394 THEORY OF HEAT. [CHAP. IX. 



whose temperature we wish to determine being x, y, z, the square of 

 the distance between these two points is (a xf + (6 y)*+ (c z} 2 ; 



and this quantity enters as a factor into the second term of -7- . 



Now the point m being very distant from the origin, it is 

 evident that the distance A from any point whatever of the heated 

 portion coincides with the distance D of the same point from the 

 origin ; that is to say, as the point m removes farther and farther 

 from the primitive source, which contains the origin of co-ordinates, 

 the final ratio of the distances D and A becomes 1. 



It follows from this that in equation (e) which gives the value 



of ^ the factor (a - xf + (b - yf + (c - zf- may be replaced by 

 dt 



$ 4. y* + or r 2 , denoting by r the distance of the point m from 

 the origin. We have then 



dv = /r^__3A 

 dt &quot; V P 2 1) 



or 



ai \ti ziy 



If we put for v its value, and replace t by -^. t in order to 



K 



re-establish the coefficient fTn w ^i ca we na( ^ supposed equal to 1, 



we have 

 dv 



GD 



394. This result belongs only to the points of the solid whose 

 distance from the origin is very great with respect to the greatest 

 dimension of the source. It must always be carefully noticed that 

 it does not follow from this condition that we can omit the varia 

 bles a, b, c under the exponential symbol. They ought only to be 

 omitted outside this symbol. In fact, the term which enters under 

 the signs of integration, and which multiplies / (a, 6, c), is the 



