For accurate measurement of the displacements of lines of stellar spectra which are produced by the relative motion of star and observer in the line of sight, a very beautiful instrument has been devised by Dr J.
By means of vibrations or shocks transmitted through the - Sub water, or by displacements in the balance or position of the animal, the otoliths are caused to impinge against the bristles of the sensory cells, now on one side, now on the other, causing shocks or stimuli which are transmitted by the basal nerve-fibre to the central nervous system.
Let x, y, z be the co-ordinates of any particle of the medium in its natural state, and, 7 7, the displacements of the same particle at the end of time t, measured in the directions of the three axes respectively.
" Let E = o,7 7 = o, =f (bt - x) be the displacements corresponding to the incident light; let O l be any point in the plane P (of the wave-front), dS an element of that plane adjacent to 01; and consider the disturbance due to that portion only of the incident disturbance which passes continually across dS.
If do is the angular displacement of the roller corresponding to displacements, dO of the disk, and ds of the point of application of P, a, and C constants, then d4) = x _ ¢Pds = C.Pds, and therefore 4) = C f 82 Pds; that is, the angular displacement of the roller measures the work done during the displacement from s 1 to s 2 .
Now if the atoms are regarded as points or spherical bodies oscillating about positions of equilibrium, the value of n+3 is precisely six, for we can express the energy of the atom in the form (9 2 v a 2 v a2v E = z(mu 2 +mv 2 +mw 2 +x 2 ax2 + y2ay2-fz2az2), where V is the potential and x, y, z are the displacements of the atom referred to a certain set of orthogonal axes.
But f o 2dx=0 since the sum of the displacements = o.
The toothed wheel being set in motion, the edge of a card or of a funnel-shaped piece of common notepaper is held against the teeth, when a note will be heard arising from the rapidly succeeding displacements of the air in its vicinity.
For this it is necessary that the total force on an element due to the sum of the displacements should be equal to the sum of the forces due to the two displacements considered separately.
If yi and y 2 are the two separate displacements and if y =3/1-4-Y2, then w = - E (dye/dx + dye/dx) =0.) i w2.
Now if yi and y 2 are the displacements due to the two trains separately, and y=yl+y2, the two separate forces are Tdy/dx and Tdy 2 /dx, while that due to y is Tdy/dx.
But, after this, the first or less rapidly vibrating note will fall behind the other, and cause a diminution in the joint displacements of the particles, till, after the lapse of onesixth of a second, it will have fallen behind the other by half a vibration.
At this moment, therefore, opposite displacements will be produced of the air particles by the two notes, and the sound due to them will be at a minimum.
Hort (Intro- duction, p. 268) has shown from a consideration of displacements in the text of the Apocalypse that it was copied from a very small MS., but this, of course, only holds good of the Apocalypse.
They also appear to contain displacements, interpolations, prefaces such as that to the Meteorologica, and appendices such as that to the Sophistical Elenchi, which may have been added.
The widening of lines does not lend itself easily to accurate measurements; more precise numerical data are obtainable by the study of the displacements consequent on increased density which were discovered and studied by W.
Curiously enough this is also approximately the ratio of the displacements found by Humphreys in the trunk series, the side branch and main branch in the order named, in cases where these displacements have been measured.
Stark discovered that in the case of the series spectrum of hydrogen and of other similar spectra the lines were displaced indicating high velocities; in other cases no displacements could be observed.
Malformations of the pelvis, accidental injuries and the diseases and displacements to which the uterus is liable, on the one hand; and, on the other, various morbid conditions of the ovum or placenta leading to the death of the foetus, are among the direct local causes.
Further, auroras are often possessed of rapid motion, so that conceivably spectral lines may receive small displacements in accordance with Doppler's principle.
Observations have shown that, in addition to the comparatively great and sudden displacements which occur in earthquakes, the ground is subject to other movements.
Vicentini of Padua, will yield excellent diagrams of the gentle undulations of earthquakes which have originated at great distances, but for local disturbances, even if the bob of the pendulum acts as a steady point, the highly multiplied displacements are usually too great to be recorded.
To avoid the possi bility of displacements clue to magnetic influences, the needle may be replaced by a brass or glass rod.
The chief function of the instrument is to measure slow displacements due to distant earthquakes.
If three plates (or three sets of exposures on one plate) are taken at intervals of six months, when the stars in the region have their maximum parallactic displacements, the first and third plates serve to eliminate the proper motion of the star, and the detection of a parallax is easy.
For the present we confine ourselves to the consideration of displacements in two dimensions, so that the body is adequately represented by a thin lamina or plate.
If, AB being held fixed, the 3 quadrilateral be slightly de formed, it is obvious that the instantaneous centre J will - be at the intersection of the - .~ straight lines AD, BC, since s the displacements of the points D, C are necessarily at right angles to AD, BC, respectively.
Hence these displacements are proportional to, JD, JC, and A therefore to DD, CC, where ~ CD is any line drawn FIG.
Hence the component displacements are ultimately &r=Xys,y=ti+x (I)
We proceed to sketch the theory of the finite displacements of a rigid body.
The analytical treatment of small displacements is as follows.
(5) The most general case is derived from this by adding the component displacements X, u, v (say) of the point which was at 0; thus Ix=X+siz1y, i 1yu+l~xEz, ~ (6)
The points whose displacements are in the direction of the resultant axis of rotation are determined by Ix: Iy: Iz = E: ~: ~, or (X +nz ly)IE = Cu + Ix ~z)/v = (e +:~y ~x)ft.
Balls Theory of Screws an analysis is made of the possible displacements of a b~dy which has respectively two, three, four, five degrees of freedom.
The analogy between the mathematical relations of infinitely small displacements on the one hand an-d those of force-systems on the other enables us immediately to convert any theorem in the one subject into a theorem in the other.
In the analogous theory of infinitely small displacements of a solid, a null-line is a line such that the lengthwise displacement of any point on it is zero.
It is also evident that the total work done in two or more successive infinitely small displacements is equal to the work done in the resultant displacement.
Considering a rigid body in any given position, we may eontemplate the whole group of infinitesimal displacements which might be given to it.
The word virtual is used because the displacements in question are not regarded as actually taking place, the body being in fact at rest.
Its special value lies in this, that by a suitable adjustment of the hypothetical displacements we are often enabled to eliminate unknown reactions.
Again, if two bodies are connectedbya string or rod, and if the hypothetical displacements be adjusted so that the distance between the points of attachment is unaltered, the corresponding stress may be ignored.
Again, the normal pressure between two surfaces disappears from the equation, provided the displacements be such that one of these surfaces merely slides relatively to the other.
Again, if the displacements be such that one curved surface rolls without sliding on another the reaction, whether normal or tangential, at the point of contact may be ignored.
We are thus able tc imagine a great variety of mechanical systems to which tht principle of virtual work can be applied without any regard tc the internal stresses, provided the hypothetical displacements be such that none of the connections of the system are violated.
For this purpose the infinitesimal displacements of W the various joints are replaced by finite lengths proportional to them, and there- FIG.
(13) The left-hand side is the kinetic energy of the whole mass, supposed concentrated at G and moving with this point, together with the kinetic energy of the motion relative to G (15); and the right-hand member represents the integral work done by the extraneous forces in the successive infinitesimal displacements into which the motion may be resolved.
The displacements of P due to small variations of these co-ordinates are or along OP, rOO perpendicular to OP in the plane ZOP, and r sinO O~ perpendicular to this plane.
This solution, taken by itself, represents a motion in which each particle of the system (since its displacements parallel to Cartesian co-ordinate axes are linear functions of the qs) executes a simple vibration of period 21r/u.
Removing the summation signs in equation (52) in order to restrict its application to two points and dividing by the common time interval during which the respective small displacements ds and ds were made, it becomes Pdsfdt = Rds/dt, that is, Pv = Rv, which shows that the force ratio is the inverse of the velocity ratio.
It is also stable as regards displacements transverse to the axis, for the film is in a state of tension, and any lateral displacement of its middle parts would produce a resultant force tending to restore the film to its original position.
Let us conceive an infinitely long circular cylinder of liquid, at rest (a motion common to every part of the fluid is necessarily without influence upon the stability, and may therefore be left out of account for convenience of conception and expression), and inquire under what circumstances it is stable or unstable, for small displacements, symmetrical about the axis of figure.
The analytical expression for the motion in the latter case involves exponential terms, one of which (except in case of a particular relation between the initial displacements and velocities) increases rapidly, being equally multiplied in equal times.
Under these conditions the equilibrium is stable for all small displacements of the surface.
This may be due to the successive divisions occurring in two or three planes instead of only across the long axis (Sarcina), or to displacements of the cells after division.
They are carried across the disk by the sun's rotation, partaking in the equatorial acceleration; they also show marked displacements of their own, whether with, or relative to, the neighbouring photosphere does not appear; at the beginning of their life they usually outrun the average daily rotation appropriate to their latitude.
The irregularities incidental to use of the spots are escaped by comparing the relative Doppler displacements of the same spectral line as given by the receding and advancing limbs of the sun.
Gabriel Cramer, in a note to his Analyse des lignes courbes algebriques (1750), gave the rule which establishes the sign of a product as plus or minus according as the number of displacements from the typical form has been even or odd.
A continuous shifting of the standpoint was in large measure substituted for the displacements of the objects viewed, which thus acquired a regularity and consistency heretofore lacking to them.
Thomas Henderson (1798-1844) had indeed measured the larger displacements of a Centauri at the Cape in 1832-1833, but delayed until 1839 to publish his result.
In mechanics, he made many researches, substituting the notion of the continuity of geometrical displacements for the principle of the continuity of matter.