It is known that there exist viscometric and non-viscometric flows in which all Rivlin-Ericksen tensors of order three and higher are zero. From Rivlin＇s work, it is also known that in these flows, the stress tensor in an incompressible viscoelastic fluid may be expressed in terms of nine material functions of which five are independent. It is shown here that only six of these functions are required, out of which only five are again independent. The well known case of a viscometric flow can also be deduced from the six term, general expansion obtained here.