Hence, if all the terms in any line or column of a determinant be multiplied by any given quantity, the determinant itself will be multiplied by the same quantity. 10. DEF. From any given determinant, other determinants may be formed, by omitting an equal number of lines and columns of the given determinants. These are termed MINORS of the given determinant, and are called first, second, &c. minors, according as one, two, &c. lines and columns have been omitted. Thus 11. To investigate the relation which must hold among the coefficients L, M, N, λ, μ, v, in order that the quadratic function La2 + MB2 + Ny2 + 2λßy + 2μya + 2vaß may be the product of two factors of the first degree in a, B, y. The given expression is identical with (La+vß + μy) a + (va + MB + λy) B + (μa + λß + Ny) y. Now, if the relation between L, M, N, λ, μ, v be such that, for all values of a, B, y, the three linear functions Τα + β + μη, να + Μβ + λγ, μα + β + Ν may bear to one another constant ratios (p: q: r, suppose), then the given expression will be the product of two factors, respectively proportional to The necessary condition is then that La + β + μη να + β + λγ μα + λβ + Ν Ρ = for all values of a, ß, y, and therefore for those which make the numerators of any two of the above fractions = 0. That is, values of a, B, y exist, which simultaneously satisfy the equations La + β + γ = 0, va + MB + λy = 0, μα + β + Ny = 0. Hence, eliminating a, B, y, we get, as the condition that the given expression may be the product of two factors, the equation the evanescence of which is the necessary condition that the given quadratic function may break up into two factors, is termed the Discriminant of that function. 12. PASCAL'S THEOREM. From the analytical result stated in Art. 6 of the present chapter, that the value of a determinant is not altered by changing its lines into columns and its columns into lines, we obtain a proof of Pascal's theorem, which asserts that If a hexagon be inscribed in a conic, and the pairs of opposite sides be produced to intersect, the points of intersection lie in the same straight line. Let AFBDCE be the conic; take ABC as the triangle of reference, and let the equation of the conic be Then, since D lies in the conic (1), we have λ +μm ̧+vn ̧ = 0, is the necessary condition that the six points A, F, B, D, C, E may lie in a conic. Again, if the pairs of opposite sides intersect in points lying in a straight line, let the equation of that straight line be pa + qẞ+ry=0. Then, since BF and CE intersect in this line, we have p+ql2+vl ̧= 0, is the condition that these points of intersection may lie in the same straight line. But (2) and (3) are identical. Hence the proposition is proved. 13. From Pascal's Theorem many interesting consequences may be deduced. Thus, if the point F coincide with A, D with B, E with C, then AF, BD, CE become the tangents at A, B, C respectively, and we obtain the theorem enunciated in Ex. 1. Chap. II. Again, by supposing D to coincide with B, and E with C, we readily obtain the following theorem: "If the opposite sides of a quadrilateral, inscribed in a conic, be produced to meet, and likewise the pairs of tangents at opposite angles of the quadrilateral, the four points of intersection will lie in the same straight line." And, by supposing F to coincide with A, we obtain a geometrical construction, by which, having given five points of a conic, we can draw a tangent at any one of them. For, since AF then becomes the tangent at A, we see that, if AE, DB be produced to meet in G, AB, EC in H, and GH intersect CD in I, then AI will be the tangent at A. 1. Prove that EXAMPLES. = (a+b+c+d) (a − b + c − d) (a − b − c + d) (a + b − c −d). a, b, c, d b, a, d, c = c, d, a, b d, c, b, a 2. and that 19 29 = a, | b,, ს,, ს |