in  the  Calculus  of  Variations.  53 
ingly  I  supposed  that  the  line  joining  the  two  ordinates  was  to 
be  a  curve ;  and  as  a  portion  of  a  circle  fulfilled  all  the  condi- 
tions of  the  question,  I  did  not  imagine  that  the  joining  line 
could  be  made  up  of  a  curve  and  one  or  more  straight  lines,  nor 
did  I  gather  from  the  enunciation  that  "  it  was  intended  the  curve 
should  be  confined  between  indefinite  straight  lines  coinciding 
in  position  with  the  extreme  ordinates."  On  turning,  however, 
to  Legendre's  memoir,  I  find  that  after  taking  account  of  the 
problem  as  above  proposed,  he  adds,  "But  if  it  be  required  that 
the  surface  ABCD  be  absolutely  contained  between  the  two 
parallels  AC,  B  D,  &c.,"  thus  introducing  a  new  condition 
which  requires  the  discussion  of  two  additional  problems.  This 
is  intelligible  enough.  But  in  the  '  History }  there  is  no  inti- 
mation that  such  a  limitation  was  expressly  made  by  Legendre. 
It  seems  to  me  that  the  investigations  in  Stegmann's  work 
(pp.  171-160),  and  those  in  Mr.  Todhunter's  (pp.  427-430),  are 
liable  to  be  misunderstood  from  the  circumstance  that  the 
authors  discuss  under  one  enunciation  three  problems  requiring 
different  enunciations  and  different  processes  of  solution.  These 
problems  might  be  proposed  as  follows : — 
Required  to  construct  a  figure  bounded  by  the  axis  of  ab- 
scissas, two  ordinates  drawn  at  given  points  of  the  axis,  and  a 
curve  joining  their  extremities,  so  that  its  contour  shall  be  of 
given  length  and  its  area  a  maximum  : 
(I.)   When  the  length  of  each  ordinate  is  given. 
(II.)  "When  the  length  of  one  of  the  ordinates  is  given. 
(III.)  When  the  length  of  neither  of  the  ordinates  is  given. 
The  first  of  these  three  problems  accords  with  that  expressed 
by  Mr.  Todhunter's  enunciation,  the  length  of  the  curve  being 
implicitly  given.  In  the  other  two  the  lengths  of  the  curves 
and  the  values  of  the  extreme  ordinates  that  are  not  given  have 
to  be  determined  by  the  solutions  of  the  problems.  In  all  the 
cases  the  curve  is  a  portion  of  a  circle.  In  Problem  (I.)  the 
circle  may  or  may  not  extend  in  the  direction  of  the  axis  of  ab- 
scissas beyond  one  or  both  of  the  given  extreme  ordinates ;  in 
Problem  (II.)  it  may  or  may  not  extend  beyond  the  given  ex- 
treme ordinate,  but  cannot  extend  beyond  the  other;  in  Problem 
(III.)  it  can  extend  beyond  neither  ordinate.  The  limits  of  the 
given  lengths  of  contour,  for  which  the  curve  is  wholly  con- 
tained within  the  extreme  ordinates  produced,  are  readily  dedu- 
cible  from  the  respective  solutions.  Mr.  Todhuntcr  says  in  two 
instances  that  a  certain  result  is  not  (( satisfactory,"  and  that  the 
problem  requires  to  be  "  modified."  These  expressions,  the 
meaning  of  which  I  did  not  apprehend,  will  be  seen  to  be  inap- 
plicable if  it  be  understood  that  there  are,  in  fact,  three  pro* 
blems,  the  separate  solution  of  each  of  which  is  perfectly  satis- 
