indeed, in some cases, be a real advantage, as pointed out by Ames in his paper on the concave grating* The amount of astigmatism is exceedingly small in an instrument of the dimensions and construction here shown, as the angle between the incident and reflected ray and the geometrical axis is less than one degree. In the first trial instrument which was constructed after this plan, the focal length of the mirror was about 175 centim. and its aperture a little less than 6 centim. The prism was of white flint of about the same aperture, with a dispersion of 3 degrees from A to H. Although the apparatus was mostly made of wood, and the adjustments were in consequence rather rough and unsatisfactory, the results obtained were even better than I had anticipated. The whole spectrum from the extreme violet to the extreme red was very bright and clean, with the lines very sharply defined. With an eyepiece having a magnifying-power of 70 and a slit-width of 01 millim. (at which width the spectrum was almost too bright for comfort), both the Ni line and the lines at 5892 between the D's could be seen, and the doubles in the tail of A were clearly distinguishable f. The freedom from diffused light may be judged from the fact that the a and A lines were both clearly seen without the aid of a colour-screen, while with this the lines below A at wave-length 8300 could be seen distinctly without taking any unusual precautions. Better evidence of this point is afforded by photographs of the lower end of the spectrum, which have been taken on an ordinary Cramer Isochromatic plate, whose maximum of sensitiveness lay in the yellowish green. If any considerable amount of diffused light had been present the plates would have been hopelessly fogged by it, as the time of exposure was in some cases nearly an hour. The linear expansion of the spectrum was so small (4.9 centim. from A to D) that some of the detail which was present was masked by reason of the coarseness of the grain of the plate, and a second mirror having a focal length nearly three times as great (470 centim.) was therefore substituted. *"Concave Grating in Theory and Practice," Phil. Mag. vol. xxvii. p. 369. † If we define the purity of the spectrum by the relation P= (Encyc. Brit. Art. Spectroscopy), where D is the width of the slit, & the angular value of the collimator aperture, as viewed from the slit, we have, in this case, for yellow light P=0.67 R, viz. a purity of nearly 70 per cent. of the theoretical resolving-power of the prism was obtained. With the longer-focus collimator a purity of nearly 80 per cent. was reached. With this new mirror the advantages of this form of instrument were still more apparent, and the results obtained leave little to be desired. With it photographs of different portions of the spectrum, extending from the violet to the red, have been obtained, which show an amount of sharpness and detail which is, I think, considerably greater than has heretofore been obtained with a single prism of the material and aperture of the one here used. Two other modifications of the form which have suggested themselves during the course of these experiments are shown in figs. 3 & 4. In the first of these the collimator is placed in the position occupied by the plane mirror, and the rays pass directly from the slit through the prism without collimation. The slit and spectral image are therefore situated at the two principal conjugate foci of the mirror. In this form, which has been given only a preliminary trial, the definition is, as might have been expected, decidedly inferior to that in the form of the instrument just described. The advantage which this form offers is simply its great simplicity and cheapness, the number of optical surfaces involved being only 4. In the second form (which is not properly a modification of the Littrow form at all, as the rays traverse each prism only once) separate concave mirrors or lenses are used for the collimator and for the view-telescope. The arrangement is shown in fig. 4. A fixed collimating telescope with slit at s and collimator at a sends the beam through a prism to a plane mirror m, by which it is reflected to a second prism p placed by the side of the first; after passage through which it falls upon the objective of the view-telescope T, which is also fixed in position, at an angle with the first equal to the angle between the incident and reflected rays on the mirror m. The prismtable on which the two prisms are mounted is connected with the arm which carries the mirror m by a minimum-deviation attachment, as in the previous forms. A little consideration will show that the central ray in the field of the T will always remain at minimum deviation as the arm carrying the mirror m is revolved. This form has not been actually tried, but would seem to offer certain advantages when it is desirable for any reason to use separate telescopes for collimating and observing. XIII. Geometrical Interpretation of log Uq. To the Editors of the Philosophical Magazine. N the notice of Dr. Molenbroek's Anwendung der Qua Internionen auf der Geometrie, your reviewer says, 66 It would probably baffle even a Hamilton to give a geometrical interpretation of log Uq" (Phil. Mag. vol. xxxvii. p. 333). As this matter has been treated of in several of my papers, I send you the interpretation required. The general quaternion q may be analysed into the product of a ratio and a versor; by Uq is meant the versor. Let a denote the axis of the versor and A its amount in T radians, then Uq=a and log Uq=Aa'; but log a = a2, therefore log Ug=Aa. A more correct definition of A is the ratio of twice the area of the sector to the square of the initial radius; for that definition applies also to a hyperbolic versor. The geometrical meaning of the above expression will become evident on considering the more general versor given by an equiangular spiral. Let a quinion be denoted by q', and let it be defined to be such that log Uq=Aa” = A cos w+A sin w.a α π π we then find that A sin w. a is the logarithm of the angle and A cos w the logarithm of the radius of an equiangular spiral of axis a and constant angle w, the initial radius being unity. Thus w is the constant angle between the radiusvector and the tangent, or rather the difference of the angle from the initial radius to the tangent and that from the initial radius to the radius-vector. In the case of the circle this difference angle is a quadrant: this is the explanation of the quadrantal versor in log Uq. In the spiral the quantity A is the magnitude of the complex logarithm, and a" gives the components of the logarithm. The expansion depends on the scalar component of the logarithm, while the rotation depends on the vector component. In the case of the circle, that is of Uq, the scalar logarithm vanishes. For further elucidation of this matter consider a hyperbolic quaternion. Let p denote such a quaternion; when the multiplier is removed we have Up=a^ and therefore log Up=iAaễ. iA π Here A is the ratio of twice the area of the hyperbolic sector π to the square of the initial radius, and a2 is the constant sum of the angle from the initial radius to the tangent and that from the initial radius to the radius-vector. This leads us to consider the hyperbolic analogue of the equiangular spiral. Let p' denote the corresponding hyperbolic quinion; then Here w denotes the constant sum of the angle between the initial vector and the radius-vector and the angle between the initial vector and the tangent. The scalar term iA cos w is the logarithm of the radius-vector, while the vector term iA sin w. a is the logarithm of the hyperbolic angle. Here i is the scalar -1; in the papers on "The Principles of Elliptic and Hyperbolic Analysis," and on "The Definitions of the Trigonometric Functions," I have shown that a quantity which is the sum of a scalar independent of i and another scalar dependent on i is represented along one straight line. ALEXANDER MACFARLANE. University of Texas, Austin, Texas, U.S.A., May 10, 1894. XIV. On a Fundamental Question in Electro-Optics. GENTLEMEN, WILL you kindly afford me space for a few remarks in connexion with Professor Quincke's letter, which appeared in the May number of this Magazine. The object of that letter was to draw attention to the fact that, in my paper on Electro-Optics which appeared in the April number, I made no mention of a paper of Professor Quincke's, published eleven years ago, which gives an account of experiments by him on the same subject and by similar methods. I think I should preface what I have to say on this matter with an expression of regret for my forgetfulness. I was well aware of the existence of that paper. I received a copy from the author, and perused it I think immediately on receipt. With regard to the electro-optic effects there described, as given by an interference-refractor, it was evident to me then, as it is now, that they were in their nature and immediate origin essentially different from those pure double refractions that are given regularly by the common polari scope and compensator as effects of electrostatic stress. They were evidently produced by mechanical disturbance of the dielectric; and effects of that kind are of no interest in electrooptics, except as hindrances to exact observation. It may be easily understood, therefore, that I omitted all reference to Professor Quincke's experiments in my paper, not from any positive intention, but because they did not occur to me as contributing in any degree to the solution or illustration of the question that I had in hand. In this connexion I may notice an old set of experiments of my own upon the same question, which did not occur to me as worth mentioning in my paper, though they had given a clear and striking exhibition of the double refraction. The dielectric was carbon disulphide, the electro-optic field was a very obtuse and very thin prism which extended from end to end of a large plate-cell, and the light (monochromatic and unpolarized) entered and left the cell respectively through the collimator and the telescope of an ordinary spectroscope, the slit being parallel to the lines of force. As the mechanically compressed prisms of glass acted in Fresnel's well-known experiment, so the electrically strained prism of CS, acted here. At high potentials the telescope gave two parallel images of the slit, clearly, though not very widely, separated from each other, and polarized in planes parallel and perpendicular to the lines of force; but the mechanical disturbance of the dielectric rendered this result useless for my purpose, as it kept the double image of the slit moving incessantly and very irregularly, so that no probable inference could be drawn as to the absolute retardations of the two component rays. I hope to have something more to say about this method and its results hereafter. There are two statements in Professor Quincke's letter which require some qualification. The first is, that his methods were identical with those followed by myself. This applies truly to the kind of instrumental means employed, and to the general conception of the arrangements, means and methods of great delicacy, for which we are indebted to Professor Jamin, by whom indeed they were put among the commonplaces of the higher experimental optics. But the method described in my paper includes something more: it deals with the chief difficulty of the subject by detaching the double refraction from the irregular and ever present effects of mechanical disturbance; and it brings out in this way the fact-clearly enough, perhaps, for a first and imperfect proofthat electric stress acts exclusively on the (Fresnel's) vibration which is directed along the line of force. Phil. Mag. S. 5. Vol. 38. No. 230. July 1894. L |
