VIS VIVA, or tiring force, a name given in mechanics to the fol lowing index of the state of a system in motion :—the eum of all the masses, each multiplied by the square of its velocity. If the system be considered as composed of a finite number of molecules, the via viva may be expressed by the symbol mrs ; but if it be a continuous mass, or a collection of continuous masses, byfrIdni, or It is enough that the mass of every particle be found in the expression, multiplied by the square of its velocity.
In the article Vravu.a. VELOCITIES we see the equation :. tars = Rif (xd.r + Ydy + 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 via viva thus :—Dividing the whole motion of the eystem, 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+ &c.) 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 via rive is receiving increase; when that which it could not take, decrease. And the via 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 via viva at the beginning of the motion. [PLITSICAL }ones, CONSERVATION OF.] The preceding equation is sometimes said to express the principle of the conservation of cis rime, which is to be understood thus: the system never acquires nor loses any quantity of via viva from the action of its parte upon each other, but only from the action of external forces. if after a certain time all external forces cease, from that moment (xdx+ &c.) is = 0, or d (Int 0, or :see remains constant.
Another remarkable property of the via viva is, that in all the cases which occur in nature, the amount of vie viva acquired in passing from one position to another depends only on the co-ordinates which settle the initial and final positions. If x, &c., be functions of co-ordinates only, it generally happens that xdx + v dy + zdt is an integrable func tion, and depends on co-ordinates only. But the force of this result is not easily seen by the beginner.
At the end of the 17th century a remarkable discussion took place on the question of the mechanical Interpretation of the via viva.
Leibnitz first gave this name : he considered force when it product* motion as via tiro, or living force; but when it is equilibrated, he called it ris inertaa, 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 twice, but four times as high as the other: accordingly, Leibnitz considered that the force which produces 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 pro 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 wrul certain that pressures were, eater's paribus, proportional to the simple velocities produced by them in a given time. John Bernoulli adopted the opinion of Leibnitz, which was opposed by various other contempod series ; and the controversy (the history of which may be seen in Montuela) continued until the publication of D'Alembert's work on dynamics, in which the question was treated as being purely one of words.