The chromatic dispersion, in the case of the object-glass, may be roughly stated to be measured by about one fiftieth of the aperture. Suppose for instance the discs, Fig. 64, to represent the image of any object, say the planet Jupiter. Then round that planet we should have a coloured fringe, and the dimensions of that coloured fringe, that is, the joint thickness of colour at A and D, will be found by dividing the diameter of the object-glass used by fifty. Now this is absolutely independent of the focal length of the telescope; therefore one way of getting rid of it is to increase the focal length of telescopes; and as the size of the image depends on focal length, and has nothing whatever to do with aperture, we may imagine that with the same sized object-glass, instead of having a little Jupiter as on the left of Fig. 64, we may have a very large Jupiter, due to the increased focal length of the telescope. Then, it may be asked, how about the chromatic aberration? It will not be disturbed. The aperture of the object-glass remains unaltered, and there is no more chromatic aberration here than in the first case; so that the relation between the visible planet Jupiter and the colour round it is changed by altering the focal length. But as we have seen, we are able by means of a combination of flint and crown glass to counteract this dispersion to a very great extent. How then about spherical aberration? Up to the present we have assumed that all rays falling on a convex lens are brought to a point or focus, but this is not strictly true, for the edges of a lens turn the rays rather too much out of their course, so that they will not come to a point; just as the rays reflected from a spherical mirror do not form a single focus. The marginal rays will be spread over a certain circular surface, just as the colour due to chromatic aberration covered a surface surrounding the focus. It was explained that for the same diameter of lens the circle of colour remained the same, irrespective of focal length, but in the case of spherical aberration this is not so; it diminishes as the square of the focal length increases; that is to say, if we double the focal length we shall not only halve, but half-halve, or quarter the aberration. Newton calculated the size of the circle of aberration in comparison with that due to colour, and he found that in the case of a lens of four inches diameter and ten feet focus, the spherical aberration was eighty-one and a half times less than that of colour. _It is found that by altering the relative curvatures of the surfaces of the lens, this aberration can be corrected without altering the focal length_; for any number of lenses can be made of different curvatures on each side but of the same thickness in the middle, so that they have all the same focal length, but the one, having one surface about three times more convex than the other, will have least aberration, so that it is the adaptation of the