Now let us consider the 3 parameter Kimura model (K3) for nucleotide substitutions. It is defined by the following infinitesimal generator
where 
> 0, 
>0, 
> 0 and 
In the Jukes-Cantor (JC) model, 
while
case 
is the Kimura 2-parameter (K2) model. It is
easy to deal with K3 as JC (at least with the following approach)
and we now do so.
The transition matrix 
associated with Q is simply 
where t is in some time units. The eigenvalues of
Q are 
.
Therefore,
In the sense of JC, 
and this simplifies to
. We conclude that the stationary
distribution is 
,
as 
has all its rows equal to this.
Notice that t is confounded with 
, 
and 
in the above expressions. Instead of studying 
or
t separately, it is the product, 
, also known as
the "amount of evolution", which is studied. More precisely,
is the quantity that gives us the expected number
of changes down a lineage over a time t.
Our interest in these models is in the table of joint probabilities
of observed pairs of nucleotides at one position, following separate
evolution from a common ancestor t time units back (e.g., 
).
Letting X denote the nucleotide at the ancestral site,
where 
is the probability of X at the ancestral node, and
, and 
. If these last two expressions have
the JC form, not necessarily with the same 
, we can easily
evaluate 
and similar expressions.
Our interest here is in obtaining simple expressions for 
,
the probability that the two nucleotides differ.
It is easy to see that we can also write
since 
is reversible, and so we can derive the expression
Suppose that 
and 
have the JC form with parameters
and 
. Then using the spectral
representation of exp tQ, we easily check that when 
(the limiting frequency),
Thus we obtain the famous Jukes-Cantor correction for multiple substitutions in the form
an estimate of 
, the expected
number of events separating the two taxa.