!4S 



NA TURE 



July 14, 1892 



this detail that the words in vacuo have been added in 

 the most recent Acts of Parliament on "Weights and 

 Measures" (41 and 42 Victoria, 1878). 



2. We now pass on to the investigation of the motion 

 set up in a body of given weight due to the action of 

 specified forces ; we use the word weight advisedly, so as 

 to agree with the terminology of the Acts of Parliament. 



As the field of force in which we live is that due to the 

 attraction of the Earth, it was natural to begin by taking 

 the attraction of the Earth on our standard weight as 

 the unit of force ; and we find that in all Statical 

 problems of architecture and engineering the unit of 

 force employed is the force with which a pound 

 weight, or a kilogramme weight, or a ton weight, is 

 attracted by the Earth. 



The engineer calls these forces the force of a pound, 

 of a kilogramme, or of a ton ; he does not add the word 

 weight, reserving the word weight to denote the quantity 

 of matter in the body which is acted upon, in accordance 

 with the language of the Act of Parliament on " Weights 

 and Measures." 



In the Dynamics of bodies on the surface of the Earth, 

 the same gravitational unit of force is universally em- 

 ployed in practice ; and now, to take a familiar problem, 

 we may investigate the motion of a train, weighing W 

 tons, on a straight level railway, pulled by an engine 

 exerting a tractive force of P tons, by the bite of the 

 driving wheels on the rails. 



Neglecting passive resistances, and the rotary inertia 

 of the wheels, the train will acquire from rest a velocity 

 V feet per second in s feet, given by 



Vs = 



W7/2 

 2^ 



(foot-tons). 



The velocity growing uniformly, the average velocity 

 will be half the final velocity v ; so that if the train 

 takes t seconds to go the first s feet, 



and 



Vt 



sit 

 Wv 



(second-tons). 



The word second-tons has been formed by analogy with 

 the -word foot-tons, to express the product of a force of P 

 tons and / seconds, the time it acts ; just as foot-tons 

 expresses the product of a force of P tons and s feet, 

 the distance through which it acts. 



While P^, the work in foot-tons done by the force P 

 tons acting through s feet, has a mechanical equivalent, 



W7/2 



-~> called the kinetic energy of the train in foot-tons ; 



so P^, which we may call the impulse in second-tons 

 of the force P tons acting for t seconds, has a mechanical 



Wz/ , 

 equivalent ~~zr, the momentum ' communicated to the 



train in second-tons. 



We merely state these theorems, with the addition of 

 the proposed new name of second-tons, as these theorems 

 are found in all dynamical treatises, being direct corollaries 

 of Newton's Second Law of Motion. 



We have measured W and P in tons, as would be natural 

 in any railway-train problem, but the same equations of 

 course hold when W and P are given ih cwt., pounds, 

 kilogrammes, or grammes ; and then impulse or mo- 

 mentum will be given in second-cwt., second-pounds, 

 second-kilogrammes or second-grammes ; while work or 

 kinetic energy will be given in foot-cwt., foot-poimds, or 

 metre-kilogrammes, or centimetre-grammes, on changing 

 to the metre or centimetre as metric unit of length, 

 and changing at the same time the numerical measure 

 of ^. 



3. The presence of g in the denominator of W in the 



NO. I 185, VOL. 46] 



dynamical equations will be remarked, and this constitutes 

 a difficulty to the student, which our teachers of Dynamics 

 have done their best to obscure. 



The quantity^ makes its appearance, not because W/^ 

 is an invariable quantity, as is generally taught, but 

 because the unit of force in which P is measured is variable, 

 being proportional to the local value of g. 



With a foot and second as units of length and time, we 

 may take the value of g at the equator as 32, increasing 

 gradually to about one-289th part more, or about \ per 

 cent, greater at the poles, in consequenc i of the Earth's 

 rotation. 



The force of a pound, meaning thereby the force with 

 which the Earth appears to attract a pound weight, is 

 thence about ^ per cent, greater at the poles than at the 

 equator ; and this does not allow for the increase in g 

 due to the ellipticity, which by Clairaut's theorem would 

 make the total increase about \ per cent. 



But to say that a body has gained in weight one-289th 

 part, or ^ per cent., in going from the equator to the pole 

 is absurd and misleading ; for if we carry our standard 

 weights and scales with us, we shall find that the body 

 weighs exactly the same. 



When the theorist tells us that a body gains or loses 

 one-289th part of its weight in being taken from the 

 equator to the pole, or back again, he means that the 

 indications on a spring balance, graduated in latitude 45° 

 by attaching standard weights, will be about ^ per cent, 

 in error at the equator and at the poles. 



But such a spring balance would be illegal if used 

 according to its graduations in any other latitude than 

 the one in which it was constructed ; and the user would 

 lose in all cases ; he would lose at the equator by selling 

 \ per cent, too much by weight ; and he would lose at 

 the poles the fines incurred from the Inspector of Weights 

 and Measures, who would test his spring balance by 

 attaching standard weights, composed of lumps of metal. 



The spring balance graduated in latitude 45', and em- 

 ployed alternately at the equator and the pole, is equiva- 

 lent to a beam balance, of which the beam stretches over 

 a quadrant of the meridian of the Earth from the equator 

 to the pole, with a fulcrum in latitude 45°, but such an 

 abnormal balance is not contemplated in the Act. 



4. If we could transport ourselves to the surface of the 

 Moon, Sun, or any planet, with our weights and scales, 

 Newton's Law of Universal Gravitation teaches us that 

 we should still find the bodyof exactly the same weight in 

 the balance, the attraction of the Moon, Sun, or planets 

 on the body and on the weights being still equal. 



The magnitudes of these equal attractions would, how- 

 ever, have changed, since the attraction is proportional to 

 the local value of ^ ; on the surface of the Moon it is 

 calculated that ^ is about 5*4 ; on the surface of the Sun 

 it is about 30 times the value on the surface of the earth, 

 while on Jupiter it is calculated that^ is about 71. 



These values of g are inferred from observation of the 

 diameter of the celestial body, and from its weight, 

 measured in terms of the weight of the Earth, or using 

 the Earth as the standard weight; and calculated by 

 Kepler's Third Law from the period and distance of a 

 satellite, compared with the period and distance of our 

 satellite, the Moon. 



The weight of the Earth itself is inferred from the 

 Cavendish Experiment, in which the attraction of gravita- 

 tion between two given weights is measured. 



According to Newton's Law of Universal Gravitation, the 

 attraction between two spherical bodies, arranged in 

 spherical strata, the Sun and Earth for instance, weighing 

 S and E g (grammes) when their centres are a cm apart, 

 will be proportional to SEa-^ ; with C.G.S. units, this 

 attraction may be expressed as CSEa~- dynes, and then 

 C is C2i\\&d.\h& constant of gravitation J and the Cavendish 

 experiment is devised for the purpose of measuring C. 



Denoting by ^ the acceleration of gravity (in C.G.S. 



