128 
Lord Kelvin on 
[r.s.e., sess. 
^(T)=£v i (T); |p 8 (T)=|p 2 (T ); etc. (5) 
and 
i M 2 (T) =4 Mi(T); 4 m ^ (t)= 4 M2(t); etc - (6) - 
§ 5. Passing now from the supposition that the temperature is 
kept constant at T and going hack to § 3, remark that whatever 
heat is taken in at temperature t imparts motivity to the apparatus 
to an amount equal to the proportion (t-T)lt of its dynamical 
equivalent. Hence, if N denote the thermal capacity of the 
apparatus with g v g 2 , etc., each constant; and if M 15 M 2 , etc., 
and P 1} P 2 , etc., denote the coefficients in (1) and (2) for any 
variable temperature t instead of the constant temperature T, we 
now have, instead of (3), 
£-T tf_T t- T 
dm — + (Pi + + (P 2 + J~ - ]H 2 )c?p 2 + etc. (7) ; 
and for the energy we have simply 
de = LNb^ + (P j + JM^)c?<7i + (P 2 + JM 2 )dg 2 + etc. (8). 
From this and (7) we have 
JT 
d(e - m) = -y(Ndt + M 1 dg 1 + M 2 dg 2 + etc.) (9). 
§ 6. From the conditions that the second members of (7) and 
(9) are complete differentials of all the independent variables, we 
now find from (9), as formerly from (4), in respect to g v g 2 , etc., 
<iM 2 c2M 3 c?M 2 
~dg[ ~ dg 2 3 dg 2 “ dg z ; e C ' 
( 10 ) 
and from these in conjunction with (7), 
^ Jpi etc. (IP 
dg 1 dg 2 ’ dg 2 dg 3 
Lastly, in respect to t, g 1 ; t, g 2 ; etc., we find from (9) and (8) 
dN _ 
dg Y dt 
and 
dg i 
J dt 
Mi , 
t ’ dg~ dt t ’ ’’ 
dPj rfN‘ = J <ai 2 dPs 
dt ’ dg 2 dt dt 
etc. 
( 12 ) 
(13). 
