PHYSICS: A. G. WEBSTER 287 
five integrals, with five arbitrary constants Xq, yo, zo, Pq, qo we make the latter 
functions of a second parameter v satisfying the equations 
Poqo-Zo = 0, — = Po—- +qo-~. (5) 
ov ov ov 
We easily obtain the five integrals, 
Po P q 
X- Xo = u, y — yo = —u, — = - , q — qo = u, 
qo Po qo 
z- Zo=-2poU+ with po = -, 
qo qo 
Instead of adopting Cauchy's form for the introduction of the arbitrary func- 
tion, we will attempt to pass the integral surface through the plane Zo = 
const., representing an isothermal. We put 
Xo = v, yo = (p (v), Po + ^0 (p'M =0, ^ = - <p'(v), 
qo 
y = <p(v) u<p'(v), (6) 
2 = Zo =*= 2 Zq(p'{v) u — (p^{v)u^, 
X = u -\-v. 
If we adopt the Clausius equation for the form of one particular isothermal, 
we may put 
Rzq a 
,{v) = 
We thus obtain finally 
X = u + v, 
_ Rzf, _ a ( Rzo 2a ") 
^ ~ ~ Zo(v +^)' [{V - ay ~ 2o(v + (8) 
z = z,^2uJjZl!o 2^ \ + «^(_gV_ 2. I 
>'°V(t;-a)^ z,{v + &y) \i.v-af z,{v + &yi 
SO that we have the parametric equation of the surface. It may be noted 
J;hat putting w = 0, zo = T we fall back on the ordinary Clausius equation 
(1) as a particular case, with (2) and the ideal gas equations as still more 
particular. 
