102 FUNDAMENTAL PRINCIPLES OP MATHEMATICS. 



intensity of the force divided by the mass of the material point; and 

 (c) the actions of two bodies upon each other are always equal and 

 take place in opposite directions. From these laws there follows, for 

 Newtonian forces at least, and with the assumption of a rigid connec- 

 tion between the points of the system, the principle of d'Alembert 

 which holds sway in the whole province of dynamics. If we designate 

 as supplied forces those which must be made to act at each point in 

 order that it should move if separated from the system as it actually 

 does move, then the principle of d'Alembert asserts that all the sup- 

 plied forces suffice to maintain equilibrium, and thus furnishes the 

 mathematician a method of determining for any moment the situation 

 of all points of the system, when the constraints of motion of the 

 points, the forces which act upon them, and the place and velocity of 

 one of the points are known for the moment under consideration. 



The advance of mechanics in this line was accompanied by the inves- 

 tigation of all the forces of nature — that is to say, the investigation of 

 all the properties of matter — for we can know nothing of these except 

 to recognize the forces which are there in play. After the discovery of 

 these principles of equilibrium and of motion, it was the endeavor of 

 scientists to obtain general laws and relations of motion. One of the 

 most important and far-reaching of these in its consequences was the 

 principle of the conservation of the so-called vis viva. If we define as 

 the vis viva of a material point one-half the product of its mass into the 

 square of its velocity, and the sum total of the vis viva for all points of 

 a system in which the separate particles are connected by ties restraining 

 their free motion as the kinetic energy of the system, then, for any system 

 subject to the conditions of the d'Alembert principle, the increase of 

 the kinetic energy attending the motion of the system from one situa- 

 tion to another is exactly equal to the work done by the various forces 

 during the time interval in which the displacement occurs. If now 

 the work done by the forces of the system during the displacement is 

 dependent only on the initial and final situations, it follows that if the 

 system returns again from the* final to the initial condition the kinetic 

 energy returns to its original value. This law is called the law of the 

 conservation of the kinetic energy, and systems to which it applies are 

 called conservative systems. A simple transformation of this law leads 

 to the most far-reaching consequences. The fact that a body by its 

 motion from one place to another does a certain amount of work neces- 

 sitates that its capacity for performing work, or, in other words, its 

 potential energy, was in its initial situation greater than in its final sit- 

 uation ; so that for conservative systems the law of the conservation of 

 kinetic energy goes over into the law of the constancy of energy. This 

 may be expressed as follows: For any conservative system the sum of 

 the potential and kinetic energies is unchangeable. It is important to 

 remember the supposition upon which this is based, namely, that the 

 work done by the motion of the system is dependent only on the initial 



