SPE'CULUM, a name frequently given to a mirror used for any scientific purpose, as in a reflecting telescope. For the astronomical bearings of the speculum we refer to TELESCOPE, but we here give the first part of the simpler mathematical description of a pencil of nirror, to which we proceed, referring to Mr. Griffin's work on Orme or further information. Let a of rays fall on the spherical nirror A B from the point r, of which rays F B is one. Let r B be into Bp. It is supposed that P is in the radius o A, which is he axis of the mirror ; o being the centre of the sphere. Let A o =r, r= u, ep=r. The nearer B is taken to A, the more nearly does the point p approach to a certain point F, at which the image of P is mid to be formed : not that any rays are actually reflected to r, but because all the rays which are reflected from points near to t. fall exceedingly near to F, which is the cusp, and brightest point of the CAUSTIC. If e P=20, the position of r is determined by the equation 1 2 1- = - — w r U The point p however is always nearer to A, or lies between -r and A, and p r, or the longitudinal aberration, is thus found : let the length of the are en be y ; then .
1 p F = — .... very nearly, if y be not very great. And for the lateral aberration r t, we have 1\ 2 (3).
Again, there is for all the rays proceeding from r, after reflection, a circle through which they all pass, as in LENS. The distance of this circle of least aberration from the focus r toward A, is the following expression :— 3 ter' /1 1\i- •... if r be the whole semi-arc of the mirror : it is therefore three-fourths of the longitudinal aberration of the extreme ray. The diameter of this circle of least aberration is 1 1 1\ e - z) • or one-half the lateral aberration of the extreme When the rays fall parallel to each other on the mirror, u is infinite, and we have - r for the value of w, y2 1 for the longitudinal aberration, 2 4r ?/3 for the lateral aberration, for the distance of the circle of 16r least aberration from focus, and for its diameter.
4 When u= r, or the incident pencil is thrown from the centre, it is returned again to the centre, and there are no aberrations.
When u is less than r, or r is between o and A, F then falls beyond o, and, as r approaches to the middle point of o A, recedes without limit. When u= r, or P IS at the middle point of o e, all the
reflected rays are parallel to one another and to the axis of the mirror.
And when e is less than 1 - w becomes negative, or the focus is on 2 the other side of the mirror, and the reflected rays diverge ; but only the latitudinal aberration alters its sign.
The formulm for a convex mirror may be found by making r nega tive in those for a concave mirror. Hence w is always negative, or the focus of every pencil is behind the mirror : the longitudinal aberra tions change sign, but not the latitudinal ones : and as w has also changed sign, the effect is that p is always nearer to the mirror than F, as before.
The image in a convex mirror is always upright ; and in a concave one always inverted, except when the object falls between the principal focus (or middle point of the radius) and the mirror.
The astronomical value of the speculum depends on the quantity of light that it can concentrate, and on the precision with which it forms the optical image of a distant object. Hence the magnitude, the cur vature, and the surface polish are all of importance. The figure may be parabolic, where every part has the same focus, or it may be an ellipsoid, where the edge is of shorter focus than the centre, or if longer an hyperboloid. The grinding and polishing of lenses in a refracting telescope, to say nothing of the difficulty of obtaining good optical glass, greatly limit the perfection of that form of telescope ; but an amount of error that would be tolerated in the best lenses, would be fatal in the speculum of a large reflecting telescope. The difficulties of forming an accurate speculum constitute a problem of the highest order, the solution of which has received the careful study of first-rate astronomers and mechanicians. The reader who is desirous of seeing how these difficulties have been overcome, will do well to consult the memoir communicated to the Royal Society of London in 1840 by,Lorcl Oxmantown (now Earl of Rosso) entitled AE Account of Experiments on the Reflecting Telescope.' We may also rays incident upon a mirror, so as to make this article a counterpart to Litss.