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 + Dæ be substituted for x, and let the corresponding value of y be expanded in a series of ascending powers of Dx, then Exy=y+a,(Dx)", + a, (Dx)"; + &c. and suppose that neither of the indices is a fraction having an even denominator : then if the numerator of my 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 m,> or <1. If the numerator of my is even, there is a maximum or minimum ordinate, if m,>1; and which is also a point of regression, or cusp, of the first species, if my < 1. If my=1, and the numerator of m, 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 my alone is even, there is a minimum abscissa, and the point is a cusp of the first species, if m, >1: if m, alone has an even denominator, there is a cusp of the first species, but no maximum or minimum ordinate: if mz alone, or any succeeding index, has an even denominator, there is a cusp of the second species. , If some terms are rendered impossible by + Dx, and some by – Dx, or if any terms become impossible for all values of Dx, 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, u? + (dou) The radius of curvature= =u.d,u. U? + 2(dou)? — u.dou u{u? + (dou)"} the chord through the pole =p.d,u. uo + 2 (dou)? — U.do u (L. C. D. 253.) (10.) The evolute of a spiral. Let U be the radius vector, and P the perpendicular on the tangent in the evolute, then U’=+*{1 + (dou)?} – 2pu.dou, (1) U? – Po=(udou-p). When the equation between u and p is given, that between U and P may be found by eliminating u, dou, 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, dau, and p may be found by eliminating U and P: the curve represented by this equation is the involute of the given curve. (11.) If, when u= 0 O 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 0= ", 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. Let y=f(w) be the equation of the given figure, y=$(a,b,x) figure sought, . then f(x)=P(a,b,x), and f'(x)=$'(a,b,x); eliminating x from these equations, we obtain an equation of the form ¥(a,b)=0, also since y(a,b) is a maximum, or minimum, V'(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 p(xy,a)=0, d.p(x,y,a)=0, we obtain an equation between & and y which is the equation of the curve required. THE CONTACT OF SURFACES. (14.) Let x, y, %, be the co-ordinates of a given surface, # 1, 41, %, those of the tangent plane; a, b, y, the angles at which the tangent plane is inclined to the planes of y%, 8%, xy, respectively. The equation of the tangent plane is %-x=d.x(x1 — «) +d,x(y1 - y). If &q, Yı, X1, are the co-ordinates of the normal, its equations are 81 - =d2%(%1-x), Yı-y=d,x(,-%). If a, b, y, are the angles which the normal makes with the axes of X, Y, %, 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.x) + (d.,x)2}}. by (15.) Let x=P(x,y) be the equation of a given surface, and %=(xx,y) the equation of a surface touching the former, and let d2%, d,x, dịx, d,d,, d,ʻz be, respectively represented P, 9, ro, t, in the first surface, and by P1, 912 12 t, in the second. %1=%, p=P 9=912 to satisfy which, the equation x=(Q2,9.) must contain three arbitrary constants. For a contact of the second order, we must also have r=r, Si=8, ta=t, to satisfy which conditions in addition to the former, the equation x=y(x1,4,) 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, 1 the tangent of the angle which the projection of the section on the plane wy makes with the axis of X, and p the radius of the sphere osculating this section, then 1 +1° +(p+92)* (1 +p+q°)}. p= pe + 2s + ta? The planes of greatest and least curvature are perpendicular to each other. Let R, and R, 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, yz, xz respectively; then if 0 is the angular distance of any section from the plane xz, and p the radius of the sphere osculating that section, R.RO R, (cos 6)* + R, (sin 6)? hence 1 1 and R. 1 р =r(sin ) + t(cos ). (16.) Let y =01()x = 02 (~), be the equations of a given curve, and y=4.(x), -%=42(x), those of the touching line; then in order that there may be a contact of the first order, we must have 0.(w)=\.(m), P2(x)=42(x), d.20.(x)=d.41(x), 0,02(x)=d, 42(X); for a contact of the second order we must also have d; P.(x)=d; 4.(w), d. 02(x)=d? 4.(v) ; and so on for the superior orders. (17.) The equations of the tangent are Yı- y=dzy(x, – X), %1-%=d7%(@; - x). Let a, ß, y, be the angles which the tangent makes with the axes of x, y, %, respectively, then (cos a)-s=1+ (dzy)", + (d.x)", (cos y) -2=1+ (d,x)2 + (dy). The length of the tangent intercepted between the curve and the plane ay=x{1+ (dux) + (dzy)}*. The equation of the normal plane is (2, -2) + dzy(.- y) + d2%(%1-x)=0. If a, b, y, are the inclinations of the normal plane to the planes y%, 8%, xy, respectively, their values are the same as above. The distances from the origin at which the plane cuts the axes of X, Y, %, respectively are 18+y.dzy+x.dxx, y + x.d,x+x.d,%, % + x.d.2 + y.day. The equation of that plane passing through the tangent in which the curvature takes place is |