832  Notices  respecting  JSew  Books. 
systematic  treatises,  mainly  in  Trench  and  German,  from  which 
Mr.  Hardy  has  drawn  much  of  his  inspiration.  Very  pointed 
reference  is  made  to  Liouville's  work  of  seventy  years  ago,  which 
seems  to  have  been  to  a  large  extent  lost  sight  of.  It  may  be  said 
in  conclusion  that  both  tracts  carry  out  admirably  the  purpose 
of  the  series,  each  of  which,  we  are  told,  is  to  be  illuminating 
rather  than  exhaustive,  is  not  to  contain  elaborate  collections  of 
problems,  and  is  not  to  be  specially  adapted  for  preparations  for 
examinations, 
Hauptsdtze  der  Differential-   und    Inter) ral-JRechnung.       Von    Dr. 
Eobeet  Fkicke.     4th  edition.     Braunschweig:  F.  Vieweg  und 
Sohn.     1905. 
Tins  is  practically  a  reprint,  with  a  few  additions,  of  the  Third 
Edition,  reviewed  in  this  Magazine  in  September  1U03. 
An  Introduction  to  the  Infinitesimal  Calculus.  By  Professor 
Cabslaw.  A  Preliminary  Course  in  Differential  and  Integral 
Calculus.  By  A.  H.  Aureus,  B.Sc.  Longmans,  Green,  &  Co. 
11)05. 
These  two  books,  each  of  about  100  pages,  embody  courses  of 
lectures  given  to  students  of  applied  science  by  the  respective 
authors,  the  one  in  Sydney,  the  other  in  Birmingham.  The 
students  are  not  supposed  to  lmowT  more  than  the  elements  of 
geometry,  algebra  and  trigonometry,  with  perhaps  a  smattering  of 
coordinate  geometry.  This  last  is,  however,  given  by  Professor 
Carslaw  in  his  opening  chapter;  and  in  Chapter  VI.  the  more 
important  Cartesian  properties  of  the  conic  sections  are  discussed 
with  the  aid  of  the  principles  of  the  calculus  established  in  the 
preceding  chapters.  The  two  books  cover  almost  exactly  the  same 
ground  in  so  far  as  the  calculus  is  concerned  ;  but  they  differ 
considerably  in  arrangement  and  mode  of  treatment.  Mr.  Angus 
writes  in  a  more  colloquial  and  what  is  regarded  in  some  quarters 
as  a  more  practical  style,  which  may  possibly  appeal  to  the  less 
highly  trained  mathematical  mind  more  powerfully  than  the  more 
academic  style  of  Professor  Carslaw.  The  use  of  the  conver- 
sational contraction  "tan  of  an  angle"  in  a  well  turned  sentence 
of  grammatical  English  does  not,  however,  commend  itself.  In 
Professor  Carslaw's  book  there  is  one  statement  which  seems  to  be 
open  to  criticism.  When  illustrating  the  idea  of  a  function  from 
physics  and  dynamics  the  author  says  that  '-when  the  pressure  is 
increased  past  a  certain  point  Boyle's  Law  ceases  to  hold,  and  the 
relation  between  p  and  v  in  such  a  case  is  given  by  van  der  Waals' 
equation  ....  a  and  b  being  certain  positive  quantities  which  have 
been  determined  by  experiment  for  different  gases."  Any  student 
reading  this  would  think  that  alter  Boyle's  Law  ceased  to  hold 
Van  der  Waals'  expression  at  once  stepped  in  with  its  two  con- 
stants a  and  b  and  continued  thereafter  to  represent  the  relation 
between  pressure  and  volume  with  an  accuracy  comparable  to  that 
with  which  Boyle's  Law  holds  within  the  limits  of  its  applicability. 
This  of  course  is  not   the  case.     The  statement  would  be  fairly 
