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on a fixed circle which encloses it. And that in any position of the two rolling circles, which roll in opposite directions, a circle may be drawn concentric with the fixed circle so as to touch both rolling circles at points such that the line joining them is the tangent to the hypocycloid.

If any three of the tangents to a three-cusped hypocycloid form an equilateral triangle, prove that the angular points of the triangle will lie on a curve whose polar equation is r = a cos 30.

Let AQ (fig. 62) be the hypocycloid traced out by the point on the rolling circle Pop.

Produce PQ to meet the fixed circle in P', and let P'O meet Qp in p', where O is the centre of the fixed circle.

The circle on P'p' as diameter will pass through Q; and if C, C' be the centres of the circles PQp, P'Qp', then, since the angles at P and P' are equal, CQCO is a parallelogram and CQ equal to OC.

Hence the arcs PQ, P'Q, PP' are similar, and as the radii PC, CO, PO.

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therefore arc PQ+ arc P'Q=arc PP', and the arc PQ being equal to the arc PA, therefore the arc P'Q is equal to the arc P'A.

Thus, if the circle P'Qp' roll on the fixed circle, it will generate the same hypocycloid AQ.

Also since Op=Op', a circle whose centre is O will touch the two rolling circles in p, p', points on the tangent at Q. [For further developments see Wolstenholme, Proceedings of the London Mathematical Society, vol. IV., p. 321.]

In the three-cusped hypocycloid, if OC=2a, CP=a, and the angle POA = 0,

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30

2

therefore the perpendicular on the tangent being a sin three tangents forming an equilateral triangle are equidistant from the centre; and if r, o be the polar coordinates of an angular point of the equilateral triangle

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in the case where P and Q are functions of x only. Prove that the variables in the differential equation

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may be separated by the substitutions, x=u+v, y = ku — v, provided the constant k be suitably chosen, and integrate the equation.

Making the substitution and reducing, the equation becomes

{(1 + k)3 uv+ ka2 — b2} du = {(1 + k)2 u2 + a2 + b2} dv. If then kab, the variables can be separated, and

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ix. Prove that in the expansion of in ascending

e-1

powers of t no odd powers except the first appear, and if

B

(-1)+1-1 be the coefficient of t",

2n

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TUESDAY, Jan. 19, 1875. 9 to 12.

MR. COCKSHOTT.

1. IF A, B be two fixed points and any plane be drawn through AB meeting a fixed plane conic in and R, prove that the locus of the point of intersection of AQ and BR will be another fixed plane conic.

The quadric cones with vertices A and B having as common plane section the given conic will intersect in another plane conic, on which AQ and BR will intersect, as also AR and BQ.

2. Prove that, if a, B, y, 8 are all different,

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+

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(a - b) (a−c) (ad) (b −c) (b-d) (b − a)

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+

(c — d) (c — a) (c — b) † (d — a) (d — b) (d — c)

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+

putting a = cos a + i sin a, b...,

α B

a

a-b=2i sin (cos ++ i sin +),

2

2

and the first term of the identity becomes

a-B a-y a-d

2

i

cos 3a + sin 3a

3a+B+y+8

8 sin

sin

sin

COS

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2

2

2

2

2

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3. If AB' CA'BC' be a regular hexagon, prove that three rectangular hyperbolas can be described, the first touching · AB, AC at B, C and touching A'B', A'C' at B', C', the second touching BC, BA at C, A, and B'C', B'A' at C', A', and the third touching CA, CB at A, B, and C'A', C'B' at A', B'; and that any one of the three is the polar reciprocal of the second with respect to the third.

Prove also that an infinite number of triangles can be described, each of which is self-conjugate to one hyperbola, whose sides touch the second, and whose angular points lie on the third.

Taking ABC or A'B'C' as the triangle of reference, the equations of the three rectangular hyperbolas will be

x2+2yz=0...(1), y2+2zx=0... (2), ≈2 + 2xy=0...(3).

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