[Illustration: FIG. 167.—The Principle of the Hartmann Test.] The arrangement of holes actually found useful is shown in Hartmann’s original papers, and also in a very important paper by Plaskett (Ap. J. _25_ 195) which contains the best account in English of Hartmann’s methods and their application. Now each hole in the screen transmits a pencil of light through the objective at the corresponding point, and each pencil comes to a focus and then diverges, the foci being distributed somewhere in the vicinity of what one may regard as the principal focus, _B_. For instance in Fig. 167 are shown five pairs of apertures _a_, _a′_, _b_, _b′_, etc., in five different zones. Now if a photographic plate be exposed a few inches inside focus as at C each pencil from an aperture in the screen will be represented by a dot on the photograph, at such distance from the axis and from the corresponding dot on the other side of the axis as the respective inclinations of the pencils of light may determine. Similarly a plate exposed at approximately equal distance on the other side of the general focus, as at _D_, will show a pattern of dots due to the distribution of the several rays at a point beyond focus. Now if all the pencils from the several apertures met at a common focus in _B_, the two patterns on the plates _C_ and _D_ would be exactly alike and for equal distance away from focus of exactly the same size. In general the patterns will not exactly correspond, and the differences measured with the micrometer show just how much any ray in question has departed from meeting at an exact common focus with its fellows. For instance in the cut it will be observed that the rays _e_ and _a′_ focus barely beyond _C_ and by the time they reach _D_ are well spread apart. The relative distance of the dots upon these corresponding plates, with the distance between the plates, shows exactly at what point between _C_ and _D_ these particular rays actually did cross and come to a focus. Determining this is merely a matter of measuring up similar triangles, for the path of the rays is straight. Similarly inspection will show that the rays _d_ and _d′_ meet a little short of _B_, and measurement of their respective records on the plates _C_ and _D_ would show the existence of a zone intermediate in focus between the focus of _e,e′_ and the general focus at _B_. The exact departure of this zone from correct focus can therefore be at once measured. A little further examination discloses the fact that the outer zone represented by the rays _a,b_, and _a′,b′_ has not quite the same focus at the two extremities of the same diameter of the objective. In other words the lens is a little bit flatter at one end of this diameter than it is at the other, so that the rays here have considerably longer focus than they should, a fault by no means unknown although fortunately not very common.