SECT. II.] ADMISSIBLE SIMPLIFICATIONS. 379 



to ascertain the variable state of the points of the line situated at 

 a great distance from the heated portion, and in order to express 



the ultimate condition suppressed also the factor e 4Jit , the 

 results which we should obtain would not be exact. In fact, 

 supposing that the heated portion extends only from a = to a=g 

 and that the limit g is very small with respect to the distance x of 

 the point whose temperature we wish to determine ; the quantity 



~~ 4kf w hi cn f rms the exponent reduces in fact to jy- ; that 



( a _ xf x z 

 is to say the ratio of the two quantities , and ^- approaches 



more nearly to unity as the value of x becomes greater with 

 respect to that of a : but it does not follow that we can replace 

 one of these quantities by the other in the exponent of e. In 

 general the omission of the subordinate terms cannot thus take 

 place in exponential or trigonometrical expressions. The quanti 

 ties arranged under the symbols of sine or cosine, or under the 

 exponential symbol e y are always absolute numbers, and we can 

 omit only the parts of those numbers whose value is extremely 

 small ; their relative values are here of no importance. To decide 

 if we may reduce the expression 



rg (a-*) 2 _^_ r g 



&/(*)* ** toe H eZa/(a), 



Jo J o 



we must not examine whether the ratio of x to a is very great, 

 but whether the terms 77- &amp;gt; -TTI are very small numbers. This 

 condition always exists when t the time elapsed is extremely great ; 



/y* 



but it does not depend on the ratio - . 





381. Suppose now that we wish to ascertain how much time 

 ought to elapse in order that the temperatures of the part of the 

 solid included between x and x = X, may be represented very 

 nearly by the reduced equation 



