where
and
The Adams-Bashforth formula is obtained by neglecting the term in (7.24) and is given as
we now make use of the following two facts about
is of constant sign in the interval and
is a continuous function of . We are thus in a position to apply the second mean value theorem of the integral calculus with the result that
where is one of the values of corresponding to values of in . By the definition of , this may be written as
in (7.24) and is given as 
is of constant sign in the interval 
and 
is a continuous function of 
. We are thus in a position to apply the second mean value theorem of the integral calculus with the result that 
is one of the values of 
corresponding to values of 
. By the definition of 
, this may be written as 
