If the system be displaced from the position of stable equilibrium by turning the pullies through an angle 0, the equation of motion is

or {M+ (ñα+b) (μ +μ')} +2g (μ-μ) 0=0,

dt2

an equation of harmonic motion, whence the time of oscillation.

x. Find the conditions of equilibrium of a fluid acted on by given forces, and prove that the .resultant force at any point of a surface of equal pressure is normal to the surface and inversely as the density at the point and the distance to the consecutive surface of equal pressure.

A quantity of homogeneous fluid which completely fills a fixed rigid spherical shell (radius c) is under the action of such a system of forces that

p-o

ρ

3 (x2 + y2 + z2)2 − 5 (x2 + y2 + z3)

a2

being the pressure at the centre, which is the origin of coordinates. Prove that the surfaces of equal pressure meet the four planes x+y+z = 0 in circles; and that the average pressure at the surface of the shell is equal to the pressure at the centre; determine also the least possible pressure at the centre.

and the surfaces of equal pressure meet the planes (1) in circles.

4

4

The average value of x+y+2 over the surface of the sphere is three times the average value of x*, and

2

Therefore the average value of 3 (x2 + y2 + z2)* − 5 (x* +y* + z*) zero, and the average value of Ρ is @.

is

The pressure is least at the points where the coordinate

axes meet the sphere and P

2c1

ρ

a2

;

therefore can

WEDNESDAY, Jan. 20, 1875. 9 to 12.

Mr. WRIGHT.

1. IF PP', QQ be diameters of an ellipse, and PR, PR' be let fall perpendiculars on P'Q, P'Q', prove that the chord of the ellipse intercepted on the straight line RR' will subtend a right angle at P.

Let PR, PR' meet the conic in S, S'; then if SQ', QS' intersect in O, ROR' will be the Pascal line of the hexagon PSQ'P'QS'Pinscribed in the conic.

Since PQP'Q' is a parallelogram, the chords SQ', QS' subtend a right angle at P, and therefore O is a fixed point on the normal at P.

Therefore the points where RR' meets the conic will subtend a right angle at P.

Adding the last row to the first row, and adding the second, third, and fourth rows to the last row, the determinants become

which vanish, because the first and last rows differ only by the factor 2.

3. If an ellipse U be described having the centre of a conic V for focus, and for axes the arithmetical and geometrical means of the axes of V, then of the common tangents to U and V, one is such that its point of contact with V lies on the auxiliary circle of U, and the line drawn through this point of contact parallel to the major-axis of V passes through one end of the major-axis of U, and the other three common tangents form a triangle whose angles are equidistant from the centre of V, and whose nine-pointic circle is the auxiliary circle of U.

If ACA', BCB' (fig. 70) be the axes of V, and LCL' the major-axis of U, C being a focus of U, then CL=CB, CL=CA; and the auxiliary circle of U touches both the auxiliary circles of V.

If LP be drawn parallel to CA to meet the auxiliary circle of U in P and CB in N, then

PN: NL=L'C: CL=CA: CB,

and therefore Plies on V.

If the tangents to U at L, L' meet CB, CA respectively in T, T', then since

CT.CN=CL2 = CB2 and CT".NP=CL"2=CA2;

therefore TT" is the tangent at P to V.

Since the angle TCT' is a right angle, therefore TT' is a tangent to U also, which proves the first part of the question.

If a circle be described with centre C and radius CA+CB, it is well known that there is an infinite series of triangles inscribed in the circle and touching V, and an infinite series of triangles inscribed in the circle and touching U.

Therefore the other three common tangents of U and V form a triangle which belongs to both series of triangles, and whose angular points are equidistant from C.

Since the feet of the perpendiculars from C on the sides of this triangle are the middle points of the sides, and also lie on the auxiliary circle of U; therefore the auxiliary circle of U is the nine-pointic circle of this triangle.

4. If one of the lines of curvature on a developable surface lie on a sphere, all the other lines of curvature, other than the rectilineal ones, lie on concentric spheres. If the common centre of these spheres lies on the surface, the surface must be a cone.

The lines of curvature on a developable are the generators and curves which cut them at right angles.

If PQRS..., P'Q'R'S'... be lines of curvature, such that PP', QQ... are generators, then PP'=QQ'=....

Now if PQRS... be on a sphere whose centre is O, since any curve on a sphere is a line of curvature; therefore the sphere and surface intersect at a constant angle.

Therefore, since OP, PP' are equal to OQ, QQ' respectively, and the angle OPP' is equal to the angle OQQ, OP' is equal to OQ'; and therefore P'Q'R'S'... is on a concentric sphere.

If O be on the surface, and if PP' be the generator through it, since OP is equal to OQ, O must be at the intersection of the generators through P and Q.

Hence, all the generators pass through the point O, and the surface must be a cone.