time looking down at a near object, as in reading or writing, I found some. times that the horizontal lines of fig. 2 crossed each other ; but they became parallel again when I had looked for some time at distant objects. In order to define the position of the corresponding points in both fields of vision, let us suppose the observer looking to the centres of the two sides of fig. 2, and uniting both pictures stereoscopically. Then planes may be laid through the horizontal and vertical lines of each picture and the centre of the corresponding eye. The planes laid through the different horizontal lines will include angles between them, which we may call through the fixed point and the horizontal meridian. The planes going through the vertical lines include other angles, which may be called angles of longitude, their zero coinciding also with the fixed point and with the apparently vertical meridian. Then the stereoscopic combination of those diagrams shows that those points correspond which have the same angles of altitude and the same angles of longitude; and we can use this result of the experiment as a definition of corresponding points. We are accustomed to call Horopter the aggregate of all those points of the space which are projected on corresponding points of the retinæ. After having settled how to define the position of corresponding points, the question, what is the form and situation of the Horopter, is only a geometrical question. With reference to the results I had obtained in regard to the positions of the eye belonging to different directions of the visual lines, I have calculated the form of the Horopter, and found that generally the Horopter is a line of double curvature produced by the intersection of two hyperboloids, and that in some exceptional cases this line of double curvature can be changed into a combination of two plane curves. That is to say, when the point of convergence is situated in the middle plane of the head, the Horopter is composed of a straight line drawn the centre of both eyes and intersecting the straight line. When the point of convergence is situated in the plane which contains the primary directions of both the visual lines, the Horopter is a circle going through that point and through the centres of both eyes and a straight line intersecting the circle. When the point of convergence is situated as well in the middle plane of the head as in the plane of the primary directions of the visual lines, the Horopter is composed of the circle I have just described, and a straight line going through that point. There is only one case in which the Horopter is really a plane, as it was supposed to be in every instance by Aguilonius, the inventor of that name, namely, when the point of convergence is situated in the middle plane of the head and at an infinite distance. Then the Horopter is a plane parallel to the visual lines, and situated beneath them, at a certain distance which depends upon the angle between the really and apparently vertical meridians, and which is nearly as great as the distance of the feet of the observer from his eyes when he is standing. Therefore, when we look straight forward to a point of the horizon, the Horopter is a horizontal plane going through our feet—it is the ground upon which we are standing. Formerly physiologists believed that the Horopter was an infinitely distant plane when we looked to an infinitely distant point. The difference of our present conclusion is consequent upon the difference between the position of the really and apparently vertical meridians, which they did not know. When we look, not to an infinitely distant horizon, but to any point of the ground upon which we stand which is equally distant from both our eyes, the Horopter is not a plane ; but the straight line which is one of its parts coincides completely with the horizontal plane upon which we are standing The form and situation of the Horopter is of great practical importance for the accuracy of our visual perceptions, as I have found. Take a straight wire-a knitting-needle for instance—and bend it a little in its middle, so that its two halves form an angle of about four degrees. Hold this wire with outstretched arm in a nearly perpendicular position before you, so that both its halves are situated in the middle plane of your head, and the wire appears to both your eyes nearly as a straight line. In this position of the wire you can distinguish whether the angle of the wire is turned towards your face or away from it, by binocular vision only, as in stereoscopic diagrams; and you will find that there is one direction of the wire in which it coincides with the straight line of the Horopter, where the inflexion of the wire is more evident than in other positions. You can test if the wire really coincides with the Horopter, when you look at a point a little more or a little less distant than the wire. Then the wire appears in double images, which are parallel when it is situated in the Horopter line, and are not when the point is not so situated. Stick three long straight pins into two little wooden boards which can slide one along the other; two pins may be fastened in one of the boards, the third pin in the second. Bring the boards into such a position that the pins are all perpendicular and parallel to each other, and situated nearly in the same plane. Hold them before your eyes and look at them, and strive to recognize if they are really in the same plane, or if their series is bent towards you or from you. You will find that you distinguish this by binocular vision with the greatest degree of certainty and accuracy (and indeed with an astonishing degree of accuracy) when the line of the three pins coincides with the direction of the circle which is a part of the Horopter. From these observations it follows that the forms and the distances of those objects which are situated in, or very nearly in, the Horopter, are perceived with a greater degree of accuracy than the same forms and distances would be when not situated in the Horopter. If we apply this result to those cases in which the ground whereon we stand is the plane of the Horopter, it follows that, looking straight forward to the horizon we can distinguish the inequalities and the distances of different parts of the ground better than other objects of the same kind and distance. This is actually true. We can observe it very conspicuously when we look to a plain and open country with very distant hills, at first in the natural position, and afterwards with the head inclined or inverted, looking under the arm or between our legs, as painters sometimes do in order to distinguish the colours of the landscape better. Comparing the aspect of the distant parts of the ground, you will find that we perceive very well that they are level and stretched out into a great distance in the natural position of your head, but that they seem to ascend to the horizon and to be much shorter and narrower when we look at them with the head inverted : we get the same appearance also when our head remains in its natural position, and we look to the distant objects through two rectangular prisms, the hypothenuses of which are fastened on a horizontal piece of wood, and which show inverted images of the objects. But when we invert our head, and invert at the same time also the landscape by the prisms, we have again the natural view and the accurate perception of distances as in the natural position of our head, because then the apparent situation of The alteration of colour in the distant parts of a landscape when viewed with inverted head, or in an inverted optical image, can be explained, I think, by the defective perception of distance. The alterations of the colour of really distant objects produced by the opacity of the air, are well known to us, and appear as a natural sign of distance; but if the same alterations are found on objects apparently less distant, the alteration of colour appears unusual, and is more easily perceived. It is evident that this very accurate perception of the form and the distances of the ground, even when viewed indirectly, is a great advantage, because by means of this arrangement of our eyes we are able to look at distant objects, without turning our eyes to the ground, when we walk. April 21, 1864. Major-General SABINE, President, in the Chair. The following communications were read :I. “On the Orders and Genera of Quadratic Forms containing more than three Indeterminates." By H. T. STEPHEN SMITH, M.A., F.R.S., Savilian Professor of Geometry in the University of Oxford. Received March 22, 1864. Let us 'represent by fi a homogeneous form or quantic of any order containing n indeterminates ; by (@(1)), a square matrix of order n; by in (al ), its ith derived matrix, i.e. the matrix of order :=I, the con stituents of which are the minor determinants of order i of the matrix (@(1)); and lastly, by fi, a form of any order containing I indeterminates, the coefficients of which depend on the coefficients of fi When fi is transformed by (a(1)), let fi be transformed by (a); if, after division or multiplication by a power of the modulus of transformation, the metamorphic of fi depends on the metamorphic of fi, in the same way in which fi depends on fi, fi is said to be a concomitant of the ith species of fi. Thus : a concomitant of the 1st species is a covariant ; a concomitant of the (n-1)th species is a contravariant; if n=2 there are only covariants; if n=3 there are only covariants and contravariants; but if n>3, there will exist in general concomitants of the intermediate species. There is an obvious difference between covariants and contravariants on the one hand, and the intermediate concomitants on the other. The number of indeterminates in a covariant or contravariant is the same as in its primitive; in an intermediate concomitant, the number of indeterminates is always greater than in its primitive. Again, to every metamorphic of a covariant or contravariant, there corresponds a metamorphic of its primitive; whereas, in the case of a concomitant of the intermediate order i, a metamorphic of the primitive will correspond, not to every metamorphic of the concomitant, but only to such metamorphics as result from transformations the matrices of which are the ith derived matrices of matrices of order n. It is also obvious that, besides the n - 1 species of concomitance here defined, there are, when n is >3, an infinite number of other species of concomitance of the same general nature. For from any derived matrix we may form another derived matrix, and so on continually; and to every such process of derivation a distinct species of concomitance will correspond. The notion of intermediate concomitance appears likely to be of use in many researches; in what follows, it is employed to obtain a definition of the ordinal and generic characters of quadratic forms containing more than 3 indeterminates. (The case of quadratic forms containing 3 indeterminates has been considered by Eisenstein in his memoir, “ Neue Theoreme des höheren Arithmetik,” Crelle, vol. xxxv. pp. 121 and 125.) Let p=n q=n тр «а PI represent a quadratic form of n indeterminates ; let (A(1)) be the symmetrical matrix of this form, and (A) the ith derived matrix of (A()); (A) will also be a symmetrical matrix, and the quadratic form p=I q=1 (A) p=1q=1 will be a concomitant of the ith species of fiIt is immaterial what P 9 principle of arrangement is adopted in writing the quadratic matrix (A1), and the transforming matrix (@(0)); provided only that the arrangement be the same in the two matrices, and that in each matrix it be the same in height and in depth. For example, if fi=a,** + a,x+ a, m+ ax + 26, X, X, + 2b, X, *x + 26,4, 5, +26, 4, 8, +265*, X, +266 XZ X, be a quadratic form containing four indeterminates, the form f,= (63-a, a) X; +(6: — a, a) x +(69–a, a) x; +2(6,6, -a, 6.)X, X, + 2(66, –a, 6) X, X, —206,65–6,6.) X, X, + 2(6,6, -a, b)X, X, +2(6.be-a,6.) X, X, is the concomitant of the second species of f. The n-1 forms defined by the formula (A), of which the first is the form f, itself, and the last the contravariant of fi, we shall term the fundamental concomitants of fii in contradistinction to those other quadratic concomitants (infinite in number) of which the matrices are the symmetrical matrices that may be derived, by a multiplicate derivation, from (A(1)).... Passing to the arithmetical theory of quadratic formsmi.e. supposing that the constituents of (A(1)) are integral numbers, we shall designate by V, V2, ... On the greatest common divisors (taken posi. tirely) of the minors of different orders of the matrix (A(1)), so that, in particular, V, is the greatest common divisor of its constituents, and in is the absolute value of its determinant, here supposed to be different from zero. By the primary divisor of a quadratic form we shall understand the greatest common divisor of the coefficients of the squares and double rectangles in the quadratic form ; by the secondary divisor we shall understand the greatest common divisor of the coefficients of the squares and of the rectangles ; so that the primary divisor is equal to, or is half of, the secondary divisor, according as the quadratic form (to use the phraseology of Gauss) is derived from a form properly or improperly primitive. It will be seen that V, V, .... On-, are the primary divisors of the forms fufa .... fn-2 respectively. We now consider the totality of arithmetical quadratic forms, contain. ing n indeterminates, and having a given index of inertia, and a given determinant. If a quadratic form be reduced to a sum of squares by any linear transformation, the number of positive and of negative squares is the same, |