Of 59 cases of Rheumatism, 34 were males, i. e. 5.54 in 100 of male Patients were rheumatic. 25 ... females, i.e. 3.52 ...... female ............ These cases were distributed equally between the Town and Country Patients Of 75 cases of Dyspepsia, 17 were males, i.e. 2.77 in 100 of male Patients. 58 ..... females, i.e. 8.18 ......... female ...... Ages of the 8 cases of Calculus (6 of which had been operated on and had recovered) were respectively 4, 5, 6*, 7*, 10, 49, 55, 58. VACCINATION, 333 individuals were vaccinated in the course of the year 1836 at the Hospital: of these, Cases in which the vaccination succeeded ....................................... 298 * Remaining for operation. t Seven of these had previously been vaccinated. REGISTER OF BAROMETER AND THERMOMETER, 1836. (Computed from the REG1st ER at the PHILosophic AL Society.) : XVII. On the Intensity of Light in the neighbourhood of a Caustic. [Read May 2, 1836, and March 26, 1838.] WHEN a great physical theory has been established originally on considerations and experiments of a simple kind, which by degrees have been exchanged for comparisons of more distant results of the theory with more complicated cases of experiment, it has always been considered a matter of great interest, to trace out accurately by mathe matical process the consequences, according to that theory, of different modifications of circumstances: which can then be compared with measures that have been made, or that may easily be made in future. It is with this view that I solicit the indulgence of the Society, for the following investigation of the Intensity of Light in the neighbourhood of a Caustic, as mathematically estimated from the Undulatory Theory. The investigation which I present here belongs, ostensibly, only to the case of reflection. The introductory part of it will, however, (with the proper modifications) apply equally well to all cases of refraction and all combinations of reflection and refraction. There seems also to be no reason why the latter part (the estimation of the intensity of light, by considering the wave of light when it leaves the last surface to be divided into a great number of small parts, whose separate effects Vol. VI. PART III. 3 C are then to be compounded,) should not apply to those cases. For though, strictly speaking, we ought to consider the wave to be thus broken up where it leaves the first surface, in order to find the intensity of vibration at every point of the second; yet it seems clear, that those reasonings which establish the definite reflection or refraction of a wave, (and which are founded upon the consideration above alluded to,) point out that there will be as to sense a mutual destruction of all vibrations at the second surface, (supposed to be not distant from the first,) excepting those which would be fully taken into account on the ordinary laws of Geometrical Optics. Where the light meets the second surface in the state of convergence, this conclusion perhaps is not so clear: but even there I believe that it may easily be shewn to be correct. I have mentioned these points because one of the most interesting cases of natural caustics (the rainbow) is affected by them; the exterior bow involving the first-mentioned condition, and the interior bow involving both the first and the second. 1. The notion of a caustic, and its mathematical definition, are essentially founded upon the laws of Geometrical Optics; and to these, therefore, we must refer in order to discover a representation of the conditions adapted to the investigations of Physical Optics. For simplicity we shall confine our diagrams to the plane of reflection, and shall consider the reflecting surface as symmetrical (to a sensible extent) with respect to that plane, so that the portion of the caustic formed by that part of the surface will be in the same plane. 2. In fig. 1., let the origin of light S be the origin of co-ordinates; a , y, the co-ordinates of a point X of the reflecting surface; p, q the co-ordinates of a point P in the reflected ray; V the length of the path of light from S to any point of the reflecting surface and thence to the point P. The ordinary law of reflection informs us that the angles of incidence and reflection are equal; and therefore, that, if we take a point X" on the reflecting surface very near to X, and join it with the origin and the point P, the lengthening ZX” of one |