same quantity, dispersed into the air, is, by the principle of the communication of heat, equal to 2πXhodt; we must therefore have at the surface the definite equation - K dv = hv. dx The nature of these equations is explained at greater length, either in the articles which refer to the sphere, or in those wherein the general equations have been given for a body of any form whatThe function v which represents the movement of heat in an infinite cylinder must therefore satisfy, 1st, the general equadv K d'o ever. tion = + 1 du), which applies whatever æ and t may dt CD\de x dv h be; 2nd, the definite equation v + = K dx 0, which is true, whatever the variable t may be, when x = X; 3rd, the definite equation v = F(x). The last condition must be satisfied by all values of v, when t is made equal to 0, whatever the variable x may be. The arbitrary function F(x) is supposed to be known; it corresponds to the initial state. SECTION IV. Equations of the uniform movement of heat in a solid prism of infinite length. 121. A prismatic bar is immersed at one extremity in a constant source of heat which maintains that extremity at the temperature A; the rest of the bar, whose length is infinite, continues to be exposed to a uniform current of atmospheric air maintained at temperature 0; it is required to determine the highest temperature which a given point of the bar can acquire. The problem differs from that of Article 73, since we now take into consideration all the dimensions of the solid, which is necessary in order to obtain an exact solution. We are led, indeed, to suppose that in a bar of very small. thickness all points of the same section would acquire sensibly equal temperatures; but some uncertainty may rest on the results of this hypothesis. It is therefore preferable to solve the problem rigorously, and then to examine, by analysis, up to what point, and in what cases, we are justified in considering the temperatures of different points of the same section to be equal. F. H. 7 122. The section made at right angles to the length of the bar, is a square whose side is 21, the axis of the bar is the axis of x, and the origin is at the extremity A. The three rectangular co-ordinates of a point of the bar are x, y, z, and v denotes the fixed temperature at the same point. The problem consists in determining the temperatures which must be assigned to different points of the bar, in order that they may continue to exist without any change, so long as the extreme surface A, which communicates with the source of heat, remains subject, at all its points, to the permanent temperature A; thus v is a function of x, y, and z. 123. Consider the movement of heat in a prismatic molecule, enclosed between six planes perpendicular to the three axes of x, y, and z. The first three planes pass through the point m whose co-ordinates are x, y, z, and the others pass through the point m' whose co-ordinates are a + dx, y +dy, z + dz. To find what quantity of heat enters the molecule during unit of time across the first plane passing through the point m and perpendicular to x, we must remember that the extent of the surface of the molecule on this plane is dy dz, and that the flow across this area is, according to the theorem of Article 98, equal dv to - K ; thus the molecule receives across the rectangle dydz dx passing through the point m a quantity of heat expressed by dv - Kdydz To find the quantity of heat which crosses the dx opposite face, and escapes from the molecule, we must substitute, in the preceding expression, + dx for x, or, which is the same thing, add to this expression its differential taken with respect to x only; whence we conclude that the molecule loses, at its second face perpendicular to x, a quantity of heat equal to we must therefore subtract this from that which enters at the opposite face; the differences of these two quantities is MICHIGAN University o SECT. IV.] STEADY MOVEMENT IN A PRISM. this expresses the quantity of heat accumulated in the molecule It is found in the same manner that a quantity of heat equal to - Kdz dx enters the molecule across the plane passing dy through the point m perpendicular to y, and that the quantity which escapes at the opposite face is the last differential being taken with respect to y only. Hence d'v the difference of the two quantities, or Kdadydz da, expresses dy' the quantity of heat which the molecule acquires, in consequence of the propagation in direction of y. Lastly, it is proved in the same manner that the molecule acquires, in consequence of the propagation in direction of z, a quantity of heat equal to K dx dy dz d'v dz Now, in order that there may be no change of temperature, it is necessary for the molecule to retain as much heat as it contained at first, so that the heat it acquires in one direction must balance that which it loses in another. Hence the sum of the three quantities of heat acquired must be nothing; thus we form the equation 124. It remains now to express the conditions relative to the surface. If we suppose the point m to belong to one of the faces of the prismatic bar, and the face to be perpendicular to z, we see that the rectangle dady, during unit of time, permits a quantity of heat equal to Vh dx dy to escape into the air, V denoting the temperature of the point m of the surface, namely what (x, y, z) the function sought becomes when z is made equal to l, half the dimension of the prism. On the other hand, the quantity of heat which, by virtue of the action of the molecules, during unit of time, traverses an infinitely small surface w, situated within the prism, perpendicular to z, is equal to -Kw, according to the theorems quoted above. Κω dv dz' This ex pression is general, and applying it to points for which the coordinate z has its complete value 1, we conclude from it that the quantity of heat which traverses the rectangle dx dy taken at the dv plete value 7. Hence the two quantities - K da dy and dz' h dx dy v, must be equal, in order that the action of the molecules may agree with that of the medium. This equality must also dv exist when we give to z in the functions. and the value -1, dz which it has at the face opposite to that first considered. Further, the quantity of heat which crosses an infinitely small surface w, dv dy' perpendicular to the axis of y, being - Kw it follows that that which flows across a rectangle dz da taken on a face of the prism perpendicular to y is K dz dx dv dy dv giving to y in the function its complete value 1. Now this rectangle dz dr permits a quantity of heat expressed by hv dx dy to escape into Kdv the air; the equation hv=- K == dy becomes therefore necessary, when y is made equal to 1 or 7 in the functions v and dv dy' 125. The value of the function v must by hypothesis be equal to A, when we suppose x = 0, whatever be the values of y and z. Thus the required function v is determined by the following conditions: 1st, for all values of x, y, z, it satisfies the general equation or, whatever x and z may be, or satisfies the equation h dv v+ =0, when z is equal to 7 or 1, whatever x and y may K dz be; 3rd, it satisfies the equation v=A, when x=0, whatever y and z may be. SECTION V. Equations of the varied movement of heat in a solid cube. 126. A solid in the form of a cube, all of whose points have acquired the same temperature, is placed in a uniform current of atmospheric air, maintained at temperature 0. It is required to determine the successive states of the body during the whole time of the cooling. The centre of the cube is taken as the origin of rectangular coordinates; the three perpendiculars dropped from this point on the faces, are the axes of x, y, and z; 2l is the side of the cube, v is the temperature to which a point whose coordinates are x, y, z, is lowered after the time t has elapsed since the commencement of the cooling: the problem consists in determining the function v, which depends on x, y, z and t. 127. To form the general equation which v must satisfy, we must ascertain what change of temperature an infinitely small portion of the solid must experience during the instant dt, by virtue of the action of the molecules which are extremely near to it. We consider then a prismatic molecule enclosed between six planes at right angles; the first three pass through the point m, whose co-ordinates are x, y, z, and the three others, through the point m', whose co-ordinates are x + dx, y+dy, z+dz. The quantity of heat which during the instant dt passes into the molecule across the first rectangle dy dz perpendicular to x, is - Kdy dz dv the molecule, through the opposite face, is found by writing x+ dx in place of x in the preceding expression, it is dv - Kdy dz - (dx) dt – K dy dz d (dv) dt, |