SECT. VII.] MOVEMENT IX THREE DIMENSIONS. 73 



In general the theorems concerning the heating of air in 

 closed spaces extend to a great variety of problems. It would 

 be useful to revert to them when we wish to foresee and regulate 

 temperature with precision, as in the case of green-houses, drying- 

 houses, sheep-folds, work-shops, or in many civil establishments, 

 such as hospitals, barracks, places of assembly. 



In these different applications we must attend to accessory 

 circumstances which modify the results of analysis, such as the 

 unequal thickness of different parts of the enclosure, the intro 

 duction of air, &c. ; but these details would draw us away from 

 our chief object, which is the exact demonstration of general 

 principles. 



For the rest, we have considered only, in what has just been 

 said, the permanent state of temperature in closed spaces. AVe 

 can in addition express analytically the variable state which 

 precedes, or that which begins to take place when the source of 

 heat is withdrawn, and we can also ascertain in this way, how 

 the specific properties of the bodies which we employ, or their 

 dimensions affect the progress and duration of the heating ; but 

 these researches require a different analysis, the principles of 

 which will be explained in the following chapters. 



SECTION VII. 



On the uniform movement of heat in three dimensions. 



92. Up to this time we have considered the uniform move 

 ment of heat in one dimension only, but it is easy to apply the 

 same principles to the case in which heat is propagated uniformly 

 in three directions at right angles. 



Suppose the different points of a solid enclosed by six planes 

 at right angles to have unequal actual temperatures represented 

 by the linear equation v = A -f ax + by + cz, x, y, z, being the 

 rectangular co-ordinates of a molecule whose temperature is v. 

 Suppose further that any external causes whatever acting on the 

 six faces of the prism maintain every one of the molecules situated 

 on the surface, at its actual temperature expressed by the general 



equation 



v A -f ax + by + cz (a), 



