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conoid, terminated by parallel planes, when compared with a cylinder of the same altitude, on a base equal to the middle section of the Frustum made by a parallel plane. The difference between the Frustum and the cylinder is always the same quantity, in different parts of the same, or of similar solids, or whatever the magnitude of the two parallel ends may be; the inclination of those ends to the axis, and the altitude of the Frustum being given; and the said constant difference is 1/4 part of a cone of the same altitude with the Frustum, and the radius of its base is to that altitude, as the fixed axis is to the revolving axis of the Frustum. Thus, if BEC be any conic section, or a right line, or a circle, whose axis, or a part of it, is AD; AB and CD the extreme ordinates, FE the middle ordinate, AF being = FD; then taking, as AD to DK, so is the whole fixed axis, of which AD is a part, to its conjugate axis; and completing the parallelogram AGHD: then if the whole figure revolve about the axis AD, the line BEC will generate the Frustum of the cone or conoid, according as it is a right line or a conic section, or it will generate the whole solid when AB vanishes, or A and B meet in the same point; likewise AGHD will generate a cylinder, and ADK a cone: then is the 4th part of this cone always equal to the difference between the said cylinder generated by AGHD and the solid or Frustum generated by ABECD; having all the same altitude or axis AD.

In the parabolic conoid, this difference and the cone vanish, and the Frustum, or whole conoid ABECD, is always equal to the cylinder AGHD, of the same altitude.

In the sphere, or spheroid, the Frustum ABECD is less than the cylinder AGHD, by 1/4 of the cone AKD. And

In the cone or hyperboloid, that Frustum is greater than the cylinder, by 1/4 of the said cone AKD, which is similar to the other cone IBCD.

It may be observed, that the same relations are true, whether the ends of the Frustum are perpendicular or oblique to the axis. And the same will hold for the Frustum of any pyramid, whether right or oblique; and such a Frustum of a pyramid will exceed the prism, of the same altitude, and upon the middle section of the Frustum, by 1/4 of the same cone.

It has been observed, that the difference, or 1/4 of the cone AKD, is the same, or constant, when the altitude and inclination of the ends of the Frustum remain the same. But when the inclination of the ends varies, the altitude being constant; then the said difference varies so as to be always reciprocally as the cube of the conjugate to the diameter AD. And when both the altitude and inclination of the ends vary, the differential cone is as the cube of the altitude directly, and the cube of the said conjugate diameter reciprocally: but if they vary so, as that the altitude is always reciprocally as that diameter, then the difference is a constant quantity.

Another general theorem for Frustums, is this. In the Frustum of any solid, generated by the revolution of any conic section about its axis, if to the sum of the two ends be added 4 times the middle fection, 1/<*> of the last sum will be a mean area, and being drawn into the altitude of the solid, will produce the content. That is, is the content of ABCD.

And this theorem is general for all Frustums, as well as the complete solids, whether right or oblique to the axis, and not only of the solids generated from the circle or conic sections, but also of all pyramids, cones, and in short of any solid whose parallel sections are similar figures.

The same theorem also holds good for any parabolic area ABECD, and is very nearly true for the area of any other curve whatever, or for the content of any other solid than those above mentioned.

FUGUE

, in Music, is when the different parts of a musical composition follow each other, each repeating in order what the first had performed.

FULCRUM

, or Prop, in Mechanics, is the fixed point about which a lever &c turns and moves.

FULGURATING Phosphorus, a term used by some English writers, to express a substance of the phosphorus kind. It was prepared both in a dry and liquid state, but the preparation it seems was not well known to any but the inventor of it. This matter not only shone in the dark in both states, but communicated its light to any thing it was rubbed on. When inclosed in a glass vessel well stopped, it sometimes would Fulgurate, or throw out little flashes of light, and sometimes fill the whole phial with waves of flame. It does not need recruiting its light at the fire, or in the sunshine, like the phosphorus of the Bolognian stone, but of itself continues in a state of shining for several years together, and is seen as soon as exposed in the dark; the solid or dry matter always resembling a burning coal of fire, though not consuming itself. Philos. Trans. N° 134.

FULIGINOUS

, an epithet applied to thick smoke or vapour replete with soot or other crass matter.

In the first fusion of lead, there exhales a great deal of Fuliginous vapour, which being retained and collected, makes what is called Litharge. And Lampblack is what is gathered from the Fuliginous vapours of pines, and other resinous wood, when burnt.

FULMINANT

, Fulminans, or Fulminating, an epithet applied to something that thunders, or makes a noise like thunder.

Aurum Fulminans. See Aurum.

Pulvis Fulminans, is a composition of 3 parts of nitre, 2 parts of salt of tartar, and 1 of sulphur.—Both the Aurum and Pulvis Fulminans produce their effect chiefly downwards; in which they disfer from gunpowder, which acts in orbem, or all around, but principally upwards. When the composition is laid in brass ladles, and so set on sire, aster fulmination, the ladles are often found perforated. It differs also from gunpowder in this, that it does not require to be confined, in order to fulminate, and it must be slowly and gradually heated. Some instants before explosion, a light blue flame appears on its surface, proceeding from the vapours beginning to kindle. No more fire or flame is perceived during the fulmination, being suffocated and extinguished by the quickness and violence of the commotion. Nor does the Fulminating powder