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XLVIII.  Notices  respecting  New  Books, 
An  Elementary  Treatise  on  Curve-Tracing.  By  Percival  Frost, 
M.A.,  formerly  Fellow  of  St.  John's  College,  Cambridgey  Mathema- 
tical Lecturer  of  King's  College.  London :  Macmillan  and  Co. 
1872  (8vo,  pp.  208). 
THIS  work  is  almost  exclusively  devoted  to  the  tracing  of  curves 
whose  equations  are  of  the  form/(,ry)=0,  where/ denotes  a 
rational  algebraical  function.  The  author  is  perfectly  aware  of  the 
limited  aim  of  his  book,  and  in  fact  says  somewhat  naively : — "  The 
student  might  expect  in  a  treatise  upon  this  subject  to  find  methods 
of  drawing  Polar  Curves,  Rolling  Curves,  Loci  of  Equations  in  Trili- 
near  Coordinates,  and  Intrinsic  Equations ;  he  might  also  expect  to 
find  interesting  Geometrical  Loci  discussed.  These,  and  many  other 
things  immediately  connected  with  the  tracing  of  curves,  have  been 
deliberately  omitted  for  reasons  which  I  consider  good"  (p.  v). 
These  omissions  are  determined  by  the  object  for  which  the  book  is 
written,  viz.  to  furnish  preliminary  exercises  in  elementary  mathe- 
matics— especially  in  Algebra  as  far  as  the  Binomial  Theorem,  the 
fundamental  parts  of  the  theory  of  Equations,  and  the  general  me- 
thods of  Algebraical  Geometry — with  a  view  to  assisting  the  student 
to  acquire  that  mastery  over  these  subjects  which  should  be  attained 
before  he  enters  on  the  higher  branches  of  mathematics,  -or  on  their 
application  to  Physical  questions. 
Of  course  the  points  in  the  theory  of  curves  which  are  commonly 
given  as  applications  of  the  Differential  Calculus  to  Geometry  occur 
here — such  as  drawing  tangents  and  asymptotes,  determining  points 
of  inflection,  cusps,  multiple  points,  radii  of  curvature,  &c. ;  but  as 
the  equations  of  the  curves  are  simple  algebraical  functions,  these 
points  can  be  discussed  by  successive  approximations  without  refer- 
ence to  the  methods  of  the  Differential  Calculus.  Thus,  let  the 
equation  to  the  curve  be  written  in  the  form 
m1  +  «8+k8+  •  •  -=0, (1) 
where  uv  w2,  u3, . . .  are  functions  of  the  1st,  2nd,  3rd  . . .  order  re- 
spectively ;  now,  when  x  and  y  are  both  small,  the  first  approximate 
value  of  (1)  is 
u=0, 
which  gives  the  tangent  to  the  curve  at  the  origin.  The  second  ap- 
proximation, 
W1  +  M2  =  0» 
gives,  generally,  a  curve  of  the  second  order  approximating  to  the 
curve  and  having  the  same  curvature  at  the  origin  as  the  given  curve. 
It  can  further  be  shown  that  there  are  generally  an  infinite  number 
of  conies  which  have  the  property  of  coinciding  with  the  curve  up 
to  the  third  order  near  the  origin,  and  from  amongst  these  the  circle 
of  curvature  can  be  easily  selected. 
If  we  suppose  the  equation  (1)  to  take  particular  forms,  such  as 
