plane parallel-sided glass bent into a curved shape.

The 4 concave lenses; 5. the Equi-concave; 6. the Unequally or Crossed concave1; 7. the Plano-concave; and 8. the Convexo-concave (or concavo-convex), are so exactly the reverse of the corresponding convex lenses, that it will be a good exercise for you to trace out the character of each till you have got it all clear before you.

We shall find much similarity between the action of the concave mirror and the convex lens, and between the convex mirror and the concave lens; as the convex lens, like the concave mirror, is the more generally useful, we shall explain it more particularly. The point where the rays meet, and from which they afterwards diverge, is, as with a mirror, the focus, real or virtual as the case may be. The focus for parallel rays is PF, and its distance from the lens is the focal length, or as it is frequently expressed, the focus of the lens. As the rays may pass through the lens either way, it has a PF on each side. Ps and axes, objects and images, are as in mirrors, only not on one but on both sides. There is however this great difference between reflection and refraction, which if we do not yet

1 Nos. 5 and 6 are both frequently called double concave.

clearly see, we ought not to go further till we have mastered it. In reflection the angles of incidence and reflection are both in the same medium, and always equal. In refraction the angles of incidence and refraction are in media of different density, and are always unequal, according to a proportion between their sines which differs in different materials. Still, the cases are many, as we shall see, in which reflection and refraction lead to similar results: the position, the inversion, the magnitude, and the brightness of the focal images follow the same rule in mirrors and lenses; but with this great difference, that in reflection the position of P, coinciding with the radius of the sphere, and the equality of the angles, are sufficient to determine every case; but in refraction we have to take into account the proportion of the sines. You might work all this out for yourselves; but I will try to make it plainer by an example, thus,—taking an equi-convex lens, and tracing the course of a ray through it—that is, of course, of any ray that can fall upon it. You will understand that if it passes along the principal axis, coinciding with P, it suffers no refraction; everywhere else it will; at the first convex surface towards P, at the second from p, and the combined effect will bring it down, as in the

prism, with an angle of deviation, to a focus, the position of which, whether principal or conjugate, on the axis, be it principal or secondary, will be determined not only by the amount of curvature (that is, the position of P) as in reflection, but also by the refractive power of the medium. It so happens that in the case of an equi-convex lens of ordinary glass this will bring

P

PF

out PF at about the distance of the radius, a convenient but accidental coincidence; with a denser medium, as diamond, the focal length with the same radii would be shorter; longer, with a rarer medium, as ice. And so as to all diverging rays, which as you know are emitted by all objects at ordinary distances; these, making greater and greater angles with P as the radiant approaches the lens, will have their divergency less and less diminished, and consequently their focal intersection with the axis

more and more deferred, till the radiant reaches the PF of rays coming in the opposite direction— when its rays on reaching the lens will pass on parallel. If the radiant is moved still nearer to the lens, the rays will pass the parallel state and diverge, but with a diminished angle, so as to point backwards to a virtual focus more distant than PF. And all this will, as you know, hold equally good reversed, if you reverse the direction of the light.

Now, in further illustration, let us suppose our equi-convex lens split in two edgeways. What will result? Two plano-convex lenses, each half as deep, or strong, that is, having half as much refraction, and therefore about twice the original focal length, which will not be affected by their facing opposite ways.

But should we get the same result from splitting an unequally convex lens? Certainly not : for though we have the same medium, and therefore the same refractive index, or power, we have different radii for the two convex faces, that is different positions of P, and consequently different angles and foci, always increasing as the curvature diminishes, and therefore longer for the one half-lens than the other. If we could imagine glass soft enough to be compressed at pleasure, and if we could gradually flatten the

convex side of either half-lens without injuring its spherical form, we should gradually lengthen the focus, till at last, when it became plane, we should have only a piece of common plate-glass left. All this of course would be reversed with concave lenses.

We shall now be able to solve some interesting problems with such lenses as may be much more easily procured than mirrors would be. Reading-glasses and eye-glasses, or old spectacle-glasses, such as may be found at an optician's or watchmaker's, will do very well. We may have a few failures at first, for want of suitable focal lengths, but perseverance is sure of its reward.

1. To find PF, the focal length, of any lens. If convex, which will be known by its magnifying objects, hold it up so that the sun may shine through it, taking care that the rays may fall upon it as perpendicularly as possible. Now trace their progress by moving a card, held parallel to the lens, steadily away from its back. You will find the circle of light gradually diminish till it ends in a very small brilliant spot, and then enlarges again. That small spot is the image of the sun in PF. No lens will show it perfectly clear from surrounding light, and the shorter the focus in proportion to the aperture,