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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 + 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,

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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,
p=u' po,

(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).
(cosa) -2=1+ (d,,x)2 + (d,x),
(cos B)-2=1+ (dzy)? + (dzy)”,
(cosy) -2=1+ (d.x) + (d,x).

If &q, , X1, are the co-ordinates of the normal, its equations are

81 - =d2%(%1-x),

-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}}.

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(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.
If these surfaces have a contact of the first order, then

%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
p=

R, (cos 6)* + R, (sin 6)?
R,=

hence

1

1

and R.

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1

р

=r(sin ) + t(cos ).
(L. D. C. 168–78; L. C. D. 313-24.)

(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 ß) -=1+(d,x)? + (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

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