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405 

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LIBRA] 


MATHEMATICAL  MONOGRAPHS 

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MATHEMATICAL  MONOGRAPHS. 

EDITED    BY 

MANSFIELD   MERRIMAN  AND  ROBERT   S.   WOODWARD. 


No.  5. 


HARMONIC  FUNCTIONS. 


WILLIAM    E.    BYERLY, 

PROFESSOR  OF  MATHEMATICS  IN  HARVARD  UNIVERSITY. 


FOURTH    EDITION,   ENLARGED. 
FIRST   THOUSAND. 


NEW  YORK: 

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COPYRIGHT,  1896, 

BY 
MANSFIELD   MERRIMAN  AND  ROBERT   S.  WOODWARD 

UNDER   THE    TITLE 

HIGHER    MATHEMATICS. 

First  Edition,  September,  1896. 
Second  Edition,  January,  1898. 
Third  Edition,  August,  1900. 
Fourth  Edition,  January,  1906. 


Engineering  & 

Mathematical 

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EDITORS'   PREFACE. 


THE  volume  called  Higher  Mathematics,  the  first  edition 
of  which  was  published  in  1896,  contained  eleven  chapters  by 
eleven  authors,  each  chapter  being  independent  of  the  others, 
but  all  supposing  the  reader  to  have  at  least  a  mathematical 
training  equivalent  to  that  given  in  classical  and  engineering 
colleges.  The  publication  of  that  volume  is  now  discontinued 
and  the  chapters  are  issued  in  separate  form.  In  these  reissues 
it  will  generally  be  found  that  the  monographs  are  enlarged 
by  additional  articles  or  appendices  which  either  amplify  the 
former  presentation  or  record  recent  advances.  This  plan  of 
publication  has  been  arranged  in  order  to  meet  the  demand  of 
teachers  and  the  convenience  of  classes,  but  it  is  also  thought 
that  it  may  prove  advantageous  to  readers  in  special  lines  of 
mathematical  literature. 

It  is  the  intention  of  the  publishers  and  editors  to  add  other 
monographs  to  the  series  from  time  to  time,  if  the  call  for  the 
same  seems  to  warrant  it.  Among  the  topics  which  are  under 
consideration  are  those  of  elliptic  functions,  the  theory  of  num- 
bers, the  group  theory,  the  calculus  of  variations,  and  non- 
Euclidean  geometry;  possibly  also  monographs  on  branches  of 
astronomy,  mechanics,  and  mathematical  physics  may  be  included. 
It  is  the  hope  of  the  editors  that  this  form  of  publication  may 
tend  to  promote  mathematical  study  and  research  over  a  wider 
field  than  that  which  the  former  volume  has  occupied. 

December,  1905. 


444680 


AUTHOR'S   PREFACE. 


THIS  brief  sketch  of  .the  Harmonic  Functions  and  their  use 
in  Mathematical  Physics  was  written  as  a  chapter  of  Merriman 
and  Woodward's  Higher  Mathematics.  It  was  intended  to  give 
enough  in  the  way  of  introduction  and  illustration  to  serve  as 
a  useful  part  of  the  equipment  of  the  general  mathematical 
student,  and  at  the  same  time  to  point  out  to  one  specially  inter- 
ested in  the  subject  the  way  to  carry  on  his  study  and  reading 
toward  a  broad  and  detailed  knowledge  of  its  more  difficult 
portions. 

Fourier's  Series,  Zonal  Harmonics,  and  Bessel's  Functions  of 
the  order  zero  are  treated  at  considerable  length,  with  the  inten- 
tion of  enabling  the  reader  to  use  them  in  actual  work  in  physical 
problems,  and  to  this  end  several  valuable  numerical  tables 
are  included  in  the  text. 

CAMBRIDGE,  MASS.,  December,  1905 


CONTENTS. 


ART.    i.  HISTORY  AND  DESCRIPTION Page     7 

2.  HOMOGENEOUS  LINEAR  DIFFERENTIAL  EQUATIONS 10 

3.  PROBLEM  IN  TRIGONOMETRIC  SERIES 12 

4.  PROBLEM  IN  ZONAL  HARMONICS .      .      .  15 

5.  PROBLEM  IN  BESSEL'S  FUNCTIONS 21 

6.  THE  SINE  SERIES 26 

7.  THE  COSINE  SERIES 30 

8.  FOURIER'S  SERIES ' 32 

9.  EXTENSION  OF  FOURIER'S  SERIES 34 

10.  DIRICHLET'S  CONDITIONS 36 

11.  APPLICATIONS  OF  TRIGONOMETRIC  SERIES 38 

12.  PROPERTIES  OF  ZONAL  HARMONICS 40 

13.  PROBLEMS  IN  ZONAL  HARMONICS 43 

14.  ADDITIONAL  FORMS 45 

15.  DEVELOPMENT  IN  TERMS  OF  ZONAL  HARMONICS         46 

1 6.  FORMULAS  FOR  DEVELOPMENT 47 

17.  FORMULAS  IN  ZONAL  HARMONICS 50 

18.  SPHERICAL  HARMONICS 51 

19.  BESSEL'S  FUNCTIONS.     PROPERTIES 52 

20.  APPLICATIONS  OF  BESSEL'S  FUNCTIONS  - 53 

21.  DEVELOPMENT  IN  TERMS  OF  BESSEL'S  FUNCTIONS 55 

22.  PROBLEMS  IN  BESSEL'S  FUNCTIONS 58 

23.  BESSEL'S  FUNCTIONS  OF  HIGHER  ORDER 59 

24.  LAME''S  FUNCTIONS 59 

TABLE      I.  SURFACE  ZONAL  HARMONICS 60 

II.  BESSEL'S  FUNCTIONS 62 

III.  ROOTS  OF  BESSEL'S  FUNCTIONS 63 

IV.  VALUES  OF  J0(xf) 63 

INDEX 65 


HARMONIC    FUNCTIONS. 


ART.  1.     HISTORY  AND  DESCRIPTION. 

What  is  known  as  the  Harmonic  Analysis  owed  its  origin 
and  development  to  the  study  of  concrete  problems  in  various 
branches  of  Mathematical  Physics,  which  however  all  involved 
the  treatment  of  partial  differential  equations  of  the  same 
general  form. 

The  use  of  Trigonometric  Series  was  first  suggested  by 
Daniel  Bernouilli  in  1753  in  his  researches  on  the  musical 
vibrations  of  stretched  elastic  strings,  although  Bessel's  Func- 
tions had  been  already  (1732)  employed  by  him  and  by  Euler 
in  dealing  with  the  vibrations  of  a  heavy  string  suspended  from 
one  end;  and  Zonal  and  Spherical  Harmonics  were  introduced 
by  Legendre  and  Laplace  in  1782  in  dealing  with  the  attrac- 
tion of  solids  of  revolution. 

The  analysis  was  greatly  advanced  by  Fourier  in  1812-1824 
in  his  remarkable  work  on  the  Conduction  of  Heat,  and  im- 
portant additions  have  been  made  by  Lame"  (1839)  ar"d  by  a 
host  of  modern  investigators. 

The  differential  equations  treated  in  the  problems  which 
have  just  been  enumerated  are 


8  HARMONIC    FUNCTIONS. 

for  the  transverse  vibrations  of  a  musical  string : 

o  * 


for  small  transverse  vibrations  of  a  uniform  heavy  string  sus- 
pended from  one  end  ; 


which  is  Laplace's  equation  ;  and 


for  the  conduction  of  heat  in  a  homogeneous  solid. 

Of  these  Laplace's  equation  (3),  and  (4)  of  which  (3)  is  a 
special  case,  are  by  far  the  most  important,  and  we  shall  con- 
cern ourselves  mainly  with  them  in  this  chapter.  As  to  their 
interest  to  engineers  and  physicists  we  quote  from  an  article 
in  The  Electrician  of  Jan.  26,  1894,  by  Professor  John  Perry: 

"  There  is  a  well-known  partial  differential  equation,  which  is 
the  same  in  problems  on  heat-conduction,  motion  of  fluids,  the 
establishment  of  electrostatic  or  electromagnetic  potential,  certain 
motions  of  viscous  fluid,  certain  kinds  of  strain  and  stress,  currents 
in  a  conductor,  vibrations  of  elastic  solids,  vibrations  of  flexible 
strings  or  elastic  membranes,  and  innumerable  other  phenomena. 
The  equation  has  always  to  be  solved  subject  to  certain  boundary 
or  limiting  conditions,  sometimes  as  to  space  and  time,  sometimes 
as  to  space  alone,  and  we  know  that  if  we  obtain  any  solution  of  a 
particular  problem,  then  that  is  the  true  and  only  solution.  Further- 
more, if  a  solution,  say,  of  a  heat-conduction  problem  is  obtained 
by  any  person,  that  answer  is  at  once  applicable  to  analogous  prob- 
lems in  all  the  other  departments  of  physics.  Thus,  if  Lord  Kel- 
vin draws  for  us  the  lines  of  flow  in  a  simple  vortex,  he  has  drawn 
for  us  the  lines  of  magnetic  force  about  a  circular  current;  if 
Lord  Rayleigh  calculates  for  us  the  resistance  of  the  mouth  of  an 
organ-pipe,  he  has  also  determined  the  end  effect  of  a  bar  of  iron 
which  is  magnetized;  when  Mr.  Oliver  Heaviside  shows  his  match- 


HISTORY    AND    DESCRIPTION,  I) 

less  skill  and  familiarity  with  Bessel's  functions  in  solving  electro- 
magnetic problems,  he  is  solving  problems  in  heat-conductivity  or 
the  strains  in  prismatic  shafts.  How  difficult  it  is  to  express  exactly 
the  distribution  of  strain  in  a  twisted  square  shaft,  for  example,  and 
yet  how  easy  it  is  to  understand  thoroughly  when  one  knows  the 
perfect-fluid  analogy!  How  easy,  again,  it  is  to  imagine  the  electric 
current  density  everywhere  in  a  conductor  when  transmitting  alter- 
nating currents  when  we  know  Mr.  Heaviside's  viscous-fluid  analogy, 
or  even  the  heat-conduction  analogy! 

"  Much  has  been  written  about  the  correlation  of  the  physical 
sciences;  but  when  we  observe  how  a  young  man  who  has  worked 
almost  altogether  at  heat  problems  suddenly  shows  himself  ac- 
quainted with  the  most  difficult  investigations  in  other  departments 
-of  physi€S,  we  may  say  that  the  true  correlation  of  the  physical 
sciences  lies  in  the  equation  of  continuity 

dt=a  \a*8  +  ay    a*'/ 

In  the  Theory  of  the  Potential  Function  in  the  Attraction 
•of  Gravitation,  and  in  Electrostatics  and  Electrodynamics,* 
V \r\  Laplace's  equation  (3)  is  the  value  of  the  Potential  Func- 
tion, at  any  external  point  (x,  y,  2),  due  to  any  distribution  of 
matter  or  of  electricity;  in  the  theory  of  the  Conduction  of 
Heat  in  a  homogeneous  solid  f  V  is  the  temperature  at  any 
point  in  the  solid  after  the  stationary  temperatures  have  been 
•established,  and  in  the  theory  of  the  irrotational  flow  of  an 
incompressible  fluid  \  V  is  the  Velocity  Potential  Function 
and  (3)  is  known  as  the  equation  of  continuity. 

If  we  use  spherical  coordinates,  (3)  takes  the  form 


=0- 


*  See  Peirce's  Newtonian  Potential  Function.      Boston. 
f  See  Fourier's  Analytic  Theory  of   Heat.     London  and  New  York,  1878 
•or  Riemann's  Partielle  Differentialgleichungen.      Brunswick. 

\  See  Lamb's  Hydrodynamics.     London  and  New  York,  1895. 


10  HARMONIC    FUNCTIONS. 

and  if  we  use  cylindrical  coordinates,  the  form 


~~ 


In  the  theory  of  the  Conduction  of  Heat  in  a  homogene^ 
ous  solid,*  u  in  equation  (4)  is  the  temperature  of  any  point 
(x,  y,  z)  of  the  solid  at  any  time  /,  and  c?  is  a  constant  deter- 
mined by  experiment  and  depending  on  the  conductivity  and 
the  thermal  capacity  of  the  solid. 

ART.  2.    HOMOGENEOUS  LINEAR  DIFFERENTIAL  EQUATIONS. 

The  general  solution  of  a  differential  equation  is  the  equa- 
tion expressing  the  most  general  relation  between  the  primi- 
tive variables  which  is  consistent  with  the  given  differential 
equation  and  which  does  not  involve  differentials  or  derivatives- 
A  general  solution  will  always  contain  arbitrary  (i.e.,  undeter- 
mined) constants  or  arbitrary  functions. 

A  particular  solution  of  a  differential  equation  is  a  relation 
between  the  primitive  variables  which  is  consistent  with  the 
given  differential  equation,  but  which  is  less  general  than  the 
general  solution,  although  included  in  it. 

Theoretically,  every  particular  solution  can  be  obtained: 
from  the  general  solution  by  substituting  in  the  general  solu- 
tion particular  values  for  the  arbitrary  constants  or  particular 
functions  for  the  arbitrary  functions  ;  but  in  practice  it  is  often 
easy  to  obtain  particular  solutions  directly  from  the  differential 
equation  when  it  would  be  difficult  or  impossible  to  obtain  the 
general  solution. 

(a)  If  a  problem  requiring  for  its  solution  the  solving  of  a 
differential  equation  is  determinate,  there  must  always  be  given 
in  addition  to  the  differential  equation  enough  outside  condi- 
tions for  the  determination  of  all  the  arbitrary  constants  or 
arbitrary  functions  that  enter  into  the  general  solution  of  the 
equation  ;  and  in  dealing  with  such  a  problem,  if  the  differen- 
tial equation  can  be  readily  solved  the  natural  method  of  pro- 


HOMOGENEOUS    LINEAR   DIFFERENTIAL    EQUATIONS.  11 

cedure  is  to  obtain  its  general  solution,  and  then  to  determine 
the  constants  or  functions  by  the  aid  of  the  given  conditions. 

It  often  happens,  however,  that  the  general  solution  of  the 
differential  equation  in  question  cannot  be  obtained,  and  then, 
since  the  problem,  if  determinate,  will  be  solved,  if  by  any 
means  a  solution  of  the  equation  can  be  found  which  will  also 
satisfy  the  given  outside  conditions,  it  is  worth  while  to  try  to 
get  particular  solutions  and  so  to  combine  them  as  to  form  a 
result  which  shall  satisfy  the  given  conditions  without  ceasing 
to  satisfy  the  differential  equation. 

(b)  A  differential  equation  is  linear  when  it  would  be  of  the 
first  degree  if  the  dependent  variable   and  all  its  derivatives 
were  regarded  as  algebraic  unknown  quantities.     If  it  is  linear 
and  contains  no  term  which  does  not  involve  the  dependent 
variable  or  one  of  its  derivatives,  it  is  said  to  be  linear  and 
homogeneous. 

All  the  differential  equations  given  in  Art.  I  are  linear  and 
homogeneous. 

(c)  If  a  value   of  the  dependent  variable  has  been  found 
which  satisfies  a  given  homogeneous,  linear,  differential  equa- 
tion,  the  product   formed   by  multiplying  this  value  by  any 
constant  will  also  be  a  value  of  the  dependent  variable  which 
will  satisfy  the  equation. 

For  if  all  the  terms  of  the  given  equation  are  transposed 
to  the  first  member,  the  substitution  of  the  first-named  value 
must  reduce  that  member  to  zero ;  substituting  the  second 
value  is  equivalent  to  multiplying  each  term  of  the  result  of 
the  first  substitution  by  the  same  constant  factor,  which  there- 
fore may  be  taken  out  as  a  factor  of  the  whole  first  member. 
The  remaining  factor  being  zero,  the  product  is  zero  and  the 
equation  is  satisfied. 

(d)  If  several  values  of  the  dependent  variable  have  been 
found  each  of  which  satisfies  the  given  differential  equation, 
their  sum  will  satisfy  the  equation  ;  for  if  the  sum  of  the  values 
in  question  is  substituted  in  the  equation,  each  term  of  the  sum 


12  HARMONIC    FUNCTIONS. 

will  give  rise  to  a  set  of  terms  which  must  be  equal  to  zero,  and 
therefore  the  sum  of  these  sets  must  be  zero. 

(e)  It  is  generally  possible  to  get  by  some  simple  device 
particular  solutions  of  such  differential  equations  as  those  we 
have  collected  in  Art.  i.  The  object  of  this  chapter  is  to  find 
methods  of  so  combining  these  particular  solutions  as  to  satisfy 
any  given  conditions  which  are  consistent  with  the  nature  of 
Jthe  problem  in  question. 

This  often  requires  us  to  be  able  to  develop  any  given  func- 
tion of  the  variables  which  enter  into  the  expression  of  these 
conditions  in  terms  of  normal  forms  suited  to  the  problem  with 
which  we  happen  to  be  dealing,  and  suggested  by  the  form  of 
particular  solution  that  we  are  able  to  obtain  for  the  differential 
equation. 

These  normal  forms  are  frequently  sines  and  cosines,  but 
they  are  often  much  more  complicated  functions  known  as 
Legendre's  Coefficients,  or  Zonal  Harmonics;  Laplace's  Coef- 
ficients, or  Spherical  Harmonics;  Bessel's  Functions,  or  Cylin- 
drical Harmonics;  Lame's  Functions,  or  Ellipsoidal  Har- 
monics; etc. 

ART.  3.    PROBLEM  IN  TRIGONOMETRIC  SERIES. 

As  an  illustration  let  us  consider  the  following  problem : 
A  large  iron  plate  n  centimeters  thick  is  heated  throughout 
to  a  uniform  temperature  of  100  degrees  centigrade;  its  faces 
are  then  suddenly  cooled  to  the  temperature  zero  and  are  kept 
at  that  temperature  for  5  seconds.  What  will  be  the  tempera- 
ture of  a  point  in  the  middle  of  the  plate  at  the  end  of  that 
time?  Given  a3  =0.185  in  C.G.S.  units. 

Take  the  origin  of  coordinates  in  one  face  of  the  plate 
and  the  axis  of  X  perpendicular  to  that  face,  and  let  ti  be  the 
temperature  of  any  point  in  the  plate  t  seconds  after  the  cool- 
ing begins. 

We  shall  suppose  the  flow  of  heat  to  be  directly  across  the 
plate  so  that  at  any  given  time  all  points  in  any  plane  parallel 


PROBLEM    IN    TRIGONOMETRIC    SERIES.  1<> 

to  the   faces  of  the  plate  will  have   the  same   temperature. 
Then  u  depends  upon  a  single  space-coordinate  x  ;  ^  -  =  o  and 

-  =  o,  and  (4),  Art.  I,  reduces  to 

dz 

*  =  <&  (i) 

3/      -  a*1 

Obviously,  u  =  100°  when  t  =  o,  (2) 

u  =       o  when  x  =  o,  (3) 

i 

and  u  =       o  when  x  =  rr ;  (4) 

and  we  need  to  find  a  solution  of  (i)  which  satisfies  the  con- 
ditions (2),  (3),  and  (4). 

We  shall  begin  by  getting  a  particular  solution  of  (i),  and 
we  shall  use  a  device  which  always  succeeds  when  the  equa- 
tion is  linear  and  homogeneous  and  has  constant  coefficients. 

Assume*  u  =  e&x+yt,  where  ft  and  y  are  constants;  substi- 
tute in  (i)  and  divide  through  by  <?£*+?'  and  we  get  y  =  c?  ft*  ; 
and  if  this  condition  is  satisfied,  u  =  e^x+y(  is  a  solution  of  (i). 

u  =  gP*+**P*  is  then  a  solution  of  (i)  no  matter  what  the 
value  of  ft. 

We  can  modify  the  form  of  this  solution  with  advantage. 
Let  ft  —  /i/,f  then  u  =  ^-"VV4**  is  a  solution  of  (i),  as  is  also 
u  —  e-We'1^. 

By  (d),  Art.  2, 

(V*»  4-  e  ~  i***} 
u  _  ,-•*»<£ — Xf 1  =  e-'W  cos  IAX  (5) 

2 

is  a  solution,  as  is  also 

(e\*-xi e~v-xt\ 

«  =  *-» Vifl    _ 1 /  =  #— ^"sin/wr;  (6) 

22  . 

and  /*  is  entirely  arbitrary. 

*  This  assumption  must  be  regarded  as  purely  tentative.     It  must  be  tested 
by  substituting  in  the  equation,  and  is  justified  if  it  leads  to  a  solution, 
f  The  letter  i  will  be  used  to  represent  4/  —  i. 


14  HARMONIC    FUNCTIONS. 

By  giving  different  values  to  jj.  we  get  different  particular 
solutions  of  (i)  ;  let  us  try  to  so  combine  them  as  to  satisfy  our 
conditions  while  continuing  to  satisfy  equation  (i). 

u  =  ^>-aV  sin  }juc  is  zero  when  x  =  O  for  all  values  of  // ;  it 
is  zero  when  x  =  n  if  yu  is  a  whole  number.  If,  then,  we  write 
u  equal  to  a  sum  of  terms  of  the  form  Ae'"3"1**  sin  mx,  where 
m  is  an  integer,  we  shall  have  a  solution  of  (i)  (see  (d),  Art.  2) 
which  satisfies  (3)  and  (4). 

Let  this  solution  be 
u  =  A^-**'  sin  x-\-A,e-*ayt  sin  2x  -f  A^-^1  sin  3*  +  ...,  (7) 

.Alt  At,  At,  .  .  .  being  undetermined  constants. 
When  /  =  o,  (7)  reduces  to 

u  —  At  sin  x  -j-  At  sin  2.x  -j-  A3  sin  ^x  -f-  •  •  .  •          (8) 

If  now  it  is  possible  to  develop  unity  into  a  series  of  the 
form  (8)  we  have  only  to  substitute  the  coefficients"  of  that 
series  each  multiplied  by  100  for  Al ,  At,  A3 .  .  .  in  (7)  to  have 
a  solution  satisfying  (i)  and  all  the  equations  of  condition  (2), 

'(3),  and.  (4> 

We  shall  prove  later  (see  Art.  6)  that 

I  —  ~    sin  x  +  -  sin  \x  -4-  —  sin  tx  4-  .  .  . 

TtL  3  5  J 

for  all  values  of  x  between  o  and  n.     Hence  our  solution  is 
u  =  -    -L--""  sin  x  +  -<T9a"  sin  3*  +  Le-&0t  sin  5 x  _j_  _  _     ^ 

To  get  the  answer  of  the  numerical  problem  we  have  only 
to  compute  the  value  of  u  when  x  =  —  and  /  =  5  seconds.  As 

there  is  no  object  in  going  beyond  tenths  of  a  degree,  four- 
place  tables  will  more  than  suffice,  and  no  term  of  (g)  beyond 

the  first  will  affect  the  result.     Since  sin  -  =  i,  we  have  to 

fi 

•  compute  the  numerical  value  of 


PROBLEM    IN    ZONAL    HARMONICS.  15 

A  OO 

- — e-"*     where     a1  =  0.185     and     t  =  5. 

71 

log  a?    =  9.2672  —  10  log  400     =  2.6021 

log  /      =  0.6990  colog  n    =  9.5059  —  10 

log  a*t  —  9.9662  —  10  colog  tf**  =  9.5982  —  10 

log  log  e     =  9.6378  —  10 

log  u         =  1.7062 

log  log  ^  =  9.6040  —  10 

log  ean  =  0.4018^  u —  5Q°-8. 

If  the  breadth  of  the  plate  had  been  c  centimeters  instead 
of  it  centimeters  it  is  easy  to  see  that  we  should  have  needed 
the  development  of  unity  in  a  series  of  the  form 

TIX  2.71X  .       iTtX 

A.  sin  —  -4-  A.  sm 4-  A,  sin +  .  . . . 

c  c  c 

Prob.  i.  An  iron  slab  50  centimeters  thick  is  heated  to  the  tem- 
perature 100  degrees  Centigrade  throughout.  The  faces  are  then  sud- 
denly cooled  to  zero  degrees,  and  are  kept  at  that  temperature  for 
10  minutes.  Find  the  temperature  of  a  point  in  the  middle  of  the 
slab,  and  of  a  point  10  centimeters  from  a  face  at  the  end  of  that 
time.  Assume  that 

4 f  •     nx    ,    i         •?  nx   .1.5  Tfx  ,       \  , 

i  —  —  sin —  sin f-  -  sin  * \- . . .    from  x  =  o  to  x  =  c. 

x\        c     '3  '5  f  I 

Ans.  84°.o;  49°, 4.       -> 

ART.  4.    PROBLEM  IN  ZONAL  HARMONICS. 

As  a  second  example  let  us  consider  the  following  problem  : 
Two  equal  thin  hemispherical  shells  of  radius  unity  placed 
together  to  form  a  spherical  surface  are  separated  by  a  thin 
layer  of  air.  A  charge  of  statical  electricity  is  placed  upon 
one  hemisphere  and  the  other  hemisphere  is  connected  with 
the  ground,  the  first  hemisphere  is  then  found  to  be  at  poten- 
tial i,  the  other  hemisphere  being  of  course  at  potential  zero. 
At  what  potential  is  any  point  in  the  "  field  of  force"  due  to 
the  charge? 

We  shall  use  spherical  coordinates  and  shall  let  Fbe  the 
potential  required.  Then  F"  must  satisfy  equation  (5),  Art.  i. 


16  HARMONIC    FUNCTIONS. 

But  since  from  the  symmetry  of  the  problem  V  is  obviously 
independent  of  0,  if  we  take  the  diameter  perpendicular  to  the 

tfV  . 
plane  separating  the  two  conductors  as  our  polar  axis,  —  —  -^  is 

zero,  and  our  equation  reduces  to 


9r  sin  0          30 

V\s  given  on  the  surface  of  our  sphere,  hence 

V  =  f(ff)     when     r=i,  (2) 

where  f(tf)  =  I  if  o  <  B  <  -,  and  /(0)  =  o  if  -  <  0  <  n. 

2  £ 

Equation  (2)  and  the  implied  conditions  that  V  is  zero  at 
an  infinite  distance  and  is  nowhere  infinite  are  our  conditions. 

To  find  particular  solutions  of  (i)  we  shall  use  a  method 
which  is  generally  effective.  Assume*  that  V  =  RQ  where/? 
is  a  function  of  r  but  not  of  0,  and  &  is  a  function  of  0  but 
not  of  r.  Substitute  in  (i)  and  reduce,  and  we  get 


„,   Dx  —jn  ,  . 

i  ra*(rK)  _  i  dOJ.  (3) 

R     dr*  ~  @  sin  0         dO 

Since  the  first  member  of  (3)  does  not  contain  0  and  the 
second  does  not  contain  r  and  the  two  members  are  identically 
equal,  each  must  be  equal  to  a  constant.  Let  us  call  this 
constant,  which  is  wholly  undetermined,  m(m-\-  i)  ;  then 

d& 


whence                             r    ^        —  m(m  -f  \}R  =  o,  (4) 

and  — — --    — -j^ •}- m(m -\- i)&  =  o.  (5) 

*  See  the  first  foot-note  on  page  175. 

X    »*? 


PROBLEM    IN    ZONAL    HARMONICS. 

Equation  (4)  can  be  expanded  into 

d*R          dR 
r'-^-j  +  2r-^r  -  m(m  +  i)R  =  o, 

and    can    be   solved    by  elementary   methods.     Its   complete 
solution  is 

R-Arm  +  Br-m~\  (6) 

Equation  (5)  can  be  simplified  by  changing  the  independ- 
ent variable  to  x  where  x  =  cos  0.     It  becomes 


an  equation  which  has  been  much  studied  and  which  is  known 
as  Legendre's  Equation. 

We  shall  restrict  m,  which  is  wholly  undetermined,  to  posi- 
tive whole  values,  and  we  can  then  get  particular  solutions  of 
(7)  by  the  following  device  : 

Assume*  that  ©  can  be  expressed  as  a  sum  or  a  series  of 
terms  involving  whole  powers  of  x  multiplied  by  constant 
coefficients. 

Let  &  =  2anxH  and  substitute  in  (7).     We  get 

2[n(n  —  i)anx"-*  —  n(n  -f-  i)anxn  -\-  m(m  +  i)anxn~]  =  o,     (8) 

where  the  symbol  2  indicates  that  we  are   to    form    all  the 
terms  we  can  by  taking  successive  whole  numbers  for  n. 

Since  (8)  must  be  true  no  matter  what  the  value  of  x,  the 
coefficient  of  any  given  power  of  x,  as  for  instance  x*,  must 
vanish.  Hence 


(k  +  2)(k  +  iX+2  —  k(k  +  i)ak  +  m(m  +  i}ak  =  o, 

m(m-\-  i)  —  k(k  -f-  i) 
a= 


If  now  any  set  of  coefficients  satisfying  the  relation  (9)  be 
taken,  ©  =  ^a^  will  be  a  solution  of  (7). 


If  k  =  m,  then  at+,  =  o,     ak+t  =  o,    etc. 

*  See  the  first  foot-note  on  page-*75^  \ 


18  HARMONIC     FUNCTIONS. 

Since  it  will  answer  our  purpose  if  we  pick  out  the  simplest 
set  of  coefficients  that  will  obey  the  condition  (9),  we  can  take 
a  set  including  am. 

Let  us  rewrite  (9)  in  the  form 


(m-.k)(m+k-  I)' 

We  get  from  (10),  beginning  with  k  =  m  —  2, 
m(m  —  i) 

"~-=       -" 


_  m(m  —  \}(m  —  2)(m  —  3) 
2.  4.  (2m-  i)(2;//-3) 

m(m  —  i}(m  —  2)(m  —  3)(;«  —  4)(m  —  5) 


"-•  ~  2.4.6.  (2m  -  i)(2/«  -  3)(  2/«  -  5)      a""  G 

If  m    is    even  we    see  that    the   set  will  end  with  a0;  if  m 
is  odd,  with  «,. 


2,(2m—l) 

m(m-  i)(m-2)(m- 


m(m-  i)(m-2)(m-3)^m_ 
2.4.(2;«  —  i)(2;«—  3) 


where  «m  is  entirely  arbitrary,  is,  then,  a  solution  of  (7).     It  is 
found  convenient  to  take  am  equal  to 

(2m  —  \)(2m  —  3)  ...  I 
~^TT  ' 

and  it  will  be  shown  later  that  with  this  value  of  am  ,  @  —  i 
when  x  =  I. 

Q  is  a  function  of  x  and  contains  no  higher  powers  of  x 
than  xm.     It  is  usual  to  write  it  as  Pm(x\ 

We  proceed  to  write  out  a  few  values  of  Pm(x)  from  the 
formula 

=  (M,  -  Qfr*  -  3)  -  -  -  1  r  .  _  >»(>>>  -  •)   „-. 
w!  L  2.  (2m—  i) 

w(«  -  i)(w  -  2)0;/  -  3)  n 

.*•        —  ... 
2  .4.(2;«  —  i)(2m  —  3)  J 


PROBLEM    IN    ZONAL    HARMONICS.  19 

We  have: 

x)  -  i  or  /'.(cos  0)  =  I, 

x)  =  x  or   /^(cos  0)  =  cos  0, 

*)  =  £(3-^  —  i)      or   /^(cos  0)  =  £(3  cos20  —  i), 

-v\   —  if  r  *-3  ?  f\     f\r      P  (m<z   H\  —  i(  C  r*  r>c3  H 2  rr>c  fi\ 

x)  —  2\ 5-*   —  3-*^   or   ^sv1-015  pj  —  1^5  C0b  c       3  cos  f7;)   <    ,     • 

—  3Ox*  -f-  3)  or 

/'.(cos  0)  =  i(35  cos40  -  30  cos'0  +  3), 

—  JQX*  +  l  Sx)    or 

/'.(cos  0)  —  1(63  cos5  0  —  70  cos30  +  15  cos  0).  J 

We  have  obtained  &  =  Pm(x)  as  a  particular  solution  of  (7), 
and  6)  =  Pm(cos  0)  as  a  particular  solution  of  (5).  /^U'j  or 
Pm(cos  0)  is  a  new  function,  known  as  a  Legendre's  Coefficient, 
or  as  a  Surface  Zonal  Harmonic,  and  occurs  as  a  normal  form 
in  many  important  problems. 

j7__  rmpm(Cos  0)  is  a  particular  solution  of  (i),  and  rmPm(cos  0) 
is  sometimes  called  a  Solid  Zonal  Harmonic. 


V  =  A,P,(cos  0)  +  AsPfros  6}  +  A,r2P,(cos  0) 

+  ^,r>/>,(cos60+...     (13) 

satisfies  (i),  is  not  infinite  at  any  point  within  the  sphere,  and 
reduces  to 
V  =  AJ>.(cos  0)  +  ^(cos  0)  +  AtPt(cos  0) 

+  .43/>3(cos0)+...     (14) 
when  r  =  i. 

r/_yJ0/>0(cos0)  ,  A^cosff)    ,  A,Pt(cos0) 
-7-  -75-  -p- 


satisfies  (i),  is  not  infinite  at  any  point  without  the  sphere,  is 
equal  to  zero  when  r  =  oo  ,  and  reduces  to  (14)  when  r  =  i. 

If  then  we  can  develop  f(ff)  [see  eq.  (2)]  into  a  series  of  the 
form  (14),  we  have  only  to  put  the  coefficients  of  this  series  in 
place  of  the  A0,  Alt  At,  ...  in  (13)  to  get  the  value  of  Ffor  a 
point  within  the  sphere,  and  in  (15)  to  get  the  value  of  Fat  a 
point  without  the  sphere. 


20        •  HARMONIC    FUNCTIONS. 

We  shall  see  later  (Art.  16,  Prob.  22}  that  if   /(#)  =  I  for 
o  <  0  <  —  and/(0)  =  o  for  -'-  <  0  <  n, 


-  •  '  jD-(cos 


Hence  our  required  solution  is 
V=  l  +  3rP>(c°S  *>  ~     '     '  r3/3'(G 


-f  —  '— 3r5/>6(cOS0)-.,  (17) 

12      2-4 


at  an  internal  point ;  and 


_|_ii..Lll       />(COS  ^)  _  .  .  . 

1    12     2.47-        v 

?t  an  external  point. 

If  r  =  -  and  6  =  0,  (17)  reduces  to 

Tr  I       ,      3        I  7         I  I         .      II          1.3          I 

L^  —  —  u  ±1.  .  --  :_  .  _  .  --  --  .  —  ^  .  -  since  P  (I)  —  i 

—   />       I       ^          ,,  O  o  ^3        I       TO          O       >  ^B     *    •    '   »      S111*-^    -1    fKV1/     -          »" 

2      44      024         12     2.  4    4 

To  two  decimal  places  F=  0.68,  and  the  point  r  =  -,  0  =  o 
is  at  potential  0.68. 


If  r  =  5  and  0  =  —  ,  (18)  and  Table  I,  at  the  end  of  this 
4 

chapter,  give 


and  the  point  r  =  5,  #  =  -  is  at  potential  0.12. 

4 

If  the  radius  of  the  conductor  is  a  instead  of   unity,  we 

f 

have  only  to  replace  r  by  —  in  (17)  and  (18). 

a 


PROBLEM    IN    BESSEL'S   FUNCTIONS.  ;:  1 

Prob.  2.  One  half  the  surface  of  a  solid  sphere  12  inches  in  di- 
ameter is  kept  at  the  temperature  zero  and  the  other  half  at  100  de- 
grees centigrade  until  there  is  no  longer  any  change  of  temperature 
at  any  point  within  the  sphere.  Required  the  temperature  of  the 
center;  of  any  point  in  the  diametral  plane  separating  the  hot  and 
cold  hemispheres  ;  of  points  2  inches  from  the  center  and  in  the 
axis  of  symmetry  ;  and  of  points  3  inches  from  the  center  in  a  di- 
ameter inclined  at  an  angle  of  45°  tp  the  axis  of  synimetry. 

Ans.  50°;  50°;  73°-9  ;  26°.!  ;  77°.!  ;  220-9. 

**   '*  ^       MO *          T-  v~- 

ART.  5.     PROBLEM  IN  BESSEL'S  FUNCTIONS. 

As  a  last  example  we  shall  take  the  following  problem  : 
The  base  and  convex  surface  of  a  cylinder  2  feet  in  diameter 
and  2  feet  high  are  kept  at  the  temperature  zero,  and  the  upper 
base  at  100  degrees  centigrade.  Find  the  temperature  of  a 
point  in  the  axis  one  foot  from  the  base,  and  of  a  point  6  inches 
from  the  axis  and  one  foot  from  the  base,  after  the  permanent 
state  of  temperatures  has  been  set  up. 

If  we  use  cylindrical  coordinates  and  take  the  origin  in  the 
base  we  shall  have  to  solve  equation  (6),  Art.  I  ;  or,  represent- 
ing the  temperature  by  u  and  observing  that  from  the  sym- 
metry of  the  problem  u  is  independent  of  0, 

tfu       I  du       tfu 

s7'  +  r^  +  ^'=0'  (I) 

subject  to  the  conditions 

u  =  o    when     z  =  o,  (2) 

u  =  o        "        r  =  I,  (3) 

U  =  IOO      "  2  =  2.  (4) 

Assume  u  =  RZ  where  R  is  a  function  of  r  only  and  Z  of 
z  only;  substitute  in  (i)  and  reduce. 

i  d*R  .     i   dR  i  d*Z 

We  get  _f  =  (5) 

R  dr        rR  dr  Z  dz 

The  first  member  of  (5)  does  not  contain  z\  therefore  the 
second  member  cannot.  The  second  member  of  (5)  does  not 


ZZ  HARMONIC    FUNCTIONS. 

contain  r  ;  therefore  the  first  member  cannot.  Hence  each 
member  of  (5)  is  a  constant,  and  we  can  write  (5) 

l^?_i-_L^-         L^- 
R~dS~^^R~dr~        ~Z~d?~-         *** 

when  yua  is  entirely  undetermined. 

Hence  ^-^Z=o,  (7) 

cTR   .    idR   . 

and  ^+7*+"*  =  a  (8) 

Equation  (7)  is  easily  solved,  and  its  general  solution  is 

Z  =  Ae**  -\-Be~  **',  or  the  equivalent  form 

Z  =  C  cosh  (us)  -f-  D  sinh  (//£).  (9) 

We  can  reduce  (8)  slightly  by  letting  /-<r  =  x,  and  it  becomes 

d*R   ,    i  dR  . 

-4-^  =  0.  (10) 

dx1    '   x  dx  n 

Assume,  as  in  Art.  4,  that  7?  can  be  expressed  in  terms  of 
whole  powers  of  x.  Let  R  =  ~2anxn  and  substitute  in  (10). 
We  get 

2[n(n  —  \]anxn  ~  '  +  nanxn  -  3  +  ajc*\  =  o, 

an  equation  which  must  be  true,  no  matter  what  the  value  of  x. 
The  coefficient  of  any  given  power  of  x,  as  xk~*,  must,  then, 
vanish,  and 

k(k  —  i)ak  +  kak  -f-  ak_  ,  =  o, 
or  ^X  +  «A-2  =  0, 

whence  we  obtain  at-*  =  —  ^at  (n) 

as  the  only  relation  that  need  be  satisfied  by  the  coefficients  in 
order  that  R  =  ^a/^  shall  be  a  solution  of  (10). 


If  £  =  o,     ak_t  =  o,     ak_t  =  o,     etc. 

We  can,  then,  begin  with  k  =  o  as  the  lowest  subscript. 


PROBLEM    IN    BESSEL  S    FUNCTIONS. 


From  (I  i)  at=  — 


Then      «.  =  -          *.  =       ;.,    «.  =  -  5,  etc. 
=  «.[,-  i  +  -        -  -,-f^  +  ...], 


Hence 

where  at  may  be  taken  at  pleasure,  is  a  solution  of  (10),  pro- 
vided the  series  is  convergent. 

Take  a0  =  I,  and  then  ^  =Jo(x)  where 

T  '     \  I  I  /        \ 

J  v\XJ  T«    I     0»     -a         Os     .a    zrz    I     -2    ,,2    /;»    o»          *  '  *      \12/ 

2  2.4  2.4«O  2.4*0.0 

is  a  solution  of  (10). 

Ja(x)  is  easily  shown  to  be  convergent  for  all  values  real  or 
imaginary  of  x,  it  is  a  new  and  important  form,  and  is  called  a 
Bessel's  Function  of  the  zero  order,  or  a  Cylindrical  Har- 
monic. 

Equation  (10)  was  obtained  from  (8)  by  the  substitution  of 
x  =  JJLT  ;  therefore 


is  a  solution  of  (8),  no  matter  what  the  value  of  jn  ;  and 
u  =J0(^r)  sinh  (fjiz)  and  u  =Jn(fA.r)  cosh  (//#)  are  solutions  of 
(i).  «=yo(jur)  sinh  (jjz)  satisfies  condition  (2)  whatever  the 
value  of  /*.  In  order  that  it  should  satisfy  condition  (3)  JA 
must  be  so  taken  that 

/.(/<)  =  o;       T,0*p  (13) 

that  is,  //  must  be  a  root  of  the  transcendental  equation  (13). 

It  was  shown  by  Fourier  that  ./„(//)  =  o  has  an  infinite  num- 
ber of  real  positive  roots,  any  one  of  which  can  be  obtained  to 
any  required  degree  of  approximation  without  serious  diffi- 
culty. Let  //,,  /*„,  //,,...  be  these  roots  ;  then 

u  =  A,J9(ns)  sinh  (pjs)  +  AJ.(^r)  sinh  (/i^) 

+  AJ0(^r)  sinh  (^2)  +  .  .  .    (14) 
is  a  solution  of  (i)  which  satisfies  (2)  and  (3). 


24  HARMONIC    FUNCTIONS. 

If  now  we  can  develop  unity  into  a  series  of  the  form 

+  BJJwr)  +  ^./.(/V)  +  .  •  •  , 
.  sinh  (  u.s)  T  .         .  J5,  sinh  (  u. 


~l 

>+-  •  J 


satisfies  (i)  and  the  conditions  (2),  (3),  and  (4). 
We  shall  see  later  (Art.  21)  that  if//*)  =  - 


dx 

I    —   9\    -/o''r~''  /       I     -'QV"«'  /    _  I      ^  ov^-s-  /  /    .r-v 

I,     /-/M\~M     7^y\^,,     /•/'»\T^<>*  V10^ 


for  values  of  r  <  i. 
Hence 


—  200l      J  °^1'  '  S'"h  ^'^    -I-     /.(^r)  sinh  (^fg)      .  /    7x 

^s«JU|  /•/..\_:.-i-/'^...\~t       ..     7"  / ..  \  _:_u  /~  ..  \     I     '  *  *   I       \    7 ) 


lifi(^i)  sinh  (2/1,)        A*«/i(^t)  sinh(; 
is  our  required  solution. 

At  the  point  r  =  o,  £  =  I  (17)  reduces  to 
sinh  /*,  .  sinh  yu2 


sn    u.  sn    u.  -\ 

u  —  200 — -I — •  —I- 

L/'./aOO  sinh  (2/^J       ^,/1(//t)  sinh  (2yw2)  ^      '  J 

i         1 


=  IOO     — 


./X/O  cosh  ^       ^7,00  cosh 
since /0(o)  =  i  and  sinh  (2*)  =  2  sinh  ^r  cosh  ;r. 

If  we  use  a  table  of  Hyperbolic  functions*  and  Tables  II 
and  III,  at  the  end  of  this  chapter,  the  computation  of  the 
value  of  u  is  easy.  We  have 

/i,  =  2.405  yua=       5.520 

/,(/*.)  =  o-5  i?p  /,W  =  -  0.3402 


colog     //,  =  9.6189  —  10    colog     //,  =       9.2581    —  10 
"    JM  =  0.2848  "    7,W=       o.4683« 

"  cosh^,=  9.2530  —  10       "  cosh^4=:       7.9037    —  10 


9.1567  —  10  7.63oi«  —  10 


*  See  Chapter  IV,  pp.   162,  163,  for  a  four-place  table  on  hyperbolic  func- 
tions. 


PROBLEM    IN    BESSEL  S    FUNCTIONS. 


»/i(/0  cosh  /O~'  =      0.1434 

~'  =  -  0-0058 


0.1376;          «  =  i3 
At  the  point  r  =  £,  z  —  I,  (17),  reduces  to 


. 

/',/,  W  cosh  /i,      /*,/,(/0  cosh;*, 

j  =  0.6698 


-  10 
,(/!,)  cosh  //,  =  9.  1  567     -  10 


8.9826    —  10; 
/.(*/«,)  =  -  0-1678 


log  /0(tM  =          9.2248W  -   10 

colog  /*j/,(/0  cosh/7,  =       7.6301;*  —  10 

6.8549   —  10; 

=  0.0961 


,)  cosh  //, 


0.0007  m 


cosh  ^      0-°968  '         «  =  9°-7 

If  the  radius  of  the  cylinder  is  a  and  the  altitude  b,  we  have 
only  to  replace  //  by  j*a  in  (13)  ;  2/1, ,  2//,,  ...  in  the  denomi- 
nators of  (15)  and  (17)  by  pj,  pj),  .  .  . ;  and  //,,  //.,,  ^us,  .  .  .  in 
the  denominators  of  (16)  and  (17)  by  /*,#,  //.,#,  //s«,  .... 

Prob.  3.  One  base  and  the  convex  surface  of  a  cylinder  20  cen- 
timeters in  diameter  and  30  centimeters  high  are  kept  at  zero  tem- 
perature and  the  other  base  at  100  degrees  Centigrade.  Find  the 
temperature  of  a  point  in  the  axis  and  20  centimeters  from  the  cold 
base,  and  of  a  point  5  centimeters  from  the  axis  and  20  centimeters 
from  the  cold  base  after  the  temperatures  have  ceased  to  change. 

Ans.   1 3°.  9;  9°.6. 


26  HARMONIC    FUNCTIONS. 

ART.  6.    THE  SINE  SERIES. 

As  we  have  seen  in  Art.  3,  it  is  sometimes  important  to  be 
able  to  express  a  given  function  of  a  variable,  x,  in  terms  of  sines 
of  multiples  of  x.  The  problem  in  its  general  form  was  first 
solved  by  Fourier  in  his  "  Theorie  Analytique  de  la  Chaleur" 
(1822),  and  its  solution  plays  an  important  part  in  most  branches 
of  Mathematical  Physics. 

Let  us  endeavor  to  so  develop  a  given  function  of  x,f(x\ 
in  terms  of  sin  x,  sin  2.x,  sin  $x,  etc.,  that  the  function  and  the 
series  shall  be  equal  for  all  values  of  x  between  o  and  n. 

We  can  of  course  determine  the  coefficients  at,  at,  a3,  .  .  .  an 
so  that  the  equation 

f(x)  =  tf,  sin  x  -f-  tf2  sin  2x  -f-  a3  sin  $x  -}-...  -f-  an  sin  nx   (i) 

shall  hold  good  for  any  n  arbitrarily  chosen  values  of  x  between 
O  and  n\  for  we  have  only  to  substitute  those  values  in  turn 
in  (i)  to  get  n  equations  of  the  first  degree,  in  which  the  n  co- 
efficients are  the  only  unknown  quantities. 

For  instance,  we  can  take  the  n  equidistant  values  Ax, 


71 

^Ax,  .  .  .  nAx,  where  Ax  =  -  ,  and  substitute  them  for  x  in 

n  -\-  i 

(i).     We  get 

f[Ax)  =    at  sin  Ax  -f-    a^  sin  2  Ax  -\-    aa  sin  3  Ax  -f-  .  . 

-f-  an  sin  nAx, 

j\2.Ax)  =  #,  sin  2Ax  -f-    #a  sin  ^Ax  -(-    a3  sin  6  Ax  -{-  .  . 

-f-    an  sin  2,nAx, 

\  (2) 
f[$Ax)  —  at  sin  3  Ax  -j-    «a  sin  6  Ax  -f-    «3  sin  <^Ax  4-  .  . 

+  an  sin  3;?  Ax, 


J\nAx}  =  #,  sin  n  Ax  -f-  a,  sin  2nAx  -j-  #3  sin 

-f-  an  sin  n*Ax, 
n  equations  of  the  first  degree,  to  determine  the  n  coefficients 

<*,-»  a*>  af  •••&*• 

Not  only  can  equations  (2)  be  solved  in  theory,  but  they 
can  be  actually  solved  in  any  given  case  by  a  very  simple  and. 


THE    SINE    SERIES.  27 

ingenious  method  due  to  Lagrange,*  and  any  coefficient  am  can 
be  expressed  in  the  form 


-^  AfA*)  sin  (K*nA*\  (3) 


K  =  l 


If  now  n  is  indefinitely  increased  the  values  of  x  for  which 
(i)  holds  good  will  come  nearer  and  nearer  to  forming  a  con- 
tinuous set  ;  and  the  limiting  value  approached  by  am  will 
probably  be  the  corresponding  coefficient  in  the  series  required 
to  represent /(.z)  for  all  values  of  x  between  zero  and  n. 

Remembering  that  (»  +  \)Ax  =  n,  the  limiting  value  in 
question  is  easily  seen  to  be 

IT 

am  =  -  Cf(x)  sin  mxdx.  (4) 

7Tt/ 
0 

This  value  can  be  obtained  from  equations  (2)  by  the  fol- 
lowing device  without  first  solving  the  equations : 

Let  us  multiply  each  equation  in  (2)  by  the  product  of  Ax 
and  the  coefficient  of  am  in  the  equation  in  question,  add  the 
equations,  and  find  the  limiting  form  of  the  resulting  equation 
as  n  increases  indefinitely. 

The  coefficient  of  any  a,  aK  in  the  resulting  equation  is 

sin  KAx  sin  mAx .  Ax  -\-  sin  2,KAx  sin  2,mAx .  Ax  -}-  ,  .  . 
-f-  sin  nKAx  sin  nmAx .  Ax. 

Its  limiting  value,  since  (n-\-  \)Ax  =  TC,  is 

ir 

/  sin  KX  sin  mx.dx\ 
but 

w  w 

I  sin  KX  sin  mx .  dx  =  \  I  [cos  (m  —  K)X  —  cos(m  -}-  K)x~\dx—Q 

0  0 

if  m  and  K  are  not  equal. 

*  See    Riemann's  Partielle    Differcntialgleichungen,  or    Byerly's    Fourier's 
Series  and  Spherical  Harmonic?. 


28  HARMONIC    FUNCTIONS. 

The  coefficient  of  am  is 

//;tr(sin2  mAx  -\-s\rf  2m  Ax  -j-  sin2  ynAx  -f-  .  .  .  -f-  sin2  nmAx\ 
Its  limiting  value  is 

IT 

71 


y»     . 
s 


sm   w;r  .x  =  —. 

2 
o 


The  first  member  is 


/(  J^r)  sin  7«z/;r  .  J^r  -\-f(2.Ax)  sin  2mAx  .  Ax  -f-  .  .  . 

-j-/(«^-^)  sin  mnAx  .  Ax, 
and  its  limiting  value  is 

/  f(x)  sin  mx  ,  dx. 

0 

Hence  the  limiting  form  approached  by  the  final  equation 
as  n  is  increased  is 

»r 

/  J\x]  sin  mx  .  dx  =  —  am. 


0 


Whence 


2  /' 

«»  =  -J  f(*}  sin  «»•<&  (5) 

7T0 

as  before. 

This  method  is  practically  the  same  as  multiplying  the 
equation 

f(x)  =  #,  sin  x  -j-  a3  sin  2^r  -f-  «s  sin  $x  -}- .  .  .  (6) 

by  sin  mx .  dx  and  integrating  both  members  from  zero  to  ft. 

It  is  important  to  realize  that  the  considerations  given  in 
this  article  are  in  no  sense  a  demonstration,  but  merely  estab- 
lish a  probability. 

An  elaborate  investigation  *  into  the  validity  of  the  develop- 
ment, for  which  we  have  not  space,  entirely  confirms  the  results 
formulated  above,  provided  that  between  x  =  o  and  x  =  n  the 

*  See  Art.  to  for  a  discussion  of  this  question. 


THE    SINE   SERIES.  Z'J 

function  is  finite  and  single-valued,  and  has  not  an  infinite  num- 
ber of  discontinuities  or  of  maxima  or  minima. 

It  is  to  be  noted  that  the  curve  represented  by  y  =  f(x) 
need  not  follow  the  same  mathematical  law  throughout  its 
length,  but  may  be  made  up  of  portions  of  entirely  different 
curves.  For  example,  a  broken  line  or  a  locus  consisting  of 
finite  parts  of  several  different  and  disconnected  straight  lines 
can  be  represented  perfectly  well  by  y  =  a.  sine  series. 

As  an  example  of  the  application  of  formula  (5)  let  us  take 
the  development  of  unity. 

Here  f(x)  =  I. 


«• 
am  =  —    /  sin  mx  .  dx  ; 

7t  i/ 


/ 
si 


cos  mx 
sin  mx  .  dx  =  --  . 


m 


V 

/I  I 

sin  mx.dx  =  —  (i  —  cos  mrr]  =  —  [i  —  (  —  iY*l 
m  m 

0 

=  o  if  m  is  even 

=  —  if  m  is  odd. 
m 

4  /sin  x    .   sin  \x  .   sin  ^x    .  sin  Jx   .         \ 
Hence  ,  =  i(—  +  _J-  +  _J-  +-J-.  +  ...).   (7) 

It  is  to  be  noticed  that  (7)  gives  at  once  a  sine  development 
for  any  constant  c.     It  is, 

c  =  4c(s\nx       sin  3*       sin  5*  \ 

n  \     i  3  5  "/' 

Prob.  4.  Show  that  for  values  of  x  between  zero  and  it 


x      sn  zx  .   sn  30:      sn 


fL\  //   \  _  4  T8^11  x      s^n  3^      s'n  S1^      s'n  7-^  j^ 

=     ""  "~     ~~    ~~ 


"30  HARMONIC  FUNCTIONS. 

if  /(x)  =  x  for  o  <  x  <  —  ,  and  f(x)  =  n  —  x  for  —  <  x  <  n. 

to  /(*)  = 

2  Fsin  x   .    2  sin  2x  ,  sin  T.X  .   sin  zx  ,   2  sin  6x  .  sin  nx   , 

-- 


if  /(.#)  —  i  for  o  <  x  <  —  ,  and  /(x)  =  o  for  —  <  a:  <  TT. 

(d}  sinh  jc  = 

2  sinh  7t  Pi     .  2     . 


nh  7t  Pi     .  2     .  i    3     •  4     • 

—  sin  x  --  sin  2X  -\-  —  sin  T.X  --  sin  AX  -+-  .  .  .    . 
n       [_2  5  rio  17  J 


(e)  x"  = 


2f/zr2       4\    .  T?    .  In*       4\  .  n*  . 

---  -,    sin  x  --  sin  2x  -\-\  ---  a  sin  -ix  --  sin  4x4-  .  . 
7rL\i        i  /  2  \3       3/  4  J 


ART.  7.    THE  COSINE  SERIES. 

Let  us  now  try  to  develop  a  given  function  of  x  in  a  series 
of  cosines,  using  the  method  suggested  by  the  last  article. 
Assume 

f(x)  =  bt  -J-  bl  cos  x  -\-b^  cos  2.x  -j-  /^s  cos  3^  -f-  .  .  .          (  i  ) 

To  determine  any  coefficient  ^m  multiply  (i)  by  cos  mx  .dx 
and  integrate  each  term  from  o  to  TT. 


cos  mx  .x  =  o. 

0 


j 

0 

IT 

/  bk  cos  kx  cos  w^r  .  dx=.Q,     if  »z  and  y^  are  not  equal. 
o 

7T 

/7T 
bm  cos"  w^r   ^  =  —  bm,     if  w  is  not  zero. 
2  . 

o 

7T 
2  /' 

Hence  £w  =  —  /  f(x)  cos  mx  .dx,  (2) 

0 

jf  /«  is  not  zero. 


THE   COSINE   SERIES.  31 

To  get  b0  multiply  (i)  by  dx  and  integrate  from  zero  to  n. 

•a 
Jb.dx  =  bjt, 

0 

IT 

/  bk  cos  kx  .  dx  =  o. 

0 

w 

Hence  b0  =  —J*f(x}dx,  (3) 

0 

which  is  just  half  the  value  that  would  be  given  by  formula  (2) 
if  zero  were  substituted  for  m. 

To  save  a  separate  formula  (i)  is  usually  written 
f(x)  =  ££0  +  b,  cos  x  +  £,  cos  2x  +  £s  cos  3*  +  .  .  .,       (4) 

and  then  the  formula  (2)  will  give  bn  as  well  as  the  other  coef- 
ficients. 

Prob.  5.  Show  that  for  values  of  x  between  o  and  n 

_TT      4  /cos*      cos  3*      cos  5*  \ 

-2~n\~^~      ~7~      ~7~         "J5 

7t         8  /COS  2*         COS  6.X    ,     COS   10 


\ 

-  -  -j, 


if  /(^c)  =  *  for  o  <  x  <  —  ,  and  f(x)  =  TT  —  x  for  —  <  A,  x  TT; 


W 


if  /(*)  =  i  for  o  <  x  <  —  ,  and  f(x)  =  o  for  —  <  x  <  TTV 

2  2 

21    I  I 

(</)  sinh  x  =  -    -(cosh  7f  —  i)  --  (cosh  n  -j-  i)  cos  * 

7T|_  2  2 

-|  —  (cosh  it  —  i)  cos  2^:  --  (cosh  n  -j-  i)  cos  30:  +  .  .  .    ; 


COS  2X         COS  t*         COS 


32  HARMONIC    FUNCTIONS. 

ART.  8.     FOURIER'S  SERIES. 

Since  a  sine  series  is  an  odd  function  of  x  the  development 
of  an  odd  function  of  x  in  such  a  series  must  hold  good  from 
x  =  —  it  to  x  =  TT,  except  perhaps  for  the  value  x  =  o,  where 
it  is  easily  seen  that  the  series  is  necessarily  zero,  no  matter 
what  the  value  of  the  function.  In  like  manner  we  see  that 
if  f(x)  is  an  even  function  of  x  its  development  in  a  cosine 
series  must  be  valid  from  x  =  —  n  to  x  =  n. 

Any  function  of  x  can  be  developed  into  a  Trigonometric 
series  to  which  it  is  equal  for  all  values  of  x  between  —  n  and  n. 

Let/(;r)  be  the  given  function  of  x.  It  can  be  expressed 
as  the  sum  of  an  even  function  of  x  and  an  odd  function  of  x 
by  the  following  device  : 

*)     A*)  -A-*) 


identically  ;  but         '  ~'-/^  --  1  is   not  changed   by  reversing 

£ 

the  sign  of  x  and  is  therefore  an  even  function  of  x\  and  when 

f(x\  —  f(—  x\ 
we  reverse  the  sign  of  x,    --  -  is  affected  only  to  the 

2 

extent  of  having  its  sign  reversed,  and  is  consequently  an  odd 
function  of  x. 

Therefore  for  all  values  of  x  between  —  it  and  n 


f(x\  -4-  /[  —  x)        i  ,         ,  , 

yv  ;    '  •        —  '  =  -£0  -(-  1>,  cos  x  -f-  £,  cos  2x  -|-  £3  cos  3*  -f-  .  .  . 

2  2 

2  rA*)  +A—  *)  ; 

where  bm  —  —  I  yv  '  —*•  cos  w^r  .  dx  ; 

7T  t/  2 


ffx\  _  ft  _  x\ 

and       -l-^—L  —  ^  -  '-  =  al  sin  ^  +  ^,  sin  2x  -f-  ^3  sin  ^x  -f-  .  .  . 


2     /y[;r)  —  /(—  ^r)     . 
where  «„  =  -  /  :iA—  -       —  -  -  sin  mx  .  dx. 

7t  »/  2 


FOURIER'S  SERIES.  33 

bm  and  am  can  be  simplified  a  little. 

2    //(*)  +  /(-•*) 
£»  =  --/  yv  '^  ^ -cos  mx.  d^ 

71  U  2 

o 

It  IT 

=  —    jf(*)  cos  mx  .  dx+Jf(—x)  cos  mx  .  dx\; 
o  o  -1 

but  if  we  replace  x  by  —  x,  we  get 

it  — »  0 

J  /(•—  #)  cos  nix .  dx=—J  f(x)  cos  mx.dx  =  J  f(x)cos  mx.dx, 

0  -IT 

»r 

and  we  have  ^w  =  —  I  /(*)  cos  w^r .  dx. 

—•a 

In  the  same  way  we  can  reduce  the  value  of  am  to 

ir 

—  /  f(x^  sin  mx .  dx. 

71  t/ 

— it 
Hence 

f(x)  =•  -  60  -\-  l>l  cos  x  -}-  bt  cos  2.x  -f-  <^s  cos  3*  -(-... 

M 

-|-  #,  sin  JT  -}"  «,  sin  2^r  -|-  «3  sin  3^r  -}-...,    (2) 

JT 

where  #„,  =  —  /  f(x)  cos  wjr .  ^,  (3) 

— n 
•n 
and  am  =  —  I  f(x}  sm  mx .  dx,  (4) 

7T  t/ 

— it 

and  this  development  holds  for  all  values  of  x  between  —  n 
and  TI. 

The  second  member  of  (2)  is  known  as  a  Fourier's  Series. 

The  developments  of  Arts.  6  and  7  are  special  cases  of 
development  in  Fourier's  Series. 

Prob.  6.  Show  that  for  all  values  of  x  from  —  n  to  n 
2  sinh  TrFi       i  i  i  i  ~| 

f*  =• COS*  H COS2X COS  $X-\ COS4.X-H... 

7T          [_2          2  '5  10  17 


54  HARMONIC    FUNCTIONS. 

2  sinh  7t  |~i     .  2  .  3     .  4    . 

H —  sin  x sin  2X  -4-  —  sin  3^ sin  AX  +  . . .    . 

7i       L_2  5  10  17  J 

Prob.  7.  Show  that  formula  (2),  Art.  8,  can  be  written 

f(x)  =  -f0  COS/?0  -j-  £,  COS  (x  —  /?j)  +  ^a  COS  (2^  —  fi^) 

+  ^3  COS  (3*  —  /?,)  -j-  .  .  .  , 

where  cm  —  (a^  +  £,„")»     and     fim  =  tan"1  -r^- 

Prob.  8.  Show  that  formula  (2),  Art.  8,  can  be  written 

T 

f(x)  =-c,  sin  ft.  +  fl  sin  (^  +  A)  +  ^  sin  (2;c  +  A) 

2 

+  ^3  sin  (3*  +  /?,)  +  .  . . , 

b 
where  cm  —  (am*  +  bm  )*     and     pw  =  tan"1  — . 

ART.  9.    EXTENSION  OF  FOURIER'S  SERIES. 

In  developing  a  function  of  x  into  a  Trigonometric  Series  it 
is  often  inconvenient  to  be  held  within  the  narrow  boundaries 
x  =  —  rt  and  x  =  n.  Let  us  see  if  we  cannot  widen  them. 

Let  it  be  required  to  develop  a  function  of  x  into  a 
Trigonometric  Series  which  shall  be  equal  to  f(x]  for  all  values 
of  x  between  x  =  —  c  and  x  =  c. 

Introduce  a  new  variable 


which  is  equal  to  —  n  when  x  =  —  c,  and  to  n  when  x  =  c. 

f(x)  = /( — z\  can  be  developed  in  terms  of  z  by  Art.  8, 
(2),  (3),  and  (4).     We  have 

/(.*"•*)  =  2  *°  ^  ^  C°S  *  ~^  ^  C°S  2Z  +  £•  cos  3*  +  . . . 

-f-  tft  sin  ^  -(-  a,  sin  2£  -J-  ^3  sin  3.2  -j-  . .  . ,     (i) 

where  bm  =  — //( — *}  cos  *»# .  afe,  (2) 

71  U          \  71     I 


EXTENSION    OF    FOURIER'S    SERIES, 

and  am  =  —Jf\—z\  sin  mz  .  dzy  (3) 

—  IT 

and  (i)  holds  good  from  z  =  —  n  to  z  =  n. 

Replace  z  by  its  value  in  terms  of  x  and  (i)  becomes 

i  nx 

/(•*•)  =  -£.  +  £,cos 

£ 

a  sin 


C>  c- 

•nx  2,nx 


^nx 
+  tf3  sin—  -+...;    (4) 


6- 

and  (2)  and  (3)  can  be  transformed  into 

c 
,  I       /*  ./v    \  WIT-*"    7 

bm  =  —J  f(x)  cos  —^-dx,  (5) 


i  r  ft  \  -   mnx  j 

am  =  —  J  f(x)  sin  —^—dxt  (6) 

—  c 

and  (4)  holds  good  from  x  =  —  c  to  x  •=  c. 

In  the  formulas  just  obtained  c  may  have  as  great  a  value 
as  we  please  so  that  we  can  obtain  a  Trigonometric  Series  for 
f(x]  that  will  be  equal  to  the  given  function  through  as  great 
an  interval  as  we  may  choose  to  take. 

It  can  be  shown  that  if  this  interval  c  is  increased  indefi- 
nitely the  series  will  approach  as  its  limiting  form  the  double 

00  00 

integral  --  /  f(\)d\  I  cos  a(h  —  x}da,  which   is   known    as  a 

—  oo  0 

Fourier's  Integral.     So  that 

+  00  ao 

f(x)  =  -W"  /(A>A  f  cos  a(\  -  x}da  (7) 

-09  0 

for  all  values  of  x. 

For  the  treatment  of  Fourier's  Integral  and  for  examples 
of  its  use  in  Mathematical  Physics  the  student  is  referred  to 
Riemann's  Partielle  Differentialgleichungen,  to  Schlomilch's 
Hohere  Analysis,  and  to  Byerly's  Fourier's  Series  and 
Spherical  Harmonics. 


36  HARMONIC    FUNCTIONS. 

Prob.  9.  Show  that  formula  (4),  Art.  9,  can  be  written 

xt     \  l  (7tX  o    \      ,  l27tX  \ 

f(x)  =  -cQ  cos  /?„  +  c,  cos  (—-  Pi)  +  f*  cos  (—  --  fl*j 


-  A  +..., 


where  c....  =  (<",,.,"    \-  bm^     and     ftm  =  tan"1  ~, 

bm 

Prob.  10.  Show  that  formula  (4),  Art.  9,  can  be  written 
/(*)  =  ^0  sin  ^o  +  ^i  sin  (-^  +  A,J  +  c,  sin  f^— ^  + 


where  ^»,  =   awa  +  £«**     and 


ART.  10.    DIRICHLET'S  CONDITIONS. 

In  determining  the  coefficients  of  the  Fourier's  Series  rep- 
resenting f(x)  we  have  virtually  assumed,  first,  that  a  series  of 
the  required  form  and  equal  to  f(x]  exists;  and  second,  that 
it  is  uniformly  convergent  ;  and  consequently  we  must  regard 
the  results  obtained  as  only  provisionally  established. 

It  is,  however,  possible  to  prove  rigorously  that  if  f(x)  is 
finite  and  single-valued  from  x  =  —  n  \.Q  x  =.  n  and  has  not 
an  infinite  number  of  (finite)  discontinuities,  or  of  maxima  or 
minima  between  x  =  —  n  and  x  —  rry  the  Fourier's  Series  of 
(2),  Art.  8,  and  that  Fourier's  Series  only,  is  equal  to  f(x} 
for  all  values  of  x  between  —  n  and  TT,  excepting  the  values  of 
x  corresponding  to  the  discontinuities  of  f(x\  and  the  values 
it  and  —  TT;  and  that  if  c  is  a  value  of  x  corresponding  to  a 
.discontinuity  of  f(x),  the  value  of  the  series  when  x  =  c  is 


«)];    and  that  when   *  =  n  or 

x  =  —  Tt  the  value  of  the  series  is  ^[/(TT)  -}-/(—  TT)]. 

This  proof  was  first  given  by  Dirichlet  in  1829,  and  may  be 
found  in  readable  form  in  Riemann's  Partielle  Differential- 
gleichungen  and  in  Picard's  Traite  d'Analyse,  Vol.  I. 


DIRICHLET'S  CONDITIONS. 


3? 


A  good  deal  of  light  is  thrown  on  the  peculiarities  of  trigo- 
nometric series  by  the  attempt  to  construct  approximately  the 
curves  corresponding  to  them. 

If  we  construct  y  —  al  sin  x  and  y  —  at  sin  2x  and  add  the 
ordinates  of  the  points  having  the  same  abscissas,  we  shall  ob- 
tain points  on  the  curve 

y  =  al  sin  x  +  ay  sin  2x. 

If  now  we  construct  y  —  as  sin  3*  and  add  the  ordinates  to 
those  of  y  —  at  sin  x  +  at  sin  2.x  we  shall  get  the  curve 

y  —  a,  sin  x  +  at  sin  2x  +  a3  sin  3^. 

By  continuing  this  process  we  get  successive  approximations  to 
y  —  a,  sin  x  +  *,  sin  2x  +  a,  sin  3*  +  a,  sin  4*  +  ... 


O/S 


II 


X 


y 

o 

CE3 

y 

Tt 
III 

! 

IY 


Let  us  apply  this  method  to  the  series 
y  -  sin  x  +  \  sin  3*  +  |  sin  *>x  + (i)  (See  (7),  Art.  6.) 

i/  =  o  when  x  —  o,  —  from  *  =  o  to  x  =  TT,  and  o  when  x  —  TT. 

4 

It  must  be  borne  in  mind  that  our  curve  is  periodic,  hav- 
ing the  period  2?r,  and  is  symmetrical  with  respect  to  the 
origin. 

The  preceding  figures  represent  the  first  four  approxima- 


444680 


38  HARMONIC    FUNCTIONS. 

tion  to  this  curve.  In  each  figure  the  curve  y  =  the  series, 
and  the  approximations  in  question  are  drawn  in  continuous 
lines,  and  the  preceding  approximation  and  the  curve  corre- 
sponding to  the  term  to  be  added  are  drawn  in  dotted  lines. 

Prob.  11.  Construct  successive  approximations  to  the  series 
given  in  the  examples  at  the  end  of  Art.  6. 

Prob.   1 2.  Construct  successive  approximations  to  the  Maclaurin's 

x3         x* 
Series  for  sinh  x,  namely  x  -\ -\ j-  +  •  •  • 

O  *  D  * 

ART.  11.    APPLICATIONS  OF  TRIGONOMETRIC  SERIES., 

(a)  Three  edges  of  a  rectangular  plate  of  tinfoil  are  kept 
at  potential  zero,  and  the  fourth  at  potential  I.  At  what  po- 
tential is  any  point  in  the  plate  ? 

Here  we  have  to  solve  Laplace's  Equation  (3),  Art.  I, 
which,  since  the  problem  is  two-dimensional,  reduces  to 


subject  to  the  conditions  V  =  o  when  x  =  o,  (2) 

V  =  o     "       x  =  a,  (3) 

V  =  o     "      y  =  o,  (4) 

V  =  i     "      j'  =  3.  (5) 

Working  as  in  Art.  3,  we  readily  get  sinh  fiy  sin  /ir, 
sinh  /?/  cos  fix,  cosh  /ty  sin  fix,  and  cosh  /?j  cos  fix  as  particu- 
lar values  of  V satisfying  (i). 

.  ,   mny    .    tmtx        .  r      ,  \    ,  \   ,  \         i  /  \ 
V  =  sinh  —  -  sin  -     -  satisfies  (i),  (2),  (3),  and  (4). 


•  •  i_  ny  •  1  ^y 

sinh  -  sinh-i-^ 

V  =  ±\    ^sin^+l-    _•    *,«=i+...          (6) 


is  the  required  solution,  for  it  reduces  to  i  when  y  =  b.     See 
(7),  Art.  6. 


APPLICATIONS    OF    TRIGONOMETRIC    SERIES.  39 

(b]  A  harp-string  is  initially  distorted  into  a  given  plane 
curve  and  then  released  ;  find  its  motion. 

The  differential  equation  for  the  small  transverse  vibrations 
of  a  stretched  elastic  string  is 

9>-    fl.9^ 

2  '    *     *  ' 


as  stated  in  Art.   i.     Our  conditions   if  we  take  one  end  of 
the  string  as  origin  are 

y  =  o  when  x  =  o,  (2) 

y  =  o    "    x  =  /,  (3) 

3J 

-a/  =  °  *  =  °>  (4) 

y  =  fx    «      t  -  o.  (5) 

Using  the  method  of  Art.  3,  we  easily  get  as  particular  solutions 
of  (I) 

y  =  sin  fix  sin  afit,  y  =  sin  fix  cos  afit, 

y  —  cos  fix  sin  «/?/,     and    y  =  cos  /?.*•  cos  a  fit. 

.    mnx        mnat  , 

y  =  sm  —  j—  cos—  -.  —   satisfies  (i),  (2),  (3),  and  (4). 

.    mnx        mrcat  /A\ 

am  sm  —  j—  cos  —  ;  —  ,  V°> 


,  /»  WTT^r      ,  /_\ 

where  am  =  j      f(x)  sin  — j— .  «* 

0 

is  our  required  solution  ;  for  it  reduces  to/(^)  when/  =  o.    See 

Art.  9. 

Prob.  13.  Three  edges  of  a  square  sheet  of  tinfoil  are  kept  at 
potential  zero,  and  the  fourth  at  potential  unity  ;  at  what  potential 
is  the  centre  of  the  sheet  ?  Ans.  0.25. 

Prob.  14.  Two  opposite  edges  of  a  square  sheet  of  tinfoil  are 
kept  at  potential  zero,  and  the  other  two  at  potential  unity  ;  at 
what  potential  is  the  centre  of  the  sheet  ?  Ans.  0.5. 

Prob.  15.  Two  adjacent  edges  of  a  square  sheet  of  tinfoil  are 


40  HARMONIC    FUNCTIONS. 

kept  at  potential  zero,  and  the  other  two  at  potential  unity.  At 
what  potential  is  the  centre  of  the  sheet  ?  Ans.  0.5. 

Prob.  16.  Show  that  if  a  point  whose  distance  from  the  end  of  a 

harp-string  is  -th  the  length  of   the  string  is  drawn  aside  by  the 
n 

player's  finger  to  a  distance  b  from  its  position  of  equilibrium  and 
then  released,  the  form  of  the  vibrating  string  at  any  instant  is  given 
by  the  equation 


mnat 


y  —  7  —  —  r~i>   ~~t  sin  —  sin  —  T  cos 

J  ^- 


7  —  —  r~i          t       —      —  T       —  ~r  ' 

(n—  \\n  ^-  \m  n  I  I 

^  '          m=l  ' 

Show  from  this  that  all  the  harmonics  of  the  fundamental  note  of 
the  string  which  correspond  to  forms  of  vibration  having  nodes  at 
the  point  drawn  aside  by  the  finger  will  be  wanting  in  the  complex 
note  actually  sounded. 

Prob.  17.*  An  iron  slab  10  centimeters  thick  is  placed  between  and 
in  contact  with  two  other  iron  slabs  each  10  centimeters  thick.  The 
temperature  of  the  middle  slab  is  at  first  100  degrees  Centigrade 
throughout,  and  of  the  outside  slabs  zero  throughout.  The  outer 
faces  of  the  outside  slabs  are  kept  at  the  temperature  zero.  Re- 
quired the  temperature  of  a  point  in  the  middle  of  the  middle  slab 
fifteen  minutes  after  the  slabs  have  been  placed  in  contact. 
Given  a3  =  0.185  in  C.G.S.  units.  Ans.  io°-3. 

Prob.  18.*  Two  iron  slabs  each  20  centimeters  thick,  one  of  which 
is  at  the  temperature  zero  and  the  other  at  100  degrees  Centigrade 
throughout,  are  placed  together  face  to  face,  and  their  outer  faces 
are  kept  at  the  temperature  zero.  Find  the  temperature  of  a  point 
in  their  common  face  and  of  points  10  centimeters  from  the  com- 
mon face  fifteen  minutes  after  the  slabs  have  been  put  together. 

Ans.  22°.8  ;  15°.  i  ;   i7°.2. 


ART.  12.  f    PROPERTIES  OF  ZONAL  HARMONICS. 

In  Art.  4,  z  =  Pm(x)  was  obtained  as  a  particular  solution  of 
Legendre's  Equation  [(7),  Art.  4]  by  the  device  of  assuming 
that  z  could  be  expressed  as  a  sum  or  a  series  of  terms  of 
Ihe  form  anx*  and  then  determining  the  coefficients.  We 

*  See  Art.  3. 

f  The  student  should  review  Art.  4  before  beginning  this  article. 


PROPERTIES   OF    ZONAL    HARMONICS.  41 

can,  however,  obtain  a  particular  solution  of  Legendre's  equa- 
tion by  an  entirely  different  method. 

The  potential  function  for  any  point  (x,  y,  z)  due  to  a  unit 
of  mass  concentrated  at  a  given  point  (x^y^  £,)  is 

=  ' 


and  this  must  be  a  particular  solution  of  Laplace's  Equation 
£(3),  Art.  i],  as  is  easily  verified  by  direct  substitution. 
If  we  transform  (i)  to  spherical  coordinates  we  get 

V=  -  l  _  =  (2) 

yr1  —  2rr1[cos  0  cos  0,  -j-  sin  0  sin  Ol  cos  (0—0,)]  -f-  r,2 

as  a  solution  of  Laplace's  Equation  in  Spherical  Coordinates 
[(5),  Art.  i]. 

If  the  given  point  (x^y^  #,)  is  taken  on  the  axis  of  X,  as  it 
must  be  in  order  that  (2)  may  be  independent  of  0,  0,  =  o,  and 

J7—  _  _  ^_  _  *  ^3) 

Vr*  —  2rr,  cos  0  -{-  r,8 

is  a  solution  of  equation  (i),  Art.  4. 
Equation  (3)  can  be  written 


(4) 

/  r  ra\-* 

and  if  r  is  less  than  rl  (1—2—  cos  0  -f-  —  J      can  be  developed 

'"i  '"i 

^ra 

into  a  convergent  power  series.     Let  5"/>OT —  be  this  series, 

rm 

pm  being  of  course  a  function  of  0.     Then  F=  —  ^pm—^  is  a 

i  i 

solution  of  (i),  Art.  4. 

Substituting  this  value  of  V  in  the  equation,  and  remem- 
bering that  the  result  must  be  identically  true,  we  get  after  a 


slight  reduction 


42  HARMONIC    FUNCTIONS. 

but,  as  we  have  seen,  the  substitution  of  x  =  cos  0  reduces  this- 
to  Legendre's  equation  [(7),  Art.  4].  Hence  we  infer  that  the 
coefficient  of  the  mth  power  of  z  in  the  development  of 
(i  —  2xz-\-z*)~*  i^  a  function  of  x  that  will  satisfy  Legendre's 
equation. 

(!   _  2**  +  *")-*  =  [I  -Z(2X  -*)]-*, 

and  can  be  developed  by  the  Binomial  Theorem  ;  the  coefficient 
of  zm  is  easily  picked  out,  and  proves  to  be  precisely  the  func- 
tion of  x  which  in  Art.  4  we  have  represented  by  Pm(x\  and 
have  called  a  Surface  Zonal  Harmonic. 
We  have,  then, 


if  the  absolute  value  of  z  is  less  than  I. 
If  x  =  i,  (5)  reduces  to 


but    (i  —  2,3r  +  *')'*  =  (i  —  *)"'  =  i+<s  +  ^  +  <s3  +  -  •  •; 

hence  ^«(0  =  i-  (6) 

Any  Surface  Zonal  Harmonic  may  be  obtained  from  the 
two  of  next  lower  orders  by  the  aid  of  the  formula 


(n  +  !)/>,  +  ,(*)  -  (2»  +  i)^^)  +  w/7,,.  ,(^)  =  o,         (7> 

which  is  easily  obtained,  and  is  convenient  when  the  numerical 
value  of  x  is  given. 

Differentiate  (5)  with  respect  to  z,  and  we  get 


(I  - 
whence 


or  by  (5) 

(i  -  2Mt  +  **)(/>,(*)  +  2^f(«)  .*+  S/'.W  •*'  '  •  -> 

+  P1(X}.z  +  P^.^+  .-0=0.     (8) 


PROBLEMS   IN    ZONAL    HARMONICS.  43 

Now  (8)  is  identically  true,  hence  the  coefficient  of  each 
power  of  z  must  vanish.  Picking  out  the  coefficient  of  z11  and 
writing  it  equal  to  zero,  we  have  formula  (7)  above. 

By  the  aid  of  (7)  a  table  of  Zonal  Harmonics  is  easily  com- 
puted since  we  have  P0(x)  =  i,  and  P^x)  =  x.  Such  a  table 
for  x  =  cos  0  is  given  at  the  end  of  this  chapter. 

ART.  13.    PROBLEMS  IN  ZONAL  HARMONICS. 

In  any  problem  on  Potential  if  Fis  independent  of  0  so 
that  we  can  use  the  form  of  Laplace's  Equation  employed  in 
Art.  4,  and  if  the  value  of  Fon  the  axis  of  X  is  known,  and 

•can  be  expressed  as  2amrm  or  as  ^>  -^qij,  we  can  write  out 
•our  required  solution  as 

F=2amrmPm(cos0)     or     V  =  ^  ^">/?">^S  ^  ; 

r 

jf or  since  Pm(i)  =  I  each  of  these  forms  reduces  to  the  proper 
value  on  the  axis ;  and  as  we  have  seen  in  Art.  4  each  of  them 
.satisfies  the  reduced  form  of  Laplace's  Equation. 

As  an  example,  let  us  suppose  a  statical  charge  of  M  units 
<of  electricity  placed  on  a  conductor  in  the  form  of  a  thin  circu- 
lar disk,  and  let  it  be  required  to  find  the  value  of  the  Poten- 
tial Function  at  any  point  in  the  "  field  of  force  "  due  to  the 
charge. 

The  surface  density  at  a  point  of  the  plate  at  a  distance  r 

from  its  centre  is 

M 


(T  = 


Vd'  -  S 


•and  all  points  of  the  conductor  are  at  potential  — .   See  Pierce's 
Newtonian  Potential  Function  (§  61). 

The  value  of  the  potential  function  at  a  point  in  the  axis 
ot  the  plate  at  the  distance  x  from  the  plate  can  be  obtained 
without  difficulty  by  a  simple  integration,  and  proves  to  be 

M  x*  -  a" 

V  —  —  cos-1-—  —  .  (i) 

2a  x   --  a 


44  HARMONIC    FUNCTIONS. 

The  second  member  of  (i)  is  easily  developed  into  a  power 
series. 

M          ,  x*  -  a* 


—  cos 


-  1 


2a  x*  -j-  «' 

MVn        x        x3         x*         x' 


"I 

•J 


. 

lfjr>* 

Hence 

y  =  Mr*  _  ^/>  (cos  0)  +  - 
#  l_2        «  3 


... 

is  our  required  solution  if  r  <  #  and  #  <  -,  as  is 

,7       M\~a  i  a*  n  ,         N        i  a1  „  , 

F  =  -[-  -   _.  -P9  (cos  60  +  -  --  P4(cos  0) 


(5) 

The  series  in  (4)  and  (5)  are  convergent,  since  they  may  be 
obtained  from  the  convergent  series  (2)  and  (3)  by  multiplying 
the  terms  by  a  set  of  quantities  no  one  of  which  exceeds  one 
in  absolute  value.  For  it  will  be  shown  in  the  next  article  that 
Pm  (cos  6)  always  lies  between  i  and  —  i. 

Prob.  19.  Find  the  value  of  the  Potential  Function  due  to  the 
attraction  of  a  material  circular  ring  of  small  cross-section. 

The  value  on  the  axis  of  the  ring  can  be  obtained  by  a  simple 

M 

integration,  and  is     .  ,  if  M  is  the  mass  and  c  the  radius  of  the 

v  £  -|-  r 

ring.     At  any  point  in  space,  if  r  <  c 

V  =  y  [/>0(cos  0)  -  I  £ 
and  if  r  >  c 


ADDITIONAL    FORMS.  45 


=  *L\£-p  o(COS  0)  -  -  -(COS  0)  +  Il£  C 

*  Lf  2  r  2  .  4  r 


ART.  14.    ADDITIONAL  FORMS. 

(a)  We  have  seen  in  Art.  12  that  Pm(x)  is  the  coefficient  of 
in  the  development  of  (i  —  2xz-\-  z*}~^  in  a  power  series. 

l    -  2X2         2*    -  1  =     I  -  ***''          *  -  •'  ^-i 


If  we  develop  (l  —  #**')-*  and  (i  —  #*-*)-*  by  the  Bi- 
nomial Theorem  their  product  will  give  a  development  for 
(  i  —  2xz  -j-  z*}  -  *.  The  coefficient  of  zm  is  easily  picked  out 
and  reduced,  and  we  get 

/>«(cos  0)  = 

1.3.5...  (2m  — 


2.  4.  6.  ..a» 

i    3.^-1) 

I  .  2  .(2W—  l)(2W  —  3) 


If  m  is  odd  the  parenthesis  in(i)  ends  with  the  term  con- 
taining cos  0  ;  if  m  is  even,  with  the  term  containing  cos  o,  but 
in  the  latter  case  the  term  in  question  will  not  be  multiplied  by 
the  factor  2,  which  is  common  to  all  the  other  terms. 

Since  all  the  coefficients  in  the  second  member  of  (i)  are 
positive,  Pm(cos  0)  has  its  maximum  value  when  6  =  o,  and  its 
value  then  has  already  been  shown  in  Art.  12  to  be  unity. 
Obviously,  then,  its  minimum  value  cannot  be  less  than  —  i. 

(b)  If  we  integrate  the  value  of  Pm(x)  given  in  (11),  Art.  4, 
m  times  in  succession  with  respect  to  x,  the  result  will  be 

1.3.5..  '(2m  —  i),  ,         .„ 

lound  to  differ  from  —  ±-^—,  —  Ti  -  (x  —  l)    by  terms  m- 

(2iri)  \ 

volving  lower  powers  of  x  than  the  mih. 

Hence  />„«  =  JL          f  -  ,)«.  (2) 


46  HARMONIC    FUNCTIONS. 

(c)  Other  forms  for  Pm(x),  which  we  give  without  demon- 
stration, are 

-  (-  0"  Q"  *  .  /x 


P  .A  - 

m 


—  I  . COS 

nj   ' 
o 

0        L  ™J 

(4)  and  (5)  can  be  verified  without  difficulty  by  expanding 
•and  integrating. 

ART.  15.    DEVELOPMENT  IN  TERMS  OF  ZONAL  HARMONICS. 

Whenever,  as  in  Art.  4,  we  have  the  value  of  the  Potential 
Function  given  on  the  surface  of  a  sphere,  and  this  value  de- 
pends only  on  the  distance  from  the  extremity  of  a  diameter, 
it  becomes  necessary  to  develop  a  function  of  6  into  a  series 
of  the  form 

AnPa(cos  ff)  +  A/3, (cos  0)  +  A,PJcos  (f)  +  . 

0        u\  /        I  *        1>  /        I  •       *\  /        I 

or,  what  amounts  to  the  same  thing,  to  develop  a  function  of 
x  into  a  series  of  the  form 

The  problem  is  entirely  analogous  to  that  of  development 
in  sine-series  treated  at  length  in  Art.  6,  and  may  be  solved  by 
the  same  method. 


Assume   f(x)  =  A.PJx)  +  Afl*)  +  AtPfc)  +  .  .  .          (i) 

for  —  i  <  x  <  i.     Multiply  (i)  by  Pn(x)dx  and  integrate  from 
—  I  to  i.     We  get 


M  C 

V 


FORMULAS    FOR    DEVELOPMENT.  47 

We  shall  show  in  the  next  article  that 

i 
I  Pm(x)Pn(x}dx  —  o,     unless  m  =  n, 


and  that 

-1 


?}}1      I       I       /> 

Hence  Am  =  —^~  J  f(x}Pm(x}dx.  (3) 

—  i 

It  is  important  to  notice  here;  as  in  Art.  6,  that  the  method 
we  have  used  in  obtaining  Am  amounts  essentially  to  deter- 
mining Am,  so  that  the  equation 

A*)  =  A.P.(x)  +  ASM  +  ASM  +  •  •  •  +  ASM 

shall  hold  good  for  n  -f-  I  equidistant  values  of  x  between  —  i 
and  i,  and  taking  its  limiting  value  as  n  is  indefinitely  in- 
creased. 

ART.  16.    FORMULAS  FOR  DEVELOPMENT. 
We  have  seen  in  Art.  4  that  z  =  Pm(x]  is  a  solution   of 

Legendre's  Equation  -j-\  (i  —  ^2)  ~      -f-  m(m  -f-  1)3  =•  o.    (i) 
ctx  L-  dx  _J 

dPm(x\~\ 

~*9)~ir  J  +w(w+  o^w  =  o,    (2) 
and       -(l-^- 


Multiply  (2)  by  Pn(x)  and  (3)  by  Pm(x),  subtract,  transpose. 
and  integrate.     We  have 

[«(«+  i)  -  n(n  + 


48  HARMONIC    FUNCTIONS. 


by  integration  by  parts, 

=  o. 


Hence  fpm(x)P«(x)dx  =  O,  (6) 


-i 
unless  m  =  «. 


If  in  (4)  we  integrate  from  x  to  i  instead  of  from  —  i  to  I, 
we  get  an  important  formula. 


y*  n y  n  y 

P  (x]P  (r}dx  = L  -*      (7) 

m(m-\-  i) i  —  n(n-\-  i)  '   v/' 

X 

and  as  a  special  case,  since  P0(x)  —  i. 


(8) 


-  -,  -  :  —  -  , 
m(m  -f-  i) 

unless  m  =  o. 
i 
To  get  flPJ^xftdx  is  not  particularly  difficult.      By  (2), 

-i 
Art.  14, 


By  successive  integrations  by  parts,  noting  that 

Jm  -  K 

-T-  —^(x1  —  i)m  contains  (x?  —  i)K  as  a  factor  if  K  <  m,  and 


FORMULAS    FOR    DEVELOPiMENT.  40 


^"'"(jtT1  _    \\m 

that  —  i—  -5  -  '—  =  (2m}\  we  get 


-  \}mdx  =  j\x  -  \y(x 


m  +  i 


' 


Hence  f\PJ(xNdx  = -t .  (u) 

/        l_         '«  \       /  J  ^  ^2    __[_     T 

-1 

1 
•Prob.  20.  Show  that  /  Pm(x}dx  =  o  if  m  is  even  and  is  not  zero 

0 
m—l  j  -      -      _  ^ 

=  (—  J)  z   -~r~    — \  •  ;.  — -T  if  m  is  odd. 

m(m+  i)    2.4.6  ...  (m—i) 


y»  i 

iPJMr^f  =  --  L~~-     Note  that 
2m  -i-  i 

0 

jc"  is  an  even  function  of  x. 


Prob.  22.  Show  that  if   f(x)  =  o  from  x  =  —  i  to  x  =  o,  and 
)  =  i  from  .r  =•  o  to  x  =  i, 


Prob.  23.  Show  that  /7(0)  =  2  S^^cos  0)     where 

>«=o 


B]  Sm0d0. 


.30  HARMONIC    FUNCTIONS. 

Prob.  24.  Show  that 

0=  -^fi  +  sfi)'/',  (cos  60  +  9(^)X(ccs  H)  +  .  .  .1. 

2    I \2  /  \2  . 4'  I 

i,  Art.  14. 


esc 
See  (i),  Art.  14. 

Prob.  25.  Show  that 


+  (..  -  ,)("  +  -)("-•)  />._(,)  +  . 

2.4  J 

1  1 

TSTote  that    /  xnPm(x)dx  =  —  -  —     I  xn—  ]—dx,  and  use  the 

«/  2mm  \    v  dxm 

-i  -i 

method  of  integration  by  parts  freely. 

Prob.  26.  Show  that  if  Fis  the  value  of  the  Potential  Function 
at  any  point  in  a  field  of  force,  not  imbedded  in  attracting  or  repel- 
ling matter;  and  if  F  =  /(0)  when  r  =  a, 

V=2Am~Pm(cos6}ifr<a 

Ur 

am+1 
and  V  =  2^m-lPm(cos  0)i(r>  a, 


where  ^4«  = I  f(6)Pm(cos  0)  sin  0^. 

2          t/ 

0 

Prob.  27.  Show  that  if 

ca  ., 

y  •=  c  when  r  =  a  ;     K  =  rur<a,    and  r  =  —  if  r  >  «. 

r 


ART.  17.     FORMULAS  IN-  ZONAL  HARMONICS. 

The  following  formulas  which  we  give  without  demonstra- 
tion may  be  found  useful  for  reference  : 


_—„_, 


SPHERICAL    HARMONICS.  51 

ART.  18.    SPHERICAL  HARMONICS. 

In  problems  in  Potential  where  the  value  of  V  is  given  on  the 
surface  of  a  sphere,  but  is  not  independent  of  the  angle  0.  we 
have  to  solve  Laplace's  Equation  in  the  form  (5),  Art.  I,  and 
by  a  treatment  analogous  to  that  given  in  Art.  4  it  can  be 

proved  that 

d*Pm(u)  d'TJi-n 

V  =  r'"  cos  nd)  sin"  (i—          •    and    V  =  r'"  sin  n<h  sin"  6  — 

dvn  <//<" 

where  //  =  cos  0,  are  particular  solutions  of  (5),  Art.  i. 

The  factors  multiplied  by  r"1  in  these  values  are  known  as 
Tesseral  Harmonics.  They  are  functions  of  0  and  0,  and  they 
play  nearly  the  same  part  in  unsymmetrical  problems  that  the 
Zonal  Harmonics  play  in  those  independent  of  0. 

FM,O,  0)  =  AtPm(p)  +n2l(Ancos  n<fy  +  Bn  sin  «0)sin"  d*-?4£ 

m=i  «/<" 

is  known  as  a  Surface  Spherical   Harmonic  of  the  wth  degree, 
and  F=r'"Fm(,u,  0)    and    V  =  -^  Ym(,.<,  0) 

satisfy  Laplace's  Equation,  (5),  Art.  I. 

The  Tesseral  and  the  Zonal  Harmonics  are  special  cases  of 
the  Spherical  Harmonic,  as  is  also  a  form  Pm(cos  y]  known  as 
a  Laplace's  Coefficient  or  a  Laplacian  ;  y  standing  for  the  angle 
between  r  and  the  radius  vector  rl  of  some  fixed  point. 

For  the  properties  and  uses  of  Spherical  Harmonics  we 
refer  the  student  to  more  extended  treatises,  namely,  to 
Ferrer's  Spherical  Harmonics,  to  Heine's  Kugelfunctionen,  or 
to  Byerly's  Fourier's  Series  and  Spherical  Harmonics. 

ART.  19.*    BESSEL'S  FUNCTIONS.    PROPERTIES. 
We  have  seen  in  Art.  5  that  z  =^J^(x]  where 


*  The  student  should  review  Art.  5  before  reading  this  article. 


52  HARMONIC    FUNCTIONS. 

is  a  solution  of  the  equation 

cfz   .    I  dz   . 

^>  +  --r  +  2  =  0''  (2) 

ax       x  dx 

and  we  have   called  Ja(x]  a  Bessel's  Function  or  Cylindrical 
Harmonic  of  the  zero  order. 

_       dj,(x]  _  xV          x*  x»  x6  -1 

~^7~   ~2l_         2T4+2.42.6      2.42.6<.8~i     ••J<2 

is  called  a  Bessel's  Function  of  the  first  order,  and 

*'=/,(*) 
is  a  solution  of  the  equation 


which  is  the  result  of  differentiating  (2)  with  respect  to  x. 

A  table  giving  values  of  J*(x)  and  /,(-*')  W1'U  De  found  at 
the  end  of  this  chapter. 

If  we  write  J9(x)  for  x  in  equation  (2),  then  multiply 
through  by  xdx  and  integrate  from  zero  to  x,  simplifying  the 
resulting  equation  by  integration  by  parts,  we  get 


dx 

or,  since  /,(*)  =  —  -^ — , 


J'xJ0(x}dx  =  xj,(x\  (5) 

0 

If  we  write  Jt(x)  for  z  in  equation  (2),  then  multiply  through 
'by  x*— j- -,  and  integrate  from  zero  to  x,  simplifying  by  inte- 
gration by  parts,  we  get 


APPLICATIONS   OF    BESSEL'S    FUNCTIONS. 

If  we  replace  x  by  jjix  in  (2)  it  becomes 

(Fz  .    i  dz    . 

^  +  ^  +  /J2  =  ° 

(See  (8),  Art.  5).     Hence  z  —  /9(f*x)  is  a  solution  of  (7). 

If  we"  substitute  in  turn  in  (7)J<>(vKx)  and  /0(/vr)  for  -,  mul- 
tiply the  first  equation  by  xj^x),  the  second  by  xJ^Kx\ 
subtract  the  second  from  the  first,  simplify  by  integration  by 
parts,  and  reduce,  we  get 


(8) 


Hence  if  //K  and  //t  are  different  roots  of  ./„(//#)  =  O,  or  of 
—  °»  or  of  wAd"*)  —  A/0(y"«)  =  o, 


=  o.  (9) 

0 

We   give  without   demonstration   the   following  formulas, 
which  are  sometimes  useful  : 


I        n 

^(x)  =  -   I  C0s(^r  COS  0)^0.  (lO) 

7ft/ 
0 

»r 
Jf     /* 

t(x\  =  -  I  sin"  0  cos  (x  cos  0)^/0.  (i  i) 

TTe/ 


They  can  be  confirmed  by  developing  cos  (x  cos  0),  inte- 
grating, and  comparing  with  (i)  and  (3). 

ART.  20.    APPLICATIONS  OF  BESSEL'S  FUNCTIONS. 

(a)  The  problem  of  Art.  5  is  a  special  case  of  the  following  : 
The  convex  surface  and  one  base  of  a  cylinder  of  radius  a 
and  length  b  are  kept  at  the  constant  temperature  zero,  the 
temperature  at  each  point  of  the  other  base  is  a  given  function 
of  the  distance  of  the  point  from  the  center  of  the  base  ;  re- 


HARMONIC    FUNCTIONS. 


quired  the  temperature  of  any  point  of  the  cylinder  after  the 
permanent  temperatures.  have  been  established. 

Here  we  have  to  solve  Laplace's  Equation  in  the  form 


(see  Art.  5),  subject  to  the  conditions 

u  =  o  when  z  =  o, 

u  =  o      "      r  =  a, 

u  =  /(r)  "      z  =  b. 
Starting  with  the  particular  solution  of  (i), 

u  =  sinh(/^r)/0OO,  (2) 

and  proceeding  as  in  Art.  5,  we  get,  if  //,,//,,  /*„,  .  .  .  are  roots 
of  Jt(t*a)  =  o,  (3) 

and        f(r)  =  AJ^r]  +  AJ^r}  +  AMM  +  •  -  -  >      (4) 


(b)  If  instead  of  keeping  the  convex  surface  of  the  cylinder 
at  temperature  zero  we  surround  it  by  a  jacket  impervious  to 
heat  the  equation  of  condition,  u  =  o  when  r  =  a,  will  be  re- 

placed by  —  =  o  when  r  =  a,  or  if  u  =  sinh  (jjufyjfyir)  by 

=  o     when  r  =  a, 


dr 
that  is,  by  — 

or  /,(/«*)  =  o.  (6) 

If  now  in  (4)  and  (5)  ;/,,  ;*,,  jw,,  .  .  .  are  roots  of  (6),  (5)  will 
be  the  solution  of  our  new  problem. 

(c]  If  instead  of  keeping  the  convex  surface  of  the  cylinder 
at  the  temperature  zero  we  allow  it  to  cool  in  air  which  is  at 
the  temperature  zero,  the  condition  u  =  o  when  r  =  a  will  be 

replaced  by  ~   '•  -\-  hu  =  o  when  r  =  a,  h  being  the  coefficient 

or 

of  surface  conductivity. 


DEVELOPMENT    IN    TERMS    OF    BESSEL's    FUNCTIONS.  55 


If  u  =  sinh  (jjiz]J  ^(JJLT']  this  condition  becomes 

—  Pj&r)  +  hJt(nr)  =  o    when  r  —  a, 
or  /"*/,(/"*)  —  ahj^d]  —  o.  (7) 

If  now  in  (4)  and  (5)  //,,//,,  //3  ,  .  .  .  are  roots  of  (7),  (5)  will 
be  the  solution  of  our  present  problem. 

It  can  be  shown  that 

SM  =  o,  (8) 

/.(•*)  =  o,  (9) 

and  */,(*)  -  A/«U')  =  0  (10) 

have  each  an  infinite  number  of  real  positive  roots.*  The 
earlier  roots  of  these  equations  can  be  obtained  without  serious 
difficulty  from  the  table  iorj^x)  and  J^x)  at  the  end  of  this 
chapter. 

ART.  21.    DEVELOPMENT  IN  TERMS  OF  BESSEL'S  FUNCTIONS. 

We  shall  now  obtain  the  developments  called  for  in  the  last 
article. 

Let  Ar)  =  AJJM-+AtfJM  +  AJ.(jis)  +  ...  (0 
/I,,;*,,  //,  ,  etc.,  being  roots  of  /„(//«)  =  O,  or  of  /,(yw«)  =  o,  or 
of 


To  determine  any  coefficient  Ak  multiply  (i)  by  rJQ(f.ikr}dr 
and  interate  from  zero  to  a.     The  first  member  will  become 


Every  term  of  the  second  member  will  vanish  by  (9),  Art. 
19,  except  the  term 


o 


o  o 

by  (6),  Art.  19. 


*  See  Riemann's  Partielle  Differentialgleichungen,  §  97. 


.50  HARMONIC    FUNCTIONS. 


Hence  Ak  =  -  /  rf(f)J <$J*ip)dr.  (2) 


The  development  (i)  holds  good  from  r  =  o  to  r  =  #  (see 
Arts.  6  and  15). 

If  yu,  ,  yua  ,  yus  ,  etc.,  are  roots  oij^a)  =  o,  (2)  reduces  to 


>  /<»  i  Ms  >  etc->  are  roots  of  /,(/*#)  =  o,  (2)  reduces  to 


If  //lt  ^s,   /i,,   etc.,   are    roots  of  jjaj^a)  —  A/0(/^)  =  o, 
(2)  reduces  to 


For  the  important  case  where  f(r)  =  i 

a  a  /J.f.a 

frf(ryt>(nkr}(tr=  frj,(fif)etr=^ 

r  o 

by  (5),  Art.  19;  and  (3)  reduces  to 


(4)  reduces  to 

^*  =  o,  (8) 

except  for  k  =  i,  when  /^  =  o,  and  we  have 

A,  =  i  ;  (9) 

(5)  reduces  to        ^  =  7——       ,          -  r.  (io'\ 


Prob.  28.  A  cylinder  of  radius  one  meter  and  altitude  one  meter 
has  its  upper  surface  kept  at  the  temperature  100°,  and  its  base  and 
convex  surface  at  the  temperature  15°,  until  the  stationary  temper- 
atures are  established.  Find  the  temperature  at  points  on  the  axis 
25,  50,  and  75  centimeters  from  the  base,  and  also  at  a  point  25 
.centimeters  from  the  base  and  50  centimeters  from  the  axis. 

*  \  O    y  O/*  O  Or> 

Ans.  29  .6;  47  .6;  71  .2;  25  .8 


DEVELOPMENT    IN    TERMS    OF    BESSEL  S    FUNCTIONS.  ,j  , 

Prob.  29.  An  iron  cylinder  one  meter  long  and  20  centimeters 
in  diameter  has  its  convex  surface  covered  with  a  so-called  non-con- 
•ducting  cement  one  centimeter  thick.  One  end  and  the  convex 
surface  of  the  cylinder  thus  coated  are  kept  at  the  temperature  zero, 
•  the  other  end  at  the  temperature  of  100  degrees.  Given  that  the  con- 
ductivity of  iron  is  0.185  an<^  °f  cement  0.000162  in  C.  G.  S.  units. 

Find  to  the  nearest  tenth  of  a  degree  the  temperature  of  the  mid- 
dle point  of  the  axis,  and  of  the  points  of  the  axis  20  centimeters 
from  each  end  after  the  temperatures  have  ceased  to  change. 

Find  also  the  temperature  of  a  point  on  the  surface  midway  be- 
tween the  ends,  and  of  points  of  the  surface  20  centimeters  from 
•each  end.  Find  the  temperatures  of  the  three  points  of  the  axis, 
supposing  the  coating  a  perfect  non-conductor,  and  again,  suppos- 
ing the  coating  absent.  Neglect  the  curvature  of  the  coating.  Ans. 
iS°.4;  4°°.  85  ;  72°.8;  15°.  3;  40°.?  ',  72°-5  5  °°-°  5  °°-°  ;  i°-3- 

Prob.  30.  If  the  temperature  at  any  point  in  an  infinitely  long 
cylinder  of  radius  c  is  initially  a  function  of  the  distance  of  the 
point  from  the  axis,  the  temperature  at  any  time  must  satisfy  the 

3«        ,  /3*«   .    i  3«\   ,  x     ... 

equation     —  =  a    \^r\  H  --  ~—  1   (see  Art.   i),  since  it  is  clearly  in- 

dependent of  z  and  0. 
Show  that 


•where,  if  the  surface  of  the  cylinder  is  kept  at  the  temperature 
.zero,  /*,  ,  jua  ,  //s  ,  .  .  .  are  roots  of  Ja(nc)  =  o  and  Ak  is  the  value 
:given  in  (3)  with  c  written  in  place  of  a  ;  if  the  surface  of.  the  cylin- 
der is  adiabatic  ju,,  //,,  yw3,  .  .  .  are  roots  of  J^c]  =  o  and  Ak  is  ob- 
tained from  (4);  and  if  heat  escapes  at  the  surface  into  air  at  the  tem- 
perature zero  yw,,  /*„,  A/,,  ...are  roots  of  HcJ^yc}  —  Ayo(/v)  =  o, 
.and  Ak  is  obtained  from  (5). 

Prob.  31.  If  the  cylinder  described  in  problem  29  is  very  long 
and  is  initially  at  the  temperature  100°  throughout,  and  the  con- 
vex surface  is  kept  at  the  temperature  o°,  find  the  temperature  of  a 
point  5  centimeters  from  the  axis  15  minutes  after  cooling  has  begun  ; 
.first  when  the  cylinder  is  coated,  and  second,  when  the  coating  is 
-absent.  Ans.  97°.  2  ;  o°.oi. 

Prob.  32.  A  circular  drumhead  of  radius  a  is  initially  slightly 
-distorted  into  a  given  form  which  is  a  surface  of  revolution  about 
.the  axis  of  the  drum,  and  is  then  allowed  to  vibrate,  and  z  is  the 
•ordinate  of  any  point  of  the  membrane  at  any  time.  Assuming  that 


58  HARMONIC    FUNCTIONS. 


z  must  satisfy  the  equation  „—  =  f{~~.  +  -     r  \>  subject  to  the  con- 
ditions z  =  o  when  r  =  a,         =  o  when  /  =  o.  and  -  =  f(r]  when 

ot 


(  =  o,  show  that  z  =-  A i_/0(A<1^)  cos  ^^ct  -f-  AJJ^Hj'}  cos  /.///  -(-  .  . . 
where  /<,,  yw,,  yU8, ,  .  .  are  roots  of  Ja(na)  —  o  and  Ak  has  the  value 
given  in  (3). 

Prob.  33.  Show  that  if  a  drumhead  be  initially  distorted  as  in 
problem  32  it  will  not  in  general  give  a  musical  note  ;  that  it  may  be 
initially  distorted  so  as  to  give  a  musical  note  ;  that  in  this  case  the 
vibration  will  be  a  steady  vibration  ;  that  the  periods  of  the  various 
musical  notes  that  can  be  given  are  proportional  to  the  roots  of 
Jn(x)  —  o,  and  that  the  possible  nodal  lines  for  such  vibrations 
are  concentric  circles  whose  radii  are  proportional  to  the  roots  of 
/.(•*)  =  o. 

ART.  22.     PROBLEMS  IN  BESSEL'S  FUNCTIONS. 

If  in  a  problem  on  the  stationary  temperatures  of  a  cylinder 
u  =  o  when  s  =  o,  21  =  O  when  z  =  b,  and  u  =  f(z)  when  r  =  a, 
the  problem  is  easily  solved.  If  in  (2),  Art.  20,  and  in  the  cor- 
responding solution  2  =  cosh  (^J^r]  we  replace  //  by  //z,  we 
can  readily  obtain  z  —  sin  (yu^)/0(//r/')  and  3  =  cos  (l*z)J <,(}*? i) 
as  particular  solutions  of  (i),  Art.  20 ;  and 

x"1         x*  x* 

Jn(xi)  =  i  +—  +  ^-r,  +  y     ,  &  4- ...  (i) 

and  is  real. 

!^°  .  •  k*iz 
f(z\  —  ^  AK  sin  —r 
J  v  '  frr  * 

^        />  J?  -ft  <? 

where  '  Ak  —-j-   I  f(z)  sin     ,-  dz  i2\ 

o 
by  Art.  9. 


Hence  u  =        Ah  sin 


l* 

u     T{* 

b    ) 
is  the  required  solution. 


*.i  -     ,  /  ^M  \ 

J\- 


LAME'S  FUNCTIONS.  5'.i 

A  table  giving  the  values  of  Jo(xi)  will  be  found  at  the  end 
of  this  chapter. 

Prob.  34.  A  cylinder  two  feet  long  and  two  feet  in  diameter  has 
its  bases  kept  at  the  temperature  zero  and  its  convex  surface  at 
100  degrees  Centigrade  until  the  internal  temperatures  have  ceased 
to  change.  Find  the  temperature  of  a  point  on  the  axis  halt  way 
between  the  bases,  and  of  a  point  six  inches  from  the  axis,  half  way 
between  the  bases.  Ans.  72. °i;  8o°.i. 

ART.  23.    BESSEL'S  FUNCTIONS  OF  HIGHER  ORDER. 

If  we  are  dealing  with  Laplace's  Equation  in  Cylindrical 
Coordinates  and  the  problem  is  not  symmetrical  about  an 
axis,  functions  of  the  form 

rn                          i-2  „« 

T(x\—  I  —  -  -  A 

2"7  X«  +  I )  L  2\H  +  I )  ~  24.  2  !(«  +  I  )(W  -f-  2 ) 

play  very  much  the  same  part  as  that  played  by  J0(.r)  in  the 
preceding  articles.  They  are  known  as  Bessel's  Functions  of 
the  «th  order.  In  problems  concerning  hollow  cylinders  much 
more  complicated  functions  enter,  known  as  Bessel's  Functions 
of  the  second  kind. 

For  a  very  brief  discussion  of  these  functions  the  reader  is 
referred  to  Byerly's  Fourier's  Series  and  Spherical  Harmonics  ; 
for  a  much  more  complete  treatment  to  Gray  and  Matthews' 
admirable  treatise  on  Bessel's  Functions. 

ART.  24.    LAME'S  FUNCTIONS. 

Complicated  problems  in  Potential  and  in  allied  subjects  are 
usually  handled  by  the  aid  of  various  forms  of  curvilinear  co- 
ordinates, and  each  form  has  its  appropriate  Harmonic  Func- 
•  tions,  which  are  usually  extremely  complicated.  For  instance, 
Lame's  Functions  or  Ellipsoidal  Harmonics  are  used  when 
solutions  of  Laplace's  Equation  in  Ellipsoidal  coordinates  are 
required ;  Toroidal  Harmonics  when  solutions  of  Laplace's 
Equation  in  Toroidal  coordinates  are  needed. 

For  a  brief  introduction  to  the  theory  of  these  functions 
see  Byerly's  Fourier's  Series  and  Spherical  Harmonics. 


60 


HARMONIC  FUNCTIONS. 


TABLE  I.     SURFACE  ZONAL  HARMONICS. 


e 

P,  (cos  0) 

P3  (cos  0) 

P3  (cos  0) 

P4  (cos  0) 

PS,  (cos  0) 

P6  (COS  0) 

I  P^  (cos  0) 

0° 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1 

.9998 

.9995 

.9991 

.9985 

.9977 

.9967 

.9955 

2 

.9994 

.9982 

.9963 

.9939 

.9909 

.9872 

.982!> 

3 

.9986 

.9959 

.9918 

.9863 

.97i)5 

.9713 

.9617 

4 

9976 

.9927 

.9854 

.9758 

.9638 

.9495 

.9329* 

5 

.9962 

.9886 

.9773 

.9623 

.9437 

.9216 

.8961 

6 

.9945 

.9836 

.9674 

.9459 

.9194 

.8881 

.8522 

7 

.9925 

.9777 

.9557 

.9267 

.8911 

.8476 

.7986 

8 

.9903 

.9709 

.9423 

.9048 

.8589 

.8053 

.7448. 

9 

.9877 

.9633 

.9273 

.8803 

.8232 

.7571 

.6831 

10 

.9848 

.9548 

.9106 

.8532 

.7840 

.7045 

.6164 

11 

.9816 

.9454 

.8923 

.H238 

.7417 

.6483 

.5461 

12 

.9781 

.9352 

.8724 

.7920 

.6966 

.5892 

.4732" 

13 

.9744 

.9241 

.8511 

.7582 

.6489 

.5273 

.3940 

14 

.9703 

.9122 

.8283 

.7224 

.5990 

.4635 

.3219' 

15 

.9659 

.8995 

.8042 

.6847 

.5471 

.3982 

.2454 

16 

.9613 

.8860 

.7787 

.6454 

.4937 

.3322 

.1699- 

17 

.9563 

.8718 

.7519 

.6046 

.4391 

.2660 

.0961 

18 

.9511 

.8568 

.7240 

.5624 

.3836 

.2002 

.0289 

19 

.9455 

.8410 

.6950 

.5192 

.3276 

.1347 

-.0443- 

20 

.9397 

.8245 

.6649 

.4750 

.2715 

.0719 

-.1072 

21 

.9336 

.8074 

.6338 

.4300 

.2156 

.0107 

-.1662 

22 

.9272 

.7895 

.6019 

.3845 

.1602 

-.0481 

-.2201 

23 

.9205 

.7710 

.5692 

.3386 

.1057 

-.1038 

-.2681 

24 

.9135 

.7518 

.5357 

.2926 

.0525 

-.1559 

-.3095 

25 

.9063 

.7321 

.5016 

.2465 

.0009 

-.2053 

-.346* 

26 

.8988 

.7117 

.4670 

.2007 

-.0489 

-.2478 

-.3717 

27 

.8910 

.6908 

.4319 

.1553 

-.0964 

-.2869 

—.3921 

28 

.8829 

.6694 

.3964 

.1105 

-.1415 

-.8211 

-.4052" 

29 

.8746 

.6474 

.3607 

.0665 

-.1839 

-.3503 

-.4114 

30 

.8660 

.6250 

.3248 

.0234 

-.2233 

-.3740 

-.4101 

31 

.8572 

.6021 

.2887 

-.0185 

-.2595 

-.3924 

-.4022 

32 

.8480 

.5788 

.2527 

-.0591 

-.2923 

—  .4052 

-.3876 

33 

.8387 

.5551 

.2167 

-.0982 

-.3216 

-.4126 

-.3670 

34 

.8290 

.5310 

.1809 

-.1357 

-.3473 

-.4148 

-.3409 

35 

.8192 

.5065 

.1454 

-.1714 

-.3691 

-.4115 

-.3096 

36 

.8090 

.4818 

.1102 

-.2052 

-.3871 

-.4031 

—  .2738 

37 

.7986 

.4567 

.0755 

-.2370 

-.4011 

-.3898 

-.2343 

38 

.7880 

.4314 

.0413 

-.2666 

-.4112 

-.3719 

—.1918 

39 

.7771 

.4059 

.0077 

-.2940 

-.4174 

-.3497 

-.146& 

40 

.7660 

.3802 

-.0252 

-.3190 

-.4197 

-.3234 

-.10031: 

41 

.7547 

.3544 

-.0574 

-.3416 

-.4181 

-.2938 

-.0534 

42 

.7431 

.3284 

-.0887 

-.3616 

-.4128 

-.2611 

—  .0065 

43 

.7314 

.3023 

-.1191 

-.3791 

-.4038 

-.2255 

.039* 

44 

.7193 

.2762 

-.1485 

-.3940 

-.3914 

-.1878 

.0846 

45° 

.7071 

.2500 

-.1768 

-.4062 

-.3757 

-.1485 

.1270 

TABLES. 


01- 


TABLE  I.     SURFACE  ZONAL  HARMONICS. 


e 

Pj  (COS  0) 

P2  cos  6) 

Ps  (cos  9) 

P4  (cos  9) 

P5  (cos  9) 

P6  (cos  9) 

P7  (cos  9) 

45= 

.7071 

.2500 

-.1768 

-.4062 

-.3757 

-.1485 

.1270 

46 

.6947 

.2238 

-.2040 

-.4158 

-.3568 

-.1079 

.1666 

47 

.6820 

.1977 

-.2300 

-.4252 

-.3350 

-.0645 

.2054 

48 

.6691 

.1716 

-.2547 

-.4270 

-.3105 

-.0251 

.2349 

49 

.6561 

.1456 

-.2781 

-.4286 

-.2836 

.0161 

.2627 

50 

.6428 

.1198 

-.3002 

-.4275 

-.2545 

.0563 

.2854 

51 

.6293 

.0941 

-.3209   -.4239 

-.2235 

.0954 

.3031 

52 

.6157 

.0686 

-.3401   -.4178 

-.1910 

.1326 

.3153 

53 

.6018 

.0438 

-.3578 

-.4093 

-.1571 

.1677 

.3221 

54 

.5878 

.0182 

-.3740 

-.3984 

-.1223 

.2002 

.3234 

55 

.5736 

-.0065 

-.3886 

-.3852 

-.0868 

.2297 

.3191 

56 

.5592 

-.0310 

-.4016 

-.3698 

-.0510 

.2559 

.3095 

57 

.5446 

-.0551 

-.4131 

-.3524 

-.0150 

.2787 

.294i> 

58 

.5299 

-.07-8 

-.4229 

-.3331 

.0206 

.2976 

.2752 

59 

.5150 

-.1021 

-.4310 

-.3119 

.0557 

.3125 

.2511 

60 

.5000 

-.1250 

-.4375 

-.2891 

.0898 

.3232 

.2231 

61 

.4848 

-.1474 

-.4423 

-.2647 

.1229 

.3298 

.1916 

62 

.4695 

-.1694 

-.4455 

-.2390 

.1545 

.3321 

.1571 

63 

.4540 

-.1908  -.4471 

-.2121 

.1844 

.3302 

.1203 

64 

.4384 

-.2117  -.4470 

-.1841 

.2123 

.3240 

.0818 

65 

'.4226 

-.2321  -.4452 

-.1552 

.2381 

.3138 

.0422 

66 

.4067 

-.2518  -.4419 

-.1256 

.2615 

.2996 

.0021 

67 

.3907 

-.2710  -.4370 

—  .0955 

.2824 

.2819 

-.0375 

68 

.3746 

-.2896  -.4305 

-.0650 

.3005 

.2605 

-.0763 

69 

.3584 

-.3074 

-.4225 

-.0344 

.3158 

.2361 

-.1135 

70 

.3420 

-.3245 

-.4130 

-.0038 

.3281 

.2089 

-.1485 

71 

.3256 

-.3410 

-.4021 

.0267 

.3373 

.1786 

-.1811 

72 

.3090 

-.3568 

-.3898 

.0568 

.3434 

.1472 

-.2099 

73 

.2924 

-.3718 

-.3761 

.0864 

.3463 

.1144 

-.2347 

74 

.2756 

-.3860 

-.3611 

.1153 

.3461 

.OV85 

-.2559 

75 

.2588 

-.3995 

-.3449 

.1434 

.3427 

.0431 

-.2730 

76 

.2419 

-.4112 

-.3275 

.1705 

.3362 

.0076 

-.2848 

77 

.2250 

-.4241 

-.3090 

.1964 

.3267 

-.0284 

-.2919 

78 

.2079 

-.4352 

-.2894 

.2211 

.3143 

-.0644 

-.294:1 

79 

.1908 

-.4454 

-.2688 

.2443 

.2990 

-  .0989 

-.2913 

80 

.1736 

-.4548 

-.2474 

.2659 

.2810 

-.1321 

-.2805 

81 

.1564 

-.4633 

-.2251 

.2859 

.2606 

-.1635 

-.2709 

82 

.1392 

-.4709 

-.2020 

.3040 

.2378 

-.1926 

-.2536 

83 

.1219 

-.4777 

-.1783 

.3203 

.2129 

-.2193 

-.2321 

84 

.1045 

-.4836 

-.1539 

.3345 

.1861 

-.2431 

-.2067 

85 

.0872 

-.4886 

-.129"! 

.3468 

.1577 

-.2638 

—.1779 

86 

.0698 

-.4927 

-.1038 

.3569 

.1278 

-.2811 

-.1460 

87 

.0523 

-.4959 

-.0781 

.3648 

.0969 

-.2947 

-.1117 

88 

.0349 

-.498-3 

-.0522 

.3704 

.0651 

-.3045 

-.073.> 

89 

.0175 

-.4995 

-.0262 

.3739 

.0327 

-.3105 

-.0381 

90° 

.0000 

-.5000 

.0000 

.3750 

.0000 

-.3125 

.0000 

62 


HARMONIC    FUNCTIONS. 


TABLE  II.     BESSEL'S   FUNCTIONS. 


X 

Jt(x) 

Jl(X) 

X 

JttX) 

J\(X) 

X 

Jo(x) 

Ji(x) 

0.0 

1.0000 

0.0000 

5.0 

-.1776 

-.3276 

10.0 

-.2459 

.0435 

0.1 

.9975 

.0499 

5.1 

-.1443 

-.3371 

10.1 

-.2490 

.0184 

0.2 

.9900 

.0995 

5.2 

-.1103 

-.3432 

10.2 

-.2496 

.0066 

0.3 

.9776 

.1483 

5.3 

-.0758 

-.3460 

10.3 

-.2477 

-.0313 

0.4 

.9604 

.1960 

5.4 

-.0412 

-.3453 

10.4 

-.2434 

-.0555 

0.5 

.9385 

.2423 

5.5 

-.0068 

-.3414 

10.5 

-.2366 

-.0789 

0.6 

.9120 

.2867 

5.6 

.0270 

-.3343 

10.6 

-.2276 

-.1012 

0.7 

.8812 

.3290 

5.7 

.0599 

-.3241 

10.7 

-.2164 

-.1224 

0.8 

.8463 

.3688 

5.8 

.0917 

-.3110 

10.8 

-.2032 

-.1422 

0.9 

.8075 

.4060 

5.9 

.1220 

-.2951 

10.9 

-.1881 

-.1604 

1.0 

.7652 

.4401 

6.0 

.1506 

-.2767 

11.0 

-.1712 

-.1768 

1.1 

.7196 

.4709 

8.1 

.1773 

-.2559 

11.1 

-.1528 

-.1913 

1.2 

.6711 

.4983 

6.2 

.2017 

-.2329 

11.2 

-.1330 

-.2039 

1.3 

.6201 

.5220 

6.3 

.2238 

-.2081 

11.3 

-.1121 

-.2143 

1.4 

.5669 

.5419 

6.4 

.2433 

—.1816 

11.4 

-.0902 

-.2225 

1.5 

.5118 

.5579 

6.5 

.2601 

-.1538 

11.5 

-.0677 

-.2284 

1.6 

.4554 

5699 

6.6 

.2740 

-.1250 

11.6 

-.0440 

-.2320 

1.7 

.3980 

.5778 

6.7 

.2851 

-.0953 

11.7 

-.021H 

-.2883 

1.8 

.3400 

.5815 

6.8 

.2931 

-.0652 

11.8 

.0020 

-.2323 

1.9 

.2818 

.5812 

69 

.2981 

-.0349 

11.9 

.0250 

-.2290 

JJ.O 

.2239 

.5767 

7.0 

.3001 

-.0047 

12.0 

.0477 

-.',284 

2.1 

.1666 

.5683 

7.1 

.2991 

.0252 

12.1 

.0697 

-.2157 

2.2 

.1104 

.5560 

7.2 

'  .2951 

.0543 

12.2 

.0908 

-.2(60 

2.3 

.0555 

.5399 

7.3 

.2882 

.0826 

123 

.1108 

-.1943 

2.4 

.0025 

.5202 

7.4 

.2786 

.1096 

12.4 

.1296 

-.1807 

2.5 

-.0484 

.4971 

7.5 

.2663 

.1352 

12.5 

.1469 

-.1655 

2.6 

-.0968 

.4708 

7.6 

.2516 

.1592 

12.6 

.1626 

-.1487 

2.7 

-.1424 

.4416 

7.7 

.2346 

.1813 

12.7 

.1766 

-.1307 

2.8 

-.1850 

.4097 

7.8 

.2154 

.2014 

12.8 

.1887 

-.1114 

2.9 

-.2243 

.3754 

7.9 

.1944 

.2192 

12.9 

.1988 

-.0912 

3.0 

-.2601 

.3391 

8.0 

.1717 

.2346 

13.0 

.2069 

-.0703 

3.1 

-.2921 

.3009 

8.1 

.1475 

.2476 

131 

.'2129 

-.0489 

3.2 

-.3202 

.2613 

82 

.1222 

.2580 

13.2 

.2167 

-.0271 

3.3 

-.3443 

.2207 

8.3 

.0960 

.2657 

13.3 

.2183 

-.0052 

3.4 

-.3643 

.1792 

8.4 

.0692 

.2708 

13.4 

.2177 

.0166 

35 

-.3801 

.1374 

8.5 

.0419 

.2731 

135 

.2150 

.0380 

3.6 

-.3918 

.0955 

8.6 

.0146 

.2728 

13.6 

.2101 

.0590 

3.7 

-.3992 

.0538 

8.7 

-.0125 

.2697 

13.7 

.2032 

.0791 

3.8 

-.4026 

.0128 

8.8 

-.0392 

.2641 

13.8 

.1943 

.0984 

3.9 

-.4018 

-.0272 

8.9 

-.0653 

.2559 

13.9 

.1836 

.1166 

4.0 

-.3972 

-.0660 

9.0 

-.0903 

.2453 

14.0 

.1711 

.1334 

4.1 

-.3887 

-.1033 

9.1 

-.1142 

.2324 

14.1 

.1570 

.1488 

4.2 

-.3766 

-.1386 

92 

-.1367 

.2174 

14.2 

.1414 

.1626 

4.3 

-.3610 

-.1719 

9.3 

-.1577 

.2004 

14.3 

.1245 

.1747 

4.4 

-.3423 

-.2028 

9.4 

-.1768 

.1816 

14.4 

.1065 

.1850 

4.5 

-.3205 

-.2311 

9.5 

-.1939 

.1613 

14.5 

.0875 

.1934 

4.6 

-.2961 

-.2566 

9.6 

-  .2090 

.1395 

14.6 

.0679 

.1999 

4.7 

-.2693 

-.2791 

9.7 

-.2218 

.1166 

14.7 

.0476 

.2043 

4.8 

-.2404 

-.2985 

9.8 

-.2323 

.0928 

14.8 

.0271 

.2066 

4.9 

-.2097 

-.3147 

9.9 

-.2403 

.0684 

14.9 

.0064 

.20«9 

5.0 

-.1776 

-.3276 

10.0 

-.2459 

.0435 

15.0 

-.0142 

.2051 

TABLES. 

TABLE  III. — ROOTS  OF  BESSEL'S  FUNCTIONS. 


63 


n 

xn  for  Jt(xn)  =  0 

xn  for  J\(xn)  =  0 

71 

xn  for  J0(xn)  =  0 

xn  for  «/iGrB)  =  0 

1 

2.4048 

38317 

6 

18.0711 

19.6159 

2 

5.5201 

7.0156 

7 

21.2116 

22.7601 

3 

8.6537 

10.1735 

8 

24.3525 

25  9037 

4 

11.7915 

13.3237 

9 

27.4935 

29.0468 

5 

14.9309 

16.4706 

10 

30.6346 

32.1897 

TABLE  IV.— VALUES  OF  Ja(xi). 


X 

J<t(xi) 

X 

J0(xi) 

X 

J0(xi) 

0.0 

1.0000 

2.0 

2  2796 

4.0 

11.3019 

0.1 

1.0025 

2.1 

2.4463 

4.1 

12.3236 

0.2 

1.0100 

2.2 

2.6291 

4.2 

13.4425 

0.3 

1.0226 

2.3 

2.8296 

4.3 

14.6680 

0  4 

1.0404 

2.4 

3.0493 

4.4 

16.0104 

0.5 

1.0635 

2.5 

3.2898 

4.5 

17.4812 

0.6 

1.0920 

2.6 

3.5533 

4.6 

19.0926 

0  7 

1.1263 

2.7 

3.8417 

4.7 

20.8585 

0.8 

1.1665 

2.8 

4  1573 

4.8 

22.7937 

0.9 

1.2130 

2.9 

4.5027 

4.9 

24.9148 

1.0 

1.2661 

3.0 

4.8808 

5.0 

27.2399 

1.1 

1.3262 

3.1 

5.2945 

5.1 

29.7889 

1.2 

1.3937 

3.2 

5.7472 

5.2 

32.5836 

1.3 

1.4963 

3.3 

6.2426 

5.3 

35.6481 

1  4 

1.5534 

3.4 

6.7848 

5.4 

39.0088 

1.5 

1.6467 

3.5 

7.3782 

5.5 

42.6946 

1.6 

1.7500 

3.6 

8.0277 

5.6 

46.7376 

1.7 

1.8640 

3.7 

8.7386 

5.7 

51.1725 

1.8 

1.9896 

3.8 

9.5169 

5.8 

56.0381 

1.9 

2.1277 

3.9 

10.3690 

5.9 

61.3766 

INDEX. 


Bernoulli!,  Daniel,  7. 
Bessel's  Functions: 

applications   to    physical    problems, 

53-55- 

development  in  terms  of,  55-56. 
first  used,  7. 

introductory  problem,  21. 
of  the  order  zero,  23. 
of  higher  order,  59. 
problems,  25,  56-59. 
properties,  51-53- 
series  for  unity,  24,  56. 
tables,  62-63. 

Conduction  of  heat,  7. 

differential  equations  for,  8,  9,  10,  13, 
21,  54,  57- 

problems,  12-15,  2I~25»  4°,  56»  57- 
Continuity,  equation  of,  9. 
Cosine  Series,  30. 

determination  of  the  coefficients,  30. 

problems  in  development,  31. 
Cylindrical  harmonics,  52. 

Differential  equations,  10. 

arbitrary     constants     and    arbitrary 
functions,  10. 

linear,  10. 

linear  and  homogeneous,  10. 

general  solution,  10. 

particular  solution,  10. 
Dirichlet's  conditions,  36. 
Drumhead,  vibrations  of,  57,  58. 


Electrical  potential  problems,    15,   39, 

4°,  43- 
Ellipsoidal  harmonics,  59. 

Fourier,  7. 

Fourier's  integral,  35. 
Fourier's  series,  32-36. 

applications  to  problems  in  physics, 
38-40. 

Dirichlet's    conditions   of    developa- 
bility,  36. 

extension  of  the  range,  34-35. 

graphical  representation,  37. 

problems  in  development,  33,  34. 

Harmonic  analysis,  7. 
Harmonics: 

cylindrical,    12,    21,    25,    51-59,    62- 

63- 

ellipsoidal,  55. 

spherical,  7,  12,  51. 

tesseral,  51. 

toroidal,  59. 

zonal,  12,  15-21,  40,  50,  60-61. 
Heat  v.  Conduction  of  heat,  7 
Historical  introduction,  7. 

Introduction,  historical  and  descriptive, 
7,  8,  9- 

Lame,  7. 

Lame's  functions,  12,  59. 

Laplace,  7. 


66  INDEX. 


Laplace's  coefficients,  12,  51. 
Laplace's  equation,  17,  41,  43,  51. 

in  cylindrical  coordinates,  10,  21. 

in  spherical  coordinates,  9,  12. 
Laplacian,  51. 
Legendre,  7. 

Legendre's  coefficients,  19. 
Legendre's  equation,  17,  40,  41,  47. 

Musical  strings,  7. 

differential  equation  for  small  vibra- 
tions, 7. 
problems,  39,  40. 

Perry,  John,  8. 

Potential  function  in  attraction: 
problems,  44,  51. 

Sine  series,  26. 

determination  of  the  coefficients,  26- 

28. 

examples,  29. 
for  unity,  12,  29. 


Spherical  harmonics,  7,  12,  51. 
Stationary  temperatures: 
problems,  21,  25,  56,  57,  59. 

Tesseral  harmonics,  51. 
Toroidal  harmonics,  59. 
Tables,  60-63. 

Vibrations : 

of  a  circular  elastic  membrane,  57,  58-. 
of  a  heavy  hanging  string,  7. 
of  a  stretched  elastic  string,  7,  39,  40. 

Zonal  harmonics: 

development  in  terms  of,  46-49. 

first  used,  7. 

introductory  problem,  15. 

problems,  21,  43,  44,  49,   50 

properties,  40,  43. 

short  table,  19. 

special  formulas,  50. 

surface  and  solid,  19 

tables,  60-6 1. 

various  forms,  45-46. 


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