August 5, 1S92.J 



SCIENCE. 



11, 



would cause it to move off with an ever-increasing velocity 

 through infinite space. This is contrary to the first law of 

 motion, which asserts that a body does not change its state 

 of motion unless acted upon by external force." That this 

 argument is based upon the assumption of the equality of 

 the action and reaction between bodies pressing against one 

 another, seems to follow from the consideration that other- 

 wise the "residual force," due to the possible inequality of 

 the action and reaction of the gravitational stress between 

 the mountain and the remainder of the earth, might be re- 

 garded as neutralized by au opposite inequality in the action 

 and reaction of the stress at their surface of contact. Even, 

 therefore, if Newton's extension of his experimental result 

 to forces acting at a distance were regarded as valid, the 

 ihird law could not be regarded as deduced from the first. 

 It would only be shown to be but partially hypothetical. 

 But since, in the present state of dynamics, the laws of mo- 

 tion must be regarded as applicable to particles, Newton's 

 argument, though valid when they were considered applica- 

 ble 10 extended bodies, can no longer be admitted; for the 

 uniformity of the motion of a body free from the action of 

 external force is itself a deduction, which can be made only 

 by assuming the third law in its most general form. 



3. Sufficiency of the Laws of Motion. 



The best test of the sufficiency of the laws of motion is the 

 question, Can they give by deduction the greatest of all 

 physical laws, the conservation of energy ? This law may 

 be proved, by the aid of the second and third laws of motion, 

 to hold in the case of any system of particles which is neither 

 giving energy to, nor receiving energy from, external bodies, 

 provided the stresses between the particles act in the lines 

 joining them and are functions of their distances. It has 

 a'so been proved by experiment to hold in a very large num- 

 ber of cases in which the laws of the forces acting are un- 

 tnown, the energy disappearing in one form and the energy 

 appearing simultaneously in another form being measured. 

 The amount of such experimental evidence is so large that 

 no doubt is now entertained that the law of the conservation 

 of energy is applicable to all natural forces. Hence the 

 fundamental hypotheses of dynamics should either include 

 this law or give it by deduction. 



Although many writers state that this law may be deduced 

 •from the laws of motion, Lodge' is the only one, so far as 

 I am aware, who claims to make the deduction. This he 

 does in a passage beginning as follows: ''All this, indeed, 

 in a much more complete and accurate form — more com- 

 plete because it involves the non destruction of energy, as 

 well as its non-creation — follows from Newton's third law 

 ■of motion, provided we make the assumptions (justified by 

 experiment)," etc. It is unnecessary to quote farther; for 

 when assumptions justiBed by experiment are called in to 

 the aid of the third law, additional fundamental hypotheses 

 are thereby selected. 



The second law of motion enables us to take the first step 

 in the deduction of the conservation of energy. The proof 

 is so well known that I may simply cite that given by Thom- 

 son and Tait,^ resulting in the familiar equation: — 



2{Xx-\-Yy-\-Zz) = 2m{xx-\-yy-\-zz), 



in which the first member represents the rate at which work 

 is being done by the forces acting on the particles of a sys- 



' Elementary Mechanics (1885), p. 82. 



2 Treatise ou Nat. Piiil. (18T9), Vol. I., Part 1, p. 269. 



tem, and the second is equal to the rate at which the kinetic 

 energy of the system is being increased. It is usually called 

 the equation of vis viva, and, having been deduced from the 

 second law of motion alone, is applicable to all forces, 

 whether conservative or not. 



Newton gave this result in the Scholium to the Laws of 

 Motion in a statement which may be paraphrased thus: 

 Work done on any system of bodies has its equivalent in 

 work done against friction, molecular forces, or gravity, to- 

 gether with that done in overcoming the resistance to accel- 

 eration. Thompson and Tait point out expressly ' that this 

 statement of Newton's, which, owing to the form he gave it, 

 is often referred to as his second interpretation of the third 

 law of motion, is equivalent to the equation given above. 

 Nevertheless, it has been interpreted as being little less than 

 an enunciation of the law of the conservation of energy it- 

 self.^ Thus Tait* says it "has been shown to require com- 

 paratively little addition to make it a complete enunciation 

 of the conservation of energy;" and "What Newton really 

 vicnted was to know what becomes of work which is spent 

 in friction." Besaut* takes the same view.' These writers 

 seem to claim that Newton's statement is equivalent to what 

 Thomson and Tait call "the law of energy in abstract dy- 

 namics," viz., "The whole work done in any time on any 

 limited material system by applied forces is equal to the 

 whole efiFect in the forms of potential and kinetic energy pro- 

 duced in the system, together with the work lost in friction." 

 Of this it may certainly be said that what it wants to make 

 it a complete enunciation of the conservation of energy 

 is a statement as to what becomes of the work spent in 

 friction. 



Compare this, however, with Newton's statement, as para- 

 phrased above, and it is at once obvious that what the latter 

 wants to make it a complete enunciation of the conservation 

 of energy, is a statement as to what becomes not only of 

 work spent in friction, but also of work done against molec- 

 ular forces and gravity, and of work done in overcoming 

 the resistance to acceleration. Newton may possibly have 

 known all this, but he does not say so; and we must there- 

 fore hold his statement to be, as Thomson and Tait point out, 

 merely a verbal expression of the equation given above. The 

 question of the interpretation of Newton's statement is of 

 more than mere historical interest; for if it would bear the 

 interpretations which have been put upon it, the law of the 

 conservation of energy would be capable of being deduced 

 from the second law of motion alone. 



To pass from the equation of vis viva to the law of the 

 conservation of energy, we require to know that the work 

 done during any change of configuration of a system of 

 particles acted upon by natural forces depends only upon 

 the changes in the positions of the particles, and not upon 

 the paths by which or the velocities with which they have 

 moved from the old positions to the new. Helmholtz' 

 showed that this deduction may be " based on either of two 

 maxims, either on the maxim that it is not possible by any 



3 Treatise on Nat. Phil. (1879), Vol. I., Part 1, p. 2T0. 



« This address was written before I had seen Professor W. W. Johnson's 

 paper on "The Mechanical Axioms, or Laws of Motion" (Bull. N. Y. Math. 

 Soc, Vol. I., No. 6, March, 1892). 



5 Properties of Matter (1885), p. 101, and Recent Advances in Physical Sci- 

 ence (1876), p. 38. 



« Dynamics (1835), p. 49. 



' Garrett (Klamentary Dynamics, 1886, p. 47) goes so tar as to say that New- 

 ton's statement " Is nothing more nor less than the enunciation of the great 

 principle of the conservation of energy.'' 



8 On the Conservaiioi of Force (1847): Taylor's Scientlfl: Memoirs. Nat. 

 Phil. (1858), p 114. 



