318 THEORY OF HEAT. [CHAP. VII. 



greater than any of the analogous fractions which enter into the 

 subsequent terms. 



Suppose now that we can observe the temperature of a point 

 on the axis of the prism situated at a very great distance x, and 

 the temperature of a point on this axis situated at the distance 

 x + 1, 1 being the unit of measure ; we have then y 0, z = 0, 

 and the ratio of the second temperature to the first is sensibly 

 equal to the fraction e~ ^ 2ni \ This value of the ratio of the tem 

 peratures at the two points on the axis becomes more exact as the 

 distance x increases. 



It follows from this that if we mark on the axis points each of 

 which is at a distance equal to the unit of measure from the pre 

 ceding, the ratio of the temperature of a point to that of the point 

 which precedes it, converges continually to the fraction e~^ 2ni z ; 

 thus the temperatures of points situated at equal distances end 

 by decreasing in geometrical progression. This law always holds, 

 whatever be the thickness of the bar, provided we consider points 

 situated at a great distance from the source of heat. 



It is easy to see, by means of the construction, that if the 

 quantity called I, which is half the thickness of the prism, is very 

 small, n { has a value very much smaller than n z , or ?? 3 , &c. ; it 

 follows from this that the first fraction e~ x ^ 2ni * is very much 

 greater than any of the analogous fractions. Thus, in the case in 

 which the thickness of the bar is very small, it is unnecessary to 

 be very far distant from the source of heat, in order that the tem 

 peratures of points equally distant may decrease in geometrical 

 progression. The law holds through the whole extent of the bar. 



330. If the half thickness Z is a very small quantity, the 

 general value of v is reduced to the first term which contains 

 e -x\/zn^^ Thus the function v which expresses the temperature of 

 a point whose co-ordinates are x, y, and z, is given in this case by 

 the equation 



(4 sin nl \ 2 -x-Jzn? 



=, . . 7 cos ny cos nz e , 



2nl + sm 2nlJ 



the arc e or nl becomes very small, as we see by the construction. 

 The equation e tan e = j- reduces then to e 2 = -r ; the first value of 



