402 THEORY OF HEAT. [CHAP. IX. 



x and and t. It follows evidently from this equation (a) that the 

 form of the surface is determined, when we give the form of the 

 vertical section in the plane which passes through the axis of x : 

 and this follows also from the physical nature of the problem ; for 

 it is evident that, the initial state of the prism being given, all the 

 subsequent states are determined. But we could not construct 

 the surface, if it were only subject tcT passing through a curve 

 traced on the first vertical plane of i and v. It would be necessary 

 to know further the curve traced on a second vertical plane 

 parallel to the first, to which it may be supposed extremely near. 

 The same remarks apply to all partial differential equations, and 

 we see that the order of the equation does not determine in all 

 cases the number of the arbitrary functions. 



401. The series (T) of Article 399, which is derived from the 

 equation 



dv d?v 



may be put under the form v = e tD&amp;lt;i &amp;lt;f&amp;gt; (x). Developing the ex- 



d* 

 ponential according to powers of D, and writing -j-. instead of D\ 



considering i as the order of the differentiation, we have 



Following the same notation, the first part of the series (X) 

 (Art. 399), which contains only even powers of x, may be expressed 

 under the form cos (x ,J D) &amp;lt;j&amp;gt; (t). Develope according to powers 



of x, and write ^ instead of D\ considering i as the order of the 



differentiation. The second part of the series (X) can be derived 

 from the first by integrating with respect to x, and changing the 

 function &amp;lt; (t) into another arbitrary function ty (t). We have 

 therefore 



v = cos (tf^- !&amp;gt;)&amp;lt;/&amp;gt;()+ W 

 and -W = I *dx cos (x J^ 



