Physics and Chemistry. 413 
Ethylene, Ca Ha -+ Ho = C1 He, Marsh gas. 
Fumaric acid, Cs H4Os + H2= Cs He Os, Succinic acid. 
Ethylene, Ca Ha + 2HO2 = Cs He Os, Glycol 
7. se 
Fumaric acid, CsHsOs-- 2HO2= Cs He Ons, Tartaric acid. 
the properties of this body are functions of the qualitative and quantita- 
tive character of the forces which determine the equilibrium of the 
These forces reside in the elements C, H, O and are more or 
property of the body CaHfOy, F an arbitrary function, a, 6, and ¢ the 
molecular forces or states of the elements C, H, O and a, 8, y the number 
of equivalents of each, we shall have :— 
U=F(a, «; 6, B<'¢,'7) 
For a second body Ca-++-4a Hb4-43 Oy differing from the first only in the 
number of equivalents of C and H, containing therefore da carbon and 
48 hydrogen more or less, the corresponding property will have another 
value U, and we shall have 
U,=F(a, «+4e; b, 6-+48; ¢, 7) 
It remains to determine the connection between U and U,. Since a, 8, 
¢ and 7 are constants we may obviously write 
U=F(«, 8) 
U ,=F(e+4e, b-+-48) 
whence by Taylor’s theorem, 
F(ef-de, B-+-48)=F(6, 6)-+4e-+-F(e, B)+-48-7,F a; A)+ 
da2 d2 2 Ag? d2 ’ 
= ad F(a, 8)--4048 PG hae qgat B)+- &e. 
> 
dU. dU , 402 d?U d2U_, 48? d?U 
In homologous series 4a=48 and we may write 4 for the increase in the 
number of equivalents of carbon and hydrogen, re Sopher 
dU dU) , 4? §d?U, ,d? 
(4) U,=U+<4 | dat @B t 26 | qa2 +27 Tt ape bo ke. 
whence for D the difference in the properties of the two bodies, 
ae dU 4? {d?U | d2U , d?U t 
(5.) pau, -v=4{245} : i + ke. 
9 
da2 'dad8' dg? 
The first differential co-efficients =. Pt express the rate at which the 
