344 THEORY OF HEAT. [df AP. IX. 



formed of two similar and alternate arcs, the integral which gives 

 the value of the temperature is 



TTV 

 



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= e~ u \ dxf(a) da e~ kqH s m qx sin qa.. 

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If we suppose the initial heat to be distributed in any manner, 

 it will be easy to derive the expression for v from the two preced 

 ing solutions. In fact, whatever the function $ (x) may be, which 

 represents the given initial temperature, it can always be decom 

 posed into two others F (x) +/(#), one of which corresponds to the 

 line FFFF, and the other to the \iueffff, so that we have these 

 three conditions 



F(x) = *(-*),/(*) = -/(- *), &amp;lt;}&amp;gt; () = F(x) +f(x). 



We have already made use of this remark in Articles 233 and 

 234. We know also that each initial state gives rise to a variable 

 partial state which is formed as if it alone existed. The composi 

 tion of these different states introduces no change into the tem 

 peratures which would have occurred separately from each of 

 them. It follows from this that denoting by v the variable tem 

 perature produced by the initial state which represents the total 

 function cf&amp;gt; (x), we must have 



-. / r r 



_ e -u M fa g-*a^ CO s qx I dot. F (a) cos qy. 



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+ 1 dq e-**** sin. qx I da/(a) sin qaj. 

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If we took the integrals with respect to a between the limits 

 oo and + oo , it is evident that we should double the results. 

 We may then, in the preceding equation, omit from the first 

 member the denominator 2, and take the integrals with respect to 

 a in the second form a = oo toa = + oo. We easily see also 



r+&amp;lt;x&amp;gt; r+oo 



that we could write I da $ (a) cos ga, instead of I da. F(a) cos qy. ; 



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for it follows from the condition to which the function /(a) is sub 

 ject, that we must have 



r+ao 

 = I daf(ot) cosqy. 



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