Page:Harold Dennis Taylor - A System of Applied Optics.djvu/400

 {| negative for rays in secondary planes, but positive for rays in primary planes. Hence the manner in which the two curves cross one another at 27 and 30 degrees respectively, after which there follows a rapid mutual separation. Now it is clear that were the ratio between the aberrations of the third order invariably 3 : 1 or any other fixed ratio between the primary and secondary planes, then such graphs as these could never arise. But since (leaving all terms containing a out of consideration, as we are dealing here with pencils of infinitely small aperture) the third order functions of - are in the ratio 3:1 in the primary and secondary planes, while the third order functions of ^ are in the ratio 5 : 1 in the primary and secondary rays, then we can clearly see that in the case of the functions of - being of the opposite sign to the functions f* of ^ we may easily have the total aberrations of the third order plus in one plane while they are minus in the other. The separation corrections to the y's, existing, as we have seen, only in the primary plane, cause a still further degree of independence between the curvature errors in the two planes. we come to deal with the images thrown by pencils of relatively large aperture, for we have seen that in the primary plane there are functions of a that are seven and six times the corresponding functions in the secondary plane. Therefore it is that, if we take the lenses we have dealt .with and open out their apertures and locate their oblique foci (by obtaining the best possible distinctness of image), we may find the curvature errors come out substantially different to those shown on Plate XXIV. It is clear, then, that it is not always practicable to determine the working character of a lens by calculating its curvature errors for progress of photographic lenses towards perfection depends chiefly upon the successful elimination of the curvature errors of the order tan 4 , and the doing of it with the simplestossible lens construction.
 * Independence of the curvature errors of the third order in the two planes.
 * the fact that the curvature errors of the order tan 4 < are decidedly
 * the fact that the curvature errors of the order tan 4 < are decidedly
 * The disturbing effect of aperture upon the curvature errors of the third order.
 * And the scope for vagaries of this sort is still more enlarged when
 * And the scope for vagaries of this sort is still more enlarged when
 * Future progress de- pends upon elimina- tion of curvature errors of third and fourth orders.
 * infinitely narrow pencils only. It will easily be seen that the future
 * infinitely narrow pencils only. It will easily be seen that the future
 * }