THE SUMS OF PERIODIC SERIES. 569 



series, which may be summed by the formula (41). The present example is a good one for 

 showing the utility of the methods contained in the present paper, inasmuch as in the Supplement 

 referred to I have pointed out the advantage of the formula contained in equation (6), with respect 

 to facility of numerical calculation, over one which I had previously arrived at by using develope- 

 ments, in series of cosines, of functions whose derivatives vanish for the limiting values of the 

 variable. 



45. Let it be required to determine the permanent state of temperature in a rectangle which 

 has two of its opposite edges kept up to given temperatures, varying from point to point, while 

 the other edges radiate into a space at a temperature zero. The rectangle is understood to be a 

 section of a rectangular bar of infinite length, which has all the points situated in the same line 

 parallel to the axis at the same temperature, so that the propagation of heat takes place in two 

 dimensions. 



Let the rectangle be referred to the rectangular axes of x, y, the axis of y coinciding with one 

 of the edges whose temperature is given, and the origin being in the middle point of the edge. 

 Let the unit of length be so chosen that the length of either edge parallel to the axis of x shall be 

 TT, and let 2/3 be the length of each of the other edges. Let u be the temperature at the point 

 (r, y), h the ratio of the exterior, to the interior conductivity. Then we have 



d' u d- u 

 dx- dy- 



dii 



— - hu = 0, when y = - /3 (80), 



dy 



du , ^ 



~~ + Jijt = 0, when y = j3 (81), 



dy 



u =fiy), when .(■ = (82), 



7/ = F(y), when x = a (83), 



f(y), P{y) being the given temperatures of two of the edges. 



According to the method by which Fourier has solved a similar problem, we should first 

 take a particular function Ke*^, where F is a function of y, and restrict it to satisfy (79)- This 

 gives V - A cos \y + B sin \y, A and B being arbitrary constants. We may of course take, still 

 satisfying (79), the sum of any number of such functions. It will be convenient to take together 

 the functions belonging to two values of \ which differ only in sign. We may therefore take, by 

 altering the arbitrary constants, 



u = 2:S^(e*<"-"'- e-*'-") + fi(6*^ -€-*")} cosXy, 



+ 2 50(6*'"-^' - £-*<'-^') + D(e*' -£-*")} sinXy (84), 



in which expression it will be sufficient to take only one of two values of \ which differ only by 

 sign, so that X, if real, may be taken positive. Substituting now in (80) and (SI) the value of // 

 given by (84.), we get either C = 0, D = 0, and 



X/3.tanX/3 = /i/3 (85), 



or else A = n, B = 0, and 



X/3.cotX/3= -k(i (86). 



It is easy to prove that the equation (85), in which X/3 is regarded as the unknown (luantity, 



TT 



has an infinite number of real positive roots lying between each even multiple of - , including zero. 



