706 
MR. A. McAULAY ON THE MATHEMATICAL 
Bb. The Entropy. 
34. If Q be the external force of type q, we have 
clA = 2 Q dq, 
where dq is the actual increment in q during the element of time. This can, of 
course, be proved directly from eq. (2). It must now be remembered that all the 
above variations are only true if we suppose the temperature of eveiy element of 
matter kept constant. In other words, the last equation must, when we do not make 
this restriction, be replaced by 
dA — d e A = t Qdq .(7), 
where d e A stands for that part of the increment in A which is due to increment in 
temperature in all elements of matter during the element of time. Let now E be the 
intrinsic energy (including under this term the ordinary kinetic energy of matter as 
well as all other forms of energy) of all the matter in space. Thus by the fundamental 
property of entropy (Tait’s ‘Heat,’ §§ 377, 378), 
cZE = tQdq +1 jj ddfds + ^0df s ds .(8), 
whence, from the last equation, 
d[— E+A+jjj Ofds + || Of/Is] = d e A + ||j fdOds + ^f s d0ds . (9). 
will be seen later on to be independent of ®. Hence 
d,A = 111 (|+ - Scto e vx) ds + f e ‘ dOds 
= {j|(|; + sv 6 v\)<» + (H' - [su^vx]. + +e*. 
[by putting d ® = VdO, and applying § 5, eq. (4)]. Thus eq. (9) becomes 
d(- E + A + {+/* + +/+ 
1 
fjj (/+1 + sv.vx) dOd, + (/, + |= - [SUx.VX]. + j)<M* 
K 10 )- 
Since the left of this equation is a perfect differential, so is the right. Hence we 
see that 
/+ 3\/00 + SV 0 V\ and /, + 0X,/00 - [SU*/ 0 V\] s + i 
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