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hended, unless the finite and more elementary properties which are illustrated are first known. The matter of the present section is collected from various sources; chiefly from the works of Cramer and Pliicker, the Journals of Liouville and Crelle, the Nouvelles Annales de Mathematiques of Terquem and Gerono, and the Cambridge and Dublin Mathematical Journal. This last Journal contains many papers on the subject by Professor Boole, Mr. Cayley, and the Rev. George Salmon. I must however add that the following is only a sketch of the most elementary parts of a subject which contains matter of the deepest reflection; and which is daily receiving large additions from the most eminent mathematicians of the present age. But it will be sufficient for the object of this work.

An algebraical equation of the nth degree in terms of two variables x and y contains a constant term, and all possible combinations of the various powers of x and y, so that the sum of the indices in no term exceeds n: thus it is of the form,

+ a^+biy

+ aixt + b2xy + c2y2

+

+ anxn + bKx"-1y + cnx'-1y2 \-... +j„xy"-1 + kHyn = 0. (45) This is the form of the equation which will generally hereafter be expressed as »(*,y) = 0. (46)

As the equation stands at present, the number of its terms is 1+2 + 3+ ... + (n + 1);

the sum of which series is ~ ;but as every term

has a coefficient the whole series may be divided by any one, and thus the number of arbitrary constants will be diminished by one; and therefore in the equation of the nth degree in terms of two variables the number of coefficients which admit of arbitrary determination is

(n + 1) (n + 2) w(« + 3)

2 1 = —2— • (47)

It is often convenient to express (45) in another form: (45) consists of a series of homogeneous expressions which are, beginning from the constant term, of the orders 0, 1, 2,... n

PRICE, VOL. i. T t

respectively: let w0, uu th,... u„ represent the homogeneous expressions which are severally of the dimensions 0, 1, 2,... it; then (45) becomes

«o + «i + M-2+ ••• +u„ = r(x,y) = 0, (48)

the number of determinable constants in which is - ^ .

If in (48) M0 = 0, that is, if there is no constant term, then the whole vanishes when x = y = 0; and thus the origin is on the curve. It is also to be observed that the degree of a curve cannot be altered by the substitution of any linear functions of new variables for the original variables; and as change of origin and change of coordinate axes require only such linear substitutions, it is evident that the degree of a curve is not affected by such substitutions.

208.] Hereafter also it will be found convenient to express (45) in another form. As it stands it is neither symmetrical nor homogeneous: the absence of these properties may appear at first sight to be of small importance; but as many conclusions, both numerical and geometrical, will be drawn from the form of expressions, it is desirable that they should be symmetrical at first, because symmetrical expressions, when operated on in differentiation, give rise to new symmetrical expressions; whereas unsymmetrical expressions do not. And if besides it is possible to make them homogeneous the results will be much simplified by means of Euler's Theorems of such functions given in Art. 82. Both these advantages may be obtained by the following method due to, I believe, M. Otto Hesse, Professor at Kcenigsberg. Let

X 11

us assume - and -, instead of x and y, to be the coordinates of a point in the plane; and thus in the equation to the plane curve let us replace x and y by - and -; as we ordinarily ex

Z Z

press a point whose coordinates are x and y by {x, y), so I shall now express it by (x, y, z); after this substitution (45) becomes

ChZn

+ («!# +*iy) z—1

+ (a2 x2 + b2xy + c2 y2) zn~2

+

+ anxH + bnxH-*y+ ... +jnxyn-1 + kny* = 0;

and (48) becomes un + Un-iZ + un-.izi + ... + u1zn-*+u0= r(x,y,z) = 0, (49)

which are homogeneous expressions of the nth degree: and to which therefore the simplifications contained in Euler s Theorem, Art. 82, are applicable. Similarly a third variable will be introduced in other cases; thus if a and b are the coordinates of

a point, I shall now take - and - to be the coordinates, and

c c

shall express the point by the notation (a, b, c); and analogously with other variables. We can pass from expressions involving three variables to the equivalent ones which involve only two, by replacing the third variable by unity. One or two examples are subjoined in which equations are given in terms of three variables.

Thus it is evident that the general form of the equation of a straight line is AX + ay + Cz = 0. (50)

The general form of the equation of a conic is

AX2 + Byt + czs + 2Tzyz + 2azx + 2Hxy = 0. (51) The equation of a straight line passing through the two points (#1, yi, Zi) and (x2, y%, zt) is

x iyiZi—ziyi) + y (Zix2—xiZt) + z (xiy%—y\Xz) = 0; (52)

the coefficients in which are the several determinants of the coordinates of the two given points taken two and two together.

The advantages arising from the symmetry of algebraical expressions in terms of three variables will be shewn hereafter. I must however here observe that as (49) is homogeneous and of n dimensions,

*(£) + y@ + Z(£) = MF(w)

= 0;

and therefore if = 0 and = 0, then also (^) = 0. Thus in (51)

= 2(nx + ny + Gz), (^) = 2(H.r + By + zz), (^) = 2(g^ + Ej/+cz); and therefore if of these three equations the first two vanish, the third also vanishes; and thus if z = l, and A,r +Hy + G = 0 = Hxi By + E, then also Gx + Ey-f c 0.

209.] Another property of homogeneous functions must also be proved, as we shall find it to be important in our subsequent investigations as to the degree of certain curves. I will here however take the problem in its most general form.

Let Fi = 0, Vi = 0, ... F„ = 0, be a system of n homogeneous equations involving n variables xu xit... x„, in the most general form and with literal coefficients; and let us suppose these equations to be respectively of the degrees m?,, ... /«„; then we have M equations involving n 1 different variables; for by reason of the homogeneity of each, the highest power of xH, whatever that is, may be divided through, and the number of variables will thereby become n 1; these equations therefore are not independent; a relation must exist amongst them; and this relation must be capable of expression in terms of the coefficients of the equations. Of this relation I propose to determine a property as to the power in which the coefficients of the several equations will be found. The relation is technically called the Resultant; and if it contains no extraneous factors, the complete resultant; and if the extraneous factors are omitted, the reduced resultant*. Let mim-j»i3 ...»»„ = M. Now as the function F] = 0 is homogeneous and of >»] dimensions, it is capable of resolution into mx linear factors, each of which is of the form axXi 4 <h#i + ••• + oHxM; and similarly is each of the other given expressions capable of resolution into w2) »«3,... m„ factors respectively of a similar form. As these equations are simultaneously true, to arrive at the most general result we must combine each of the component factors of Fj = 0 with each and every one of F2 = 0, F3 = 0,... F„ = 0; let us take one of each set; then eliminating the unknown quantities, which are n 1 in number, between these n equations, we obtain a condition of the nth degree into which each coefficient enters in the first degree; and this process may be performed in M different ways, and the product of these M different quantities is the complete resultant, and is evidently homogeneous in terms of the original coefficients. The coefficients of ?! = 0,

* See Crelle, Band XXXIV, p. 30.

*

enter into each term in the degree —. those of F. = 0 in the degree —. and so with the others: to prove this, let us fix our thoughts by assuming mi = 3; so that

*i = («i *i + (Hx-i + ... + a„ xn) (6i #1 +... + b„x„) (cxXi +... + cHx„)

= KX\3-\-*XliX-l+ ...

In the product of the M different results of elimination au bu C\, say, will enter symmetrically; and therefore each one will

enter in the power ^- only, and therefore in the product, A o

which is equal to at bi C\, will enter in the power —; and the

o

same is true of all the other coefficients, and of the coefficients of the other equations; so that the degree of the complete resultant is j j j

M ( h — + ... + — )•

vz»! m-i mn' Thus much at present as to curves expressed by equations in terms of three variables, and homogeneous. We shall return to the subject hereafter. Now I shall proceed with the investigations of other properties of curves as expressed by equations in terms of two variables.

210.] The curve which is expressed by an equation of the nth degree is said to be of the nth degree or of the nth order, the two words order and degree being used indifferently. Suppose now that we have two curves, one of the nth and the other of the mth degree; these will generally intersect in mn points; for let us take the equation to the curve as given in (45); then if y is eliminated from it and from a similar equation of the mth degree, the result will be an expression in terms of x of the mnth degree; the roots of which will be the abscissae to the mn points in which the two curves intersect; the points in which they intersect, I say; because if all the roots are real, the curves will intersect in mn points in the plane of reference; and if the roots are imaginary, the points of intersection will be imaginary so far as they are exhibited in the plane of reference; but the equation of the mnth degree will just as truly have mn roots. Thus two curves of the second degree may intersect in four points; a curve of the second degree may intersect a curve of the third degree in six points.

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