SECT. II.] NUMERICAL APPLICATION. 381 



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 = O l. 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 and the 

 point in, 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 tem 

 perature nearly to less than one hundredth part, when the value 

 of the time elapsed exceeds one month. In order that the ap 

 proximation may be admissible, it is necessary in general, 1st that 



2 2ft _ ft 2 



the quantity - -^ - should be equal to but a very small fraction 



4/Lfc 



as T~AA or TAAA or ^ ess j 2nd, that the error which must follow 

 1UU 



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. 



