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XXVII.  Notices  respecting  New  Books. 
Researches  on  the  Calculus  of  Variations,  principally  on  the  Theory  of 
Discontinuous  Solutions  ;  an  Essay  to  which  the  Adams  Prize  was 
awarded  in  the  XJyiiversity  of  Cambridge  in  1871.  By  I.  Tod- 
hunter,  M.A.,  F.R.S.,  late  Fellow  and  Principal  Mathematical 
Lecturer  of  St.  John's  College,  Cambridge.  London  and  Cambridge  : 
Macmillan  and  Co.     1871.     (Pp.  278.) 
THE  terms  in  which  the  subject  of  the  essay  was  originally  pre- 
scribed are  these  : — "A  determination  of  the  circumstances 
under  which  discontinuity  of  any  kind  presents  itself  in  the  solution 
of  a  problem  of  maximum  or  minimum  in  the  calculus  of  variations, 
and  applications  to  particular  instances.  It  is  expected  that  the 
discussion  of  the  instances  Should  be  exemplified  as  far  as  possible 
geometrically,  and  that  attention  be  especially  directed  to  cases  of 
real  or  supposed  failure  of  the  calculus."  As  far  as  we  can  venture 
to  sum  up  the  result  of  Mr.  Todhunter's  researches  in  a  few  words, 
it  is  this : — that  the  discontinuity  arises  from  conditions  imposed, 
either  explicitly  or  implicitly,  which  limit  the  generality  of  the  ques- 
tion. Thus,  suppose  it  is  required  to  determine  the  curve  which, 
taken  between  fixed  limits  (A  and  B),  shall  give  \<l>(q)doc  a  maxi- 
mum or  minimum.  There  is  no  difficulty  in  showing  that  the  re- 
quired curve  is  a  parabola,  and  for  some  forms  of  $  it  will  be  a  maxi- 
mum, for  others  a  minimum  ;  and  there  is  no  discontinuity.  But 
now  suppose  that  we  introduce  the  condition  that  the  curve  shall 
pass  through  a  third  point  (D).  If  D  is  in  the  parabola  already  de- 
termined, there  is  again  no  discontinuity.  But  if  D  is  not  in  this 
parabola,  the  required  curve  will  consist  of  two  parabolic  arcs,  one 
passing  from  A  to  D  and  the  other  from  D  to  B ;  or  it  may  be  one 
passing  from  A  to  B,  and  the  other  from  B  to  D.  This  is  a  case  of 
discontinuity  introduced  by  a  condition  consciously  imposed  ;  it  may, 
of  course,  be  introduced  by  a  condition  unconsciously  imposed. 
The  fundamental  principle,  which  is  extensively  applied  through- 
out the  essay,  is  this  ; — "  Let  there  be  an  integral  \  tydoc  which  is  re- 
quired to  be  a  maximum  or  a  minimum,  where  0  is  a  known  function 
of  y  and  its  differential  coefficients  with  respect  to  cc.  Change  y  into 
y  +  hy ;  then  in  the  usual  way  we  obtain  for  the  variation  of  the 
integral  to  the  first  order  an  expression  of  the  form 
\j+§M.lydx, 
where  L  depends  on  the  values  of  the  variables  and  the  differential 
coefficients  at  the  limits  of  the  integration.  Now  if  ty  may  have 
either  sign,  we  must  have  M=0  as  an  indispensable  condition  for  a 
maximum  or  a  minimum  ;  and  moreover  we  must  have  L=0.  These 
statements  are  universally  admitted  to  be  true. 
"  Suppose,  however,  that  owing  to  some  condition  in  the  problem 
we  cannot  always  give  hy  either  sign ;  for  example,  suppose  that 
