﻿1875.] 
  On 
  the 
  Integration 
  of 
  Algebraical 
  Functions. 
  279 
  

  

  an 
  important 
  record 
  for 
  reference 
  in 
  the 
  future 
  progress 
  of 
  seismology, 
  

   I 
  have 
  thought 
  it 
  desirable 
  that 
  it 
  should 
  be 
  presented 
  to 
  the 
  Royal 
  

   Society, 
  with 
  a 
  view 
  to 
  it 
  being 
  preserved 
  in 
  the 
  Archives 
  of 
  the 
  Society 
  ; 
  

   and 
  I 
  would 
  beg 
  to 
  be 
  informed 
  whether 
  the 
  Council 
  may 
  think 
  fit 
  to 
  

   accept 
  the 
  deposit. 
  

  

  I 
  remain, 
  dear 
  Sir, 
  

  

  Truly 
  yours, 
  

  

  Robekt 
  Mallet. 
  

  

  The 
  thanks 
  of 
  the 
  Society 
  were 
  given 
  to 
  Mr. 
  Mallet 
  for 
  his 
  valuable 
  

   Present. 
  

  

  The 
  following 
  Paper 
  was 
  read 
  : 
  — 
  

  

  "On 
  the 
  Integration 
  of 
  Algebraical 
  Functions, 
  with 
  Illus- 
  

   trations 
  in 
  Mechanics." 
  By 
  W. 
  H. 
  L. 
  Russell, 
  F.R.S. 
  

   Received 
  December 
  17, 
  1874. 
  

  

  (Abstract.) 
  

  

  The 
  profound 
  researches 
  of 
  Weierstrass, 
  of 
  Riemann, 
  of 
  Clebsch, 
  and 
  

   Gordan 
  on 
  the 
  higher 
  integrals 
  have 
  of 
  late 
  attracted 
  the 
  attention 
  and 
  

   been 
  the 
  admiration 
  of 
  mathematicians. 
  There 
  is, 
  however, 
  this 
  differ- 
  

   ence 
  between 
  these 
  researches 
  and 
  the 
  corresponding 
  investigations 
  in 
  

   elliptic 
  functions 
  — 
  in 
  the 
  latter 
  we 
  investigate 
  the 
  properties 
  of 
  the 
  inte- 
  

   grals 
  themselves 
  ; 
  in 
  the 
  former 
  we 
  investigate 
  the 
  properties 
  of 
  certain 
  

   differential 
  equations, 
  involving 
  these 
  integrals, 
  and 
  with 
  more 
  than 
  one 
  

   variable. 
  How 
  the 
  values 
  of 
  the 
  integrals 
  themselves 
  are 
  to 
  be 
  found 
  

   from 
  these 
  equations 
  is 
  difficult 
  to 
  see, 
  and 
  at 
  all 
  events 
  must 
  be 
  a 
  subject 
  

   of 
  enormous 
  complexity. 
  Accordingly 
  it 
  becomes 
  desirable 
  to 
  ascertain, 
  

   if 
  possible, 
  a 
  more 
  simple 
  method 
  of 
  evaluating 
  the 
  integrals 
  themselves. 
  

   This 
  is 
  what 
  I 
  have 
  attempted 
  in 
  the 
  first 
  section 
  of 
  this 
  paper. 
  I 
  express 
  

   the 
  values 
  of 
  irrational 
  algebraic 
  quantities 
  by 
  means 
  of 
  linear 
  differential 
  

   equations 
  with 
  rational 
  coefficients, 
  and 
  then 
  express 
  their 
  integrals 
  by 
  

   means 
  of 
  converging 
  series. 
  

  

  In 
  the 
  second 
  section 
  I 
  consider, 
  to 
  a 
  certain 
  extent, 
  the 
  inverse 
  pro- 
  

   blem 
  — 
  namely, 
  to 
  ascertain 
  under 
  what 
  circumstances 
  linear 
  differential 
  

   equations 
  of 
  the 
  second 
  order 
  are 
  satisfied 
  by 
  irrational 
  functions. 
  This 
  

   problem 
  I 
  have 
  already 
  considered, 
  although 
  in 
  an 
  incomplete 
  manner, 
  in 
  

   the 
  Proceedings 
  of 
  the 
  Royal 
  Society. 
  

  

  In 
  the 
  third 
  section 
  I 
  illustrate 
  the 
  principles 
  enunciated 
  in 
  the 
  

   first 
  section 
  by 
  the 
  solution 
  of 
  dynamical 
  problems. 
  I 
  show 
  that 
  

   the 
  principle 
  of 
  vis 
  viva 
  enables 
  us 
  to 
  resolve 
  these 
  problems 
  to 
  a 
  great 
  

   extent 
  by 
  means 
  of 
  hyperelliptic 
  functions 
  and 
  the 
  higher 
  transcendents. 
  

  

  Altogether 
  I 
  venture 
  to 
  hope 
  that 
  the 
  memoir 
  which 
  I- 
  have 
  the 
  

   honour 
  to 
  lay 
  before 
  the 
  Society 
  will 
  be 
  read 
  with 
  interest 
  by 
  mathema- 
  

   ticians. 
  

  

  