TABLE OF CONTENTS. XI 



SECTION VIII. 



APPLICATION OF THE GENERAL EQUATIONS. 



ART. PAGE 



155, 156. In applying the general equation (A) to the case of the cylinder 

 and of the sphere, we find the same equations as those of Section III. 

 and of Section II. of this chapter 123 



SECTION IX. 



GENERAL BEMARKS. 



157162. Fundamental considerations on the nature of the quantities 

 x, t, r, K, h, C, D, which enter into all the analytical expressions of the 

 Theory of Heat. Each of these quantities has an exponent of dimension 

 which relates to the length, or to the duration, or to the temperature. 

 These exponents are found by making the units of measure vary . . 126 



CHAPTER III. 



Propagation of Heat in an infinite rectangular solid. 



SECTION I. 



STATEMENT OF THE PROBLEM. 



163166. The constant temperatures of a rectangular plate included be 

 tween two parallel infinite sides, maintained at the temperature 0, are 



expressed by the equation -^ + -^=0 131 



167 170. If we consider the state of the plate at a very great distance from 

 the transverse edge, the ratio of the temperatures of two points whose 

 coordinates are a^, y and x z ,y changes according as the value of y 

 increases ; x l and x. 2 preserving their respective values. The ratio has 

 a limit to which it approaches more and more, and when y is infinite, 

 it is expressed by the product of a function of x and of a function of y. 

 This remark suffices to disclose the general form of v, namely, 



^ = S): i V~ (2&amp;lt; ~ 1)a: . cos(2i-l).y. 



It is easy to ascertain how the movement of heat in the plate is 

 effected 134 



