# How to use *Singularities* in a sentence

There are other infinite

**singularities**of detail; but the above are more than sufficient to establish the point.For example, in the curve for gold-aluminium, ignoring minor

**singularities**, we find two intermediate summits, one at the percentage Au 2 A1, and another at the percentage AuAl 2.He goes on to observe also that in this area are found many of the most remarkable forms - all the red Lories, the great cockatoos, the pigmy Nasiternae and other

**singularities**.Among his most remarkable works may be mentioned his ten memoirs on quantics, commenced in 1854 and completed in 1878; his creation of the theory of matrices; his researches on the theory of groups; his memoir on abstract geometry, a subject which he created; his introduction into geometry of the "absolute"; his researches on the higher

**singularities**of curves and surfaces; the classification of cubic curves; additions to the theories of rational transformation and correspondence; the theory of the twenty-seven lines that lie on a cubic surface; the theory of elliptic functions; the attraction of ellipsoids; the British Association Reports, 1857 and 1862, on recent progress in general and special theoretical dynamics, and on the secular acceleration of the moon's mean motion.The work falls into two parts, which treat of the asymptotes and

**singularities**of algebraical curves respectively; and extensive use is made of the method of counting constants which plays so large a part in modern geometrical researches.AdvertisementThe ordinary

**singularities**, arranged according to a cross division, are Proper.The

**singularities**(I) and (3) have been termed proper**singularities**, and (2) and (4) improper; in each of the first-mentioned cases there is a real singularity, or peculiarity in the motion; in the other two cases there is not; in (2) there is not when the point is first at the node, or when it is secondly at the node, any peculiarity in the motion; the singularity consists in the point coming twice into the same position; and so in (4) the singularity is in the line coming twice into the same position.First, if the equation be in point-co-ordinates, (3) and (4) are in a sense not

**singularities**at all.The theory of compound

**singularities**will be referred to farther on.In regard to the ordinary

**singularities**, we have m, the order, n „ class, „ number of double points, Cusps, T double tangents, inflections; and this being so, Pliicker's ” six equations ” are n = m (m - I) -2S -3K, = 3m (m - 2) - 6S- 8K, T=Zm(m -2) (m29) - (m2 - m-6) (28-i-3K)- I -25(5-1) +65K-1114 I), m =n(n - I)-2T-3c, K= 3n (n-2) - 6r -8c, = 2n(n-2)(n29) - (n2 - n-6) (2T-{-30-1-2T(T - I) -1-6Tc -}2c (c - I).AdvertisementTo complete Pliicker's theory it is necessary to take account of compound

**singularities**; it might be possible, but it is at any rate difficult, to effect this by considering the curve as in course of description by the point moving along the rotating line; and it seems easier to consider the compound singularity as arising from the variation of an actually described curve with ordinary**singularities**.The most simple case is when three double points come into coincidence, thereby giving rise to a triple point; and a somewhat more complicated one is when we have a cusp of the second kind, or node-cusp arising from the coincidence of a node, a cusp, an inflection, and a double tangent, as shown in the annexed figure, which represents the

**singularities**as on the point of coalescing.So that, in fact, Pliicker's equations properly understood apply to a curve with any

**singularities**whatever.By means of Pliicker's equations we may form a table - The table is arranged according to the value of in; and we have m=o, n= r, the point; m =1, n =o, the line; m=2, n=2, the conic; of m = 3, the cubic, there are three cases, the class being 6, 4 or 3, according as the curve is without

**singularities**, or as it has 1 node or r cusp; and so of m =4, the quartic, there are ten cases, where observe that in two of them the class is = 6, - the reduction of class arising from two cusps or else from three nodes.The cases may be divided into sub-cases, by the consideration of compound

**singularities**; thus when m= 4, n= 6, S = 3, the three nodes may be all distinct, which is the general case, or two of them may unite together into the singularity called a tacnode, or all three may unite together into a triple point or else into an oscnode.AdvertisementThe system has

**singularities**, and there exist between m, r, is and the numbers of the several**singularities**equations analogous to Pliicker's equations for a plane curve.Among other

**singularities**of habitat, not the least curious is the freedom with which some small species, especially in the genus Dichelaspis, occupy the very jaws of large crustaceans.Allied to the matter just mentioned was Plucker's discovery of the six equations connecting the numbers of

**singularities**in algebraical curves (see Curve).However, as I said earlier, I shall take it that such

**singularities**are harmless.All these and other

**singularities**do not hinder the poem from being a very spirited one.AdvertisementThese orbifold

**singularities**in the backgrounds could be resolved, by replacing a small neighborhood of the nut, by an ALE metric.The two new chapters describe numerical experiments on two- and three-dimensional rotors, and phase

**singularities**in the heart wall.We apply the method to compressible and incompressible linear elasticity problems, including problems with stress

**singularities**.Although computer simulations can help, many algorithms fail when they address regions near black hole

**singularities**where the gravitational fields theoretically approach infinity.