SECT. IV.] SUPERPOSITION OF EFFECTS. 163 



We proceed to shew that the variable state of the third solid 

 is determined by the equation v = (f&amp;gt;(x, y, t} + &amp;lt;!&amp;gt;(#, y, ) 



In fact, the temperature of a point m of the third solid varies, 

 because that molecule, whose volume is denoted by M, acquires 

 or loses a certain quantity of heat A. The increase of tempera 

 ture during the instant dt is 



the coefficient c denoting the specific capacity with respect to 

 volume. The variation of the temperature of the same point in 



the first solid is ~^ dt, and ^dt in the second, the letters 



d and D representing the quantity of heat positive or negative 

 which the molecule acquires by virtue of the action of all the 

 neighbouring molecules. Now it is easy to perceive that A 

 is equal to d + D. For proof it is sufficient to consider the 

 quantity of heat which the point m receives from another point 

 m belonging &quot;to the interior of the plate, or to the edges which 

 bound it. 



The point ??&,, whose initial temperature is denoted by f v 

 transmits, during the instant dt, to the molecule m, a quantity of 

 heat expressed by qj.f^ f)dt t the factor q l representing a certain 

 function of the distance between the two molecules. Thus the 

 whole quantity of heat acquired by in is S.q^f^f^jdt, the sign 

 2 expressing the sum of all the terms which would be found 

 by considering the other points m z , m 5 , ??? 4 &c. which act on m ; 

 that is to say, writing q 2 ,/ 2 or ^ 3 ,/ 3 , or q^ / 4 and so on, instead of 

 q v f v In the same manner ^q l (F l F)dt will be found to be 

 the expression of the whole quantity of heat acquired by the 

 same point in of the second solid ; and the factor q l is the same 

 as in the term 2$\C/i f)dt, since the two solids are formed of 

 the same matter, and the position of the points is the same; we 

 have then 



d = *?,(./; -/)* and D = Sfc(F, - F)dt, 

 For the same reason it will be found that 



112 



, A d T) 



hence A = d + D and -^ = ;, -f -j-, . 



cM cM cM 



