be performed mentally. It is generally more advantageous to use equation (3) than equation (2), because (3) gives the result in its simplest form, whereas if (2) be used it is often necessary to reduce by substitutions from the equation of the curve. 171. Application to General Rational Algebraic Curve. If the equation of the curve be written in the form f(x, y) = Un+Un-1+Un-2+...+u2+u1+u1 = 0 (where u, represents the sum of all the terms of the pth degree), then when made homogeneous by the introduction where necessary of a proper power of z we shall have f(x, y, z)=un+Un-12+Un-222+.. and +u2zn-2+u2zn-1+u12z", Un-1+2Un-2z+3un-322 + ... +(n−2)μ¿¿1−3+(n−1)u ̧22¬2+nu zn−1, and therefore substituting in (3) and putting z=1, the +(n−2)u2+(n − 1)u,+nu = 0...... (4) 172. NORMAL. DEF.- The normal at any point of a curve is a straight line through that point and perpendicular to the tangent to the curve at that point. Let the axes be assumed rectangular. The equation of the normal may then be at once written down. For if the equation of the curve be y=†(x), L dy dx(x − x), the tangent at (x, y) is Y-y=x and the normal is therefore (X − x) + ( Y − y) dy = 0. If the equation of the curve be given in the form This requires 22 in the last term to make a homogeneous equation in x, y, and z. We have then Ex. 2. Take the general equation of a conic ax2+2hxy+by2+2gx+2fy+c=0. When made homogeneous this becomes ax2+2hxy+by2+2gxz+2fyz+cz2=0. The equation of the tangent is therefore X(ax+hy+g)+Y(hx+by+f)+gx+fy+c=0, and that of the normal is 1. Find the equations of the tangents and normals at the point (x, y) on each of the following curves :— 2. Write down the equations of the tangents and normals to the curve y(x2+a2)= ax2 at the points where y α y= 4 X -1 touches the curve y=be at the point a b Hence write down the polar equation of the locus of the foot of the perpendicular from the origin on the tangent to this curve. Examine the cases of an ellipse and of a rectangular hyperbola. 5. Prove that, if the axes be oblique and inclined at an angle w, the equation of the normal to y=f(x) at (x, y) is It will be shown by a general method in a subsequent article (254) that in the case in which a curve, whose equation is given in the rational algebraic form, passes through the origin, the equation of the tangent or tangents at that point can be at once written down; the rule being to equate to zero the terms of lowest degree in the equation of the curve. Ex. In the curve x2+y2+ax+by=0, ax+by=0 is the equation of the tangent at the origin; and in the curve (x2+y2)2=a2(x2 — y2), x2 - y2-0 is the equation of a pair of tangents at the origin. It is easy to deduce this result from the equation of the tangent established in Chapter II. That equation is Y―y=m(X-x) where m= At the origin this becomes dy dx Y=mX, where the limiting value or values of m are to be found. Let the equation of the curve be arranged in homogeneous sets of terms, and suppose the lowest set to be of the nth degree. The equation may be written Dividing by a", and putting y = mx, and then x=0 and y=0, the above reduces to the form fr(m)=0, an equation which has r roots giving the directions in which the several branches of the curve pass through the origin. If m,, M2, M3, ... m, be the roots, the equations of the several tangents are y = mx, y=M2X, y = mxx. These are all contained in the one equation f(2) = 0 ; and this is the result obtained by "equating to zero the terms of lowest degree" in the equation of the curve, thus proving the rule. In this manner all the trouble of differentiation is avoided, and the result written down by inspection. GEOMETRICAL RESULTS. 174. Cartesians. Intercepts. From the equation Y-y=dy (x − x) X it is clear that the intercepts which the tangent cuts off from the axes of x and y are respectively for these are respectively the values of X when Y=0 Let PN, PT, PG be the ordinate, tangent, and normal to the curve, and let PT make an angle 0 with the axis |