419 
where w# is to be computed along the initial adiabatic of the gas, 
Eqs. 4.4 and 4.5 constitute a system of differential equations for b and 
U_ , the solution of which together with the initial values of these func- 
tions determines }> and on the gas sphere as functions of time, the veloc- 
ities @,C ,C*, and ¢ ¥ being determined as functions of fp and of U along 
the initial adiabatics of the two fluids by their respective equations of 
state. In the present discussion, we shall suppress the subscript a used 
to denote values of functions on the surface a(t) in Section 3, since all 
equations of this section refer to this surface alone, 
We now introduce the approximations @#U and Cu for © and 
c* into Eqs. 4.4 and 4.5. The error introduced in the rarefaction wave in 
the gas by approximating o  with c-u is of the same relative magnitude as 
the corresponding approximation in water, although perhaps numerically a 
little larger due to the fortunate cancalation of certain terms in the re- 
mainder for water. With these approximations, Eqs. 4.4 and 4.5 reduce to 
Se) ee ee ee eee 
Peet ai De oe 
+1. DP. _ Dw . p*uc 
P*e" Dt jae Dro 
aire boar 2 Cru ) 
* ' *_ + 2 3 
tee et ee 
Dp VR eine Oh Bea 
ee seri! Amica Pera a 
Du - Jc’ - ap (407) 
t per pre” 
32 
