April 28, 1887] NA TURE 



605 



naturally expressed thus, V = L'T- On the same principle 

 we should have — 



A = V/T=L/TT, U = MV-MLrr, A = MA = ML/TT, 

 E = LA = LML/TT 



This may be simplified, using the analogy of the fluxional 

 notation, by writing L for L'T and L for L/TT- Then L, L 

 may take the place of V ^^<^ A. ^nd we have — 



U = ML A = ML E = LA= LML 



Names for the several units are hardly needed in the general 

 system. 



In the C.G. S. system the units of length, mass, and time are 

 respectively the centimetre, gramme, and second, denoted by 

 C> G' S- Then if Uc Ao. E(; denote the units of momentum or 

 impulse, force, and energy or work, we have, on the same prin- 

 ciple as before, C> C 'o denote the speed of a centimetre per 

 second and the acceleration of a centimetre per second per 

 second respectively, and the equations — 



Ug= go. Ao = GO. E<, = OA .= OGC 



If names for all these units are required, we may use these : 

 vei, eeli morn, dyne, erg ; and we may say, a mom js a gramtne- 

 vel, a dyne is a gramme-eel, and an erg a centimetre-dyne. 



In the F.P.S., or British system, the units of length, mass, 

 and time are respectively the foot, pound, and second, denoted 

 by F, P, S- Then, if (J A Aa Ep denote the units of momen- 

 tum or impulse, force, and energy respectively, we have, on the 

 same principles, p, p to denote the speed of a foot per second 

 and the acceleration of a foot per second per second respectively, 

 and the equations — 



\ip = PR A/'= PR Ep^ ¥Lp= FPR 



I propose as names for these units : footz'el, footcel, poundem, 

 poundal, poitnderg. I should have called the unit of force a 

 poiind-dyne or poundyn, but that poundal has already obtained 

 general acceptance. 



The foregoing are all absolute units. The corresponding 

 (Greenwich) gravitation-units are the pound-weight = 32"I9 

 poundals, the foot-pound = 32' 19 poundergs, and the second- 

 pound (as it has been proposed by Prof. Unwin to name the 

 gravitation-unit of momentum, or time-integral of a pound- 

 weight through one second) = 32"I9 poundems ; so that to 

 convert absolute F.P. S. units into gravitation-units, or vice 

 versd, it is only necessary to divide or multiply by 32"19, since 



the acceleration due to gravity at Greenwich = 32'I9 p. 



It is, I think, comparatively unimportant whether the names 

 above suggested are, all of them, accepted or not ; but the 

 notation will, I believe, be found a great aid to the beginner in 

 fixing in his mind the dimensions of the different magnitudes, 

 and an effective safeguard against the too common confusion of 

 units of force, impulse, and work. I doubt whether in speaking 

 much would be gained by saying "footvel" instead of " foot per 

 second," or "fojteel" instead of " foot per second per second," 



while in writinr, the symbols p and p might be always used 

 and read either way. ROBT. B. Hayward 



Harrow, April 12 



Units of Weight, Mass, and Force 



Far be it from me to interfere between Mr. Alfred Lodge 

 and Prof. Greenhill ; but, whilst leaving Mr. Lodge to his fate 

 and Prof. Greenhill, perhaps, as an engineer, I may be per- 

 mitted to offer a few remarks on the general question. First, 

 then, it appears to me that Prof. Tail's thoroughgoing condem- 

 nation of certain phrases of the engineering vernacular, and of 

 the grave errors of certain writers, has been strained in some 

 quarters to mean a general charge against engineers of inability 

 to think or write clearly on the physical laws which lie at the 

 root of their every-day practice. Such an unqualified charge is 

 on the face of it absunl ; for otherwise we must confess that the 

 lives' work of Thomson and Tait has been a total failure as 

 regards its influence on the truly practical men of their generation. 

 Can this be so ? 



Prof. Greenhill's remarks on the abominable semi-numerical 

 IV 

 equation IV — Mg, or — = M, 1 most heartily welcome [where 



IF is a mass and g a numeric ; the moment writers on dynamics, 

 who use the gravitation system, pass from merely proportional 

 equations to their physical interpretation, then must we face 

 with them this most wretched equation]. But perhaps it will 

 be at once a surprise and a gratification to Prof. Greenhill to 

 know that a whole (academic) generation of Scottish University 

 engineering students has been taught to eschew this same equa- 

 tion as an unclean thing, and to adopt the mode of thought 

 clearly set forth in the last two paragraphs of his letter of 

 February 28. Some five or six years ago I was myself so 

 taught by one who is now, alas ! no more. The possibility of 

 thus clearing of cant not only the engineering but the purely 

 mathematical mind seems to me, as indeed Prof. Greenhill 

 indicates, to be a direct consequence of the acceptance by 

 Thomson and Tait of the British Imperial pound as the unit of 

 mass or quantity of matter. 



Having fixed, then, for good and all, the unit of mass, and 

 taking the British foot and the second as the units of time and 

 length respectively, the unit of force defines itself in virtue of 

 Newton's Second Law. To this unit — the British unit of force 

 — Prof. James Thomson has given, as nearly everyone knows, 

 the name "poundal." Now the most convenient practical unit 

 of force, for physicists as well as engineers, is not the poundal, but 

 the gravitation at the earth's surface of the unit of mass, a 

 quantity which is not absolutely constant, the inconvenience so 

 arising being, however, practically unimportant, or at most 

 involving a reduction to an arbitrary standard. To pass, then, 

 from a force expressed in poundals to a force expressed in units 

 of gravitation of the Imperial pound at the standard place, one 

 simply wants to know how many poundals go to the gravitation 

 of the Imperial pound at that place — in other words, the change 

 ratio. The answer is simple : the numeric g for the standard 

 place ; for a force equal to the gravitation of the Imperial pound 

 at the standard place acting upon a mass of one Imperial pound 

 would generate a momentum per second of g (numeric) pounds 

 mass X feet per sec. per sec. ; and the poundal a momentum per 

 second of one pound mass X feet per sec. per sec. ; and by New- 

 ton's Second Law the ratio of the forces is, therefore, the numeric 

 g. What more does the physicist or engineer want to know ? 

 How many poundals go to the gravitation of a ton mass at the 

 standard place ? Answer, the numeric 2240 "-. Could any- 

 thing be simpler? The difficulty is to find the difficulty, or to 

 assign the raison d'etre oi the so-called gravitation unit of mass. 



When the old unnatural gravitation unit of mass is abandoned, 

 and the transition from the natural unit to the gravitation 

 unit of force made by means of the change ratio, the 

 vicious use of the word mass to denote the result of dividing 

 by the numeric g the mass of a body in standard pounds 

 (viciously on the same system called the weight and denoted by 

 ff) ceases ; and the dire confusion between weight and mass 

 becomes a thing of the past. The emancipation of the term 

 weight from its liondage to mass would appear to have afforde 1 

 opportunity for its use wherever it might be of service in sug- 

 gesting or denoting gravitation. For example, there are three 

 units, each called a pound, viz. the Imperial British unit of 

 mass, the gravitation of the same mass at any the same place 

 on the earth's surface, and the pound sterling. Physicists, like 

 Sir Wm. Thomson and Prof. Tait, use terms such as "pound 

 weight," "gramme weight " ; similarly we have "pound mass" 

 coming into use ; and probably we shall soon hear of "pound 

 money." Pro^. Greenhill, with his strong engineering sym- 

 pathies, objects to the time-honoured "pounds per square inch " 

 being rendered " pounds weight per square inch '' ; and if I may 

 presume to offer an opinion, the old phrase is already cumbrous 

 enough. Still, if Prof. Tait and Prof. Greenhill ultimately 

 agree that there is anything to be gained in perspicuity, for the 

 sake of the weaker members perhaps it might be well for us to 

 put in practice, on some occasions, the injunction to sacrifice all 

 rather than cause our brother to offend. 



Some one might possibly step in to draw attention to the fact 

 that the pages of even our own great high priest of exact applied 



science are disfigured by — ; but sure I am that were Rankine 



now with us he would lead the way with Prof. Greenhill in a 

 crusade against the apologists for the ob.^curity of which U'lg is 

 the symbol. Take, for example, Rankine's bold introduction of 

 the dynamical unit of quantity of heat. Take the of)inion of 



