SECT. V.] VAIIIED MOVEMENT IX A CUBE. 103 



129. It remains to form the equations which relate to the 

 state of the surface, which presents no difficulty, in accordance 

 with the principles which we have established. In fact, the 

 quantity of heat Avhich, during the instant dt : crosses the rectangle 



dz dy, traced on a plane perpendicular to x } is K dy dz -v- dt. 



This result, which applies to all points of the solid, ought to hold 

 when, the value of x is equal to I, half the thickness of the prism. 

 In this case, the rectangle dyds being situated at the surface, the 

 quantity of heat which crosses it, and is dispersed into the air 

 during the instant dt, is expressed by hvdydz dt, we ought there 

 fore to have, when x = l } the equation hv = K-j-. This con- 



CL*k 



dition must also be satisfied when x = I. 



It will be found also that, the quantity of heat which crosses 

 the rectangle dz dx situated on a plane perpendicular to the axis 



of y being in general Kdz dx -j- , and that which escapes at the 

 surface into &quot;the air across the same rectangle being hvdzdxdt, 

 we must have the equation hu + K-j- = Q, when y l or L 



U 



Lastly, we obtain in like manner the definite equation 



dz 



which is satisfied when z = I or L 



130. The function sought, which expresses the varied move 

 ment of heat in the interior of a solid of cubic form, must therefore 

 be determined by the following conditions : 



1st. It satisfies the general equation 



2nd. It satisfies the three definite equations 



, , 



dx ay 



which hold when x= 1, y = 1, z= 1; 



