1080 19 2616-7573-25 47151 20548 18686 1-6050 1.53762-089621-037621-52308 18272 38820 2-4446 32007 506933-2132 8022 15 58 20569 1-6058 8029 53083 51655 20 70 36 330-63 59-37 40425 22814 16532 1-6910 1.4633 1-48840 1.02962 86318 1-93610 16424 1-4596 17272 33804 2.1779 8525 48 51 23075 1.7012 8506 53547 1.72874 1.98352 9627 06455 20622 1.6078 925753 226323 1-8333 9166 190 60.0 08316 · X, 90°-x in degrees and decimals of a degree, to correspond with Legendre's Tables. Nat. Log F(b, 0). Add log Ec. (2) Nat. ☎ +(1)−(2). Do. in (°). Log Eic -log cos x. Nat. do., = length. * Prof. Cayley on the Geodesic Lines 338 where the columns marked with an * show respectively the longitude of the node, and the length (or distance of node from vertex), for the geodesic lines belonging to the different values of the argument l'. The remarks which follow have reference to the stereographic projection of the figure on the plane of the equator, the centre of projection being the pole (say the South Pole) of the spheroid. It is to be remarked that if a point P of the spheroid is projected as above, by means of an ordinate into the point Q of the sphere radius OK(=OA), then projecting stereographically as to the spheroid and the sphere from the south poles thereof respectively, the points P and Q have the same projection. And it is hence easy to show that an azimuth a at a point of the meridian (parametric latitude N', normal latitude λ, and therefore tan λ= projected into an angle (a) such that C tan λ) is In fact in fig. 3, if we take therein OK, OC for the axes of x, z respectively, and the axis of y at right angles to the plane of the paper, and if we have at P on the surface of the spheroid an element of length PR at the inclination to the meridian PK, then if x, y, z are the coordinates of P, and x+8x, y+dy, z+dz those of R, we have Now, if the meridian and the points P, R are referred by lines parallel to Oz to the surface of the sphere radius O A, the only difference is that the ordinates z are increased in the ratio C: A; so that if the projected angle be (a), we have and then projecting the sphere stereographically from its south pole, the angle in the projection is (a). And according to the foregoing remark, the angle (a) thus obtained is also the projec = tion of a from the south pole of the spheroid. We have thus determine in the stereographic projection the inclination (a) to the radius, or projection of the meridian, of the geodesic line (parametric latitude of vertex =) at the point the parametric latitude of which is =N; viz. they enable the construction (in the projection) of the direction of the successive elements of the geodesic line. There would be no difficulty in performing the construction geometrically; but it would, I think, be more convenient to calculate (a) numerically for a given value of and for the successive values of X. Observe that for λ=0 we have (as above) 90°-a-l', and then C sin sin λ tan N C = = conse tan X A Α' quently tan (a)=cotl': but we have also cot l'= C that this equation becomes tan (a) = cotl, or we have 90° — (a)=l; viz., in the projection, the geodesic line cuts the equator at an angle the normal latitude of the vertex of the geodesic line. The preceding formulæ and results have enabled me to construct a drawing, on a large scale, of the stereographic projection of the geodesic lines for the spheroid, polar axis = equatorial axis. XLIV. Reply to Mr. Templeton's "Remarks suggested by Mr. Douglas's Account of a New Optometer." By J. C. DouGLAS, Government Science Teacher, Assistant Superintendent East-India Government Telegraph Department*. UNE [NFORTUNATELY Mr. Templeton has misunderstood my description of the optometer principle; his proof that the indications of an instrument on this principle are fallacious is therefore not applicable. I did not give exact measurements, as such were not necessary to a description of the principle; I * Communicated by the Author. merely indicated a principle on which an instrument might be constructed for accurately defining the region of accommodation. I did this after innumerable experiments had convinced me of the existence of the phenomena described, and careful consideration had led me to the explanation of the same. After I had arrived at my results I referred to Brewster's Journal; I found there the experiment of Le Cat, to which, I presume, Mr. Templeton refers when he states, "the case is simply an extreme example of the experiment described half a century ago in Brewster's Journal." In my former paper I have referred to Brewster's Journal (vol. iv. p. 89); but the experiment there described is not that described by me, as comparison will prove; it is not applied to any useful purpose; and, as I have already insisted, the explanation given is not the true one*. The first production of multiple images by two or more orifices is due to Father Scheiner (1685); the application of Father Scheiner's experiment to optometry to Porterfield (1759)†. Helmholtz recommends as a means of finding the far point the use of a point of light, a small opaque body being passed close to the eye, and the light moved to or from the eye until the opaque body is no longer seen in transit. I have suggested the use of two or more luminous points; and instead of moving these luminous points, I proposed to move the opaque body to or from the eye until it is seen single. The proposed method has several advantages: it is easily applied, more strictly defines the distance required, particularly in the case of an inexperienced observer; for the several shadows are very distinct, and their coalescence is therefore easily traced, easier than the gradual disappearance in Helmholtz's experiments; and the multiplication of shadows may be applied to purposes other than optometrical; for example, if several stars appear to be one only, in consequence of the encroachment on each other of their circles of confusion, by passing a small object close to the cornea, the number of shadows seen will indicate the number of luminous points. Mr. Templeton has referred to hypotheses of lens-adjustment, the most difficult portion of physiological optics. Rather than repeat what has been written elsewhere, I would refer Mr. Templeton to the chapters treating of this subject and the history of its development in Professor Helmholtz's work referred to already. The treatment is very full, and the several theories proposed are tested by experiment. Mr. Templeton's experiment with the ruler may be explained on the same principle as the formation of images by a pinhole, the frowning of myopes, &c. Mr. Templeton states, "physicists have assumed, on quite in* Another refutation of the explanation referred to appeared in the Philosophical Magazine for June last, in a paper by Mr. J. L. Tupper. + Helmholtz, Physiologische Optik. Leipzig, 1867. sufficient evidence, that the eye is simply a camera obscura, with a lens to form a picture, a retinal surface to receive a picture, and an adjusting-power for distance." That inaccuracies exist in works on "physics" is not to be denied; but text-books on experimental physics are scarcely the sources from which to obtain authoritative statements on difficult points in physiological optics. The comprehensive work of Professor Helmholtz*, the works of Donders, C. Stellwag, von Cariou, &c., would be truer representatives of the opinions of those who are building up this branch of knowledge. I do not find such physicists consider the eye any more like a camera obscura than it really is. Helmholtz, for instance, states in one place, the camera obscura is the optical instrument most similar to the eye in giving real images of objects; in another place, the eye acts on incident light essentially as a camera obscura (Physiologische Optik); but he fully elucidates the differences, and calls in the analogy where it assists his description; he discusses the various hypotheses of accommodation; considers not the lens only, but also the other refractive media, the several curvatures, the great relative thickness of the refractive media, &c. Mr. Templeton is no doubt aware images may be viewed through the sclerotica both in the dead and living eye, and in the latter by the ophthalmoscope; while accommodation is universally experienced, is observable with the ophthalmoscope, producible in the dead eye by galvanism, and measured in the living eye more or less accurately with the optometer. In production and reception of the image and in adjustment is not the analogy proved? Comparative study of the visual apparatus would be as efficient in assisting to a correct knowledge of the eye in man as the study of comparative physiology has been in the development of human physiology; but observations to be useful must, I submit, be accurate, and the publication of experiments which do not even give any qualitative result, positive or negative, is undesirable. Mr. Templeton states he has endeavoured to find the principal focus of the crystalline lens without success by experiments which "carry with them a suspicion of inexactness." I submit such experiments are useless, as no conclusion can be drawn from them: they do not afford even a useful degree of probability, the matter they are instituted to decide remaining as doubtful after as before their institution. The principal focus of the crystalline lens has been measured by Helmholtz, a reference to whose experiments† will point out the precautions to be taken, and may explain why Mr. Templeton's experiments were not decisive. * Physiologische Optik. |