Reciprocally, every surface resulting from one of these two modes of generation is a hyperbolic paraboloid.
The simplest equations of the ellipsoid, of the hyperboloid of one sheet and of two sheets, of the elliptical and the hyperbolic paraboloid, of cones and of cylinders of the second order.
Through each point of the surface of the hyperbolic paraboloid two right lines may be traced, whence results the generation of the paraboloid by two systems of right lines.
Thus as in hyperbolic geometry the theory of similarity does not hold, and the elements of a triangle are determined when its three angles are given.
It may be shown that all motions in a hyperbolic plane consist, in a general sense, of rotations; but three types must be distinguished according as the centre is real, imaginary or at infinity.
From these and a few other remarks it appears that Gauss possessed the foundations of hyperbolic geometry, which he was probably the first to regard as perhaps true.
As he gave, by an elegant method, a convenient Euclidean interpretation of hyperbolicplane geometry, his results will be stated at some length[14].
Thus the investigation of the type (elliptic, hyperbolic or parabolic) of this special congruence-group is a perfectly definite problem, to be decided by experiment.
The existence of a natural unit of length is a peculiarity common both to hyperbolic and elliptic geometries, and differentiates them from Euclidean geometry.
Many of those properties of Euclidean parallels, which do not hold for Lobatchewsky's parallels in hyperbolic geometry, do hold for Clifford's parallels in elliptic geometry.
In an hyperbolic involution any two conjugate points are harmonic conjugates with regard to the two foci.
Hence we may write-- In anhyperbolic involution, OA.
But these Notes were written nigh forty years ago, so we may hope that by this time he has cast out, or at least subdued by diligent exorcism, that same hyperbolic fiend which entered in and rent him at certain seasons of his youth.
He plunged into his speech, forgetful of what he had written, with a passion queerly hyperbolic in view of the subject.
It was a fighting, enthusiastic, hyperbolic speech, glowing with the divine fire of youth.
It was a fighting, enthusiastic, hyperbolic speech, glowing, as did the young face of the speaker, with the divine fire of youth.
That a concussion would not add to its velocity is certain, and the departure in a hyperbolic orbit would be contrary to the law of gravitation.
Subsequent observations best accorded with a hyperbolic orbit; and it was in view of this anomaly, that the late Sears C.
It is true that seeming evidence of hyperbolic eccentricity is sometimes afforded by observations and regarded by some astronomers as sufficient.
It is conceivable that a comet might revolve in a hyperbolic orbit.
The acceptance of this seemingly inevitable conclusion leads to another: that no comet yet known moves in a really hyperbolic orbit, but that the limit of eccentricity must be regarded as 1, or that of the parabola.
Again and again are their bodily charms dwelt on rapturously, as is customary in the poems of all Orientals with all sorts of quaint hyperbolic comparisons, some of which are poetic, others grotesque.
The time when alone it flourishes with its mental purity, its minute sympathies, its gallant attentions and sacrifices, its hyperbolic adorations, and mixed moods of agonies and ecstasies, is during the period of courtship.
Now there is nothing to hinder us from admitting these ideas to possess an objective and hyperbolic existence, except the cosmological ideas, which lead reason into an antinomy: the psychological and theological ideas are not antinomial.
Comets with parabolic and hyperbolic orbits may be regarded as stray objects which visit our system once, and depart never to return again.
Some observers are disinclined to admit the accuracy of this statement; whilst others believe that the stars have executed a hyperbolic sweep round their common centre of gravity and are now separating.
This applies to an elliptic or hyperbolic orbit; the case of the parabolic orbit may be examined separately or treated as a limiting case.
It is easily seen graphically, or from a table of hyperbolic tangents, that the equation u tanh u = 1 has only one positive root (u = 1.
Lexell's Comet, for instance, rushed in its hyperbolic path too near to Jupiter, and was caught in the attraction of its mass, and made to dance attendance on the sun through two successive elliptical revolutions.
At the end of the second, the influence that had impounded the comet came, however, into play oppositely, and restored it again to its wandering life and hyperbolic courses.
Hyperbolic functions are extremely useful in every branch of pure physics and in the applications of physics whether to observational and experimental sciences or to technology.
The principal advantage arising from the use of hyperbolic functions is that they bring to light some curious analogies between the integrals of certain irrational functions.
Whenever mechanical strains are regarded great enough to be measured they are most simply expressed in terms of hyperbolic functions.
Checked in their proud hyperbolic sweep, made captive in a planetary net, deprived of their trains, these quondam free-lances of the heavens are now mere shadows of their former selves.
In many of these instances the hyperbolic gold and silver of the writers would undoubtedly be bronze.
It may be that there is reason to adopt a hyperbolic theory of time and a corresponding hyperbolic theory of space.
But no such hyperbolic language, when thus applied to man as we know him, or as we can even conceive him here below, can, of course, be kept up.
But these are more or less of an Agadic (legendary) character, and thus lend themselves to exaggeration and hyperbolic language.
With the power Ken speaks of, I believe the hyperbolic form could carry an effective wave into the stratosphere.
On the roof of Science Hall there was being erected a massive, 30-foot, hyperbolic reflector whose metal surface had been beaten out of aluminum chicken-shed roofs.
Names may also be used for the different forms of infinite branches, but we have first to consider the distinction of hyperbolic and parabolic.
The two legs of a hyperbolic branch may belong to different asymptotes, and in this case we have the forms which Newton calls inscribed, circumscribed, ambigene, &c.
The epithets hyperbolic and parabolic are of course derived from the conic hyperbola and parabola respectively.
And we thus see that the two hyperbolic legs belong to a simple intersection of the curve by the line infinity.
Omega] passes through a crunode we have pairs of hyperbolic legs belonging to two parallel asymptotes.
I have omitted to mention thehyperbolic form of government.
By an appeal to a Higher Authority than that of earthly kings and potentates was this rule exercised; but its hyperbolic form is fast passing away, and degenerating into that of a circle with indefinitely small radius.
We may hence deduce the following property of the corresponding hyperbolic State.
We shall not, therefore, discuss the complex polemical problems which a hyperbolic State suggests.
The Ducal Palace furnishes three anomalies in the arch, dogtooth, and dentil: it has a hyperbolic arch, as noted above, Chap.
I have never myself met with it on a large scale; but it occurs in the niches of the later portions of the Ducal palace at Venice, together with a singular hyperbolic arch, a in Fig.
He turned to look, and saw another one thrown from the corn-field near by ascending in a hyperbolic curve and then coming down so near to his head that he moved out of the way.
The scarcity ofhyperbolic comets can be similarly explained.
Divide the length of stroke through which the steam expands, by the length of stroke performed with full pressure, which last call 1; the hyperbolic logarithm of the quotient is the increase of efficiency due to expansion.
In a similar way he accounted for the hyperbolic bands of colour observed by Grimaldi within the shadow of a square near its corners.
When a tuning-fork is vibrating there are certain hyperbolic lines along which the disturbance caused by one prong is exactly neutralized by that due to the other prong.
The above list will hopefully give you a few useful examples demonstrating the appropriate usage of "hyperbolic" in a variety of sentences. We hope that you will now be able to make sentences using this word.
