=mx-a-38-2y-3v-4w-8H; secondly, the v(x−2r+9) points on the lines v; thirdly, the points H each counting as 4 cusps. For first consider a generating line meeting the curve x in B and the curve x in A; if we imagine on the curve a point which approaches and ultimately coincides with B, the generating line through meets the torse in the neighbourhood of its cuspidal edge in two points which come ultimately to coincide with the point A, and we thus see that A is a stationary point on the (r-2) curve.

54. Secondly, observing that the line v is a cuspidal line on the torse, and considering in like manner a generating line of the cone, which approaches and comes ultimately to coincide with one of the x- 2r+9 points, we see that this is a stationary point on the x (r− 2) curve. And thirdly, any line through a point H meets the torse in this point counting 4 times, and in r-4 other points. Hence considering the generating line of the x cone, which travelling along any one of the four partial branches of the x curve comes ultimately to coincide with H, 2 of the r-2 points on such generating line come to coincide at the point H; and we have thus the point H as a singular point on the x(r-2) curve; viz. it reckons as a stationary point once in respect of each of the four partial branches of the curve x (it must be assumed that this is so, but a further proof is required), that is as 4 cusps on the x (r2) curve.

55. By what precedes we have

x (r-2) (r-3)=

n(x-2r+8)

+3 {(mx− a−3ẞ – 27 - 3v - 4w - 8H) + v(x − 2r + 9) + 4H}, which is the true theoretical form in which the equation for x (r−2) (r−3) was obtained by Cremona.

DEFORMATION OF AN ELASTIC SPHERE PRESSED BETWEEN TWO PARALLEL PLANES.

By R. HOPPE.

IN the Philosophical Transactions of the Royal Society of London, Vol. 153, Mr. (now Sir) W. Thomson has resolved the general problem: A shell consisting of isotropic elastic material, and bounded by two concentric spherical surfaces being given, and displacements being arbitrarily produced or forces arbitrarily applied over its whole bounding surface, to find the displacement of every point of its substance.

It is obvious that the condition concerning the surface does not include every possible case, since it requires that the displacements or forces are applied to the whole surface. What occurs most frequently, is that a part of the surface is deformed in a given manner, and the effect experienced by every particle of the substance is to be determined. As the most simple case of this sort, I propose to find the displacements of all points of a solid sphere pressed between two parallel planes of a given distance, a little smaller than the diameter. The material is likewise supposed isotropic, the planes perfectly rigid and without friction. To realize the exact conditions, we may think an unlimited range of spheres, all of equal size, and pressed against each other; each of them will be in the state to be supposed here.

The given conditions for the surface are the following: The normal and tangential strains are in general zero, except for two small areas around the poles, where the normal strains are unknown, and instead of them the two planes are given, into which all points are transferred.

It is, however, known beforehand that the displacement of any point is independent of its azimuth, and that any azimuth remains unvaried by the deformation. If, therefore, the original coordinates of an element, referring to the centre as origin and to the normal of the pressing planes as axis of x, x = K COSλ, y = к sinλ, z,

receive by the deformation the increments §, n, s, we shall have

=к COSλ, n = şinλ,

and, will not depend on λ.

Let now denote the increment of any unity of volume, and m the ratio of the transversal and longitudinal elasticity,

that means, that a cubical element longitudinally compressed is shortened m times as much as it is stretched in either transversal direction. Then the conditions of internal equilibrium are

The equilibrium on the surface requires (1), that the tangential strains are throughout zero, which is expressed by

the equations

(2) that the normal strain normal to the surface vanishes at the free surface, which is expressed by the equation

(3) that at the flattened parts of the surface,

z + 3 = ±(c − y),

where c is the radius of the sphere, c-y the half distance of the planes.

The equations for internal equilibrium may be satisfied

by the values

where V is an homogeneous function of x, and z satisfying the equations

+

dy dz

=n√ ̧‚; √.=1,

8-1

=0,

.(3).

+

dx

dy dz

+2 : = n√.

Besides, if we put

v will be a spherical function of one variable, completely determined by the recurrent relation

nv„-, − (2n + 1) — v„ + (n + 1) o ̧‚s = 9........................ (4).

On account of the symmetry of the figure each of the components of displacement is to change sign with the corresponding coordinate, while the two other components as well as remain unchanged. Hence, it follows that n can only have even values. Moreover, continuity at the centre requires that no negative power of r enters into the expressions, and for this reason 2 is the least value of n. Collecting together all particular solutions and passing to meridional coordinates k, z, we obtain the following expressions:

b

'n n

C for odd values of n are zero, while all other

c as well as b, remain arbitrary constants, and

n

2

an

where a and c

The superficial equations (2) reduced to meridional co

ordinates give

2kL=2

+2

dk

Introducing the values (5) expressed by spherical functions,

we find

The latter equation admits of an immediate solution. Putting for brevity

we may make each term of the first part of the series destroy the subsequent term of the second part, by establishing the relation

The equations (7), (9), (10) are sufficient to reduce all constant coefficients to b, c, C19 C6) ••

....

expressions, let

VOL. XI.

Y