B, C, D, E, enlarged.) branched rootstock from which spring slender aerial shoots which are green, ribbed, and bear at each node a whorl of leaves reduced to a toothed sheath.
In other cases the root structure of the stele continues up to the cotyledonary node, though the hypocotyl is still to be distinguished from the primary root by the character of its epidermis.
In many of those ants whose third abdominal segment forms a second " node," the basal dorsal region of the fourth segment is traversed by a large number of very fine transverse striations; over these the sharp hinder edge of the third segment can be scraped to and fro, and the result is a stridulating organ which gives rise to a note of very high pitch.
The motion of the node of this plane is found with great exactness from observaMass, of the g tions of the transits of Venus.
Its weak point is that the apparent motion of the node depends partly upon the motion of the ecliptic, which cannot be determined with equal precision.
From the observed motion of the node of Venus, as shown by the four transits of 1761, 1769, 1874 and 1882, is found Mass of (earth +moon) _Mass of sun 332600 In gravitational units of mass, based on the metre and second as units of length and time, Log.
We may represent the displacement due to one of the trains by y l =a sin 2 i (24) where x is measured as in equation (16) from an ascending node as A in fig.
The ordinate of the curve changes sign as we pass through a node, so that successive sections are moving always in opposite directions and have opposite displacements.
It is obvious that the nodes are alternately in compression and extension, or vice versa, and that for 4X on each side of a node the motion is either to it on both sides or from it on both sides.
He used a tube of variable length and determined the length resounding to a given fork, (1) when the closed end was the first node, (2) when it was the second node.
There is a node in the middle.
When the pipe is blown softly the fundamental is very predominant, and there is a node at the middle point.
Each of the first few harmonics may be easily obtained by touching the string at the first node of the harmonic required, and bowing at the first loop, and the presence of the nodes and loops may be verified by putting light paper riders of shape A on the string at the nodes and loops.
36, for the fundamental and the first harmonic. When a string is struck or bowed at a point, any harmonic with a node at that point is absent.
The next mode has a node in the middle and two others each 0.132 from the end.
The fundamental mode has that node only.
The next mode has a second node 0.226 from the free end; the next, nodes at 0.132 and 0 .
The lower end of the jet tube, being open, is a loop, and the node may be regarded as in an imaginary prolongation of the jet tube above the nozzle.
If the jet tube is somewhat longer than half the sounding tube there will be a node in it, and now the condition of equality of pressure requires opposite motions in the two at the nozzle, for their nodes are situated on opposite sides of that point.
The form of the limacon depends on the ratio of the two constants; if a be greater than b, the curve lies entirely outside the circle; if a equals b, it is known as a cardioid; if a is less than b, the curve has a node within the circle; the particular case when b= 2a is known as the trisectrix.
When a single leaf with opposite stem with alteris produced at a node, and leaves.
In the lowest leaf is directly the point of insertion of the pair one leaf is in above the first, leaf in the node, dividing front and the other and commences the leaf into similar halves, at the back; in the the second cycle.
31, leaf 1 arises from a node n; leaf 2 is separated from it by an internode m, and is placed to the right or left; while leaf 3 is situated directly above leaf 1.
A, The branch with the leaves numbered in their order, n being the node and m the internode; b is a magnified representation of the branch, showing the points of insertion of the leaves and their spiral arrangement, which is expressed by the fraction or one turn of the spiral for two internodes.
Oedogonium sp., oogonium antheridium at a node on at moment of fertilization a lateral appendage.
From the first node arise rhizoids; from the second a lateral bud, which becomes the new plant.
In the higher normal modes there are nodes or points of rest (y = 0); thus in the second mode there is a node at a distance .1901 from the lower end.
In such cases they are very generally given off from just above each node (often in a circle) of the lower part of the stem or rhizome, perforating the leaf-sheaths.
They are solitary at each node and arranged in two rows, the lower often crowded, forming a basal tuft.
The spike of an inflorescence bears whorls of flowers at each node in the axils of concrescent bracts accompanied by numerous sterile hairs (paraphyses); in a male inflorescence numerous flowers occur at each node, while in a female inflorescence the number of flowers at each node is much smaller.
Longitude of ascending node (1908.0), 74° 28.6'.
Each node of a trabecula may be simple, i.e.
The double point larities - cusp or spinode; or node; Line-singu5 3.
The node: the point may in the course of its motion come to coincide with a former position of the point, the two positions of the line not in general coinciding.
It may be remarked that we cannot with a real point and line obtain the node with two imaginary tangents (conjugate or isolated point or acnode), nor again the real double tangent with two imaginary points of contact; but this is of little consequence, since in the general theory the distinction between real and imaginary is not attended to.
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.
The curve (1 x, y, z) m = o, or general curve of the order m, has double tangents and inflections; (2) presents itself as a singularity, for the equations dx(* x, y, z) m =o, d y (*r x, y, z)m=o, d z(* x, y, z) m =o, implying y, z) m = o, are not in general satisfied by any values (a, b, c) whatever of (x, y, z), but if such values exist, then the point (a, b, c) is a node or double point; and (I) presents itself as a further singularity or sub-case of (2), a cusp being a double point for which the two tangents becomes coincident.
The expression 2 is that of the number of the disposable constants in a curve of the order m with nodes and cusps (in fact that there shall be a node is I condition, a cusp 2 conditions) and the equation (9) thus expresses that the curve and its reciprocal contain each of them the same number of disposable constants.
But it can be shown, analytically or geometrically, that if the given curve has a node, the first polar passes through this node, which therefore counts as two intersections, and that if the curve has a cusp, the first polar passes through the cusp, touching the curve there, and hence the cusp counts as three intersections.
But, as is evident, the node or cusp is not a point of contact of a proper tangent from the arbitrary point; we have, therefore, for a node a diminution and for a cusp a diminution 3, in the number of the intersections; and thus, for a curve with 6 nodes and K cusps, there is a diminution 26+3K, and the value of n is n= m (m - I)-26-3K.
But if the given curve has a node, then not only the Hessian passes through the node, but it has there a node the two branches at which touch respectively the two branches of the curve; and the node thus counts as six intersections; so if the curve has a cusp, then the Hessian not only passes through the cusp, but it has there a cusp through which it again passes, that is, there is a cuspidal branch touching the cuspidal branch of the curve, and besides a simple branch passing through the cusp, and hence the cusp counts as eight intersections.
The node or cusp is not an inflection, and we have thus for a node a diminution 6, and for a cusp a diminution 8, in the number of the intersections; hence for a curve with 6 nodes and cusps, the diminution is = 66+8K, and the number of inflections is c= 3m(m - 2) - 66 - 8K.
Thirdly, for the double tangents; the points of contact of these are obtained as the intersections of the curve by a curve II = o, which has not as yet been geometrically defined, but which is found analytically to be of the order (m-2) (m 2 -9); the number of intersections is thus = m(rn - 2) (m 2 - 9); but if the given curve has a node then there is a diminution =4(m2 - m-6), and if it has a cusp then there is a diminution =6(m2 - m-6), where, however, it is to be noticed that the factor (m2 - m-6) is in the case of a curve having only a node or only a cusp the number of the tangents which can be drawn from the node or cusp to the curve, and is used as denoting the number of these tangents, and ceases to be the correct expression if the number of nodes and cusps is greater than unity.
Hence, in the case of a curve which has 6 nodes and K cusps, the apparent diminution 2(m 2 - m-6)(26+3K) is too great, and it has in fact to be diminished by 2 1(25(5 - I) +66K+ 2 K(K - I)1, or the half thereof is 4 for each pair of nodes, 6 for each combination of a node and cusp, and 9 for each pair of cusps.
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.
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.
A property independent of the particular axes of co-ordinates used in the representation of the curve by its equation; for instance, the curve may have a node, and in order to this, a relation, say A = o, must exist between the coefficients of the equation; supposing the axes of co-ordinates altered, so that the equation becomes u' = o, and writing A' = o for the relation between the new coefficients, then the relations A = o, A' = o, as two different expressions of the same geometrical property, must each of them imply the other; this can only be the case when A, A' are functions differing only by a constant factor, or say, when A is an invariant of u.
Similarly a cubic through the two circular points is termed a circular cubic; a quartic through the two points is termed a circular quartic, and if it passes twice through each of them, that is, has each of them for a node, it is termed a bicircular quartic. Such a quartic is of course binodal (m = 4, 6= 2, K = o); it has not in general, but it may have, a third node or a cusp. Or again, we may have a quartic curve having a cusp at each of the circular points: such a curve is a " Cartesian," it being a complete definition of the Cartesian to say that it is a bicuspidal quartic curve (m= 4, 6 = o, K= 2), having a cusp at each of the circular points.
.,, a given node 5.
„ node on given line 6.