Substituting the distribution functions, equations (4.107) and
(4.108) into the constraint equations (4.109) and
(4.110) and using equations (4.37) - (4.41)
we can find the coefficients.
The conservation of mass, 
, gives
Considering the coefficients of 
and 
separately we see
and
The conservation of momentum, 
, gives
Considering the coefficients of and 
we find
and
The constraint 
gives
Setting 
and considering the coefficients of 
and
gives
and
respectively which can be solved to find
and
Considering the coefficient of for the two cases and
gives
and
which have solution
and
The coefficient of with 
gives directly
The values of 
and 
can now be found from
equations (4.119)
and (4.120) to be
and
The constraints 
and
give
and
in the same way as equations (4.119), (4.120),
(4.122) and (4.123).
The constraint 
gives
The coefficient of with and
gives
and
These have solution
and
Setting 
and 
gives
The coefficient of 
gives
when and . Setting and
and substituting
the expression found for d in equation (4.146) gives
Finally we find the value of 
and 
from equations (4.136)
and (4.137):
and
This gives all the coefficients in equations (4.107) and (4.108).
