384 THEORY OF HEAT. [CHAP. IX. 



and the variables a, /3, 7 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+ p, one of which is very small 

 with respect to the other. But from the fact that the ratio 



^ is a very small fraction, we cannot conclude that the ex 



ponential e H+ * becomes equal to e M , 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 JJL, but only the absolute 

 value of yLt. In order that we may be able to reduce the exact 

 integral (j) to the. equation 



e m 



=jB 



it is necessary that the quantity 



2ao; + 2ffy + fyz - a* - ft 2 - 7* 



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 . 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. 



