what is the quantity which, during a given time, passes across the base A and replaces that which flows into the cold masses B and C; we must consider that the flow perpendicular to the axis of y is expressed by - K The quantity which during dv the instant dt flows across a part dy of the axis is therefore and, as the temperatures are permanent, the amount of the flow, during unit of time, is K dv ddy. This expression must be integrated between the limits y=- and y=+, in order to ascertain the whole quantity which passes the base, or which is the same thing, must be integrated from y=0 to y=π, and dv dx the result doubled. The quantity is a function of y, x and in which a must be made equal to 0, in order that the calculation may refer to the base A, which coincides with the axis of y. The expression for the expenditure of the source of heat is therefore 2 The integral must be taken from y = 0 to 2 (- Kdv dy). dv y=1; if, in the function XC is not supposed equal to 0, dx' but xx, the integral will be a function of a which will denote the quantity of heat which flows in unit of time across a transverse edge at a distance x from the origin. 193. If we wish to ascertain the quantity of heat which, during unit of time, passes across a line drawn on the plate parallel to the edges B and C, we employ the expression - K dv dy and, multiplying it by the element de of the line drawn, integrate with respect to a between the given boundaries of the line; thus -Kada) shews how much heat flows across the the integral ((— K = dv dy whole length of the line; and if before or after the integration we make yπ, we determine the quantity of heat which, during unit of time, escapes from the plate across the infinite edge C. We may next compare the latter quantity with the expenditure of the source of heat; for the source must necessarily supply continually the heat which flows into the masses B and C. If this compensation did not exist at each instant, the system of temperatures would be variable. 194. Equation (2) gives -3.c (e cos y - e cos 3y+e cos 5y — e cos 7y+ &c.); multiplying by dy, and integrating from y = 0, we have If y be made = 1⁄2π, and the integral doubled, we obtain as the expression for the quantity of heat which, during unit of time, crosses a line parallel to the base, and at a distance x from that base. From equation (2) we derive also (e sin ye sin 3y+e sin 5y—e" sin 7y+&c.): hence the integral – K (1) dæ, taken from x = 0, is 4K {(1 − e ̄) sin y − (1 − e ̃31) sin 3y + (1 − e) sin 5y − (1 − e1x) sin 7y + &c.}. If this quantity be subtracted from the value which it assumes when a is made infinite, we find 4K π 1 -3r 1 (6 sin ye sin 5y + sin 5y — &c.) ; 5 and, on making y, we have an expression for the whole quantity of heat which crosses the infinite edge C, from the point whose distance from the origin is a up to the end of the plate; namely, which is evidently equal to half the quantity which in the same time passes beyond the transverse line drawn on the plate at a distance x from the origin. We have already remarked that this result is a necessary consequence of the conditions of the problem; if it did not hold, the part of the plate which is situated beyond the transverse line and is prolonged to infinity would not receive through its base a quantity of heat equal to that which it loses through its two edges; it could not therefore preserve its state, which is contrary to hypothesis. 195. As to the expenditure of the source of heat, it is found by supposing x = 0 in the preceding expression; hence it assumes an infinite value, the reason for which is evident if it be remarked that, according to hypothesis, every point of the line A has and retains the temperature 1: parallel lines which are very near to this base have also a temperature very little different from unity: hence, the extremities of all these lines contiguous to the cold masses B and C communicate to them a quantity of heat incomparably greater than if the decrease of temperature were continuous and imperceptible. In the first part of the plate, at the ends near to B or C, a cataract of heat, or an infinite flow, exists. This result ceases to hold when the distance a becomes appreciable. 196. The length of the base has been denoted by π. If we assign to it any value 27, we must write instead of y, and multiplying also the values of a by, we must write i instead of x. Denoting by A the constant temperature of the This equation represents exactly the system of permanent temperature in an infinite rectangular prism, included between two masses of ice B and C, and a constant source of heat. 197. It is easy to see either by means of this equation, or from Art. 171, that heat is propagated in this solid, by separating more and more from the origin, at the same time that it is directed towards the infinite faces B and C. Each section parallel to that of the base is traversed by a wave of heat which is renewed at each instant with the same intensity: the intensity diminishes as the section becomes more distant from the origin. Similar movements are effected with respect to any plane parallel to the infinite faces; each of these planes is traversed by a constant wave which conveys its heat to the lateral masses. The developments contained in the preceding articles would be unnecessary, if we had not to explain an entirely new theory, whose principles it is requisite to fix. With that view we add the following remarks. 198. Each of the terms of equation (2) corresponds to only one particular system of temperatures, which might exist in a rectangular plate heated at its end, and whose infinite edges are maintained at a constant temperature. Thus the equation v=ecos y represents the permanent temperatures, when the points of the base A are subject to a fixed temperature, denoted by cosy. We may now imagine the heated plate to be part of a plane which is prolonged to infinity in all directions, and denoting the co-ordinates of any point of this plane by x and y, and the temperature of the same point by v, we may apply to the entire plane the equation vecos y; by this means, the edges B and C receive the constant temperature 0; but it is not the same with contiguous parts BB and CC; they receive and keep lower temperatures. The base A has at every point the permanent temperature denoted by cos y, and the contiguous parts AA have higher temperatures. If we construct the curved surface whose vertical ordinate is equal to the permanent temperature at each point of the plane, and if it be cut by a vertical plane passing through the line A or parallel to that line, the form of the section will be that of a trigonometrical line whose ordinate represents the infinite and periodic series of cosines. If the same curved surface be cut by a vertical plane parallel to the axis of x, the form of the section will through its whole length be that of a logarithmic curve. 199. By this it may be seen how the analysis satisfies the two conditions of the hypothesis, which subjected the base to a temperature equal to cos y, and the two sides B and C to the temperature 0. When we express these two conditions we solve in fact the following problem: If the heated plate formed part of an infinite plane, what must be the temperatures at all the points of the plane, in order that the system may be self-permanent, and that the fixed temperatures of the infinite rectangle may be those which are given by the hypothesis? We have supposed in the foregoing part that some external causes maintained the faces of the rectangular solid, one at the temperature 1, and the two others at the temperature 0. This effect may be represented in different manners; but the hypothesis proper to the investigation consists in regarding the prism as part of a solid all of whose dimensions are infinite, and in determining the temperatures of the mass which surrounds it, so that the conditions relative to the surface may be always observed. 200. To ascertain the system of permanent temperatures in a rectangular plate whose extremity A is maintained at the temperature 1, and the two infinite edges at the temperature 0, we might consider the changes which the temperatures undergo, from the initial state which is given, to the fixed state which is the object of the problem. Thus the variable state of the solid would be determined for all values of the time, and it might then be supposed that the value was infinite. The method which we have followed is different, and conducts more directly to the expression of the final state, since it is founded on a distinctive property of that state. We now proceed to shew that the problem admits of no other solution than that which we have stated. The proof follows from the following propositions. 201. If we give to all the points of an infinite rectangular plate temperatures expressed by equation (2), and if at the two edges B and C we maintain the fixed temperature 0, whilst the end A is exposed to a source of heat which keeps all points of the line A at the fixed temperature 1; no change can happen in the dev d2v dx dy2 state of the solid. In fact, the equation + F. H. = 0 being 11 |