In the companion paper (Holmquist et al. 1988), we concluded that there is no agreement on either the correct branching order or differential rates of evolution among the higher primates, and we examined in depth why this uncertainty in the evolutionary understanding of our closest living relatives persists. Recently, Lake developed two novel methods, based on group properties of transition and transversion operators, that (a) permit, in principle, objective resolution of problems of the above type and (b) attach a statistical significance level to the conclusions drawn. In the present paper, we develop formulas for using these two methods in tandem and apply them to study transversion differences in (1) nuclear DNA for a 7-kb segment of the psi eta-globin locus and a 3-kb intergenic region between the psi beta- and delta-globin loci and (2) mitochondrial DNA for the 896-bp fragment of Brown et al. Although each of these nucleotide sequence regions has its characteristic tempo and mode of evolution, the nuclear and mitochondrial data together, comprising a total of 10,939 base positions, support a Homo/Pan clade at the 97% confidence level. If we calibrate the divergence point for humans and chimpanzees at 5 Myr, consideration of the transversion branch lengths for the combined nuclear data indicates that the gorilla lineage branched off 600,000-900,000 years prior to that, although the 2 sigma sampling errors do not preclude either a temporal trifurcation for the three species or a considerably more ancient branch point for the gorilla. To resolve the length of this central branch to a relative accuracy of 25% and 30% will require a factor of 16 and nine times more data, respectively--i.e., in excess of 100,000 homologous nucleotides for each of the four primates. For the nuclear genes, heterogeneity in evolutionary rates between different parts of the genome is mostly restricted to the human lineage for these two segments. The lineage leading to chimpanzees has evolved 0.4 (3-kb fragment) to 3.5 (7-kb segment) times as rapidly as the lineage leading to humans, and that leading to the gorilla has evolved approximately one-fifth to one-half as rapidly as that leading to chimpanzees. Thus, even local molecular clocks can "tick" badly. As significant is the fact that virtually contiguous parts of the genome tick at markedly different rates.(ABSTRACT TRUNCATED AT 400 WORDS)