SECT. I.] INFINITE RECTANGULAR SOLID. 133 



temperature 1, whilst each of the two infinite sides B and C, 

 perpendicular to the base A, is submitted also at every point 

 to a constant temperature 0; it is required to determine what 

 must be the stationary temperature at any point of the plate. 



It is supposed that there is no loss of heat at the surface 

 of the plate, or, which is the same thing, we consider a solid 

 formed by superposing an infinite number of plates similar to 

 the preceding : the straight line Ax which divides the plate 

 into two equal parts is taken as the axis of x, and the co-ordinates 

 of any point m are x and y ; lastly, the width A of the plate 

 is represented by 21, or, to abridge the calculation, by IT, the 

 value of the ratio of the diameter to the circumference of a 

 circle. 



Imagine a point m of the solid plate B A (7, whose co-ordinates 

 are x and y, to have the actual temperature v, and that the 

 quantities v, which correspond to different points, are such that 

 110 change can happen in the temperatures, provided that the 

 temperature of every point of the base A is always 1, and that 

 the sides B and C retain at all their points the temperature 0. 



If at each point m a vertical co-ordinate be raised, equal to 

 the temperature v, a curved surface would be formed which 

 would extend above the plate and be prolonged to infinity. 

 We shall endeavour to find the nature of this surface, which 

 passes through a line drawn above the axis of y at a distance 

 equal to unity, and which cuts the horizontal plane of xy along 

 two infinite straight lines parallel to x. 



166. To apply the general equation 



di CD \dx 2 dy 2 d 

 we must consider that, in the case in question, abstraction is 



72 



made of the co-ordinate z, so that the term -y-n must be omitted ; 



az 



with respect to the first member -=- , it vanishes, since we wish to 

 determine the stationary temperatures ; thus the equation which 



