11. Prove that the equation of the line passing through the feet of the perpendiculars from a point a,, B, Y,, of the circle aßy+bya + caẞ= 0, on the sides of the triangle of reference, may be put in the form, 12. Shew that the axis of the parabola, whose equation is a'a' 4bcẞy, is given by the equation (c + b cos 4) B − (b + c cos 4) y = } ( − ;) aa. Ꮳ - 13. The equation of the directrix of the parabola, which touches the sides of the triangle of reference, and also the straight line la+mẞ+vy = 0 is (la)§ + (mß)1 + (ny)$ = 0 represent a parabola, the equation of its axis is CHAPTER III. ON ELIMINATION BETWEEN LINEAR EQUATIONS. 1. BEFORE entering upon the discussion of the conic represented by the general equation of the second degree, it will be necessary to devote a few pages to the subject of elimination between homogeneous linear equations, and to explain some of the terms recently introduced in connection with this branch of analysis. We shall, however, only state and prove such elementary theorems as will be necessary in our future investigations; referring the reader who may be desirous of fuller information to Salmon's Lessons on the Higher Algebra; Spottiswoode, On Determinants (the second edition of which will be found in Crelle's Journal, t. 51, pp. 209, 328), and to the original memoirs communicated to various scientific Journals by Messrs Boole, Sylvester, Cayley, and others. 2. If we have given n homogeneous linear equations, connecting n unknown quantities x, x,... x, such as the quantities x, x,... x, can be eliminated between them, and the result of the elimination may be expressed by omitting x, x,... x, and writing the coefficients only in the order in which they appear in the given equations, thus The left-hand member of this equation is what is called the determinant of the given system of equations. We proceed to investigate the law of its formation. 3. First, suppose we have two equations, Multiply the first by b2, the second by a,, and subtract, and we get Hence We may remark in passing that we shall obtain the same result by eliminating A, A, between the equations A like theorem will be proved to be true for all determinants. DETERMINANTS OF THREE ROWS. 61 Multiply these equations in order by the arbitrary multipliers A, A, A, and add them together. Let the two ratios λλλ be determined by the conditions that the coefficients of x and x, in the resulting equation shall each be zero, i.e. let Multiply the first of equations (A) by a,, the second by a2, and subtract, we then get (a,b,—a,b) λ, + (c‚¤ ̧ — câα2) λ ̧=0, Hence, dividing each term of (B) by the corresponding member of (C) we get It will be seen that the above process is really equivalent to that of eliminating A, A, A, between the equations (A) and (B). Hence To effect the elimination, multiply the equations in order by λ, λ1⁄2, λ3, λ, add them, and equate the coefficients of X, X, X severally to zero. We shall then have aλ1 + b2λ + c2 λ + d2 λ=0 | which equations involve as a consequence = 2 (A'), To determine the three ratios λ: λ1⁄2: λ ̧: λ, multiply equations (A') in order by M, M, M, add, and equate to zero the coefficients of λ,, . We thus get Now, treating equations (C') as equations (A) were treated, we see that |