On a quantity that is analogous to potential and a theorem that relates to it
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1 Su une quantité analogue au potential et su un théoème y elatif C R Acad Sci 7 (87) On a quantity that is analogous to potential and a theoem that elates to it By R CLAUSIUS Tanslated by D H Delphenich One knows that a lage pat of mathematical physics is based upon the potential theoy Since knowledge of that theoy is not as widespead as one might desie I believed that it would be useful to publish a small teatise in which I pesented the most essential popeties of the potential function and the potential Folie although occupied with some vey impotant geometic eseach in his own ight and who has aleady published some inteesting esults has nonetheless been kind enough to take the touble to poduce a Fench tanslation of that teatise and it is not necessay to add that it was accomplished with the geatest cae and popiety Pemit me to pay homage to the name of that tanslato befoe this academy I shall take this occasion to communicate a theoem to the Academy that I discoveed in my eseach into the mechanical theoy of heat and which petains to the subject that I teated in that book In a pape that I had the hono of communicating to the Academy I stated the following theoem: The foce that acts upon heat is popotional to the absolute tempeatue ( ) Since accoding to my way of thinking heat is nothing but a motion I have no doubt that this theoem coesponds to a geneal theoem in mechanics that allows one to deive the equations of motion in the same way that the pinciple of the equivalence of heat is only a special case of the pinciple of the equivalence of the vis viva and mechanical wok Hee is that theoem which efes to stationay motion of an abitay system of mateial points ie to a motion in which the positions and velocities of the points do not always change in the same sense but emain within cetain limits Let a system of mateial points m m m be given with coodinates x y z ; x y z ; x y z ; which ae subject to foces whose components ae X Y Z ; X Y Z ; X Y Z ; Fom the sum: m dy dz + + o the sum (which is known by the name of the vis viva of the system): ( ) Poggendoff Annalen v CXCI Jounal de Liouville () v VIII Théoie mécanique de la Chaleu tanslated by F Folie v I pps 57 6 and 34
2 Clausius On a quantity that is analogous to potential and a theoem that elates to it when one denotes the velocities of the points by v v v and futhe fom the sum: ( Xx + Yy + Zz) to whose mean values I popose to give the name of the viial of the system (in Geman viial fom the Latin wod vis fo foce) We will then have the theoem: The mean vis viva of the system is equal to its viial If we distinguish the mean value of a quantity fom its vaiable value by putting an oveba on the fomula that epesents the vaiable quantity then ou theoem can be expessed by the following equation: = ( Xx + Yy + Zz) As fo the value of the viial it will take some vey simply foms in the most impotant cases in natue When thee ae two points m and m that ae sepaated by a distance of and which exet an attactive o epulsive foce upon each othe that is epesented by the function ϕ() and which we will suppose to be positive o negative accoding to whethe the foce is attactive o epulsive espectively we will have: x x x x ( ) X x + X x = ϕ( ) x + ϕ( ) x = ( ) x ϕ x and as a esult: (X x + Y y + Z z + X x + Y y + Z z ) = ϕ () When that esult is extended to an abitay numbe of points that ae subject to only attactive o epulsive foces that they exet upon each othe one will have: (X x + Y y + Z z) = ϕ () in which the sum on the ight elates to all pai-wise combinations of given points The viial of the system of points will then have the expession: ϕ( ) in this case One easily ecognizes the analogy between that quantity and anothe known quantity If we intoduce the function Φ () by setting:
3 Clausius On a quantity that is analogous to potential and a theoem that elates to it 3 then we will have: Φ () = ϕ( ) d (X x + Y y + Z z) = d Φ( ) In the special case whee the attactive o epulsive foces ae invesely popotional to the squaes of the distances the sum Φ( ) up to sign is called the potential of the system Since that quantity has not futhe been given a name in the geneal case I popose the name of egon fo it (fom the Geek wod ἔργον fo wok) whose Geman fom is egal but it might be ponounced egiel in Fench The known theoem of the equivalence of vis viva and mechanical wok is then expessed vey simply and in ode to show moe clealy the analogy between that theoem and the one that concens the viial I will juxtapose two theoems: The sum of the vis viva and the egon is constant The mean vis viva is equal to the viial In ode to apply ou theoem to heat conside a body to be a system of mateial points in motion Those points act upon each othe and in addition they ae subject to extenal foces We can then sepaate the viial into two pats one of which efes to the intenal foces and the othe to the extenal foces which we will call the intenal viial and the extenal viial espectively The intenal viial is epesented by the fomula that was cited befoe: ϕ () in which the oveba is no longe necessay because due to the lage numbe of atoms that move iegulaly the value that the sum possesses at a cetain time will be equal to its mean value As fo the extenal viial in the most common case in which the only extenal foce that acts is a unifom pessue that is nomal to the suface one can expess it by the following fomula in which p epesents the pessue and v epesents the volume: 3 pv If we futhe denote that vis viva of the motion that we call heat by h then we will have: h = ϕ () + 3 pv It emains fo us to pove the stated theoem about the viial That poof is vey easy The equations of motion of a mateial point m ae: d x = X d y = Y d z = Z
4 Clausius On a quantity that is analogous to potential and a theoem that elates to it 4 Now one has: d ( x ) = d dx x = d x + x Upon multiplying that equation by m / 4 and putting X in place of obtain: m = m d ( x ) Xx + 4 d x one will so upon integating this and dividing by t one will infe that: m t t t m d( x ) d( x ) = X x t + 4t d( x ) in which The fomulas: epesents the initial value of t t and d( x ) t X x t when taken ove a lage value of time t epesent the mean values of and Xx which we have denoted by and X x espectively Fo a peiodic motion the last tem in the equation will be equal to zeo at the end of each peiod because d (x ) / d( x ) will take on its initial value If the motion is not egulaly peiodic but iegula like the motion of the atoms in the inteio of a body then the diffeence d( x ) d( x ) will not egulaly epesent the value zeo but nonetheless that value will pesent itself fom time to time and othe than that the diviso t will make the last teanish when time t becomes vey lage Hence upon suppessing that tem we can wite: m = X x
5 Clausius On a quantity that is analogous to potential and a theoem that elates to it 5 Since the same equation will be tue fo the othe coodinates we will get: o moe biefly: m dx dy dz + + = = ( X x + Y y + Z z) ( X x + Y y + Z z) and fo a system with an abitay numbe of points: = ( X x Y y Z z) + + Ou theoem has then been poved and we will likewise see that it is not only tue fo the entie system of points and the thee coodinates when taken togethe but also fo each point and each coodinate when taken sepaately
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