Crucial to the understanding of surface-wave propagation in an anisotropic elastic solid is the notion of transonic states, which are defined by sets of parallel tangents to a centred section of the slowness surface. This study points out the previously unrecognized fact that first transonic states of type 6 (tangency at three distinct points on the outer slowness branch S
1
) indeed exist and are the rule, rather than the exception, in so-called C
3
cubic media (satisfying the inequalities
c
12
+
c
44
>
c
11
-
c
44
> 0); simple numerical analysis is used to predict orientations of slowness sections in which type-6 states occur for 21 of the 25 C
3
cubic media studied previously by Chadwick & Smith (In
Mechanics of solids
, pp. 47-100 (1982)). Limiting waves and the composite exceptional limiting wave associated with such type-6 states are discussed.