# How to use Singularities in a sentence

singularities
• 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.

• The 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).

• To 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.

• The 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.