144 THE EEV. S. EAENSHAW THE MATHEMATICAL THEOEX OE SOEXD. 
Tc being the ratio of the specific heat of the gas under a constant pressure, to its specific 
heat under a constant volume. The dynamical equation takes for this case the following 
form to be used instead of that in art. 1, 
\dx) ' dx^' 
This equation being integrated as explained in art. 1, gives 
k-\ 
(14.) 
33. From these we obtain 
U-. 
_L+ a 
izi 
2 
For the same reasons as before we shall suppose w=0 and to be simultaneous 
equations ; which gives 
P I 2 X'V 
A-1 ’ 
and 
A:-l 
\qo) “b 2 ' 
(15.) 
This equation gives the relation between density and velocity ; from which that between 
pressure and velocity is easily found. 
34. The general integral (14.) may be expressed in terms of the original genesis pre- 
cisely in the same manner as was employed in art. 4 ; and the result is 
y=Y+(^^U+y^;<.)(<-T) (16.) 
Ar + 1 
(^""T) (17.) 
w=U, andp=^o(f) (18-) 
These equations, with (15.), are those from which the properties of the motion are to 
be deduced. The degree of modification of former results required by these forniulie 
will be in most cases sufficiently evident, and need not therefore to be particularly 
pointed out. 
35. The result of art. 10 takes the following form — 
/c— 1 k—l A:— 1 
e~+e7=-s7- 
