Between each grain and its neighbours there is a true surface, the tension of which probably depends in some measure upon the relationship of their axes to each other. Now we know that a by no means unimportant portion of the energy of a solid body depends upon its surface area; and I would suggest also that it depends in some measure upon the surfaces separating the crystal units, of which it is composed, from each other, even when they are of similar composition. In a block of glacier-ice, therefore, or in a block of granite, the total energy depends, not only upon its temperature and volume, but also upon the energy of the surface-tensions of its component grains, whether they be similar or dissimilar in composition. Now it is an important theorem of Dynamics that, for the stable equilibrium of a system, the potential energy of the whole must be a minimum. For instance, when immiscible liquids are intimately mixed, work is done in increasing their surface-energy, and the energy thus rendered potential serves to again separate them. Glacier-ice is in much the same condition. Its potential energy is always tending to a minimum, a condition which can only be reached by the disappearance of the interfaces between the crystal-grains. Being viscous, it is unable to do more than delay the disappearance of the interfaces, and the grains would continue to grow larger and larger with the lapse of time were it not for the fact that they are broken in the production of the ribboned or veined structure. For the same reason masses of all liquids possessing structure, whatever their viscosities may be, eventually become single crystalline particles. Yours truly, R. M. DEELEY. XLV. On the Kinetic Interpretation of the Dissipation Function. By Dr. LADISLAS NATANSON, Professor of Natural Philosophy, University of Cracow*. 1. N the following paper we consider a fluid medium from Maxwell's point of view (see the paper "On the Dynamical Theory of Gases," 'Scientific Papers,' ii. 26); we suppose it to be composed of a great number of moving molecules. Let έ, n, be the components of the individual (or "molecular ") velocity of a molecule; and let u, v, w be *Translated from " Rozprawy" (Transactions) of the Cracow Academy of Sciences, Math. and Phys. Section, vol. xxix. Communicated by the Author. the components of the mean (or "molar ") velocity within an element de dy dz of volume; nda dy dz may represent the number of molecules within that element. The mass of a molecule being M, p= Mn being the density of the medium, the kinetic energy of a molecule is ↓ M { (u + §)2 + (v + n)2 + (w + 5)2 } ; (1) and the total kinetic energy of a portion of the medium consists of the two following parts: (1) the kinetic energy of the visible motion, K = 1⁄2 SSS p(u2 + x2 + w2) dx dy dz ; . . . (2) it is this energy that, in Hydrodynamics, is taken into account; (2) the molecular or heat-energy, E = } } p(§2 + n2 + 32) dx dy dz. (3) per In these expressions the integrations are supposed to be formed throughout the volume occupied by the medium; and and similar symbols represent the mean values of §2 &c. for the molecules within an element. We have At the point (x, y, z) we have the normal component (5) (6) Pyz=Pzy=pm; Pzx=Prz=p; Pzy=Pyz=Pn. Let Q be any property of the molecule which can be expressed as a function of (u+§), (v+n), and (w+5). Then writing d/dt for the actual or total variation of Q, and 8/dt for any alteration of Q that can be due to the mutual interference of molecules, (EQp) + 2y (nQp) + Z2 (5Qp) SQ aQ да =P St +pX ди ди X, Y, Z denote here the components, at (x, y, z), of acceleration due to external forces. From this fundamental equation (7) the equation of continuity, as well as the equations of motion, du a P + (p2) + dt immediately follow. Let us now put Q = (u + §)2 + (v + n)2 + (w + 5)2; (8) (9) (u2 +v2+w2 + §2+n2 + (2) a ду a (2up §2+2vp5n+2wp§Ÿ+p§3 +p§n2+p§52) · (2upn§+2vpn2+2wpn5+pn§2+pn3+pn52) (2up 55+2vp5n+2wp52 +p552+p5n2+pŸ3) =2p(uX+vY+wZ) + p = (u2 + v2 +w2 +§2 + n2 +52). (10) δι All these equations are quite independent of any view we may entertain as to the intimate nature of molecules and as to the law of force between them; accordingly they may be said to belong to the Kinematical Part of the Theory of constitution of fluids. 2. If we suppose that the molecules of the fluid do not contain any internal energy (or that they contain such energy in absolutely constant amount) the energy of a molecule will be only that which depends on the velocity of its centre of inertia. Assuming this equation (11) may be transformed. Put for brevity, where 0=a+b+c. Multiplying by da dy dz and integrating throughout the volume occupied by the medium, we have then ӘЕ + + SSp (52 + n2 + 52) (lu+mv+nw)dS at + [[[[p¥3a+pn2b+pš2c+pñ¢A+pš§B+p&n C]dx dy dz=0; (15) here the direction-cosines of the normal to the element ds of әк at −SSS [p¥3a+ pñ3b+ p i.e. molar energy is subject to change from two sources, the d'K • SSS [p2 2 a +pm2 + p} 2c+pn$A+p¢§B+p§nC]dx dy dz.(18) == F=(p−pš2)a+(p−pn2)b+ (p−pš2)c−pnšA—pšВ—p§nC = — pr2 (a — 30) — pn2 (1 − }0) — pš2 (c— }0)—pnŠA—pš§В—p§ŋC =(p-pš2) (a−30)+(p−pn2)(b−30)+(p−pš2)(c−30) (20) the molecular energy E is therefore subject to change from Put in (7) Q=(u+)2; Q=(v+n) (w+5); .. (22) if the disturbance is not very violent, we shall find 1 and other equations which may be written down from F= - 18 ́4p &t { (p − p¥3)2 + (p −pn2)2+(p−p53)2 +2(pn5)2+2(p§§)2+2(p§n)2}. • (25) From this it appears that the direction of that transformation If now we suppose the following equations to be given : : (26) . (27) (the symbolμ denoting a constant, the coefficient of viscosity), F = = { } (p—p ̧2)2 + {( p −pn2)2 + $('p−p52)2 + (p»5)2 + (p5¥)2 + (p¥7)2}, =μ{2(a−}0)2+2(b−30)2+2(c−30) 2 + A2 + B2+C2} 1 = ¦ { ¿(pn2 −p52)2 + ¿(p52 —p¥3)2 + }{p ̧2 − pn2)2 + (pn?)2 + (p?Ej2 + (p£n)2 } μ =μ{} (b−c)2 + 3 (c—a)2 + }(a−b)2 + A2 + B2+C2}, (28) |