LIGHT BY A PRISM.* FRANCIS E. NIPHER. An elementary proof of the law of minimum deviation of light by a prism may be obtained as follows: Let and r represent the angles of incidence and refraction at the point where the light enters the prism, and let ¿' and r' represent these angles where the same ray leaves the prism. The angles i and r' are in air and r and are in the glass. Then by geometry, calling d the angle of deviation. of the ray, dir+r' — i' = i + r' -A (1) where A =ri', is the made zero we shall have These traces make an angle of 45° with the axes which they intersect, and the distance of each point of intersection, from the origin, is A. The position of this plane is wholly A independent of the refractive properties of the prism. Any two prisms having equal FIG. 1. i Read before Regular Academy Meeting, November 4, 1895. Approved by Council, November 18, 1895. angles A, the ordinates d would terminate in points upon a common plane whose equation is (1). If any two prisms have unequal angles A, the ordinates d would terminate in different, but parallel planes. This plane is in all cases symmetrically placed with respect to the axes i and r'. Its position is shown in Fig. 1. There is, however, another condition depending on the index of refraction. We have sin i = n sin r sin 'n sin i' = n sin (A—r) n sin A cos rn cos A sin r = n sin AV1 — sin2 r—cos A sin i or finally sin r' sin An 2-sin 2 i cos A sin i.........(2) By considering the physical conditions, it is easily seen that the quantities' and i must be symmetrical in equation (2). If the light be made to reverse its direction, it will retrace its path through the prism. The angles r' and i will then replace each other. The same result will be obtained by solving (2) for sin i, which gives sin i sin A Vn 2 sin 2 r' cos A sin r'........(3) Equation (2) or (3) may be used in the computation of simultaneous values of i and r'. When '90, we have The values of that will be physically possible must lie between the value determined in the last equation, and 90°. These values of r' and i determine a curve on the plane of r', i of Fig. 1. This curve is convex toward the axes r' and i, and it is symmetrical with respect to them. This curve is a projection in a direction parallel to the axis d, of points on the plane represented by (1), which must represent the relation between d, r' and i. The conditions of symmetry involved in equations (1) and (2), both of which must be satisfied, show that the minimum ordinate d must lie in a plane symmetrically located with respect to the axes r' and i. This plane is determined by the condition r'i, which makes the entering and emergent rays symmetrical with respect to the bounding surfaces of the refracting angle A. Putting this condition in (2) it reduces to The angles' within the prism then become independent of n, their value being dependent on A only. If the sines are regarded as the variables, equations (2) and (3) represent an ellipse. Calling sin' sin ix, those equations become, 2 y and When the angle of the prism becomes zero the ellipse becomes the diagonal of a square whose sides are 2 n, the last equation being y=-x. When A 90° the ellipse becomes a circle whose equation is y2+ x2 = n2. For intermediate x2 n2. values of A the ellipse has the square whose side is 2 n as an envelope, the major axis lying in the line whose equation is The minor axis always lies in the line whose equation is yx, which involves the condition ir'. In its general form (2) becomes -= y= -x. The line yn sin A laid off on the axis y and the line whose equation is are conjugate diameters of the ellipse. Those portions of the ellipse corresponding to values of x or y greater than unity have no corresponding values of i and r', but within the limits where physical interpretation is possible, the symmetry of the curves i = f (r') with respect to the axes i andr' of Fig. 1 is evident. RELATIONS OF SALIX MISSOURIENSIS, BEBB, TO S. CORDATA, MUHL. BY N. M. GLATFELTER, M. D. The great variability of that species of Salix formerly included under the name of cordata has long been known. Indeed, so impressed am I with the extent of this variation, I have come to feel that no simple, straight statement regarding any one of its characters, can represent a truth: it is for this reason that the usual descriptions render but a part of a truth, and, by too much inclusion, become virtually errors. Looking at the different forms of leaves and stipules as represented on the plates accompanying this paper, one will be struck with the apparent indifference of the plant as to what kind of leaf or stipule it produced. Certain of its forms have, long ago, been named as varieties. The matter of erecting one of these forms into a species under the name of Missouriensis, judging from the language of its projector, Mr. M. S. Bebb (see Garden and Forest, p. 373), seems, as yet, a somewhat open one. Considering it important to know whether we really have here a new species, and what its specific limits might be, I have collected during the season just past, nearly two hundred specimens. The range of the territory of my collections is limited westward to Forest Park, Ferguson and Creve Coeur lake, northward to the Missouri river, eastward to the Illinois bluffs and Cahokia. For convenience of handling my subject-matter, I shall assume, from the first, but one species under the name S. Cordata, the reasons for which assumption will be apparent later on. North of the city along most of the ravines it grows abundantly, extending almost to the very heads of the ravines or gullies, most of which, in dry seasons, become perfectly dry, as is the case at the present writing. These ravines and small creek valleys were all covered over with that earliest alluvium |