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(11) If from every point in a curve of the second order pairs of tangents be drawn. to another curve of the second order, find the curve which is constantly touched by the chord of contact.

Let us suppose for simplicity that the second curve is an ellipse referred to its centre, its equation being

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Let the co-ordinates of a point from which a pair of tangents to (I) is drawn be a, b, then the equation to the chord of contact is

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a, ß are supposed to be the co-ordinates of a point which is always in a curve of the second order : they are therefore connected by the equation

Aa+ 2 Baß + CBS + 2 Da + 2 EB + 1 = 0, III. Now to find the curve which is constantly touched by (II) differentiate (II) and (III) with respect to a and ß. ada ydß

(1) (Aa + BB+D) da +(Ba + CB + E') = 0: (?) ^ (1) + (2) = 0 gives us

+ Aa + BB+ D = 0,

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1 +Ba+ CB + E = 0. Multiply by a, ß and add, then by (II) and (III)

= Da + + 1.
Substituting in the preceding equations we have

(Da + EB+1) + da + BB + D = 0,
(Da + EB + 1) + Ba+ CB + E = 0.

Between (II), (3), and (4) we can eliminate a, ß, and we obtain the final equation

(C-E") - 2 (B DE) + (1 - D') + 2 (CD - BE) – + 2 (AE BD) + AC B= 0. IV.

This being of the second order, it appears that the locus of the ultimate intersections of (I) is a conic section. This is a case of the general problem of reciprocal polars. The curve (I) is called the directrix, the point a, ß its pole ; and the line (II) the polar with reference to (a, b). The curves (III), (IV) are the reciprocal polars, and possess a great number of corresponding properties of considerable interest, but the nature of this work precludes us from entering on that subject. The reader who is curious in such matters will find memoirs on these related curves by Poncelet, in the Annales de Gergonne, Vol. viii. p. 201, and Bobillier, Ib. Vol. xix. p. 106, and p. 302. He will also find these questions along with others of a similar kind very ingeniously treated, in a short tract on “ Tangential Co-ordinates," by J. Booth of Trinity College, Dublin. The method employed by that author does not come within the scope of the present work, but it merits attention, as affording a ready solution of many curious problems which yield with difficulty to the power of ordinary analysis.

(12) A plane whose equation is

- + +-= l,

a b c a, b, c being subject to the condition

abc = m', will always touch the surface whose equation is

m AYU =

(13) To find the envelop of the system of spheres determined by the equations

(v – a)” + (y - 6) + z" = r, a+ b* = co.

Differentiating with respect to a and b, (x - a) da + (y - b) db = ( (1), ada + bdb = 0 (2); 1 (2) + (1) = 0 gives on equating to zero the coefficients of each differential, na + (20 – a) = 0 (3) 16+ (y – b) = 0 (4).

whence ay = bæ, or = a (3) + 6(4) gives

1c + ax + by - c = 0.

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Substituting this value of 1 in (3) and (4), squaring and adding,

{c+ (202 + yê)?} = (x – a)* + (y b) = yox – , by the original equation; and this is the equation to the envelop.

(14) To find the surface always touched by a plane which cuts off from a right cone an oblique cone of constant volume.

Taking the vertex of the cone as origin, and its axis as the axis of >, the equation to the cone is x + y = c*x?

(1) where c is the tangent of the half angle of the cone. The equation to the cutting plane is

2 + mx + ms = , l, m, n being the cosines of the angles which it makes with the co-ordinate planes, so that 1 + ma + n° = 1,

(3) and v being the perpendicular from the origin on the plane.

(2) Extracting the square root of (1) and substituting in it the value of x from (2), we have

(1x2 + y)2 =

cu c(l x + my)

n which is the equation to the projection on (vy) of the section of (1) by (2); and as the radius vector is a rational function of w and y, the origin, that is, the vertex of the cone, must be the focus of the projection. Comparing it with the general equation to a conic section referred to its focus

a (1 – e)

**T+e cos (8 – a)' or (x + y')! = a (1 – e“) – e cosax – e sin ay, we find

C (1? + mo) a (1 – e?) = ; e =¢¢*); whence the area of the projection is

ancév?

{n? – (? (+ m%)}} and the area of the section is therefore

пс? о?

{no – c* (1* + m')}i
The volume of the oblique cone cut off is

C

203

3 {na - c* (1+ mo)}1' which is to be constant. Neglecting the constant multiplier and extracting the cube root, we may put

(4) Tmj2 (121 m2) 20 = , or v= a {no - c* (1? + m')}4. We therefore have the cquation

1x + my + nz= a {n’ – co (r + m2)}\, l, m, n being connected by the equation

* + m + m^ = 1.

The result of the elimination of l, m, n is

x 2 + y - co ** = ac {cox – (x2 + yo)}},
or {co? – (x2 + y2)}[ac + {come – (x2 + y')}'] = 0.

The factor ac +{c** – (x2 + y')}} = 0, is the equation to the required envelop. Transposing and squaring, this becomes

dxi - (x2 + y) = a*c*, the equation to a hyperboloid of revolution of two sheets, the possible axis of which coincides with the axis of .

If the theory of reciprocal polars given in Ex. 11, be applied to the surfaces of the second order, it will be found that the reciprocal polar of a surface of the second order is also a surface of the second order ; and that when the one surface can be generated by the motion of a straight line, the other can be so generated also. For the properties of reciprocal polars in surfaces the reader may consult the memoirs indicated in Ex. 11, and also one by Brianchon, Journal de TEcole Polytechnique, Vol. vi. p. 308.

(15) Find the surface traced out by the ultimate intersections of the planes which touch the ellipsoid

along the curve made by its intersection with the plane

1 + m + 8 = 5. If s', y', s' be the current co-ordinates of the tangent plane, its equation is

where x, y, % are supposed to vary subject to the previous conditions. Differentiating we have

a da ydy xdx

a + beta = 0, Ida + mdy + ndx = 0,

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