In Part III of this series (Lighthill 1947
c
) it was shown that the problem of finding a plane steady adiabatic compressible flow round a body which reduces to a given incompressible flow (with or without circulation) when the Mach number tends to zero can be solved, in the subsonic region at least, if certain functions
ψ
n
(ז), whose properties are set out at length in Part II, are known, with their derivatives, for positive integral
n
and for ז < (
γ
– 1)/(
γ
+ 1), where
γ
is the adiabatic index. (The functions
ψ
n
(ז) depend on
γ
.) In the supersonic region the same functions may be needed for higher values of ז; but other values of
n
would also be needed. In Part I (Lighthill 1947
a
) it is shown how a knowledge of the
ψ
n
(ז) may help in the design of symmetrical channels. It is also well known (see, for example, Chaplygin 1904) that the exact solution of ‘free streamline’ problems in subsonic compressible flow may be achieved by means of the functions
ψ
n
(ז). Finally, it seems likely that future theories of compressible flow will use the functions
ψ
n
(ז), especially for positive integral
n
. These considerations have led us to tabulate the functions
ψ
n
(ז) and
ψ
'
n
(ז) for values of ז between 0 and ½ at intervals of 0∙02 and for the values 1, 2, 3, . . . , 15 of
n
taking
γ
= 1∙4 (
T
= 1/6 then corresponds to Mach number 1 and
T
= ½ and to Mach number √5.) We have taken
γ
= 1∙4 for air rather than
γ
= 1∙405, because, while both values have been widely adopted, the little experimental evidence relating to this rather variable quantity points to the lower value as nearer the truth; it is also considerably simpler to use.
ON THE HODOGRAPH TRANSFORMATION FOR HIGH-SPEED FLOW: II. A FLOW WITH CIRCULATION