I- 



Vis 



VISIINT. 



in the diagram. Thrre will be equilibrium, but the principle of virtua 

 relocitie* will nut bo true, evcu in the extended form which we hve 

 u**d. The moment* belonging to the two possible motion* are pod 

 tire, but they are equal. On which aide U the descent to take place f 



The mathematician bu a warning in luch caiei, which may be easily 

 and briefly expressed. The expression Sprfp, before it U uped, require* 

 that the quantities p t . p v Ac.. ahould be reduced to the smallest niirn 

 ber of independent variable*. Let 7,, o, ... be these variables, and let 

 the sum of the momenta, when reduced to terms of these variable*, be 

 Q.rfo, + At, or XQdy. The principle is then intelligible when, for al 

 the virtual motions, dp., rf<;,, Ac., hare finite ratios to one another 

 But if there be any poutiou in which for a certain virtual mo- 

 or more of the set rf/>,, dp,, Ac., become infinite with respect to those 

 of the set </o,, </(/,, *c., the equation becomes incapable of being used 

 For if we take the actual virtual rtlocitia, and attempt to reduce 



14 - ? to its equivalent Zr^, 

 at tit 



the first side, which may be made finite, is equated to an expression 

 in which infinite terms occur, which i* always a warning to expect 

 the possibility of cases of exception. Circumstances of this sort have 

 never received sufficient investigation, and in all probability there are 

 numerous varieties of the cases of equilibrium which arise out of them 

 and which cannot be treated by the ordinary principle. So much we may 

 certainly say, that if there be different virtual motions with the sums 

 of the moments positive and maxima, either there must be equilibrium, 

 or the test for determining which of the motions will ensue is wholly 

 unknown. 



On the history* of virtual velocities, there is not much to say. 

 Uuido Ubaldi saw it in some cases, Galileo in some others; Walfis 



adopted it as a principle, and after him John Bernoulli, who gave it in 

 the most general form. Lagrange made it the foundation of his 

 ' Mecanique Analytique,' and since his time it has formed part of every 

 well-constituted treatise on mechanics. It was in the ' Mecanique 

 Analytique' that the principle given by D'Alerabert was first joined to 

 that of virtual velocities in such a manner as to give the science of 

 dynamics its present uniformity of system. 



VIS INERTI.E. [INERTIA.] 



VIS VIVA, or lirimg force, a name given in mechanics to the fol- 

 lowing index of the state of a system in motion : the sum of all the 

 missfs, each multiplied by the square of its velocity. If the system 

 be considered as composed of a finite number of molecules, the vis viva 

 may be expressed by the symbol 2mr* ; but if it be a continuous mass, 

 or a collection of continuous masses, by_/"r'dnj, or 2_/"t*<im. It is 

 enough that the mass of every particle be found in the expression, 

 multiplied by the square of its velocity. 



In the article VIBIUAL VELOCITIES we see the equation 



2 ntr* = Sm f (lit + Tidy + zdz), 



the integral being taken for each molecule over the whole path which 

 it has described since the beginning of the motion. 



Presuming a knowledge of the article cited, we may describe the vis 

 viva thus : Dividing the whole motion of the system, from the 

 beginning to the time under consideration, into an infinite number of 

 infinitely small changes of place, each of those changes is one of the 

 virtual motions which come under consideration in the principle of 

 virtual velocities. And each motion has, generally speaking, its con- 

 trary ; and one of these two the system would tend to take, and to 

 refuse the other, if its motion were for an instant restricted, so that it 

 could only choose between those two. The one which it would tend 

 to take is that for which 2 m (x d x + Ac.) is positive. Now, it appears 

 in the preceding equation that whenever the infinitely small motion 

 which is taking place for the time being is that which (when restricted 

 as above) the system would take, the vis viva is receiving increase ; 

 when that which it could not take, decrease. And the vis viva is the 

 balance, so to speak, of all the sums of moments,' each with its proper 

 sign, added, also with its proper sign, to the vis viva at the beginning 

 of the motion. [PHYSICAL FORCE, COMSERVATIOM OK.] 



The preceding equation is sometimes said to express the principle of 

 the emuerralium ofru rira, which is to be understood thus: the system 

 never acquires nor loses any quantity of vis viva from the action of its 

 parts upon each other, but only from the action of external forces. If 

 after a certain time all external forces cease, from that moment 

 2m (xdx + Ac.) is = 0, or of (Star*) 0, or 2me remains constant 



Another remarkable property of the vis viva is. that in all the cases 

 which occur in nature, the amount of vis viva acquired in passing from 

 one position to another depends only on the co-ordinates which settle 

 the initial and final positions. If x, Ac., be functions of co-ordinates 

 only, it generally happens that xdx + idy + zrfz is an integrable func- 

 tion, and depends on co-ordinates only. But the force of this result is 

 not easily teen by the beginner. 



At the end of the 17th century a remarkable discussion took place 

 on the question of the mechanical interpretation of the vis viva. 



On Ihb point, and manT others connected with the hlilorjr of mrchnnici, 

 Ik* nadrr will find prcUle account* and riluabl* refcmun In Walton'* 

 CoUrtUos) of Problem* on Theoretical Mechanic*.' 



.: first gave this name : he considered force whrn it produces 

 motion as ril rira, or living force; but when it is equilibrated, he 

 called it ri. marina, or dead force; and he measured the effect of living 

 force by the mass multiplied into the square of the velocity. To take 

 the simple case which was mostly appealed to : If two equal weights 

 be thrown up in vacuo, the one with a velocity double that of the 

 other, it is well known that the one will rise, not ttciet, but four times 

 u high as the other : accordingly, Leibnite considered that the force 

 which produce* the double velocity is four times as effective as the 

 other force. Various other instances were produced in which the 

 duplication of the velocity is the quadruplication of the effect pr.. 

 duced. It was accordingly argued that, for a given mass, the square 

 of the velocity is the proper measure of the force necessary to destroy 

 or to create the velocity. But, on the other hand, it was very well 

 known that, whatever might be adopted as the measure of force, it was 

 certain that pressures were, caterit paribiu, proportional to the *im].l- 

 velocities produced by them in a given time. John Bernoulli adopted 

 the opinion of Leibnitz, which was opposed by various other contempo- 

 raries ; and the controversy (the history of which may be seen in 

 Montucla) continued until the publication of D'Alembert's work on 

 dynamics, in which the question was treated as being purely one of 

 words. 



It was objected to the opinion of Leibnitz, that though the double 

 velocity would give four times the ascent, it ought not to be forgotten 

 tliat it required twice the time : so that in a given time double the 

 \ 1- -ity would produce only, double the ascent, one part of the ascent 

 with another. Thin argument was never satisfactorily answered ; and 

 while we cannot help thinking that it ought to have been decisive of 

 the question, we draw from it a conclusion different from that of 

 D'Alembert ; we cannot think the dispute a mere question of words. 

 It must be granted that, for all purposes in which time is not an ele- 

 ment, the measure of the effect of a force may be the square of the 

 velocity, as exemplified iu the instance cited. But when is it that a 

 mechanical effect can be properly estimated without reference to the 

 time in which it is produced / The definition of the words measure 

 and effect may thus without doubt be accommodated either to the idea 

 of Leibnitz or of his opponents; and those who disputed on the ques- 

 tion without requiring exact definitions might degenerate into a mere 

 question of words. But it ought to have been a question as to what 

 was the proper meaning of the word effect, in the fundamental phrase 

 ' effect of a force," the proper explanation of which muut precede all 

 good reasoning in mechanics. If pressure be denned as that which 

 produces a certain effect [PRESSURE] on our senses, undoubtedly it i* 

 a known fact that uncounteracted pressure produces motion ; but it JH 

 only when allowed to act for a finite time : consequently, the element 

 of time is as essential to the conception of the phenomenon as that of 

 pressure or motion. Height iu a rectangle gives area; but it would 

 iiot therefore be allowable to measure that area by the height ; for 

 here must be a base, or there is no rectangle at all. But if pressure 

 merely considered as the cause of motion, and called force iu that 

 sense, it is very difficult to see why the cause, which is only known by 

 he effect, is to be measured by anything but the simple effect. Pro- 

 >ably this discussion gave rise to the chapter of the ' Mecanique 

 Celeste,' in which Laplace speculates upon what the laws of m 

 would have been if force had been as a function of the velocity, in 

 of as the simple velocity. We have never met with any one \\h<. 

 could give us an intelligible account of the meaning of this investi- 

 ;ation. 



VISCIN. [BIRDLIME.] 



VISCOUNT, the name of a dignity which ranks fourth in the 

 erage, immediately above that of baron. It is the most recent 

 inglish title, having, it is said, its origin in the time of Henry VI., 

 who, in 1440, created by letters patent John, Lord Beaumont, Viscount 

 feaumont In Scotland the title of Viscount was first granted by 

 Tames VI. 



Camden observes that, although this is a new title of dignity, yet it 

 * an ancient one of office : viscount, victromet, tho deputy of the 

 count or earl, is the Latin name for the sheriff of a county [SHERIFF], 

 an office in ancient times held by persons of the highest rank. Whether 

 he title of viscount was suggested by that office it is difficult to say ; 

 >ut Spelman mentions that William the Conqueror made Baldwin 

 lereditary Viscount (rirr-cvmltcm) of Devon and Baron of Okehampton; 

 nd " he made Ursus or Urso Abtot viscount of Worcester, but Roger 

 lis son was deprived of the title by Hmry I., because he had killed a 

 ertain servant of the king ; the office, however, was transferred 

 hrough bis sister to the Beaumonts." Spelman seems in these passages 

 o consider this title as one of dignity before Henry VI.'s time, and as 

 Living been distinct from that of sheriff: in the first instance he joins 

 t to the title of baron and gives it precedence ; in the second, he treats 

 he Beaumonts, who are usually deemed the first viscounU, as only 

 restored to a title which had been in abeyance or forfeited for three 

 enturies. In the British peerage, in 1861. there were 22 viscounts; 

 nd there were 41 Irish viscount*, of whom 10 hold British peerages 

 also, with 5 Scotch viscounts, of whom 3 held British peerages. 



(Spelman, title fiet-comet, nomrn diynilalit; Camden ' JIHtannia 

 dough's), i., cxciv. ; 2,299; 4,24.) 



VISHNU (from CM', "to enter," or "to pervade") occupies the 

 econd place in the Triuiurtti, or Triad of the Hindus, and is the 



