336 THEORY OF HEAT. [CHAP. IX. 



more simple, suppose the initial form of the curve to be composed 

 of two symmetrical parts, so that we have the condition 



F(x)=F(-x}. 



JL-i. HL - 



CD~ CDS~ 

 dt^ 



in the equation ~rr k-j 2 hv, make v = e~ ht u, and we have 



du -, d*u 



dt dz* Jc v 



\ 



Assume a particular value of u, namely, a cos qx e&quot;^ 1 ; a and q 

 being arbitrary constants. Let q v q 2 , q 3 , &c. be a series of any 

 values whatever, and a l9 a 2 , a 3 , &c. a series of corresponding 

 values of the coefficient Q, we have 



u = a l cos fax) e~*&amp;lt;zi 2&amp;lt; + a 2 cos faai) e~ kq ^ + a a cos fax) e-^* + &c. 

 Suppose first that the values q lt q^, q s , &c. increase by infinitely 

 small degrees, as the abscissa q of a certain curve ; so that they 

 become equal to dq, 2dq, 3dq&amp;gt; &c. ; dq being the constant differen 

 tial of the abscissa; next that the values a^ a 2 , a 3 &amp;gt; &c. are pro 

 portional to the ordinates Q of the same curve, and that they 

 become equal to Q^dq, Q^dq, Q 3 dq, &c., Q being a certain function 

 of q. It follows from this that the value of u may be expressed 

 thus : 



u = Idq Q cos qx e~ ktjH } 



Q is an arbitrary function f(q), and the integral may be taken 

 from q Q to q=vo. The difficulty is reduced to determining 

 suitably the function Q. 



346. To determine Q, we must suppose t in the expression 

 for u, and equate u to F (x). We have therefore the equation of 

 condition 



If we substituted for Q any function of q, and conducted the 

 integration from q = to q = oo, we should find a function of x : 



it is required to solve the inverse problem, that is to say, to 

 ascertain whatranctioii of q, after being substituted for Q, gives 

 as the result the function F(x) t a remarkable problem whose 

 solution demands attentive examination. 



