I!T J. E. ’TREYOR
C h i page 523 of tlie present 1-oluine of this Journal, I lireseiited A calculation of the forms which the thermal and d>-namic coefficients of isotropic one-phase s!-stenis of constant mass and c-miposition assume when the!- are expressed as functions of the partial derii-ati1.e.i;of each of the foiir fiiiiclaiiieiital therinod>-nnniic fiinctions E. F,G,H. I t was there noticed that tlie results, as regards their fonn, arrange tliemsel\-es in a striking iiiaiiiier i l i tliree p o u p s . T h e question arises \\-li>. this is so. A\ closer esaniiiiatioii of the matter makes clear that tlit. eiitl actually souglit \\-as a series of expressions for all the possible partial tleri\-atives,
(a?) aiJ
(a:,)8 35
( ;;)H a/,
);(
,,
(a:;)p ae
(27) ae ae
~
(”-) a-
,.
):(;
0
~-
7
P
0
of tlie entropJ- and of tlie voluine. ( I t slioulcl not be overlooked tliat the last two cleril-atives in either row are the reciprocals of tlie last t n o in the other.) Lookiiig at tlie matter i n this \\-a>.. it lieconies oli\-iotis that in the most general form of the prolilein w e are required to find all tlie twel\.e mutual ratios of the differentials of the quantities, $, i’,8,? J , which appear in the energ! eqiiati on dE -$d;,--~ ed?. ~
TYheii we do so, it will he seen a t oiict’ that tlie results fall all?. into three groups of four each.
iiatiir-
It1 tlie problem as set, tlie quantities j , i', 8, 77 in the initial differeritial equation are corinected by two independent relatioils, sa!.[-(\
e)
p.
+(17%
i'.
6,
0.
so that any two are functioiis of the other two. TYe are to find the twel\.e miitiial ratios of dj,dis, de, d ~ iii , terms of the derivatives of the function E. T h e prohleiii fails naturally into
two parts, it1 one of which the extensities and 7 . in the other t h e intensities j and 8, are successively iiiaintaiiied constant. I n its development in general form I am indebted to niy colleagne, Professor James Mcllahon, for very helpful suggestions. In tlie first of these parts we set out with
and, further, 1)). division,
yylCJ*Jil t 7 /
(2
Jld
L!>J1lZ
(2 111 I'C
c~Oe$CI'&Jl
[J
575
I n attacking the second part of the prohleiii, \re set oiit
For coilstant 4 (1. e. for +=o) \\e have, re\vritiiig the first of these equation\, aiid eliniinatiiig di ( o r do)bet\\ eeii liotli, d7
fir,
de
i II
i
T h e equations !I) to (12)furnish the desired expressions of all the ratios of the differentials dj? dzc, de, 0'7. They show it to be a necessar!. cotiseqiieiice of the initial equation and the two atteiidaiit conditions, that these expressions should fall, as regards their form, into tlie three groups : G I I O r P I.
GKOCF
11.
Eqn a t'ioiis (;RC>rI' 111.
R.I'I'IOS
Equ at 1011s '
SECOSI) D I I K I V h T I \ - E S
RATIOS O F
swoxrj
(5)
(61
DERIVATIVES
(71
iIO)
O F S E C O S D DERIT~.~'I'IVEST O 'I'HE HESSIAS
!Si
(91
(11
1
(12)
T o obtaiii the correspoiidiiig expression of these twelve partial derivatives, in teriiis of the second derivatives of each of tlie other fundatneiital functions F, G , H, xve inerel). require, in eacli case, to treat the corresponding initial differential equation in esactlj. the iiiaiiiier eniploj-ed above. TI-ith the equations ( I ) to ( 1 2 1 before lis, the desired results can be foiiiitl by inspection. Coi.ne/I L1iii,er,sifj,,St;ifeiiibL,r, ,399
