52  Prof.  Challis  on  the  Solutions  of  Three  Problems 
determinations  can  be  made  by  a  very  delicate  movement  of  the 
reflecting -prism . 
2.  Description  of  the  Reversion- Ocular, 
The  object  of  the  reversion-ocular,  and  the  principle  of  its 
operation,  are  the  same  as  those  of  the  reversion-objective ;  but 
while  the  latter  presupposes  a  divided  objective,  in  the  employ- 
ment of  the  reversion-ocular  this  is  not  the  case. 
That  is  to  say,  it  contains  the  moveable  reflection-prism,  on 
a  correspondingly  diminished  scale,  close  in  front  of  the  collec- 
tive lens  of  the  ocular,  so  that  the  field  appears  half  covered  by 
this  prism,  and  thus  the  two  spectra  are  moveable  in  juxtaposi- 
tion in  opposite  directions.  Since  no  parallel  rays  fall  on  the 
reflection-prism,  the  corresponding  correction  of  the  focal  dis- 
tance is  provided  for  by  the  moveable  half  of  a  concave  lens  in 
the  part  of  the  ocular  not  covered  by  the  prism. 
With  the  reversion-ocular  the  contact  of  the  two  spectra  is 
far  less  sharply  defined  than  with  the  reversion-objective.  This 
inconvenience,  however,  can  be  partially  obviated  by  the  em- 
ployment of  a  cylindrical  lens  in  front  of  the  ocular,  by  which 
the  lines  are  lengthened,  and  at  the  same  time  the  dark  divisions 
are  weakened.  I  make  bold  to  propose  generally  such  an  em- 
ployment of  cylindrical  lenses  immediately  in  front  of  the  cover 
of  the  ocular  wherever  cross  lines  arising  from  dust  or  other  in- 
equalities of  the  slit  produce  disturbance  in  delicate  measure- 
ments. These  lines  are  hereby  completely  neutralized,  and, 
indeed,  vanish  when  their  thickness  is  not  too  great,  while  the 
lines  of  the  spectrum,  perpendicular  to  them,  lose  nothing  of 
their  sharpness, 
VII.  On  the  Solutions  of  Three  Problems  in  the  Calculus  of  Va- 
riations, in  reply  to  Mr.  Todhunter.  By  Professor  Challis, 
M.A.,  LL.D.,  F.R.S.,  F.R.A.S* 
I  UNDERSTOOD  the  problem  which  is  the  subject  of  Mr. 
Todhunter's  remarks  in  the  December  Number  in  the 
only  sense  that  the  terms  in  which  it  was  enunciated  admit  of, 
and  therefore  I  do  not  consider  myself  accountable  for  any  mis- 
apprehension of  what  was  intended.  The  enunciation,  as  given 
in  the  '  History  of  the  Calculus  of  Variations '  (p.  427)  and  re- 
peated in  the  above-mentioned  remarks,  is  as  follows : — "  Re- 
quired to  connect  two  fixed  points  by  a  curve  of  given  length  so 
that  the  area  bounded  by  the  curve,  the  ordinates  of  the  fixed 
points,  and  the  axis  of  abscissae  shall  be  a  maximum."  Accord- 
Communicated  by  the  Author. 
