The magnetic field along a hole bored through a sequence of dipping layers with varying magnetization and planar interfaces is calculated. The emphasis is on recovering remanent magnetization polarity transitions for magnetostratigraphic applications (dating, correlations), although intensity of remanent magnetization can in itself be a useful rock property. Results are presented for various cases of geologic interest in the form of axial vertical profiles and section maps of the holes at varying distances from polarity reversal interfaces. The vertical profiles demonstrate a resolving power of about six times the hole radius; meaningful magnetostratigraphies can be expected for rocks with a magnetization larger than 10 (super -3) Am (super -1) , for instruments with a sensitivity of 0.1 nT. In a number of natural occurrences, it may not be necessary to resort to progressive demagnetization to recover the polarity sequence. Depending on which magnetic field and magnetization component one looks at, the section maps display characteristic patterns, in which, for instance, the direction of magnetization and the dip of the layers interfere. These maps are discussed in some detail. They can be used as guidelines to build a multisensor vector magnetometer (downhole magnetostratigraphic tool), whose output should be coupled with measurement of magnetic susceptibility for reduction of induced magnetization and with output from a surface instrument in a differential operation mode to reduce transient magnetic variations.