SECT. IV.] COEXISTENCE OF PARTIAL STATES. 1,55 



This value of v satisfies the equation -j t + -^ = ; it becomes 



nothing when we give to y a value equal to \TT or JTT ; lastly, 

 it is equal to unity when x is nothing and y is included between 

 ^TT and + |TT. Thus all the physical conditions of the problem 

 are exactly fulfilled, and it is certain that, if we give to each 

 point of the plate the temperature which equation (a) deter 

 mines, and if the base A be maintained at the same time at the 

 temperature 1, and the infinite edges B and C at the tempera 

 ture 0, it would be impossible for any change to occur in the 

 system of temperatures. 



191. The second member of equation (a) having the form 

 of an exceedingly convergent series, it is always easy to deter 

 mine numerically the temperature of a point whose co-ordinates 

 os and y are known. The solution gives rise to various results 

 which it is necessary to remark, since they belong also to the 

 general theory. 



If the point m, whose fixed temperature is considered, is very 

 distant from the origin A, the value of the second member of 

 the equation (a) will be very nearly equal to e~ x cos y it reduces 

 to this term if x is infinite. 



4 

 The equation v = - e~ x cos y represents also a state of the 



solid which would be preserved without any change, if it were 

 once formed ; the same would be the case with the state repre- 



4 



sented by the equation v ^ e 3x cos %y, and in general each 



O7T 



term of the series corresponds to a particular state which enjoys 

 the same property. All these partial systems exist at once in 

 that which equation (a) represents ; they are superposed, and 

 the movement of heat takes place with respect to each of them 

 as if it alone existed. In the state which corresponds to any 

 one of these terms, the fixed temperatures of the points of the 

 base A differ from one point to another, and this is the only con 

 dition of the problem which is not fulfilled ; but the general state 

 which results from the sum of all the terms satisfies this special 

 condition. 



According as the point whose temperature is considered is 



