SECT. I.] INFINITE RECTANGULAR PLATE. 135 



constant quantity, and as it is proposed only to find a particular 

 value of v, we deduce from the preceding equations F(x) = e~ mx } 

 /(?/)= cos my. 



168. We could not suppose m to be a negative number, 

 and we must necessarily exclude all particular values of v, into 

 which terms such as e mx might enter, m being a positive number, 

 since the temperature v cannot become infinite when x is in 

 finitely great. In fact, no heat being supplied except from the 

 constant source A y only an extremely small portion can arrive 

 at those parts of space which are very far removed from the 

 source. The remainder is diverted more and more towards the 

 infinite edges B and C, and is lost in the cold masses which 

 bound them. 



The exponent m which enters into the function e~&quot; lr cosmy 

 is unknown, and we may choose for this exponent any positive 

 number: but, in order that v may become nul on making 

 y = | TT or y = + |- TT, whatever x may be, m must be taken 

 to be one of the terms of the series, 1, 3, 5, 7, &c. ; by this 

 means the second condition will be fulfilled. 



169. A more general value of v is easily formed by adding 

 together several terms similar to the preceding, and we have 



le~ 3x cos 3j/ -f- ce~ 5x cos 5y + de~ lx cos 7y + &c. . . f. . . 



It is evident that the function v denoted by $ (x, y) satis! 

 the equation -^ + -=- = 0, and the condition &amp;lt;f&amp;gt; (x, J TT) = 0. 



A third condition remains to be fulfilled, which is expressed thus, 

 &amp;lt;f&amp;gt; (0, y) = 1, and it is essential to remark that this result must 

 exist when we give to y any value whatever included between 

 \ TT and -f J TT. Nothing can be inferred as to the values 

 which the function &amp;lt;f&amp;gt; (0, y) would take, if we substituted in place 

 of y a quantity not included between the limits J TT and -f J TT. 

 Equation (b) must therefore be subject to the following condition : 



1 = a cos y + b cos 3^ + c cos 5y + d cos 7y + &c. 



The coefficients, a, b, c, d, &c., whose number is infinite, are 

 determined by means of this equation. 



The second member is a function of y, which is equal to 1 



