50 
MR. J. J. WATERSTON ON THE PHYSICS OF MEDIA COMPOSED OF 
x = — ) = A 4 , we have ( x du = 3 — 3A = 3 — 7 —--—— area CKHB in 
\i + y) J (i + yY 
terms of ABCN unity. In this, y being made infinite, we have the area of the whole 
asymptotal space = 3 ABCN, which accords with § 19, as the asymptotal area repre¬ 
sents the collective force of expansion from the original volume to infinity. 
If, daring the expansion from B to H, vis viva were supplied to the medium so as to 
maintain the original quantity unimpaired, the point K in the hyperbola x = 
would coincide with M, the point in the common hyperbola x = ^ - , or MH 
— th-CB. Now, it is evident that MH : KH : : -—— : \ : : 3 : —r—- —- 1 ; 
ah 1 + y \1 + y) (1 +yY 
3 
but the preceding integral gave ^ ^ ^ 1 = 3ABCN — CKHB, therefore 
MH : KH : : 3AB.CB : 3AB.CB — CKHB = ratio of original vis viva of the 
medium to the force remaining after expansion from B to H. Thus, MH : MK : : 
original vis viva : decrement of vis viva owing to expansion; and three times the 
area KNAS is equal to the asymptotal area on the other side of KH. These 
relations evidently hold good in whatever part of the conic hyperbola the point C 
may be taken. 
Suppose the medium is maintained at its original vis viva while it expands from 
C to E, it will exert the mechanical force CEDB = p, and absorb the vis viva 
CEDB, the original quantity in the medium being 3AB.CB. From E let it expand to 
F without being supplied with vis viva; then, as before, FG — 
.ED and ~ is 
IG 
the proportion of original vis viva expended represented by EFGD = (m), its 
equivalent mechanical force exerted. Let it now be compressed from F to K, the vis 
viva communicated to the medium being continually withdrawn. The amount 
withdrawn and the force exerted is represented by the area FGHK = q. From K 
let the medium be compressed without withdrawing the vis viva generated until the 
original tension CB and density AB are regained. The force of compression and 
vis viva communicated to the medium in the last operation is represented by the area 
CKHB = n. For shortness put the area KQDH = s, CEFK = 8 , and EQF = e. It 
is evident since the molecular vis viva through CEL is constant and through 
o O 
KQF constant, that 
LF MK 
LG “ ME 
and 
MH 
LG 
MK 
LF' 
Also, 
MH 
LG 
AG _ MK 
AH ~ LF 5 
AG.LF — AH.MK = SK.MK = n = m. But q-\-e = s-\-m=s-\-n, and 
s + n + 8 — e = p. In this equation substitute for s + n its equal, q + e, and we 
have 5 , + e-bS“e~ 9 '+S= :: P- Thus, the curvilinear area 8 , or CEFK, is the 
excess of the force exerted by the medium expanding from C to E at the higher 
constant temperature, over the force exerted upon the same, compressing it from 
F to K at the lower. constant temperature. It is also the excess of the vis viva 
absorbed in the first part of the process over the vis viva given out in the last part. 
