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denote by p(x, y) = 0, has two equal roots x: y, if l, m, n be still the co-ordinates of a tangent to the curve f; an equality which obviously corresponds to the coincidence of two intersections of that line with that curve. Conversely, if we seek by the usual methods the condition of equality of two roots x:y of the homogeneous equation of the pth degree,

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by eliminating the ratio x:y between the two derived homogeneous equations, 0 = Dp, 0 Dyp, we shall in general be conducted to a result of the dimension 2p(p-1) in l, m, n, and of the form,

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0 = nP(p-1) F (l, m, n);

and so, by the rejection of the foreign factor no(P-1), introduced by this elimination, we shall obtain the tangential equation F=0, which will be in general of the degree p(p-1); such being generally the known class (56) of the curve of which the order (46) is denoted by p: with (of course) a similar mode of passing, reciprocally, from a tangential to a local equation.

61. As an example, when the function fhas the cubic form assigned in 54, we are thus led to investigate the condition for the existence of two equal roots in the cubic equation,

0 = p(x, y) = {(n − 1) x + (m − 1 ) y } 3 + 27n2xy (lx + my),

by eliminating : y between two derived and quadratic equations; and the result presents itself, in the first instance, as of the twelfth dimension in the tangential co-ordinates l, m, n; but it is found to be divisible by n, and when this division is effected, it is reduced to the sixth degree, thus appearing to imply that the curve is of the sixth class, as in fact the general cubic is well known to be. A further reduction is however possible in the present case, on account of the conjugate point o (54), which introduces (comp. 57) the quadratic factor,

Compare the method employed in Salmon's Higher Plane Curves, page 98, to find the equation of the reciprocal of a given curve, with respect to the imaginary conic, x2+ y2+z2 = 0. In general, if the function F be deduced from ƒ as above, then F(xyz) = 0, and ƒ(xyz) = 0 are equations of two reciprocal curves.




(l + m + n)2 = 0;

and when this factor also is set aside, the tangential equation is found to be reduced to the biquadratic form* already assigned in 59; the algebraic division, last performed, corresponding to the known geometric depression of a cubic curve with a double point, from the sixth to the fourth class. But it is time to close this Section on Plane Curves; and to proceed, as in the next Chapter we propose to do, to the consideration and comparison of vectors of points in space.



SECTION 1.-On Linear Equations between Vectors not Complanar.

62. When three given and actual vectors oa, Ob, oc, or a, ß, y, are not contained in any common plane, and when the three scalars a, b, c do not all vanish, then (by 21, 22) the expression aa + bẞ+cy cannot become equal to zero; it must therefore represent some actual vector (1), which we may, for the sake of symmetry, denote by the symbol do: where the new (actual) vector d, or oD, is not contained in any one

* If we multiply that form = 0 (59) by z2, and then change nz to - lx - my, we obtain a biquadratic equation in :m, namely,

0 = ↓ (1, m) = (1 − m)2 (lx + my)2 + 2lm (l + m) (lx + my) z + 12m2z2 ;

and if we then eliminate 1: m between the two derived cubics, 0 = Di, 0 = Dm¥, we are conducted to the following equation of the twelfth degree, 0=x3y3z3ƒ (x, y, z), where ƒ has the same cubic form as in 54. We are therefore thus brought back (comp. 59) from the tangential to the local equation of the cubic curve (54); complicated, however, as we see, with the factor x3y31⁄23, which corresponds to the system of the three real tangents of inflexion to that curve, each tangent being taken three times. The reason why we have not here been obliged to reject also the foreign factor, 212, as by the general theory (60) we might have expected to be, is that we multiplied the biquadratic function F only by z2, and not by z^.


of the three given and distinct planes, BOC, COA, AOB, unless some one, at least, of the three given coefficients a, b, c, vanishes; and where the new scalar, d, is either greater or less We shall thus have a linear equation between four

than zero.


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parallelepiped, whereof the sum, OD or 8, is the internal and co-initial diagonal (comp. 6). Or we may project D on the three planes, by lines DA", DB", DC" parallel to the three given lines, and then shall have OA" = OB' + Oc'=bB+cy,


= OD = OA' + OA" = OB′ + OB′′ = oc' + oc".

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And it is evident that this construction will apply to any fifth point D of space, if the four points OABC be still supposed to be given, and not complanar: but that some at least of the three ratios of the four scalars a, b, c, d (which last letter is not here used as a mark of differentiation) will vary with the position of the point D, or with the value of its vector d. For example, we shall have a = 0, if D be situated in the plane Boc; and similarly for the two other given planes through o.

63. We may inquire (comp. 23), what relation between these scalar coefficients must exist, in order that the point D




may be situated in the fourth given plane ABC; or what is the condition of complanarity of the four points, A, B, C, D. Since the three vectors DA, DB, DC are now supposed to be complanar, they must (by 22) be connected by a linear equation, of the form

a (a − d) + b(ß − S) + c (y − 8) = 0 ;

comparing which with the recent and more general form (62), we see that the required condition is,

a+b+c+d= 0.

This equation may be written (comp. again 23) as

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and, under this last form, it expresses a known geometrical property of a plane ABCD, referred to three co-ordinate axes OA, OB, oc, which are drawn from any common origin o, and terminate upon the plane. We have also, in this case of complanarity (comp. 28), the following proportion of coefficients and areas:

a:b:c:- d.= DBC: DCA: DAB: ABC;

or, more symmetrically, with attention to signs of areas,

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where Fig. 18 may serve for illustration, if we conceive o in that Figure to be replaced by D.

64. When we have thus at once the two equations,

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so that the four co-initial vectors a, B, y, & terminate (as above) on one common plane, and may therefore be said (comp. 24) to be termino-complanar, it is evident that the two right lines, DA and BC, which connect two pairs of the four complanar points, must intersect each other in some point a' of the plane, at a finite or infinite distance. And there i no difficulty in perceiving, on the plan of 31, that the vectors of the three

points a', B', c' of intersection, which thus result, are the fol

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expressions which are independent of the position of the arbitrary origin o, and which accordingly coincide with the corresponding expressions in 27, when we place that origin in the point D, or make 8=0. Indeed, these last results hold good (comp. 31), even when the four vectors a, ß, y, d, or the five points o, a, B, C, D, are all complanar. For, although there then exist two linear equations between those four vectors, which may in general be written thus,

a' a + b2ß +cy + dd = 0, a′′a + b′′ß + c′′y + d′′S = 0,

without the relations, a'+ &c. = 0, a" + &c. = 0, between the coefficients, yet if we form from these another linear equation, of the form,

(a" + ta') a + (b" + tb′)ß + (c" + te') y + (d" + tď) d = 0, and determine t by the condition,

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we shall only have to make a = a" + ta', &c., and the two equations written at the commencement of the present article will then both be satisfied; and will conduct to the expressions assigned above, for the three vectors of intersection: which vectors may thus be found, without its being necessary to employ those processes of scalar elimination, which were treated of in the foregoing Chapter.

As an Example, let the two given equations be (comp. 27, 33),
aa + b + cy = 0, (2a + b + c)a" - aa =
= 0;

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