286 THEORY OF HEAT. [CHAP. V. 



satisfy the fundamental condition of the instrument, which is, that 

 any two intervals on the scale which include the same number of 

 degrees should contain the same quantity of mercury. For the 

 rest we omit here several details which do not directly belong to 

 the object of our work. 



301. We have determined in the preceding articles the tem 

 perature v received after the lapse of a time t by an interior 

 spherical layer at a distance x from the centre. It is required 

 now to calculate the value of the mean temperature of the sphere, 

 or that which the solid would have if the whole quantity of heat 

 which it contains were equally distributed throughout the whole 



mass. The volume of a sphere whose radius is x being Q , 



o 



the quantity of heat contained in a spherical envelope whose 

 temperature is v, and radius x } will be vdl-^-J. Hence the 

 mean temperature is 



PrS 



J n 



or 



the integral being taken from x to x = X. Substitute for v 

 its value 



e~ kniH sin n.x + e~ kn * H sin njx + e~ kn ** sin njc -f etc. 



X X X 



and we shall have the equation 



We found formerly (Art. 293) 



2 sin n t X n,X cos n,X 



a.= -- - i . 



