SECT, i.] FOURIER S THEOREM. 351 



whence we conclude 



/- -+00 



TT [F(x) +/(#)] = TT&amp;lt;J&amp;gt; (x) = dq sin qx \ cZa/(a) sin qy. 



JO&quot; J -oo 



/. /+&amp;lt; 



4- I dq cos # I dzF (a) cos ^or, 



JO J - oo 



and TT&amp;lt; (a?) = I d^ g i n %% I dx(j&amp;gt; (a) sin qa. 



JO J-oo 



.00 -+W 



+ dg cos &amp;lt;?# I dz&amp;lt;j) (a) cos x, 

 Jo / -* 



or w$(#) = | di&amp;lt;f&amp;gt;(a)l dq(8mqx6 



J - 00 / t &quot;* 



or lastly 1 , f (*) -&amp;gt;~ f d *4&amp;gt; W f c!qcosq(x-a) 



TTj-oo JO &quot; 



The integration with respect to q gives a function of x and 

 a, and the second integration makes the variable a disappear. 



Thus the function represented by the definite integral Idqcosq (x a) 



has the singular property, that if we multiply it by any function 

 &amp;lt;/&amp;gt; (a) and by dx, and integrate it with respect to a between infinite 

 limits, the result is equal to TTCJ) (x) ; so that the effect of the inte 

 gration is to change a into a?, and to multiply by the number IT. 



362. We might deduce equation (E) directly from the theorem 



1 Poisson, in his Memoire sur la Theorie des Ondes, in iheMemoires de V Academic 

 dcs Sciences, Tome i. , Paris, 1818, pp. 85 87, first gave a direct proof of the theorem 



1 00 -(-so 



f(x) = - r dq r da e~ k ^ cos (gx - qa)f(a), 



in which k is supposed to be a small positive quantity which is made equal to 

 after the integrations. 



Boole, On the Analysis of Discontinuous Functions, in the Transactions of the 

 fioyal Irish Academy, Vol. xxi., Dublin, 1848, pp. 126130, introduces some ana 

 lytical representations of discontinuity, and regards Fourier s Theorem as unproved 

 unless equivalent to the above proposition. 



Deners, at the end of a Note sur quelques integrates definies &c., in the Bulletin 

 des Sciences, Societe Philomatique, Paris, 1819, pp. 161 166, indicates a proof of 

 Fourier s Theorem, which Poisson repeats in a modified form in the Journal Pobj- 

 technique, Cahier 19, p. 454. The special difficulties of this proof have been 

 noticed by De Morgan, Differential and Integral Calculus, pp. 619, 628. 



An excellent discussion of the class of proofs here alluded to is given by 

 Mr J. W. L. Glaisher in an article On sinac and cos oo , Messenger of Mathematics, 

 Ser. i., Vol. v., pp. 232244, Cambridge, 1871. [A. F.] 



