In measuring the normal stress exerted by a fluid on a surface it is often convenient to puncture the surface with a ‘small’ hole leading to a larger cavity. An estimate of the stress is then made by measuring the hydrostatic pressure at the bottom of this cavity with a pressure transducer. Such a procedure, expedient though it is in practice, can result in large systematic errors in the estimate of the stress exerted on the surface, as has been shown experimentally by Kaye, Lodge, and Vale. The present paper gives a discussion of these errors based on kinematic considerations. The value of an approach of this kind is that it gives a physical understanding of the source of the error and suggests how such errors may arise even with rather complicated fluids. Three different configurations are examined. In the first example we consider a shear flow past a deep two-dimensional slot normal to the flow; for the particular case of a second-order fluid the intrinsic error is a quarter of the primary normal-stress difference, in agreement with the previous calculation of Tanner and Pipkin. In the second case, the rectilinear motion along a slot aligned with the flow is considered, and the result obtained by Kearsley is recovered. However, the present analysis is applicable to the rectilinear motion of any material along a slot and, accordingly, a new method is suggested by which direct measurements of the secondary normal-stress difference might be made. The third configuration comprises a shear flow past a circular hole and it is suggested, in this case, that the intrinsic error is approximately a sixth of the difference between the primary and the secondary normal-stress differences of the material.