442      On  the  Sonorous  Undulations  of  Stringed  Instruments. 
where  a  is  constant,  <f>,  fa,  fa  are  arbitrary  functions,  and 
a. 
u  =  v+  -; 
P 
and  since  the  expression  for  p  may  be  put  under  the  form 
p  =  av  —  ci\v+  -J  +  <f)(v-\-  -) 
it  is  clear  that,  if  these  results  are  to  be  received,  in  the  case  of 
aerial  motion  in  one  direction  at  least,  the  portion  of  the  pressure 
due  to  the  velocity  may  be  very  considerable,  while  that  due  to 
the  density  is  inappreciable. 
"With  regard  to  the  results  themselves,  I  may  observe  that 
they  flow  as  certainly  from  the  equations  of  motion  as  the  pro- 
positions in  Euclid  result  from  the  definitions  and  axioms.  Hence 
any  objection  to  the  results  must  apply  equally  to  the  equation 
from  which  they  are  derived.  If  the  one  be  untrue,  or  true  only 
in  a  qualified  sense,  the  other  must  be  equally  untrue,  or  its 
truth  must  be  equally  qualified.  The  results  can  only  be  re- 
jected on  the  ground  of  there  being  in  the  background  some 
controlling  principle  which  is  not  taken  into  account  in  forming 
the  equation  of  motion.  Until  the  existence  of  such  a  principle 
has  been  demonstrated,  which,  so  far  as  I  am  aware,  has  never 
been  so  much  as  suspected,  we  are  bound  to  admit  every  fact 
derivable  from  the  solution. 
In  the  foregoing  observations  I  have  endeavoured  to  indicate, 
in  a  very  general  way,  the  mode  in  which,  as  I  conceive,  the  so- 
lution of  the  problem  before  us  is  to  be  sought.  Before  that 
object  can  be  accomplished  many  circumstances  will  need  to  be 
taken  into  account  which  I  have  not  touched  upon*;  indeed  a 
solution  of  the  equations  of  aerial  motion  in  two  directions,  such 
as  that  above  given  for  the  equation  applicable  to  motion  in  one 
direction,  would  seem  to  be  an  indispensable  preliminary. 
Such  a  solution  I  am  in  a  position  to  afford,  but  I  must  seek 
another  opportunity  for  its  exhibition,  and  for  the  discussion  of 
the  results  flowing  from  it. 
I  desire  to  express,  in  conclusion,  my  conviction  that  the  com- 
plete resolution  of  this  problem  will  be  found  to  involve  that  of 
others  of  very  great,  and  in  one  instance  of  transcendent  im- 
portance. 
6  New  Square,  Lincoln's  Inn, 
May  18,  1872. 
*  For  instance,  the  lateral  action  by  which  I  have  supposed  the  particles 
outside  the  cylinder  spoken  of  to  be  moved  upwards  necessarily  implies  a 
corresponding  horizontal  action. 
