MECHANICS 715 mpul- These give the values of -^ and -J- , and so completely solve the problem. 163. In a somewhat similar way we may treat the effects of a slight disturbance, made once for all, in the hangeof nio tion of a particle describing a definite path under given >city> forces. A single example must suffice. Thus, we have in an elliptic orbit about the focus, 144 (9), 1 2 = JL -. T y r 2a At the end of the major axis farthest from the focus this becomes y 2 = _M_ 1-g a li-e Now if at this point V be made V + 8V, without change of direc tion, we have the condition that in the new orbit a(l+c) shall have the same value as in the old, since this will still be the apsidal distance. Hence S(V 2 ) = 8( = ) ,
a l+ej
and 8J.(l+e) [=0 ; i ) 2V5V=--- - - , a l + e and
Sa = - a /! o -(l-e*) 8V, Stress which determine the increase of the major axis and the diminution of the excentricity ; and the same method is applicable to more complicated cases. A very excellent series of examples of the elementary geometrical treatment of disturbed orbits is to be found in Airy s Gravitation. Third Law Kinetics of Tivo or More Particles, 164. We have, by means of the first two laws, arrived )etween at a definition and a measure of force, and have found how articles. ^ compound, and therefore how to resolve, forces, and also how to investigate the conditions of equilibrium or motion of a single particle subjected to given forces. But more is required before we can completely understand the more complex cases of motion, especially those in which we have mutual actions between or amongst two or more bodies, such as, for instance, tensions or pressures or transference of energy in any form. This is perfectly supplied by the third law, on which Newton comments nearly as follows. Newton s 165. If one body presses or draws another, it is
- om " pressed or drawn by this other with an equal force in the
Mrd" U oppo s i te direction. If any one presses a stone with his aw> finger, his finger is pressed with an equal force in the opposite direction by the stone. A horse, towing a boat on a canal, is dragged backwards by a force equal to that which he impresses on the towing-rope forwards. By whatever amount, and in whatever direction, one body has its "motion" changed by impact upon another, this other body has its " motion " changed by the same amount in the opposite direction ; for at each instant during the impact they exerted on each other equal and opposite pressures. When neither of the two bodies has any rotation, whether before or after impact, the changes of velocity which they experience are inversely as their masses. When one body attracts another from a distance, this other attracts it with an equal and opposite force. Stress, 166. We shall for the present take for granted that the mutual action between two particles may in every case be imagined as composed of equal and opposite forces in the straight line joining them, two such equal and opposite forces constituting a " stress " between the particles. From this it follows that the sum of the quantities of motion, G onser- parallel to any fixed direction, of the particles of any vation of system influencing one another in any possible way, ^ remains unchanged by their mutual action; also that the O f moment sum of the moments of momentum of all the particles of momen- round any line in a fixed direction in space, and passing turn. through any point moving uniformly in a straight line in any direction, remains constant. From the first of these propositions we infer that the centre of mass of any system of mutually influencing particles, if in motion, continues moving uniformly in a straight line, except in so far as the direction or speed of its motion is changed by stresses between the particles and some other matter not belonging to the system ; also that the centre of mass of any system of particles moves just as all their matter, if concentrated in a point, would move under the influence of forces equal and parallel to the forces really acting on its different parts. From the second we infer that the axis of resultant rotation through the centre of mass of any system of particles, or through any point either at rest or moving uniformly in a straight line, remains unchanged in direc tion, and the sum of moments of momentum round it remains constant, if the system experiences no force from without, or onlyforces whose resultant passes through the centre of inertia of the system. This principle is some times called "conservation of areas," a very misleading designation. 167. Newton s scholium, which we treat as a fourth Conse- law, points out that resistances against acceleration are to queucesof be reckoned as reactions equal and opposite to the actions ^wto . . , , , . . u , , ri ., . , scholium, by which the acceleration is produced, llms, it we consider any one material point of a system, its reaction against acceleration must be equal and opposite to the resultant of the forces which that point experiences, whether by the actions of other parts of the system upon it, or by the influence of matter not belonging to the system. In other words, it must be in equilibrium with these forces. Hence Newton s view amounts to this, that all the forces of the system, with the reactions against acceleration of the material points composing it, form groups of equilibrating systems for these points considered individually. Hence, by the principle of superposition of forces in equilibrium, all the forces acting on points of the system form, with the reactions against acceleration, an equilibrating set of forces on the whole system. This is the celebrated D Alem- principle first explicitly stated and very usefully applied bevt s by D Alembert in 1742, and still known by his name. principle. 168. Thus Newton lays, in an admirably distinct and Abstract compact manner, the foundations of the abstract theory of theory of " energy," which recent experimental discovery has raised enei SJ - to the position of the grandest of known physical laws. He points out, however, only its application to mechanics. The actio agentis, as he defines it, which is evidently equivalent to the product of the effective component of the force into the velocity of the point at which it acts, is simply, in modern English phraseology, the rate at which the agent works, called the " power " of the agent. The subject for measurement here is precisely the same as that for which Watt, a hundred years later, introduced the practical unit of a "horse-power," or the rate at which an Horse- agent works when overcoming 33,000 times the weight of P ower - a pound through the distance of a foot in a minute, that is, producing 550 foot-pounds of work per second. The unit, however, which is most generally convenient is that which Newton s definition implies, namely, the rate of doing work in which the unit of work or energy is produced in the unit of time. 169 Looking at Newton s words in this light, we see that they may be converted into the following :
"Work done on any system of bodies (in Newton s