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 ground is again the plane of the Horopter of our eyes.

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 f, a homogeneous form or quantic of any order containing n indeterminates; by (a(1)), a square matrix of order n; by

(a), its ith derived matrix, i. e. the matrix of order

stituents of which are the minor determinants of order i of the matrix (a(1)); and lastly, by fi, a form of any order containing I indeterminates, the coefficients of which depend on the coefficients of f1. When fis transformed by (a(1)), let ƒ¡ 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 f, in the same way in which f; depends on f1, 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, 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

represent a quadratic form of n indeterminates; let (A()) 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

will be a concomitant of the ith species of f. It is immaterial what

principle of arrangement is adopted in writing the quadratic matrix (A©), and the transforming matrix (a); 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.

2

For example, if ƒ1=a, x2+a, x2+ a ̧x2+ a ̧x2+ 2b, x, x2+2b, x, x3 + 2ь ̧x ̧x ̧+2b ̧ x ̧x ̧+2b ̧ x ̧x ̧+2b ̧ x, x be a quadratic form containing four ⚫ indeterminates, the form f2 =

=

6

(b2—a ̧ a ̧) X2 + (b2−a, a,) X2 + (b3−a, a ̧) X3
+(b2−a, a,) X2 + (b3—a, a ̧) X3 + (b2—a ̧ a ̧) X2

The n-1 forms defined by the formula (A), of which the first is the form f, itself, and the last the contravariant off,, we shall term the fundamental concomitants of f; 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 forms—i. e. supposing that the constituents of (A(1)) are integral numbers, we shall designate by V1, V2... Vn the greatest common divisors (taken positively) of the minors of different orders of the matrix (A), so that, in particular, v, is the greatest common divisor of its constituents, and V, 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 V1, V2, · V-1 are the primary divisors of the forms ff....fn-1 respectively.

....

We now consider the totality of arithmetical quadratic forms, containing 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,

whatever be the real transformation by which the reduction is effected. For the index of inertia we may take the number of the positive squares; it is equal to the number of continuations of sign in a series of ascending principal minors of the matrix of the quadratic form, the series commencing with unity, i. e. with a minor of order 0, and each minor being so taken as to contain that which precedes it in the series (see Professor Sylvester "On Formulæ connected with Sturm's Theorem," Phil. Trans. vol. cxliii. p. 481). The distribution of these forms into Orders depends on the following principle:

"Two forms belong to the same order when the primary and secondary divisors of their corresponding concomitants are identical."

Since, as has been just pointed out, there are, beside the fundamental concomitants, an infinite number of other concomitants, it is important to know whether, in order to obtain the distribution into orders, it is, or is not, necessary to consider those other concomitants. With regard to the primary divisors, it can be shown that it is unnecessary to consider any concomitants other than the fundamental ones; i. e. it can be shown that the equality of the primary divisors of the corresponding fundamental concomitants of two quadratic forms, implies the equality of the primary divisors of all their corresponding concomitants. And it is probable (but it seems difficult to prove) that the same thing is true for the secondary divisors also.

Confining our attention (in the next place) to the forms contained in any given order, we proceed to indicate the principle from which the subdivision of that order into genera is deducible.

1

2

If F, be any quadratic form containing r indeterminates, and F, be its concomitant of the second species, we have the identical equation

minants, y

y

k=1

(a) indicates that the deter

y

are to be taken for the indeterminates of

F2, the order in which they are taken being the same as the order in which the determinants of any two horizontal rows of the matrix

=

1

of F, are taken in forming the matrix of F. Let Of, for every

value of i from 1 to n 1; it will be found that, if we form the concomitant of the second species of 0, its primary divisor is the quotient Vi+Vi, which, as has been shown elsewhere (see Phil. Trans. Vi Vi-1

vol. cli. p. 317) is always an integral number. Let ; be any uneven

prime dividing Vi+1; we infer from the identity (B) that the

Vi V8-1

numbers prime to d;, which can be represented by 0,, are either all quadratic residues of ', or all non-quadratic residues of d;. In the former case we attribute to f; the particular character

'Ai
di

Ꮎ ;

= +1; in the latter

the particular character )=-1. If v1=1, i. e. if the form f1 itself

do not admit of any primary divisor beside unity (which is the only Vn Vn-1 Vn-1 Vn-2 X ÷ Vn-2/ Vn-2 Vn-3/

important case), the product (

Vn

is equal to ; whence, inasmuch as every prime that divides Vn also

divides Vn

Vn-1

", it appears that a primitive quadratic form will always have

Vn-1

one particular character, at least with respect to every uneven prime dividing its determinant, and will have more than one if the uneven Vi+l prime divide more than one of the quotients Vi

Vi
Vi-1

The subdivision of an order into genera can now be effected by assigning to the same genus all those forms whose particular characters coincide. But it remains to consider whether the above enumeration of particular characters is complete. It is evident that we might apply the theorem (B) to other concomitants besides those included in the fundamental system; and it might appear as if in this manner we could obtain other particular characters besides those which we have given. But it can be shown that such other particular characters are implicitly contained in ours; i. e. it can be shown that two quadratic forms, which coincide in respect of the particular characters deducible from their fundamental concomitants, will also coincide in respect of the particular characters deducible from any other concomitant. Again, it will be found that if the determinant be uneven, there are no particular characters with respect to 4 or 8. For this case, therefore, our enumeration is complete. But when the determinant is even, besides the particular characters arising from its uneven prime divisors, there may also be particular characters with regard to 4 or 8. There is no difficulty in enumerating these particular characters; nevertheless we suppress the enumeration here, not only because it would require a detailed distinction of cases, but also because there appears to be some difficulty in showing that the characters with regard to 4 or 8, which may arise from the excluded concomitants, are virtually included in those which arise from the concomitants of the fundamental set.