160 MR W. J. M. RANKINE ON THE 
=-87R' # QDy@D,7) = ip o") au 
R 
‘ OR. bile 6D 4mR°D . 
which, because B34. ce and 3 =M is equal to 

2 , oD ) 
+QM. oie Y (uD, 7) w? (Sy — 3) aw. 
We must suppose that the velocity of oscillation is equalised throughout the 
atomic atmosphere, by a propagation of motion so rapid as to be practically 
instantaneous. 
Then if the above expression be integrated with respect to du, from w=0 to 
u=1, the result will give the whole increase of heat in the atom arising from the con- 
densation 6D; and dividing that integral by the atomic weight M, we shall obtain 
the corresponding development of heat in unity of weight. This is expressed by 
the following equation :— 
oD! 
dq=2q tt {> é du . uw? )(u, D, 7) 
-3 fan . wdus(u,D, 7) } Bd (2) 
The letter Q/ is here introduced to denote, when negative, that heat which is 
consumed in producing changes of volume and of molecular arrangement, and 
when positive, as in the above equation, the heat which is produced by such 
changes. 
The following substitutions have to be made in Equation (1.) of this Section. 
For Q is to be substituted its value, according to Equation XII. of the Intro- 
duction ; or abbreviating Cn into K:— 
5007 A... LON) 
The value of the first integral in Equation (2.) of this Section is 
sf du .wwtb(u,D, ")=5 
0 
The value of the second integral 
-3f- du .u Oud (uD, T) 
remains to be investigated. The first step in this inquiry is given by the condition, 
that whatsoever changes of magnitude a given spherical layer undergoes, the por- 
tion of atmosphere between it and the nucleus is invariable. This condition is 
expressed by the equation 
0= (dup td7z-+OD zp) fan UAE Te) Sh eounl(4s) 
from which it follows that 
1 d d u 
Ou=— aa DF) (d7 5 +3Dan) f, du. w(u, D, 7) 

