  The alternating ternary sum in an associative algebra, $abc - acb - bac + bca
+ cab - cba$, gives rise to the partially alternating ternary sum in an
associative dialgebra with products $\dashv$ and $\vdash$ by making the
argument $a$ the center of each term: $a \dashv b \dashv c - a \dashv c \dashv
b - b \vdash a \dashv c + c \vdash a \dashv b + b \vdash c \vdash a - c \vdash
b \vdash a$. We use computer algebra to determine the polynomial identities in
degree $\le 9$ satisfied by this new trilinear operation. In degrees 3 and 5 we
obtain $[a,b,c] + [a,c,b] \equiv 0$ and $[a,[b,c,d],e] + [a,[c,b,d],e] \equiv
0$; these identities define a new variety of partially alternating ternary
algebras. We show that there is a 49-dimensional space of multilinear
identities in degree 7, and we find equivalent nonlinear identities. We use the
representation theory of the symmetric group to show that there are no new
identities in degree 9.
