Notices  respecting  New  Books,  225 
throughout  the  whole  range  of  the  integration  8y  is  essentially  posi- 
tive, then  it  is  no  longer  necessary  that  M  should  vanish.  If  M  is 
positive  through  the  whole  range  of  the  integration,  we  are  sure  of 
a  minimum  ;  and  if  M  is  negative  through  the  whole  range  of  the 
integration,  we  are  sure  of  a  maximum.  We  assume,  of  course, 
that  we  are  able  to  satisfy  the  condition  L=0,  or  to  ensure 
that  L  shall  be  positive  in  the  former  case  and  negative  in  the  latter 
case. 
"  Next  suppose  that  £y  may  have  either  sign  through  part  of  the 
range  of  the  integration,  but  that  it  is  essentially  positive  during 
the  remainder  of  the  range.  Then,  if  M  vanishes  through  the  former 
part  and  is  positive  through  the  latter  part  of  the  range,  we  are  sure 
of  a  minimum ;  and  if  M  vanishes  through  the  former  part  and  is 
negative  through  the  latter  part  of  the  range,  we  are  sure  of  a  maxi- 
mum."— P.  13. 
After  exemplifying  the  discontinuity  arising  from  conditions 
in  several  simple  cases,  Mr.  Todhunter  goes  on  to  discuss — mainly 
by  the  aid  of  the  above  principle — various  cases  of  the  questions  of 
chief  historical  importance  on  the  subject,  viz.  minimum  surface  of 
revolution,  maximum  solid  of  revolution,  brachistochrone  under  the 
action  of  gravity,  problem  of  least  action,  solid  of  minimum  resist- 
ance, area  between  a  curve  and  its  evolute.  To  each  of  these  ques- 
tions a  chapter  is  devoted.  The  way  in  which  they  are  discussed 
may  be  judged  of  from  the  contents  of  a  single  chapter,  viz.  that  on 
the  brachistochrone  (c.  7).  The  well-known  general  solution  is  first 
mentioned,  viz.  that,  when  the  body  falls  from  a  fixed  point  (A)  to 
another  fixed  point  (B),  the  curve  is  a  cycloid  with  a  cusp  at  A  and 
a  horizontal  base.  The  following  particular  cases  are  then  consi- 
dered in  order  : — (a)  when  the  path  must  pass  through  a  third  point 
C  either  fixed  or  on  a  given  curve ;  (6)  when  an  obstacle  with  an 
aperture  of  given  size  is  interposed  between  A  and  B  ;  (c)  when  C 
is  on  a  given  surface  ;  (d)  when  the  condition  is  that  the  moving 
point  must  not  descend  below  a  horizontal  line  through  B  ;  (e)  when 
it  must  not  pass  outside  a  circular  arc  of  which  B  is  the  lowest 
point,  and  AB  does  not  exceed  a  quadrant ;  (f)  when  it  must  not 
pass  inside  the  circular  arc  ;  (g)  when  the  condition  is  that  the  radius 
of  curvature  shall  never  be  less  than  a  given  constant,  and  that  there 
is  no  abrupt  change  of  direction.  With  regard  to  the  discussion  of 
the  cases  which  we  have  marked  (d),  (e),  (f),  (g),  Mr.  Todhunter 
says  (p.  146)  that  they  completely  illustrate  the  general  principles 
laid  down.  In  each  case  the  discontinuity  arises  from  a  condition 
or  conditions  imposed.  This  is,  in  fact,  the  theme  which  he  has. 
illustrated'in  the  present  essay  with  the  utmost  fulness  and  in  a  man- 
ner worthy  of  his  high  reputation  as  a  mathematician.  To  students 
of  the  Calculus  of  Variations  (we  fear  they  are  but  few)  the  work 
cannot  fail  to  prove  highly  instructive,  as  it  comprises  a  thorough 
investigation  of  a  class  of  cases  which,  when  they  occur,  have  been 
commonly  passed  over  with  the  remark  that  "  the  process  of  the  Cal- 
culus of  Variations  fails  in  this  case."     It  was,  in  fact,  in  reference  to 
Phil.  Mag.  S.  4.  Vol.  43.  No.  285.  March  1872.  Q 
