  distinctive  &f 
the  phases  of  nature  in  them  snore  clearly,  shortly,  and  impres- 
sively"— Saturday  Review. 

Warington.— THE  WEEK  OF  CREATION;  OR,  THE 
COSMOGONY  OF  GENESIS  CONSIDERED  IN  ITS 
RELATION  TO  MODERN  SCIENCE.  By  (Jkoroe  War- 
ington, Author  of  **  The  Historic  Character  of  the  Pentateuch 
Vindicated.'*    Crown  8vo.     4/.  6d, 

Wilson.— RELIGIO  CHEMICL  By  the  late  George  Wilson, 
M.D.,  F.R.S.E.,  Regius  Professor  of  Technology  in  the  University 
of  Edinburgh.  WiUi  a  Vignette  beautifully  engraved  after  a 
design  by  Su:  Noel  Paton.     Crown  8vo.     8j.  6d, 

Wilson  (Daniel).— CALIBAN :  a  Critique  on  Shakespeare's 
"Tempest"  and  "Midsummer  Night's  Dream."  By  Daniel 
Wilson,  LL.D,  Professor  of  History  and  English  Literature  in 
University  College,  Toronto.     8vo.     lou.  6d. 

**  TTie  whole  volume  is  ^nost  rich  in  the  eloquence  of  thought  and 
imagination  as  well  as  of  words.  It  is  a  choice  contribution  at 
once  to  science,  theology,  religion,  and  li/trature,^*—Bniish. 
Quarterly  Review. 

\Vright.— METALS  AND  THEIR  CHIEF  INDUSTRIAL 
APPLICATIONS.  By  C.  Alder  Wright,  D.Sc,  &c..  Lec- 
turer on  Chemistry  in  St.  Mary's  Hospital  SchooL  Extra  fcap. 
8vo.    ys,  6d. 

Wlirtz.— A  HISTORY  OF  CHEMICAL  THEORY,  from  the 
Age  of  Lavoisier  down  to  the  present  time.  By  Ad.  Wurtz. 
Translated  by  Henry  Watts,  F.R.S.     Crown  8vo.     6/. 

"  The  discourse,  as  a  resume  of  chemical  theory  and  research,  unites 
singular  luminousness  and  grasp.  A  few  judicious  notes  are  added 
by  the  translator."— VzW  Mall  Gazette.  **  The  treatment  of  the 
subject  is  admirable,  and  the  translator  has  evidently  done  his  duty 
most  efficiently ," — Westminster  Review. 


22  SCIENTIFIC  CATALOGUE. 


SCIENCE    PRIMERS    FOR    ELEMENTARY 

SCHOOLS. 

•Under  the  joint  Editorship  of   Professors  HuxLEY,   RoscOB,  and 

Balfour  Stewart. 

Introductory.  By  Professor  Huxley,  F.R.S.        [Nearly  ready. 

Chemistry — By  H.  E.  Roscoe,F.R.S.,  Professor  of  Chemistry 
in  Owens  College,  Manchester.  With  numerous  Illustrations. 
l8mo.     IJ.     New  Edition.     With  Questions. 

Physics.—  By  Balfour  Stewart,  F.R.S. ,  Professor  of 
Natural  Philosophy  in  Owens  College,  Manchester.  With  numer- 
ous Illustrations.     i8mo.     \s.    New  Edition.     With  Questions. 

Physical  Geography .— By  Archibald  Geikie,   F.R.S., 

Murchison  Professor  of  Geology  and  Mineralogy  at  Edinburgh. 
With  numerous  Illustrations.  New  Edition  with  Questions. 
i8mo.     I  J. 

Geology— By  Professor  Geikie,  F.R.S.  With  numerous  Illus- 
trations.    New  Edition.     i8mo.  doth.     \s. 

Physiology — By    Michael    Foster,    M.D.,    F.R.S.      Wit 
numerous  Illustrations.     New  Edition.     i8mo.     \s, 

Astronomy—By  J.  Norbian  Lockyer,  F.R.S.  With  numerous 
Illustrations.     New  Edition.     i8mo.     \s. 

Botany — By  Sir  J.  D.  Hooker,  K.C.S.I..  C.B.,  F.R.S.  With 
numerous  Illustrations.     New  Edition.     i8mo.     xs. 

Logic — By  Professor  Stanley  Jevons,  F.R.S.  New  Edition. 
i8mo.     \s. 

Political  Economy — By  Professor  Stanley  Jevons,  F.R,S. 
l8ma     I  J. 

Others  in  freparadim, 

ELEMENTARY  SCIENCE  CLASS-BOOKS. 

Astronomy.— By  the  Astronomer  Royal.  POPULAR  AS- 
TRONOMY.  With  Illustrations,  By  Sir  G.  B.  Airy,  K.CB.. 
Astronomer  Royal.     New  Edition,     lomo.     \s,  6d, 

Astronomy.— ELEMENTARY  LESSONS  IN  ASTRONOMY. 
With  Coloured  Diagram  of  the  Spectra  of  the  Sun,  Stars,  and 
Nebulae,  and  numerous  Illustrations.  By  J.  Norman  Lockykr. 
F.R.S.    New  EdiUon.     Fcap.  Sra     $/.  & 


SCIENCE  CLASS-BOOKS.  23 

Elementary  Science  Class-books — continued, 

QUESTIONS  ON  LOCKYER'S  ELEMENTARY  LESSONS 
IN  ASTRONOMY.  For  the  Use  of  Schools.  By  John 
Forbes  Robertson.    iSmo,  cloth  Ump.    \s,  6d, 

Physiology.— LESSONS  in  elementary  physiology. 

With  numerous  lUustrations.  By  T.  H.  Huxley,  F.R.S.,  Pro- 
fessor of  Natural  History  in  the  Royal  School  of  Mines.  New 
Edition.     Fcap.  8vo.     4;.  6d, 

QUESTIONS  ON  HUXLEY'S  PHYSIOLOGY  FOR 
SCHOOLS.     By  T.  Alcock,  M.D.     i8mo.     u.  td. 

Botany — lessons  in  elementary  botany.    By  D. 

Oliver,  F.R.S.,  F.L.S.,  Professor  of  Botany  in  University 
CoU^e,  London.  With  nearly  Two  Hundred  Illustrations.  New 
Edition.     Fcap.  8vo.    4;.  6d, 

Chemistry — lessons  IN  elementary  chemistry, 

INORGANIC  AND  ORGANIC.  By  Henry  E.  Roscoe, 
F.R.S.,  Professor  of  Chemistry  in  Owens  College,  Manchester. 
With  numerous  Illustrations  and  Chromo-Litho  of  the  Solar 
Spectrum,  and  of  the  Alkalies  and  Alkaline  Earths.  New  Edition. 
Fcap.  Svo.    4J.  6</. 

A  SERIES  OF  CHEMICAL  PROBLEMS,  prepared  with 
Special  Reference  to  the  above,  by  T.  E.  Thorpe,  Ph.D., 
Professor  of  Chemistry  in  the  Yorkshire  College  of  Science,  Leeds. 
Adapted  for  the  preparation  of  Students  for  the  Government, 
Science,  and  Society  of  Arts  Examinations.  With  a  Preface  by 
Professor  Roscoe.     New  Edition,  with  Key.     iSrao.    2s, 

Practical  Chemistry — ^the  owens  college  junior 

COURSE  OF  PRACTICAL  CHEMISTRY.  By  Francis 
Jones,  F.R.S.E.,  F.C.S.,  Chemical  Master  in  the  Grammar  School, 
Mai^chester.  With  Preface  by  Professor  Roscoe,  and  Illustrations. 
New  Edition.     iSmo.     zr.  6d^. 

Chemistry. — questions  on.  a  Series  of  Problems  and 
Exercises  in  Inorganic  and  Organic  Chemistry.  By  F.  Jones, 
F.R.S. E.,  F.C.S.     i8mo.    31. 

Political  Economy POLITICAL  ECONOMY  FOR   BE- 

GINNERS.  By  Millicent  G.  Fawcett.  New  Edi.ion. 
i8mo.    2s.  6d, 

Logic ELEMENTARY  LESSONS  IN  LOGIC  ;  Deductive  and 

Inductive,  with  copious  Questions  and  Examples,  and  a  Vocabulary 
of  Logical  Terms.  By  W.  Stanley  Jevons,  M.A„  Professor  of 
Political  Economy  in  University  College,  London.  New  Edition. 
Fcap.  Svo,    y,  6d, 


24  SCIENTIFIC  CATALOGUE^ 

ElemcDtary  Science  Class-books — ce^nued. 

Physics — LESSONS  IN  ELEMENTARY  PHYSICS.  By 
Balfour  Stewart,  F.R.S.,  Professor  of  Nalural  PfailoFophy  in 
Owens  College,  Manchester.  With  numerous  Illnstrations  and 
Cbromo-Liiho  of  the  Spectra  of  the  Sod,  Stars, 'and  NebnlK.  New 
Edition.     Fcap.  Svo.     41.  bd. 

Anatomy LESSONS  IN  ELEMENTARY  ANATOMY.     By 

St.  George  Mivart,  F.R.S.,  Lecrurer  in  Corapanlive  Anatomy 
at  St.  Maiy's  Hospital.  With  upwards  of  400  Illiutrations.  Fcap. 
Svo.    61.  fni. 

KS    ELEMENTARY  TREATISE.      By  A.   B. 
Tfity 

ttOTl. 

Steam — an  elementary  treatise.  By  John  Perry, 
Profesfor  of  Engineering,  Imperial  Coll^^  of  Engineerii^r,  Yedo. 
'Wiih  numerous  Woodcuts  and  Numerical  Examples  and  I^eicise$. 
iSmo.     4'.  hd. 

Physical  Geography.  _  elementary  LESSONS  IN 
PHYSICAL  GEOGKAPHY.     By  A.  Geikie,   F.R.S.,  Mnrchi- 

son   Professor  of   Geology,   &c.,    Edinbui^h.      With    nnmerous 
Ittuslralion^.    Fcap.  Svo.     4/.  61/. 
QUESTIONS  ON  THE  SAME.     u.  U. 

Geography.—CLASSBOOK  OF  GEOGRAPHY.  By  C,  B. 
Clarke,  M.A..  F.R.G.S.     Fcap.  Svo.     2/.  &/. 

Natural  Philosophy.- natural   philosophy    for 

BEGINNERS.     By  I.   Todhuntbr,   M.A.,   F.R.S.      Part   I. 

The  Properties  of  Solid  and  Fluid  Bodies.  iSmo.  y.^.  Fart 
II.  Sound,  Light,  and  Heat.     iSmo.     y.  6d, 

Sound — an   elementary  treatise.     By  Dr.  W.   H. 

Stone.     With  1 11  usl  rat  ions.     iSmo.     31.  6J. 

Others  in  FrrfaraHon. 


MANUALS  FOR  STUDENTS. 


Dyer  and  Vines.-THE  STRUCTURE  OF  PLANTS.  Hy 
Professor  Thiselton  Dyer,  F.K.S.,  assisted  by  Svdkev 
Vines,  B.Sc,  Fellow  aiwi  Lecturer  of  Christ's  Collie,  Cambridge. 
With  numerous  Illustrations.  [In fttfaralu  11. 


AfANUALS  FOR  STUDENTS.  25 


Manuals  for  Students — continued, 

FaWCCtt— A  MANUAL   OF    POLITICAL   ECONOMY.     By 
'  Trofcssor  Fawcett,  M.P.     New  Edition,  revl  ed  and  enlarged. 
Crown  8vo.     lis,  (id. 

Fleischer— A  SYSTEM  OF  VOLUMETRIC  ANALYSIS. 
Tramlated,  with  Notes  and  Additions,  from  the  second  German 
Edition,  by  M.  M.  rATTisoN  MuiR,  F.R.S.E.  With  lUustra- 
tions.     Crown  8vo.     71.  dd. 

Flower  (W.  H.).--an  introduction  to  the  oste- 
ology OF  THE  MAMMALIA.  Being  the  Substance  of  the 
Course  of  Lectures  delivered  at  the  Royal  CoUeire  of  Saigeons  of 
EngUnd  in  1870.  By  Professor  W.  H.  Flower,  F.R.S., 
F.R.C.S.  With  numercus  Illustrations.  New  Edition,  enlarged. 
Crown  8vo.     ioj.  6</. 

Foster  and  Balfour the  elements  of  embry- 
ology. By  Michael  Foster,  M.D.,  F.R.S.,  and  F.  M. 
Balfour,  M.A.     Part  I.  crown  8vo.     is,  (kf. 

Foster  and  Langley.— a  COURSE  OF  elementary 

PRACTICAL  PHYSIOLOGY.     By  Michael  Foster,  M.D., 
F.R.S.,  and  J.  N.  Langley,  B.A.  New  Edition.  Crown  8vo.  6j. 

Hooker  (Dr.)— the  students  flora  of  the  British 

ISLANDS.     By  Sir  J.   D.  Hooker,  K.C.S.I.,  C.B.,  F.R.S., 
M.D.,  D.C.L.  .  New  Edition,  revised.     Globe  8vo.     loj.  6d, 

Huxley — PHYSIOGRAPHY.  An  Introduction  to  the  Study  of 
Nature.  By  Profesor  Huxley,  F.R.S.  With  numerous 
Illustrations,  and  Coloured  Plates.  New  Edition.  Crown  8vo. 
is.dd. 

Huxley  and  Martin — a  COURSE  OF  PRACTICAL  in- 
struction IN  ELEMENTARY  BIOLOGY.  By  Professor 
Huxley,  F.R.S.,  as^bted  by  H.  N.  Martin,  M.B.,  D.Sc.  New 
Edition,  revised.     Crown  8vo.     dr. 

Huxley  and  Parke r._ELEMENTARY  biology,  part 

11.      By  Professor  HuXLEY,  F.R.S.,   assiaed  by  —  Parker. 
With  Illustrations.  [///  preparation. 

Jevons.—THE  PRINCIPLES  OF  SCIENCE.  A  Treatise  on 
Logic  and  Scientific  Method.  By  Professor  W.  Stanley  Jevons, 
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. 

]^  Professor  Daniel  Oliver,  F.R.S.,  F.L.S.,  Keeper  of  the 
Herbarium  and  Library  of  the  Royal  Gardens,  Kew.  With 
numerous  Illustrations.    Extra  fcap.  8vo.    6r.  6^. 

Parker  and  Bettany.—THE  MORPHOLOGY  OF  the 

SKULL.  By  Professor  Parker  and  G,  T.  Bettany.  Illus- 
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}. 

ARISTOTLE  ON  FALLACIES;  OR,  THE  SOPHISTICI 
ELENCHI.  With  a  Translation  and  Notes  by  Edward  Poste, 
M.A.,  Fellow  of  Oriel  College,  Oxiord.    8vo.    8x.  dd. 

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' 
tion  of  Moral  Philosophy,  These  are  the  Certainty  and  Dignity 
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 
well  as  comprehensvueness,  of  the  author's  views.** — Dr.  Domer,  of 
Berlin. 


MENTAL  AND  MORAL  PHILOSOPHY,  ETC.     31 

M'Cosh — canHmteJ, 

AN  EXAMINATION  OF  MR.  J.  S.  MILL'S  PHILOSOPHY: 
Bein^  a  Defence  ol  Fundamental  Truth.  Second  edition,  with 
additions.     loif.  6d, 

**Such  a  work  greatly  needed  to  be  dom^  and  the  author  was  the  man 
to  doit.  This  volume  is  important,  not  merely  in  reference  to  thi 
views  of  Mr.  Mill,  but  of  the  whole  school  of  loriters,  past  and 
present,  British  and  Continental,  he  so  ably  represents,** — Princeton 
Review. 

THE  LAWS  OF  DISCURSIVE  THOUGHT :   Being  a  Text- 

book  of  Formal  Logic     Crown  8vo.     5^. 

"  The  amount  of  summariud  information  which  it  contains  is  very 
great;  and  it  is  the  only  work  on  the  very  important  subject  with 
which  it  deals.  Never  was  such  a  work  so  much  needed  as  in 
the  present  day.** — London  Quarterly  Review. 

CHRISTIANITY  AND  POSITIVISM  :  A  Series  of  Lectures  to 
the  Times  on  Natural  Theology  and  Apologetics.  Crown  8vo. 
is.  td. 

THE  SCOTTISH  PHILOSOPHY  FROM  IIUTCHESON  TO 
HAMILTON,  Biographical,  Critical,  Expository.    Royal  8vo.  idr. 

MaSSOn.— RECENT  BRITISH  PHILOSOPHY:  A  Review 
with  Criticisms ;  including  some  Comments  on  Mr.  Mill's  Answer 
to  Sir  William  Hamilton.  By  David  Masson,  M.A.,  Professor 
of  Rhetoric  and  English  Literature  in  the  University  of  Edinburgh. 
Third  Edition,  with  an  Additional  Chapter.    Crown  8vo.     6s 

"  We  can  noiuhere  point  to  a  work  which  gives  so  clear  an  exposi* 
(ion  of  the  course  of  philosophical  speculation  in  Britain  during 
the  past  century,  or  which  indicates  so  instructively  the  mutual  in* 
fluences  of  philosophic  and  scientific  thought.  ** — Fortnightly  Review. 

Maudsley. — Works  by  H.  Maudsley,  M.D.,  Professor  of  Medical 
Jurisprudence  in  University  Collie,  London. 

THE  PHYSIOLOGY  OF  MIND  ;  being  the  First  Part  of  a  Third 
Edition,  Revised,  Enlarged,  and  in  ereat  part  Re-written,  of  "The 
Physiology  and  Pathology  of  Mind.  '    Crown  8vo.     lar.  6d. 

THE  PATHOLOGY  OF  MIND.  Revl-ed,  Enlarged,  and  in  great 
part  Re- written.     8vo.     i8f. 

BODY  AND  MIND  :  an  Inquiry  into  their  Connexion  and  Mutual 
Influence,  specially  i^nth  reference  to  Mental  Disorders.  An 
Enlarged  and  Revised  edition.  To  which  are  added,  Psycholc^cal 
Essays.     Crown  8vo.    6s,  6d. 


33  SCIENTIFIC  CA  TALOGUE.  -  .         -  -  - 

Maurice. — Works  by  the  Rev.  Frederick  Denison  MAHJacK^ 
M.A.,  Professor  of  Mpral  Philosophy  in  the  University  of  C^un- 
bridg^.  (For  other  Works  by  the  same  Author,  see  THs6L6(XbSiL 
Cataixx;ue.) 

SOCIAL  MORALITY.  Twenty-one  Lectures  deliVereS  ift"  the 
University  of  Cambridge.  New  and  Clieaper  Edition*  Crown  8vo. 
lox.  6d, 


it 


Whilst  reading  U  we  are  charmed  by  ihefi-ee^m/rom  exclmweness 
and  prejudice^  the  large  charity^  the  loftiness  of  thought^  the  eager" 
ness  to  recognize  and  appreciate  whatever  there  is  of  real  worth 
extant  in  the  worlds  whuh  aninuUes  it  Jrom  one  ind' tQ.Vu^^f^, 
Wcgain  new  thoughts  and  Hew  ways  of  viewing  tM/ngs^  emn-  more^ 
perhaps^  from  being  brought  for  a  time  under  the  influence  of  so 
noble  and  spiritual  a  mind,   — Athenaeum. 

THE  CONSCIENCE  :  Lectures  on  Casuistry,  delivered  in  the  Uni- 
versity of  Cambridge.   New  and  Cheaper  Edition.   Crown  8vo.   51. 

The  Saturday  Review  says:  **JVe  rise  from  them  with  detestation 
of  all  that  is  selfish  and  mean^  and  with  a  living  impression  that 
there  is  such  a  thing  as  goodness  after  all" 

MORAL  AND  METAPHYSICAL   PHILOSOPHY.      Vd.  I. 
Ancient  Philosophy  from  the  First  to  the  Thirteenth  Centuries ; 
VoL  II.  the  Fourteenth  Century  and  the  French  Revolution,  with 
a    glimpse  into    the  Nineteenth    Century.      New  Edition   and - 
Preface.     2  Vols.     8vo.   25X. 

Morgan.— ANCIENT  SOCIETY  :  or  Researches  in  the  Lines  of 
Human  Progress,  from  Savagery,  through  Barbarism  to  Civilisation. 
By  Lewis  II.  Morgan,  Member  of  the  National  Academy  of 
Sciences.     8vo.     i6j. 

Murphy,— THE  SCIENTIFIC  BASES  OF  FAITH.  By 
Joseph  John  Murphy,  Author  of  "  Habit  and  InteUigence.^' 
8vo.     i4r. 

**  The  book  is  not  without  substantial  valtie ;  the  writer  cotttiitues,  ike 
work  of  the  best  apologists  of  the  last  century^  it  may  be  witk  less 
force  and  clearness^  but  still  with  commendable  persuasiveness  and 
tact;  and  with  an  intelligent  feeling  for  the  changed  conditiom  of 
the  problem,^* — Academy. 

Paradoxical  Philosophy. — ^A  Sequel  to  "The  Unseen  Uni- 
verse."   Crown  8vo.     is,  dd, 

PictOn,— THE  MYSTERY  OF  MATTER  AND  OTHER 
ESSAYS.  By  J.  Allanson  Picton,  Author  of  "  New  Theories 
and  the  Old  Faith."  Cheaper  issue  with  New  Preface.  Crown 
8vo.    6s, 


MENTAL  AND  MORAL  PHILOSOPHY,  ETC,    33 


Picton — continued. 

Contents  :—  The  Mystery  of  Matter--  The  Philosophy  of  Igno- 
rance—  The  Antithesis  of  Faith  and  Sif^ht — The  Essential  Nature 
of  Keli^ioft — Christian  Pantheism. 

Sidgwick.—THE  METHODS  OF  ETHICS.  By  Henry 
SiDGWiCK,  M.A.,  Praelcctor  in  Moral  and  Political  Philosophy  in 
Trinity  College,  Cambridge.  Second  Edition,  revised  throughout 
with  important  additions.     8vo.     I4r. 

A  SUPPLEMENT  to  the  First  Edition,  containing  all  the  important 
additions  and  alterations  in  the  Second.     8vo.     zr. 

**  This  excellent  and  very  welcome  volume.  ....  Leaving  to  meta- 
physicians any  further  discussion  thai  may  he  needed  respecting  the 
already  over-discussed  problem  of  the  origin  of  the  moral  faculty ^  he 
takes  it  for  granted  as  readily  as  the  geometrician  takes  space  for 
granted,  or  the  physicist  the  existence  of  matter.  But  he  takes  little 
else  for  grantea,  and  defining  ethics  as  *  the  science  of  comluct,*  be 
carefully  examines,  not  the  various  ethical  systems  that  have  been 
propounded  by  Aristotle  and  Aristotle  s  followers  dowtewards,  but 
the  principles  upon  which,  so  far  as  they  confine  themselves  to  the 
strict  province  of  ethics,  they  are  based.^ — Athenaeum. 

Thornton.— OLD-FASHIONED  ETHICS,  AND  COMMON- 
SENSE  METAPHYSICS,  with  some  of  their  Applications.     Bv 
William  Thomas  Thornton,  Authorof  **  A  Treatise  on  Labour. 
8vo.     for.  6d. 

The  present  volume  aeals  with  problems  which  are  agitating  the 
minds  of  all  thoughtful  men.  The  follmving  are  the  Contents : — 
/.  Ante- Utilitarianism.  II,  History  s  Scientific  Pretensions.  HI. 
Daind  Hume  as  a  Metaphysician.  IV.  Huxleyism.  V,  Red  nt 
Phase  of  Scientific  Atheism.    VI,  Limits  of  Demonstrable  Theism. 

Thring  (E.,  M. A.).— thoughts  ON  lifescience. 

By  Edward  Thring,  M.A.  (Benjamin  Place),  Head  Master  of 
Uppingham  School  New  Edition,  enlarged  and  revised.  Crown 
8vo.    7^.  6«/. 

Venn. — the  logic  of  chance  :  An  Essay  on  the  Founda- 
tions  and  Province  of  the  Theory  of  Probability,  with  esoecial 
reference  to  its  logical  bearings,  and  its  application  to  Moral  and 
Social  Science.     By  John  Venn,  M.A.,  bellow  and  Lecturer  of 
Gonville  and   Caius    College,    Cambridge.     Second  Edition,  re- 
written and  greatly  enlarged.     Crown  8vo.     lOc  dd. 
"  One  of  the  most  thoti^htful  and  philosophical  treatises  on  any  sub- 
ject connccteti  with  logic  and  evidence  which  has  been  produced  in 
this  or  any  other  country  for  many  years, ^^ — Mill's  Logic,  vol  ii. 
p.  77.     Seventh  Edition. 

C 


NATURE    SERIES. 


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