From the nodes spring whorls of similar but more slender branches.
The line of nodes rotates in a period of 18.612 years and the line of apsides in a period of 8.84 years.
The plant is readily propagated by cuttings, a piece of the stem bearing buds at its nodes will root rapidly when placed in sufficiently moist ground.
This section includes a number of families characterized by the backward extension of the prothorax to the tegulae and distinguished from the ants by the absence of " nodes " at the base of the abdomen.
Now we may resolve these trains by Fourier's theorem into harmonics of wave-lengths X, 2X, 3A, &c., where X=2AB and the conditions as to the values of y can be shown to require that the harmonics shall all have nodes, coinciding with the nodes of the fundamental curve.
(25) The sum of the disturbance is obtained by adding (24) and (25) y = y l +y 2 = 2a cos Ut s i n 57 x, (26) At any given instant t this is a sine curve of amplitude 2a cos (27r/A)Ut, and of wave-length A, and with nodes at x = o, a A, A, ..., that is, there is no displacement at these nodes whatever the value of t, and between them the displacement is always a sine curve, but of amplitude varying between +2a and - 2a.
The points A, B, C, D are termed " nodes," and the points half-way between them " loops."
Further, the original nodes are always at rest, and the intervening sections vibrate to and fro.
At the nodes A, B, C, D, E there is no displacement, but there are maximum volume and pressure changes.
Then the middle point is a loop, and the middle flame is only slightly affected, while the other two, now being at nodes, vibrate strongly.
Stationary waves are formed in the air in the dust-tube if the length is rightly adjusted by the closely-fitting piston, and the lycopodium dust collects at the nodes in little heaps, the first being at the fixed end and the last just in front of the piston on the sounder.
Since the nodes are always at rest we may represent the vibration of a given string by the length between any two nodes.
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.
When the harmonic is sounded the riders at the loops are thrown off, while those at the nodes remain seated.
For a bar free at both ends the fundamental mode of vibration has two nodes, each 0.224 of the length from the end.
The third mode has four nodes 0.094 and 0.357 from each end, and so on.
The next mode has a second node 0.226 from the free end; the next, nodes at 0.132 and 0 .
Bells may be regarded as somewhat like circular plates vibrating with radial nodes, and with the edges turned down.
Placing the sensitive flame at different parts of this train, he found that it was excited, not at the nodes where the pressure varied, but at the loops where the motion was the greatest and where there was little pressure change.
It is evident that the pressure condition will be fulfilled only if the motions in the two tubes are in the same direction at the same time, closing into and opening out from the nodes together.
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.
88), although now discarded, received countenance from the finding of Juno by Harding, and of Vesta by himself, in the precise regions of Cetus and Virgo where the nodes of such supposed planetary fragments should be situated.
This view is expressed by Laelius Felix, a lawyer probably of the age of Hadrian, when he writes "Is qui non universum populum, sed partem aliquam adesse jubet, non comitia, sed concilium edicere debet" (Gellius, Nodes Atticae, xv.
The points on the stem at which leaves appear are called nodes; the part of the stem between the nodes is the internode.
The pairs alternate leaves, and the nodes are separated so are placed at right ranged in a penthat each leaf is placed at angles alternately, tastichous or different height on the stem, or in what is called quincuncial manthe leaves are alternate
Besides the Nodes Vaticanae, to which he appears to have contributed, the only literary relics of this intrepid and zealous reformer are some homilies, discourses and sermons, with a collection of letters.
The antheridia and oogonia are formed at the nodes of the appendages.
Although in his sixty-fourth year, he undertook to observe the moon through an entire revolution of her nodes (eighteen years), and actually carried out his purpose.
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.
The slender stem is hollow, and, as generally in grasses, has well-marked joints or nodes, at which the cavity is closed by a strong diaphragm.
Similarly, the corresponding epitrochoids will exhibit three loops or nodes (curve b), or assume the form shown in the curve; c. It is interesting to compare the forms of these curves with the three forms of the cycloid.
For the reciprocal curve these letters denote respectively the order, class, number of nodes, cusps, double tangent and inflections.
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.
With regard to the demonstration of Pliicker's equations it is to be remarked that we are not able to write down the equation in point-co-ordinates of a curve of the order m, having the given numbers 6 and of nodes and cusps.
We can only use the general equation (*fix, y, z) m = o, say for shortness u= o, of a curve of the mth order, which equation, so long as the coefficients remain arbitrary, represents a curve without nodes or cusps.
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.
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.
520) is that every singularity whatever may be considered as compounded of ordinary singularities, say we have a singularity =6' nodes, cusps, double tangents and c' inflections.
The ten cases may be also grouped together into four, according as the number of nodes and cusps (5+ic) is = o, r, 2 or 3.
The septa of modern perforate corals are shown to have a structure nearly identical with that of the secondary forms, but the trabeculae and their nodes are only apparent on microscopical examination.