is therefore ac+1662=0, or b=-ivac. Substituting this value, the equations become cμ22+i√ca(μ+v)2+a=0 cv2X2+i √ca(v+λ)2+a=0 2 cλ2μ2+i √ca(λ+μ)2+a=0. The first and second of these are A=(−i√ã+μ2 √c)i √a, H=i√ caμ, B= (i √ ã+ μ2 √ ĕ) V č A'=(—i √ā+λ2 √c)i √a, H'=i √ caλ, B'=(i √/a+ λ2 √c) No AB'+A'B-2HH'=2i √ ac(a—i √ acλμ+cλ2μ2) And the result of the elimination therefore is (a—i √ acλ2+cλ1) (a-i √ acμ2+cμ3)—(a−i √ acλμ+cλ2μ2), viz. 2 √ ca(λ—μ)2 (cλ3μ2 + i √ca(λ+μ)2+a)=0; which agrees, as it should do, with the third equation. To find the condition that it may be possible in the conic x2+ y2+z2=0 to inscribe an infinity of triangles, each of them circumscribed about the conic ax2+by2+cz2=0. Let the equations of the sides be 1 √ ax+m √ by +n √ cz=0 l' √ ax+m' √by+n' √ cz=0 1′′ √ax+m" √by+n" √ cz=0. Then the conditions of circumscription are 12 + m2 + n2 =0 112 + m12 + n12=0 1112 + m112 + n'12=0. And the conditions of inscription are bc(m'n" ―m"n')2+ca(n'l" —n"l')2 + ab(l'm" —l'"m')2=0 Now (mn' —m'n)2 = (m2 +n2) (m12 + n'2) — (mın' + nn')2 = − (ll' + mm' + nn') (—ll' + mm' + nn'); and making the like change in the analogous expressions, and putting for shortness −bc+ca+ab=a bc-ca+ab=B bc+ca-ab=Y, the conditions in question become (l'l" +m'm" +n'n") (al'l" +ßm'm" +yn'n")=0 (!"'l + m'm+n'n) (al"l + ßm"m + yn"n) =0 The proper solution is that given by the system of equations 12 + m2 + n2 =0 The first of which systems expresses that the points (f, g, h), (f', g′, h'), (f", g', h") are points in the conic Ax2+By2+Cz2=0; and the second condition expresses that each of the points in question is the pole with respect to the conic x2+ y2+z2=0 of the line joining the other two points, i. e. that the three points are a system of conjugate points with respect to the last-mentioned conic. The problem is thus reduced to the following one:To find the condition in order that it may be possible in the conic Ax2+ By2+Cz2=0 to inscribe an infinity of triangles such that the angles are a system of conjugate points with respect to the conic x2+ y2+z2=0. Before going further it is proper to remark that if, instead of assuming all" +Bm'm" + yn'n"=0, we had assumed l'l"+m'm"+n'n" =0, this, combined with the equations 112 112 +m22 + n22=0, 1112+m"12+n"2=0, would have given l': m': n'=l":m": n", i. e. two of the angles of the triangle would have been coincident: this obviously does not give rise to any proper solution. Returning now to the system of equations in f, g, h, &c., since the equations give only the ratios f:g: h; f': gh'; f":g": h", we may if we please assume ƒ2 +g2 +h2 =1 f12 +gl2 + h12=1 which, combined with the second system of equations, gives A+B+C=A(f2+ƒ12 +ƒ112)+B(g2+g12 +g!!2) + C(h2 + h12+h112) =(Af® +Bg® +Ch®) + (Af® +Bg? +Ch®)+(Afl2+Bg" + Ch"), i. e. 12 A+B+C=0, for the condition that it may be possible in the conic to describe an infinity of triangles the angles of which are conjugate points with respect to the conic a2+y2+22=0. The equation of the conic Ax2+ By2+ Cz2=0 may be written in the form (b2c2 — c2a2—a2b2+2a2bc) x2+(+c2a2 — a2b2 —b2c2+2ab2c)y2 + (a2b2 — b2c2 — c2a2 +2abc2)z2=0, which gives the values of A, B, C; or again in the form 2(bc+ca + ab) (bcx2+cay2+abz2) − (bc+ca+ab)2(x2 + y2+z2) +4abc(ax2+by2+cz2)=0; where it should be observed that bcx2+ cay2+abz2=0 is the equation of the conic which is the polar of ax2+by2+cz2=0 with respect to a2+y2+z2=0. It is very easy from the last form to deduce the equation of the auxiliary conic, when the conics ax2+by2+cz2=Ō, x2+y2+22=0 are replaced by conics represented by perfectly general equations. The condition A+B+C=0 gives, substituting the values of A, B, C, b2c2+c2a2+ a2b2-2abc(a+b+c)=0; or in a more convenient form, (bc+ca+ab)2-4abc(a+b+c)=0, as the condition in order that it may be possible to inscribe in the conic x2+y2+z2=0 an infinity of triangles, the sides of which touch the conic ax2 + by2+cz2=0: this agrees perfectly with the general theorem. It is convenient to add (as a somewhat more general form of the equation A+B+C=0), that the condition in order that it may be possible in the conic Ax2 + By2+ Cz2=0 to inscribe an infinity of triangles the angles of which are conjugate points with respect to the conic A1x2+ B1y2+C1≈2=0, is be But the problem to find the condition in order that it may possible in the conic a2+y2+2=0 to inscribe an infinity of triangles the sides of which touch the conic ax2 + by2+cz2=0, may, by the assistance of the geometrical theorem to be presently mentioned, be at once reduced to the problem, ~ To find the condition in order that it may be possible in the conic 2+ y2+22=0 to inscribe an infinity of triangles the sides of which are conjugate points with respect to a conic A122+B1y2+C1≈2=0. The theorem referred to is as follows: : Theorem. If the chord PP' of a conic S envelope a conic σ, the points P, P' are harmonics with respect to a conic T which has with S, σ, a system of common conjugate points. Take for the equation of S, x2 + y2+z2=0 ; and for the equation of σ, ax2+ by2+cz2=0. Then if (x, y, 1), (X, Y, Z2) are the coordinates of the points P, P' respectively, we have And the condition in order that the chord may touch the conic σ is bc(Y1~2—Y1⁄2Z1)2 + ca(≈ ̧¤ ̧—≈‚¤ ̧)2+ab{(x ̧¥2−X2Y1)2=0. But we have (Y12—Y271)2=(y12+z,2) (y22+2,2—(Y1Y2+%1~2)2 = (X1X2+Y1Y2+≈1%2) (X1Xq—Y1Y2—Z1%q). And making the like change in the analogous quantities, and putting for shortness y= bc+ca-ab, the condition in question becomes (X1X2+Y1Y2+Z1≈2) (αX1X2+B¥1Y2+ y21≈q)=0. 2 2 But the equation x12+Y12+Z12=0 must be rejected, as giving with the equations x2+y12+12=0, x22 + y22+22=0 the relation x111 = X2 Y2 v2; we have therefore : which implies that the points (a,, y1, 71) and (X2, Y2, 22) are harmonics with respect to the conic ax2+By2+yz2=0, which is a conic having with S, σ, a system of common conjugate points. The equation may also be written ́(-bc+ca+ab) x2 + (bc−ca+ab)y2 + (bc+ca−ab)z2=0; or, as it may also be written, (bc+ca+ab)(x2 + y2+z2) −2(bcx2 + caz2 + abx2). And, as before remarked, bcx2+cay2+abz2=0 is the equation of the conic which is the polar of ax2+ by2+ cz2=0 with respect to x2+ y2+x2=0. The condition in order that there may be inscribed in the conic x2+ y2+z2=0 an infinity of triangles the angles of which |