the branches which pass through this point have a common tangent: if all the roots are impossible, there is, as before, a conjugate point. (8.) Singular points at which y or any of its differential coefficients become infinite. Let x + Da be substituted for x, and let the corresponding value of y be expanded in a series of ascending powers of Da, Ꭰ, then Exy=y+a1(Dx)TM1 +α ̧(Dx)TM2 + &c. and suppose that neither of the indices is a fraction having an even denominator: then if the numerator of m1 is odd, there is a point of contrary flexure, the tangent at that point being parallel to the axis of y, or x, according as m1> or <1. If the numerator of m1 is even, there is a maximum or minimum ordinate, if m1>1; and which is also a point of regression, or cusp, of the first species, if m1<1. ·If m1=1, and the numerator of m2 is odd, there is a point of contrary flexure; the tangent at that point being inclined to the axis at a finite angle. If the denominator of m, alone is even, there is a minimum abscissa, and the point is a cusp of the first species, if m1 >1 : if m, alone has an even denominator, there is a cusp of the first species, but no maximum or minimum ordinate: if m, alone, or any succeeding index, has an even denominator, there is a cusp of the second species.. If some terms are rendered impossible by + Da, and some by - Da, or if any terms become impossible for all values of Da, there is a conjugate point. (L. D. C. 160—7.) CURVES REFERRED TO POLAR CO-ORDINATES. (9) Let p be the perpendicular on the tangent from the pole, The radius of curvature= the chord through the pole = (10.) The evolute of a spiral. Let U be the radius vector, and P the perpendicular on the tangent in the evolute, then U2=u2{ 1 + (d ̧u)2 } — 2pu.d ̧u, P2 = u2 — p2, U2 — P2 = (u du — p)2. (1) When the equation between u and p is given, that between U and P may be found by eliminating u, du, and p, between the given equation, and two of the equations (1); the equation thus obtained is the equation of the evolute. When the equation between U and P is given, then that between u, du, and p may be found by eliminating U and P: the curve represented by this equation is the involute of the given curve. u= (11.) If, when u∞, and the subtangent have finite values, the curve admits of a rectilinear asymptote, which may be determined by drawing u at the given finite angle 0, and making the subtangent equal to the given value; then a parallel to u through the extremity of the subtangent will be the asymptote required. If, when = ∞, u has a finite value, the spiral admits of an asymptotic circle, of which that value of u is the radius. If dup=0, there is a point of contrary flexure. CIRCUMSCRIBING FIGURES. (12.) Method of determining the maximum inscribed, or minimum circumscribing figure. be the equation of the given figure, Let y=f(x) then ·ƒ(x)=4(a,b,x), and f'(x)=p'(a,b,x); eliminating from these equations, we obtain an equation of also since (a,b) is a maximum, or minimum, '(a,b) =0, from which equations a and b may be determined. (13.) Method of determining the curve which circumscribes any number of curves described according to a given law. Let the equation p(x,y,a)=0 represent a system of curves depending on the value of the parameter a; then by eliminating a between the equations we obtain an equation between x and y which is the equation of the curve required. THE CONTACT OF SURFACES. (14.) Let x, y, z, be the co-ordinates of a given surface, 1 y1,1, those of the tangent plane; 19 19 a, B, y, the angles at which the tangent plane is inclined to the planes of yx, xx, xy, respectively. The equation of the tangent plane is If x1, y1, ≈1, are the co-ordinates of the normal, its equations are If a, ẞ, y, are the angles which the normal makes with the axes of x, y, z, respectively, their values are the same as above. The length of the normal intercepted between the surface, and the plane of xy=x{ 1+(d,≈)2 + (d,x)2}*. (15.) Let x=(x,y) be the equation of a given surface, and ≈1=√(x ̧‚y1) the equation of a surface touching the former, and let d≈, d≈, d2≈, d ̧d~, dž≈ be, respectively represented by P, q, r, t, in the first surface, and by P1 919 r19 819 8, t1, in the second. If these surfaces have a contact of the first order, then to satisfy which, the equation 1 = √(x,y1) must contain three arbitrary constants. For a contact of the second order, we must also have to satisfy which conditions in addition to the former, the equation ≈1=√(x11) must contain six arbitrary constants: hence an osculating sphere cannot be found generally, as a circle of curvature to a plane curve. If however we suppose a section of the surface to be made by a plane passing through the normal, λ the tangent of the angle which the projection of the section on the plane xy makes with the axis of a, and p the radius of the sphere osculating this section, then 1 2 The planes of greatest and least curvature are perpendicular to each other. Let R, and R2 be the radii of greatest and least curvature, and let the co-ordinates be so transformed, that the tangent plane, and the planes of greatest and least curvature may become the planes of xy, yx, xx respectively; then if is the angular distance of any section from the plane xx, and the radius of the sphere osculating that section, ρ (16.) Let y =1(x), ≈ =42(x), be the equations of a given curve, and y1 =¥1(∞1), ≈1=¥2(1), those of the touching line; then in order that there may be a contact of the first order, we must have for a contact of the second order we must also have d21(∞)=d2\1(x), d2(x)=d2¥,(x); 1 ß, Let a, ẞ, y, be the angles which the tangent makes with the axes of x, y, z, respectively, then (cos a)2=1+ (d ̧y)2, + (d≈)o, The length of the tangent intercepted between the curve and the plane y=≈ {1+ (d ̧≈)2 + (d2y)2}}. The equation of the normal plane is If a, ẞ, y, are the inclinations of the normal plane to the B, planes yx, xx, xy, respectively, their values are the same as above. The distances from the origin at which the plane cuts the axes of x, y, z, respectively are x+y.d2y+x.d ̧z, y+x.d„x+z.d,s, x+x.d ̧x+y.dy. The equation of that plane passing through the tangent in which the curvature takes place is |