the  Mechanical  Theory  of  Heat.  113 
Carnot's  proposition  ....  established,  on  a  new  foundation,  in 
the  dynamical  theory  of  heat." 
Very  soon  after  the  publication  of  those  propositions,  appeared 
an  essay  by  Rankine  "  On  the  Reconcentration  of  the  Mechanical 
Energy  of  the  Universe  "  *,  wherein  a  view  was  stated  which  is 
in  opposition  to  them,  namely  that  by  concentration  of  the  heat- 
rays  the  lost  differences  of  temperature  might  be  restored.  I 
have  shown,  however,  in  a  special  memoir  f,  that,  even  by  radia- 
tion, under  no  circumstances  can  heat  pass  from  a  cooler  into 
a  warmer  body  and  thereby  increase  an  existing  difference  of 
temperature. 
In  other  memoirs  I  have  endeavoured  to  reduce  to  a  definite 
quantity  the  universal  tendency  in  nature  to  transformation,  the 
value  of  which  quantity  can  only  alter  in  one  direction  and  not 
in  the  contrary.  In  a  memoir  in  1854  J  I  introduced  a  quantity, 
N,  relative  to  any  circular  processes,  naming  it  the  uncompensated 
transformation,  and  defining  it  by  the  equation 
N 
"J  T' 
in  which  dQ  denotes  a  heat-element  given  out  by  a  variable  body 
to  a  heat-reservoir,  and  T  its  absolute  temperature.  I  showed 
that  the  quantity  N  can  only  be  positive,  while  0  forms  the  limit- 
ing value  relative  to  reversible  circular  processes. 
In  a  paper  that  followed  soon  after,  on  Steam-engines,  I  made 
use  of  the  same  quantity,  in  order  to  obtain — instead  of  the 
usual  determination  of  the  work  of  a  steam-engine,  in  which  the 
quantities  of  work  done  during  the  various  processes  are  singly 
determined  and  then  added  up — an  opposite  method,  in  which 
we  start  from  the  maximum  of  work,  and  then  deduct  from  it 
the  loss  of  work  occasioned  by  the  imperfections  of  the  process 
(incomplete  expansion,  vicious  space,  less  vapour-pressure  in  the 
cylinder  than  in  the  boiler,  &c).     The  amount  of  heat  represent- 
ing  this  loss  of  work  I  exhibited  by  the  product  T0N  or  T0  1  -„-, 
where  T0  signifies  a  temperature  (occurring  in  the  process)  at 
which  heat  is  given  out§. 
The  very  same  formula  is  developed  by  Tait  in  his  f  Sketch  of 
Thermodynamics/  p.  100;  but  instead  of  quoting  it  as  mine,  he 
says  : — "  This  is  Thomson's  expression  for  the  amount  of  heat 
*  Phil.  Mag.  S.  4.  vol.  iv.  p.  358. 
t  Pogg.  Ann.  vol.  cxxi.  p.  1  ;  and  Mechanical  Theory  of  Heat,  p.  290. 
X  Pogg.  Ann.  vol.  xciii.  p.  499 ;  Phil.  Mag.  S.  4.  vol.  xii.  p.  81 ;  Me- 
chanical Theory  of  Heat,  p.  127. 
§  Pogg.  Ann.  vol.  xcvii.  p.  452;  Phil.  Mag.  S.  iv.  vol.  xii.  p.  241  ;  Me- 
chanical Theory  of  Heat,  p.  146. 
Phil.  Mag.  S.  4.  Vol.  43.  No.  284.  Feb.  1872.  I 
