20  Lord  Kelvin  on 
r^=  [** +y(l  +  2i2)]x/l  +  i2     .     .     .  (120). 
Hence,  by  differentiation  on  the  supposition  of  x,  y,  r  constant, 
we  find 
du 
chjr 
r^=[.<l  +  2^)+^(5  +  60]\/l:M2    .     (121). 
By   (120)  we  find   for  tm.  which  makes  u  a  maximum  or 
minimum, 
*.+*(1+*l)=9 (122); 
a  quadratic  equation  which,  when  (y/x)2  <  §,  has  real  roots  as 
follows, — 
And  substituting  tm,  (either  o£  these;)  for  t  in  (121)  we  find 
r(^)m«[«(l-4)  +  ^.]v^+£   •     •  (124), 
or  with  simplification  by  (119), 
Klp)„=2™»-*<1+'»)3/2-  •  -(m)'- 
Eliminating  £^  from  the  first  factor  of  (124)  by  (122)  we 
find 
which,  with  m=l,  and  m  =  2,  gives  /3,  and  /32  by  (116). 
§  86.  Using  (123)  we  see  that  {dlu\d^2\x  vanishes  when 
x=y^/%}  and  that  it  is  negative  for  £1?  and  positive  for 
t2,  when  x  >y  \/S.  Hence  tx  makes  d2u/d-yjr2  negative.  There- 
fore ux  is  the  maximum ;  and  t2  makes  it  positive.  Therefore 
u2  is  the  minimum  ;  and  (119)  gives  for  these  maximum  and 
minimum  values 
ru1=(/v+yt1)Vl-^-t12,     ru2  =  (x  +  yt2)^l  +  t22 .  (125). 
By  (122),  (123)  we  see  that  when  y/x=0,  we  have  —^=+00, 
and  —  t2  —  Q.  If  we  increase  y  from  0  to  +x/v/S,  —U  falls 
continuously  from  00  to  *J\,  and  —t2  rises  continuously 
from  0  to -y/J.  Thus  — 1\  and  — 12  become,  each  of  them,  v'i  5 
which  is  the  tangent  of  35°  16'. 
