SECT. VII.] GENERAL SURFACE EQUATION. 11? 



we have also 



,^-s Bi a &s 2 =-(m 2 



or a. = ^ &z , denoting by q the quantity (m 2 + n* + p 2 ) &quot; , 



w vfdv dv , cfaA 1 

 hence - = [m -, + n-j-+p-j- 1- ; 



a \ dx dy L dzj q 



consequently the equation 



becomes the followin 





dv dv 



This equation is definite and applies only to points at the 

 surface ; it is that which must be added to the general equation of 

 the propagation of heat (A), and to the condition which deter 

 mines the initial state of the solid ; m, n, p, q, are known functions 

 of the co-ordinates of the points on the surface. 



148. The equation (B) signifies in general that the decrease of 

 the temperature, in the direction of the normal, at the boundary of 

 the solid, is such that the quantity of heat which tends to escape 

 by virtue of the action of the molecules, is equivalent always to 

 that which the body must lose in the medium. 



The mass of the solid might be imagined to be prolonged, 

 in such a manner that the surface, instead of being exposed to the 

 air, belonged at the same time to the body which it bounds, and 

 to the mass of a solid envelope which contained it. If, on this 

 hypothesis, any cause whatever regulated at every instant the 

 decrease of the temperatures in the solid envelope, and determined 

 it in such a manner that the condition expressed by the equation 

 (B) was always satisfied, the action of the envelope would take the 



1 Let .ZV be the normal, 



the rest as in the text. [B. L. E.] 



dv m dv 



-7T7 = -T- + &c. ; 



&amp;lt;LV q dx 



