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moments

moments Sentence Examples

  • It was one of those break-through moments.

  • What felt comfortable only moments ago now seemed bold and foolish.

  • After a few moments Jonathan answered.

  • In those last moments when it was too late, she had decided to go with them.

  • For a few moments the car hung there, the back half solidly on the ground, the front hanging precariously over the cliff.

  • There she sat for a few moments, gasping for breath.

  • Had her family known the same terror in their last moments of life?

  • She paused for no more than a few moments, but apparently Giddon hadn't heard.

  • Closing her eyes for a few precious moments, she breathed deeply and let her breath out in a long, relaxing sigh.

  • After a few moments, he glanced up at her.

  • He gnawed on the inside of his mouth, studying her for a few moments.

  • He was silent for a few moments and she thought he was listening to the radio.

  • Adrienne expected a sharp response from Mr. Marsh, but he only regarded his wife thoughtfully for a few moments.

  • He stared at her for a few moments.

  • The line was silent for a few moments and then Mary let loose with a heavy sigh.

  • He watched her intently for a few moments longer and then turned his attention to his food.

  • Mary contemplated her soberly for a few moments and then spoke gently.

  • He shrugged nonchalantly, but his gaze rested on her thoughtfully for a few moments before he finished his meal.

  • He glanced at her sharply and regarded her thoughtfully for a few moments.

  • Mary was silent again for a few moments and when she spoke it was in a controlled voice.

  • They were both silent for a few moments, remembering that emotion packed morning - and another one.

  • I took it all in for a few moments and it petrified me!

  • He tried to speak but it took a few moments before the words were understandable.

  • I then left the building and waited outside for the others who arrived moments later.

  • It was a hang-up but when she returned moments later, she heard a car leaving the alley behind her house.

  • The street he was on looked as it had on the television he said, but the houses were absent numbers so it took him a few moments to locate the correct place.

  • A few moments passed before he told me to go ahead.

  • In a few moments I heard a gruff male voice ask, "Is this the swami?"

  • While Martha was on the phone trying to get our air conditioning system checked, I had a few moments alone with my wife.

  • Brennan took a few moments to reply.

  • Whenever we needed a few moments alone, a dog walk was suggested and eagerly accepted.

  • He grunted as he shoved it in front of the door moments before the door bucked under the force of some otherworldly being.

  • The two spoke for a few moments before Xander bowed his head and walked away.

  • She cried until she was too tired to cry more and drifted into a vision, reliving the few moments she spent with Jilian.

  • Alarms sounded a few moments before the doors exploded off their hinges.

  • "There are so many moments where I don't think I can do this," she said.

  • But the brief exercise failed to dispel the glumness for more than a few moments.

  • She was silent for a few moments and he thought the conversation was over.

  • A few moments later, Dickinson Faust stepped into sight.

  • It was only moments later when his fears were realized by the gnashing, booming, ripping sound of metal on rock, echoing across the valley like a clap of thunder, repeating and repeating, as if car after car had met a similar fate, further and further away.

  • She hugged him again but after a few moments he pulled back.

  • It took the group only moments to confirm the year—1961.

  • By the time Dean strolled by a few moments later, red-faced Fitzgerald was getting an ear full from a half dozen tourists and an elderly local, known for his unwavering opinions and surly disposition.

  • They were quiet for a few moments.

  • There were moments when he didn't know what the human side of her was thinking.

  • It was one of those uncomfortable moments that left her wondering if the odometer was what he was interested in.

  • For a few moments they stood that way.

  • They stood that way, inches apart, for a few moments.

  • It was moments like these.

  • All the way to the hospital she kept rehashing the attack and their last moments together.

  • The words echoed in her head, and she walked blindly for several moments, until the cold burning her lungs made her stop.

  • Even as he thought of his last moments with Kris, he felt his anger turn to resolve.

  • They sat dutifully, sharing the pillow, and were calm for several moments before a discreet elbow match broke out between them.

  • She hesitated, but Talal entered and reappeared several moments later with a small bowl of water.

  • She spoke to him for several moments before his eyes rose and lingered on Kiera.

  • Within moments, they burst into the chilly desert night.

  • He could picture her, holding the hem of her gown, climbing onto the chair, perhaps even smiling, before kicking it away, and waiting the few agonizing moments until death took her hand and led her to its darkness.

  • Edith looked up at the ceiling once more but in a few moments there was only silence.

  • After a few moments of mutual silence, both men rose and returned to their rooms.

  • At other moments, I am as distressed as the sinner that I am, being dragged to hell's fire beyond.

  • They spoke for a few moments and she promised to call again in the evening when, perhaps, she'd have more information.

  • He donned biking clothes, packed a jacket and sweater in his pannier and set off in pursuit of a few peaceful moments in one of his favorite worlds.

  • And yet, Cynthia was disturbed about something moments after the incident.

  • Fred O'Connor strolled into the kitchen a few moments later, all smiles, carrying his ever-present notebook.

  • The Quincy sisters strolled by moments later, arm in arm.

  • He could picture her climbing onto the velvet chair, perhaps even smiling, before kicking it away, and waiting the few agonizing moments until death set her free.

  • Tell them that naked Edith Shipton came out of his room and moments later hanged herself.

  • They were silent for a few moments before he asked, "If you were that close to where Shipton fell, you must have seen Edith."

  • During the hurried explanations of her mother's condition, her trip home and hushed comments on the recent happenings, it was moments before Dean noticed Janet O'Brien sitting in the far corner.

  • He conferred with the other five, then left the chamber for a few moments, returning with a goblet and a knife.

  • The dark eyes searched her face suspiciously for a few more moments and then he shrugged.

  • They stood that way for a few moments and finally he gently pulled her toward him.

  • Just because you sacrificed your virtue for a few moments of pleasure doesn't mean I have tossed aside my innocence.

  • Lana lay gasping, unable to catch her breath for several moments.

  • After all, it was where he and Katie spent the few good moments they'd had, where he'd found something worth living for.

  • He hesitated and then held out a hand.  She took it.  His warm hands were rough and large.  He squeezed hers.  He led her away from the courtyard and lights into the dark night.  They walked hand in hand for a few moments, alone under the full moon.  She'd walked with him before, but this night, it was different.  She felt the shift between them.

  • He had awakened a few moments before the usual time, ordinarily a good sign, but after rubbing open his eyes, he discovered it was a white day, hazy and sultry, without a speck of blue in the sky.

  • He held the phone for a few moments while Mrs. Riley checked.

  • Dean dutifully paused a few moments until Fred took off his glasses, slammed down the book on a table and said with a broad smile, I knew it!

  • Neither man said anything for a few moments.

  • It took a few moments before the man focused, looked down at it and sighed.

  • He was sitting there a few moments later with Cynthia Byrne still unconscious when the attendant reappeared with Mr. Cole, a young intern, in tow.

  • They drove along in silence for a few moments.

  • He took a few moments to sign the guest book but didn't bother looking at the names before his.

  • He listened for a few moments with Dean standing by the doorway as the boy's mother talked.

  • They were both quiet for a few moments, breathing in the cool night air before Dean spoke.

  • Jonathan looked at him for several moments, recognizing Dean was serious.

  • Such wonderful moments they had when he was leaving and no harm could be done.

  • He pulled her close again and for a few moments they stood in silence again.

  • For a few moments he teased her with light quick kisses, until she captured his neck with her arms and forced him to hold still.

  • It was one of those moments when the world ceased to exist around them.

  • Alex studied her for a few moments.

  • He watched her silently for a few moments.

  • For a few electrified moments they exchanged hungry looks.

  • July passed into August without any defining moments.

  • They gazed at each other for a few moments in silence.

  • For a few moments she resisted the temptation to surrender, but his embrace was electrifying and she found herself passionately returning his affection.

  • Alex watched her reflectively for a few moments and then his eyes registered comprehension.

  • They walked along in silence for a few moments and then Carmen gazed up at Alex.

  • Sitting there alone in Josh's living room while Lori slept, all Carmen could think about was the last moments for Josh.

  • They sparred hard for several moments, until Darian's upper body carried a sheen of sweat.

  • The black memories that made him wake up screaming at night, the fear he could still taste in moments of despair, were softened by the sense of stillness that settled into him.

  • Her hand went to her chest, where the necklace had been just moments before.

  • This room was smaller than her quarters within the city's walls, but she was glad to have a private place to be alone for a few moments.

  • He had never felt such a blinding combination of fury and terror as he did in those few moments before she ran.

  • Within moments after finishing the drink, he grew too drowsy to stay awake and retreated to the bed.

  • His gaze became distant in thought for a few moments.

  • For a few moments they sipped in silent reflection.

  • Gerald was silent for a few moments.

  • Moments later it clattered on the rocks below.

  • For a few moments he stood looking over the edge at the rocks below.

  • Probably the same thing you wanted from Gerald – a few moments of companionship with a friend.

  • He rubbed his jaw and eyed her reflectively for a few moments.

  • For a few moments he simply held her.

  • He was quiet for a few moments and then he stroked her arms as he spoke.

  • After a few moments of silence, he stopped and stared at her again.

  • He reappeared a few moments later in a t-shirt.

  • A few moments later, he heard the sounds of groggy protest as Jessi roused his dinner and led her out of the apartment.

  • She let the cousins talk at him for a few moments before clapping her hands.

  • Before applying the solution to a mathematical investigation of the present question, it may be well to consider the matter for a few moments from a more general point of view.

  • He had fits of superstition which in healthier moments he despised.

  • In such moments of baffled inquiry he would leave his books, perform the requisite ablutions, then hie to the mosque, and continue in prayer till light broke on his difficulties.

  • Of this our true individual life, our present life is a glimpse, a fragment, a hint, and in its best moments a visible beginning.

  • The dissensions which broke out among them within a few months of the accession of their party to power never afterwards disappeared, except at rare moments when it became necessary to unite in preventing the return of the Conservatives.

  • There are three stadia, or moments, in this process of nature - (i) the mechanical moment, or matter devoid of individuality; (2) the physical moment, or matter which has particularized itself in bodies - the solar system; and (3) the organic moment, or organic beings, beginning with the geological organism - or the mineral kingdom, plants and animals.

  • In the districts bordering on the coast the thermometer seldom falls below 37°; and only for a few moments and at long intervals has it been known to rise as high as 105°.

  • The necessary elements of a Hindu sacrifice are: (I) the sacrificer, who provides the victim, and is affected, directly or indirectly, by the sacrifice; he may or may not be identical with (2) the officiant, who performs the rite; we have further (3) the place, (4) the instruments of sacrifice and (5) the victim; where the sacrificer enjoys only the secondary results, the direct influence of the sacrifice is directed towards (6) the object; finally, we may distinguish (7) three moments of the rite - (a) the entry, (b) the slaughter, (c) the exit.

  • But in our present state of existence the moments of this ecstatic union must be few and short; " I myself," says Plotinus simply,.

  • Further, two ducal councillors were appointed to assist the doge, and he was compelled, not merely permitted, to seek the advice of the more prominent citizens at moments of crisis.

  • To the Romanist " Catholic " means " Roman Catholic "; to the high Anglican it means whatever is common to the three " historic " branches into which he conceives the church to be divided - Roman, Anglican and Orthodox; to the Protestant pure and simple it means either what it does to the Romanist, or, in expansive moments, simply what is " universal " to all Christians.

  • But that view which admits a life of God that is not benumbed in an unchangeable sameness will be able to understand his eternal co-working as a variable quantity, the transforming influence of which comes forth at particular moments and attests that the course of nature is not shut up within itself.

  • The period as a whole had some anxious moments; emigration to the gold-fields and the strife which afflicted Wesleyan Methodism brought loss and confusion between 1853 and 1860.

  • He could jest, it was said, even in his last moments.

  • At such moments of crisis it almost excelled human comprehension; the mind seems to have gathered to itself and summed up the balance of all human passions arranged for and against him, and to have calculated with unerring exactitude the consequences of each decision.

  • This final moult is effected soon after the insect's appearance in the winged form; the creature seeks a temporary resting-place, the pellicle splits down the back, and the now perfect insect comes forth, often differing very greatly in colours and markings from the condition in which it was only a few moments before.

  • In the preceding century reliance was placed entirely on the observed moments at which Venus entered upon or left the limb of the sun, but in 1874 it was possible to determine the relative positions of Venus and the sun during the whole course of the transit.

  • If two magnets having moments M, M' are arranged at right angles to each other upon a horizontal support which is free to rotate, their resultant R will set itself in the magnetic meridian.

  • The effect of the field upon the speed of the revolving electrons, and therefore upon the moments of the equivalent magnets, is necessarily a very small one.

  • But by Green's transformation f flpdS = f f PPdxdydz, (2) thus leading to the differential relation at every point = dy dp The three equations of equilibrium obtained by taking moments round the axes are then found to be satisfied identically.

  • for a few moments, on its way to a steam and juice separator, where the steam due to the superheated juice flashes off, and is either utilized for aiding the steam supplied to the multiple effect evaporators, or for heating cold juice on its way to the main heater, or it is allowed to escape into the atmosphere.

  • Few particulars are extant concerning the real condition of the town; but we occasionally find Pisa mentioned, almost as though it were an independent city, at moments when Italy was overwhelmed by the greatest calamities.

  • Savonarola waited a few moments and then went away.

  • In a few moments his decision was taken.

  • It was almost dark when the Prussians approached the French position between Rezonville and the woods to the northward, and the troops soon lost direction in the smoke and became involved in the direst confusion; the firing again blazed out for a few moments, only to die away as utter exhaustion at length put an end to the Prussian advance.

  • He continued by every means in his power to work for the independence of Hungary, especially at moments of European war, such as 1854, 1859 and 1866, at which an appeal to arms seemed to him to promise success.

  • If now, after a few moments' interval to allow some air to diffuse into the cylinder, a taper again be applied, an explosion takes place, due to a mixture of carbon monoxide and air.

  • and therefore T, are measured by the algebraical sum of their individual moments with respect to the axis.

  • The moments of the components of these actions and reactions in a plane to which the axis of rotation is at right angles are the two aspects of the torque acting, and therefore the torque acting on B through the shaft is measured by the torque required to hold A still.

  • Meanwhile, the same considerations had not been applied to time, so that in the days of Zeno of Elea time was still regarded as made up of a finite number of ` moments,' while space was confessed to be divisible without limit.

  • But Jansen, as he said, did not mean to be a school-pedant all his life; and there were moments when he dreamed political dreams. He looked forward to a time when Belgium should throw off the Spanish yoke and become an independent Catholic republic on the model of Protestant Holland.

  • David's revolutionary ideas, which led to his election to the presidency of the Convention and to the committee of general security, inspired his pictures "Last Moments of Lepelletier de Saint-Fargeau" and "Marat Assassinated."

  • Moments and Centroids.

  • of its distance from the line or plane with regard to which the moments are being taken.

  • moments by the total volume, area or length of the figure, we get the mean distance, mean square of distance, mean cube of distance, ...

  • We sometimes require the moments with regard to a line or plane through the centroid.

  • are its moments with regard to a line in its plane, the moments M 1, M2, ...

  • These formulae also hold for converting moments of a solid figure with regard to a plane into moments with regard to a parallel plane through the centroid; x being the distance between the two planes.

  • This also holds for higher moments, provided that the edges of the elementary strips or prisms are parallel to the line or plane with regard to which the moments are taken.

  • We have now to consider the extension of formulae of this kind to other figures, and their application to the calculation of moments and volumes.

  • from a line drawn through 0 parallel to the ordinates) is equal to the mean distance (§ 32) of the trapezette from this axis; moments with regard to the central ordinate are therefore sometimes described in statistics as " moments about the mean."

  • Moments.

  • The formulae can then be applied to finding the moments of certain volumes.

  • moments of a trapezette with regard to the axis of u.

  • Moments of a Trapezette.

  • - The above methods can be applied, as in §§ 59 and 60, to finding the moments of a trapezette, when the data are a series of ordinates.

  • The determination of the moments in these cases involves special methods, which are considered in the next two sections.

  • , to calculate the moments on this assumption, and then to apply certain corrections.

  • moments, so calculated, before correction are denoted by p 1, P21

  • -[- x P „,IA r _ These are called the raw moments.

  • Then, if the true moments are denoted by Pi, v 2, ..., their values are given by vi ?Pl P2 - 11.

  • To find the moments with regard to the central ordinate, we must use this approximate value, and transform by means of the formulae given in § 32.

  • If the transformation is made first, and if the resulting raw moments with regard to the (approximate) central ordinate are o, 72, 71-3, ..., the true moments µ1, u2, /13, ...with regard to the central ordinate are given by Lo=o 1 i / h2,3 83.

  • On calculation of moments of graphs, see W.

  • For mechanical methods of calculating areas and moments see CALCULATING MACHINES.

  • A few moments afterwards Zasulich ordered the retreat.

  • It appears not only at great moments of the history such as the Baptism (i.

  • - In the case of braced structures the following method is convenient: When a section of a girder can be taken cutting only three bars, the stresses in the bars can be found by taking moments.

  • Let s be the perpendicular from 0, the join of C and T on the direction of S; t the perpendicular from A, the join of C and S on the direction of T; and c the perpendicular from B, the join of S and T on the direction of C. Taking moments about 0, Rx - W 1 (x+a) - W 2 (x+2a) =Ss; taking moments about A, R3a-W 1 2a-W 2 a =Tt; and taking moments about B, Rea-W I a = Cc.

  • Or generally, if M 1 M2 M3 are the moments of the external forces to the left of 0, A, and B respectively, and s, t and c the perpendiculars from 0, A and B on the directions of the forces cut by the section, then Ss=M11 Tt=M2andCc=M3.

  • The lower portion of the figure is the curve of bending moments under the leading load.

  • Till W 1 has advanced a distance a only one load is on the girder, and the curve A"F gives bending moments due to W 1 only; as W1 advances to a distance a+b, two loads are on the girder, and the curve FG gives moments due to W 1 and W2.

  • GB" is the curve of moments for all three loads WI+W2+W3.

  • As might be expected, the moments are greatest in this case at the sections under the 15-ton load.

  • With short bridges it is best to draw the curve of maximum bending moments for some assumed typical set of loads in the way just described, and to design the girder accordingly.

  • For longer bridges the funicular polygon affords a method of determining maximum bending moments which is perhaps more convenient.

  • Now, except for very short bridges and very unequal loads, a parabola can be found which includes the curve of maximum moments.

  • This parabola is the curve of maximum moments for a travelling load uniform per ft.

  • run which would produce the maximum moments represented by this parabola.

  • the same ordinate at one-quarter span as the curve of maximum moments, agrees with it closely enough for practical designing.

  • Let M, and M a be those moments.

  • Then the bridge is designed, so far as the direct stresses are concerned, for bending moments due to a uniform dead load and the uniform equivalent load we.

  • Miaoulis, for all his high character and courage, was often unable to prevent his captains from sailing home at critical moments, when pay or booty failed.

  • He left none of the usual legacies for masses or other clerical purposes, and was not attended by any priest or confessor in his last moments.

  • Jeffrey's own contributions, according to a list which has the sanction of his authority, numbered two hundred, all except six being written before his resignation of the editorship. Jeffrey wrote with great rapidity, at odd moments of leisure and with little special preparation.

  • In his last moments he cursed both the Bourbons and Admiral Sir Sidney Smith for having induced him to engage in this reckless adventure (1806).

  • By them he was escorted to the praetorian camp, where, after a few moments of surprise and indecision, he was saluted imperator.

  • The part played by the judiciary has at some moments been of special importance, while at others it has been little noticed.

  • (21st of September) and gave him extreme unction; then raised a curious controversy as to whether Charles, in his last moments, had been infected with Lutheranism.

  • Gregory himself, in his last moments, seems to have felt that it was impossible to maintain them, for Didier, abbot of Monteregularly enthroned on the 24th of the same month.

  • It is from this time that we find the popes in moments of crisis transporting themselves to Capetian territory, installing their governments and convening their councils there, and from that place of refuge fulminating with impunity against the internal and external foe.

  • Bishop Alexander reluctantly assented to receive him once more into the bosom of the church, but before the act of admission was completed, Arius was suddenly taken ill while walking in the streets, and died in a few moments.

  • The baboon took it, held it in her hands for a few moments, and then let it escape uninjured without trying to taste it.

  • To the Cyrenaics pleasure was of moments; to Epicurus it extended as a habit of mind through life.

  • In 1831 Offutt made him clerk of his country store at New Salem, a small and unsuccessful settlement in Menard county; this gave him moments of leisure to devote to self-education.

  • Of the betrayed chiefs, many were laid low in a few moments; some, dismounting, and throwing off their outer robes, vainly sought, sword in hand, to return, and escape by some other gate.

  • His counsels were marred by his reluctance to appeal to arms at the critical moments of action, and by the slenderness of his own resources, but they deserve attention for their broad common sense and spirit of tolerance.

  • They, again, constantly intrigued with Spain, and there were moments when James, driven desperate by the preachers, listened to their projects.

  • Only the chief moments in the struggle between Charles I.

  • Every person, or thing, or god, is therefore a putting together, a compound; and in each individual, without any exception, the relation of its component parts is ever changing, is never the same for two consecutive moments.

  • In his last moments he refused under torture to disclose his father's hiding-place.

  • The scientific instrument of the other, with its moments of analysis and construction, metodo risolutivo and metodo compositivo, engineers the road for the advance.

  • Three moments are involved.

  • Stuart Mill's System of Logic marked a fresh stage in the history of empiricism, for the reason that it made the effort to hold an even balance between the two moments in the thought of the school.

  • Different moments in it were emphasized, with a large diversity of result.

  • They must be exhibited as distinguishable moments within a unity which can at one and the same time be seen to be the ground from which the distinction springs and the ground in virtue of which it is over-ruled.

  • clarification of concepts which are wholly abstract, so that they are not merely not ultimate realities, but also in no sense actual moments of our concrete thinking.

  • same is true of physical quantities such as potential, temperature, &c., throughout small regions in which their variations are continuous; and also, without restriction of dimensions, of moments of inertia, &c. Hence, in addition to its geometrical applications to surfaces of the second order, the theory of quadric functions of position is of fundamental importance in physics.

  • 0E6s, god, and Qoybia, wisdom), a term used to denote those forms of philosophic and religious thought which claim a special insight into the Divine nature and its constitutive moments or processes.

  • He now proceeded to distinguish three moments in God, the first of which is the pure indifference which, in a sense, precedes all existence - the primal basis or abyss, as he calls it, in agreement with Boehme.

  • - Beethoven's Sonatas; Weber's Concertstack and Sonatas; Schubert Fantasia, 4 Sonatas, Impromptus, Valses and Moments musicaux.

  • The sum of the moments of two forces about any point 0 is equal to the moment of their resultant (P. Varignon, 1687).

  • It is now evident that in the process of reduction of a coplanar system no change is made at any stage either in the sum of the projections of the forces on any line or in the sum of their moments about any point.

  • Again it is necessary and sufficient for equilibrium that the sum of the projections of the forces on each of two perpendicular directions should vanish, and (moreover) that the sum of the moments about some one point should be zero.

  • - the sum of the moments of the forces about each of the three points which are not collinear must be zero.

  • If F, Q, R, be any three forces acting along BC, CA, AB, respectively, the line of action of the resultant is determined by the consideration that the sum of the moments about any point on it must vanish.

  • The sum of the moments of the two forces of a couple is the same about any point in the plane.

  • Thus in the figure the sum of the moments about 0 is P. OAP. OB or P. AB, which is independent of the position of o 0.

  • 20 replaced by a single couple whose moment is the sum of their moments.

  • A system of forces represented completely by the sides of I plane polygon taken in order is equivalent to a couple whosc moment is represented by twice the area of the polygon; this is proved by taking moments about any point.

  • The force R is the geometric sum of the given forces, and the moment (G) of the couple is equal to the sum of the moments of the given forces about 0.

  • If the remaining forces acting on the portion of the structure on either side of P are known, then resolving vertically we find F, and taking moments about P we find M.

  • which is the moment of F about P. If the given forces are all parallel (say vertical) OM is the same for all, and the moments of the several forces about P are represented on a certain scale by the lengths intercepted by the successive pairs of sides on the vertical through P. Moreover, the moments are compounded by adding (geometrically) the corresponding lengths HK.

  • Hence if a system of vertical forces be in equilibrium, so that the funicular polygon ii closed, the length which this polygon intercepts on the vertical through any point P gives the sum of the moments about P of all the forces on one side of this vertical.

  • Since two concurrent forces and their resultant obviously project into two concurrent forces and their resultant, we see that the sum of the moments of two concurrent forces about any axis HK is equal to the moment of their resultant.

  • Hence, in whatever way one system of forces is by successive steps replaced by another, no change is made in the sum of the moments about any assigned axis.

  • The length of the vector must be proportional to the moment of the couple, and its sense must be such that the sum of the moments of the two forces of the couple about it is positive.

  • the line of intersection (B); the arms (AB,BC) will then be proportional to the respective moments.

  • Hence the force Xi can be transferred from P1 to 0, provided we introduce couples of moments z~X~ about Oy and yiX1, about 01.

  • It is seen that L~, Mi, Ni are the moments of the original force at P~ about the co-ordinate axes.

  • In words: the sum of the projections of the forces on each of the co-ordinate axes must vanish; and, the sum of the moments of the forces about each of these axes must vanish.

  • For by giving the body (in imagination) a displacement of translation we learn that the sum of the resolved parts of the forces in any assigned direction is zero, and by giving it a displacement of pure rotation we learn that the sum of the moments about any assigned axis is zero.

  • 51) be any force acting on the joint C, its virtual work will be equal to the moment of F~ about C; the equation of virtual work is thus transformed into an equation of moments.

  • The equation of virtual work is then formed by taking moments about C, D, E, F of the extraneous forces FIC 2 which act at C, D, E, F, respectively.

  • We proceed to the theory of the plane, axial and polar quadratic moments of the system.

  • Th~ axial moments have alone a dynamical significance, but the others are useful as subsidiary conceptions.

  • If we divide any of the above quadratic moments by the total mass ~(m), the result is called the mean square of the distances of the particles from the respective plane, axis or pole.

  • If we take rectangular axes through any point 0, the quadratic moments with respect to the co-ordinate planes are I,, = Z(mxi), I,,= Z(my1), I, = ~(mz2); (9) those with respect to the co-ordinate axes are Ii,, = ~lm(y~+z2)~, I,, = ~tm(z2+x2)l, I,, ~tm(x2+y1)j; (10) whilst the polar quadratic moment with respect to 0 is 10 = ~tm(x2+y2+z1)}.

  • The distance between the planes of and of will be of the second order of small quantities, and the quadratic moments with respect to of and co will therefore be equal, to the first order.

  • Since the quadratic moments with respect to w and of are equal, it follows that w is a plane 01 stationary quadratic moment at P, and therefore a principal plane of inertia at P. In other words, the principal axes 01 inertia at P are the normals to the three confocals of the systen (3,~) which pass through P. Moreover if x, y, z be the co-ordinates of P, (33) is an equation to find the corresponding values of 0; and if Of, 02, 03 be the roots we find Oi+O2+81r1a2$-7, (35)

  • To find the relations between the moments of inertia about different axes through any assigned point 0, we take 0 as origin.

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