SECT, iv.] LAPLACE S FORM OF THE INTEGRAL. 399 



If we employ instead of a another unknown quantity ft 



making = ft we find 



%Jt 



(8). 



This form (8) of the integral l of equation (a) was given in 

 Volume VIII. of the Memoires de VEcole Poly technique, by M.Laplace, 

 who arrived at this result by considering the infinite series which 

 represents the integral. 



Each of the equations (/3), (7), (8) expresses the linear diffusion 

 of heat in a prism of infinite length. It is evident that these are 

 three forms of the same integral, and that not one can be con 

 sidered more general than the others. Each of them is contained 

 in the integral (a) from which it is derived, by giving to R an 

 infinite value. 



infm 



r s 



399. It is easy to develope the value of v deduced from 

 equation (a) in series arranged according to the increasing powers 

 of one or other variable. These developments are self-evident, 

 and we might dispense with referring to them; but they give rise 

 to remarks useful in the investigation of integrals. Denoting by 



&amp;lt;j&amp;gt;, &amp;lt;&quot;, (f&amp;gt;&quot;, &c., the functions -7- &amp;lt;(#), -j 2 $(#&quot;)&amp;gt; ~T~3 $( x }&amp;gt; & c -&amp;gt; we 

 have i 



dv /, , r 7 // T~&quot;* \^ 



-77 = v , and v = c + 1 at v ; 

 1 A direct proof of the equivalence of the forms 



tt 



t 

 F&amp;lt;f&amp;gt; (x + 2/3^) and e dic2 $ (x), (see Art. 401), 



has been given by Mr Glaisher in the Messenger of Mathematics, June 1876, p. 30. 

 Expanding &amp;lt;(&amp;gt;(x+2pJt) by Taylor s Theorem, integrate each term separately: 

 terms involving uneven powers of &amp;gt;Jt vanish, and we have the second form ; 

 which is therefore equivalent to 



]_ /*&amp;gt; [3 



~ I da I 



T y- Jo 



from which the first form may be derived as above. We have thus a slightly 

 generalized form of Fourier s Theorem, p. 351. [A. F.] 



