and that 0 and g may be the limits of the portion originally K (1), (y), CD k denoting the value of the conducibility. In order that the equation (y) may be substituted for the preceding equation (i), it 2a.x-a2 is in general requisite that the factor et, which is that which we omit, should differ very little from unity; for if it were 1 or we might apprehend an error equal to the value calculated or to the half of that value. Let then e 4kt 1+w, w being a small 2ax_a3 = 1000; from this we derive the condition and if the greatest value g which the variable a can receive is very small with respect to x, we have t = 1 gx w 2k' We see by this result that the more distant from the origin the points are whose temperatures we wish to determine by means of the reduced equation, the more necessary it is for the value of the time elapsed to be great. Thus the heat tends more and more to be distributed according to a law independent of the primitive heating. After a certain time, the diffusion is sensibly effected, that is to say the state of the solid depends on nothing more than the quantity of the initial heat, and not on the distribution which was made of it. The temperatures of points sufficiently near to the origin are soon represented without error by the reduced equation (y); but it is not the same with points very distant from the source. We can then make use of that equation only when the time elapsed is extremely long. Numerical applications make this remark more perceptible. 382. Suppose that the substance of which the prism is formed is iron, and that the portion of the solid which has been heated is a decimetre in length, so that g=01. If we wish to ascertain what will be, after a given time, the temperature of a point m whose distance from the origin is a metre, and if we employ for this investigation the approximate integral (y), we shall commit an error greater as the value of the time is smaller. This error will be less than the hundredth part of the quantity sought, if the time elapsed exceeds three days and a half. In this case the distance included between the origin 0 and the point m, whose temperature we are determining, is only ten times greater than the portion heated. If this ratio is one hundred instead of being ten, the reduced integral (y) will give the temperature nearly to less than one hundredth part, when the value of the time elapsed exceeds one month. In order that the approximation may be admissible, it is necessary in general, 1st that the quantity 1 1 as or 2xx 4kt a2 should be equal to but a very small fraction or less; 2nd, that the error which must follow 100 1000 should have an absolute value very much less than the small quantities which we observe with the most sensitive thermometers. When the points which we consider are very distant from the portion of the solid which was originally heated, the temperatures which it is required to determine are extremely small; thus the error which we should commit in employing the reduced equation would have a very small absolute value; but it does not follow that we should be authorized to make use of that equation. For if the error committed, although very small, exceeds or is equal to the quantity sought; or even if it is the half or the fourth, or an appreciable part, the approximation ought to be rejected. It is evident that in this case the approximate equation (y) would not express the state of the solid, and that we could not avail ourselves of it to determine the ratios of the simultaneous temperatures of two or more points. 383. It follows from this examination that we ought not to 1 0 (a-x)2 conclude from the integral v = 2 Jkt dif (2) e- that the law of the primitive distribution has no influence on the temperature of points very distant from the origin. The resultant effect of this distribution soon ceases to have influence on the points near to the heated portion; that is to say their temperature depends on nothing more than the quantity of the initial heat, and not on the distribution which was made of it: but greatness of distance does not concur to efface the impress of the distribution, it preserves it on the contrary during a very long time. and retards the diffusion of heat. Thus the equation only after an immense time represents the temperatures of points extremely remote from the heated part. If we applied it without this condition, we should find results double or triple of the true results, or even incomparably greater or smaller; and this would not only occur for very small values of the time, but for great values, such as an hour, a day, a year. Lastly this expression would be so much the less exact, all other things being equal, as the points were more distant from the part originally heated. 384. When the diffusion of heat is effected in all directions, the state of the solid is represented as we have seen by the integral If the initial heat is contained in a definite portion of the solid mass, we know the limits which comprise this heated part, and the quantities a, B, y, which vary under the integral sign, cannot receive values which exceed those limits. Suppose then that we mark on the three axes six points whose distances are + X, + Y, +Z, and X, Y, Z, and that we consider the successive states of the solid included within the six planes which cross the axes at these distances; we see that the exponent of e under the sign of 2xx increases without limit. In fact, the terms such as and 4kt a2 4kt receive in this case very small absolute values, since the numerators are included between fixed limits, and the denominators increase to infinity. Thus the factors which we omit differ extremely little from unity. Hence the variable state of the solid, after a great value of the time, is expressed by The factor (d2fdß [dvƒ (2, B, 7) represents the whole quantity of heat B which the solid contains. Thus the system of temperatures depends not upon the initial distribution of heat, but only on its quantity. We might suppose that all the initial heat was contained in a single prismatic element situated at the origin, whose extremely small orthogonal dimensions were w1, w,, w,. The initial temperature of this element would be denoted by an exceedingly great number f, and all the other molecules of the solid would have a nul initial temperature. The product www.f is equal in this case to the integral Whatever be the initial heating, the state of the solid which corresponds to a very great value of the time, is the same as if all the heat had been collected into a single element situated at the origin. 385. Suppose now that we consider only the points of the solid whose distance from the origin is very great with respect to the dimensions of the heated part; we might first imagine that this condition is sufficient to reduce the exponent of e in the general integral. The exponent is in fact MICHIGAN Juiversity of and the variables a, B, y are, by hypothesis, included between finite limits, so that their values are always extremely small with respect to the greater co-ordinate of a point very remote from the origin. It follows from this that the exponent of e is composed of two parts M+, one of which is very small with respect to the other. But from the fact that the ratio μ M is a very small fraction, we cannot conclude that the exponential e+ becomes equal to e, or differs only from it by a quantity very small with respect to its actual value. We must not consider the relative values of M and μ, but only the absolute value of μ. In order that we may be able to reduce the exact integral (j) to the equation whose dimension is 0, should always be a very small number. If we suppose that the distance from the origin to the point m, whose temperature we wish to determine, is very great with respect to the extent of the part which was at first heated, we should examine whether the preceding quantity is always a very small fraction w. This condition must be satisfied to enable us to employ the approximate integral but this equation does not represent the variable state of that part of the mass which is very remote from the source of heat. It gives on the contrary a result so much the less exact, all other things being equal, as the points whose temperature we are determining are more distant from the source. The initial heat contained in a definite portion of the solid mass penetrates successively the neighbouring parts, and spreads itself in all directions; only an exceedingly small quantity of it arrives at points whose distance from the origin is very great. |