ON THE THEORY OF INTEGRAL EQUATIONS. 349 



Keymond. 1 The conditions were simplified by C. Neumann, 2 Dini, 3 and 



C. Jordan. 4 The last-named proved equation (3) on the supposition 



that i/.(.c) is a bounded function of limited total fluctuation such that 

 i 



CO 



is convergent. These have practically remained the standard conditions 

 until the present time, but the results have been recently established 

 under slightly different conditions by Hilb, 5 Weyl, 6 Orlando, 7 Hobson, 8 

 and Pringsheim. 9 A good account of the present state of the theory is 

 given in Pringsheim's papers. The formulae have been discussed by 

 Hobson from the point of view of the Lebesgue integral. Cauchy 10 and 

 Poisson u discussed the formulae (3) by introducing a convergence 

 factor e" ,; \ changing the order of integration and finally making k tend 

 to zero. This enables us to use the formula in cases when the integrals 

 are not convergent without the factor e ~*\ The method has been dis- 

 cussed by Boole '- and other writers, and has been extended to other 

 integral formulae by Sommerfeld, 13 Hardy, 14 and Orr.' 5 



3. Applications of Fourier's Formula. 



Fourier's formulae are of very great importance in all branches of 

 mathematical physics in which the phenomena are expressed by means 

 of linear partial differential equations. Whenever the equations possess 

 solutions of the form 



F(rt,£,?7 . . .) cos (at + e) 



the solution may be generalised by Fourier's theorem. The method 

 consists in multiplying by an arbitrary function of a and integrating 

 between o and oo. The introduction of the boundary conditions peculiar 

 to the problem then leads to an integral equation for the unknown 

 function, and this may be solved by Fourier's inversion formula. This 

 method of reducing a problem in partial differential equations to the 

 solution of an integral equation is of very wide application, and the 

 integral equations which occur in this way are of primary importance. 

 It also happens that in many cases the inversion formula can be 

 expressed in a concise form. 



Cauchy's method of solving partial differential equations by means 

 of definite integrals is founded on an application of Fourier's theorem, 

 or at least the analogous theorem in several variables. 



1 Math. Ann., Bd. 4, pp. 362-390 ; Crelle, Bd. 79. 



2 Ueber die naeh Kreis-, Kvyel-, und Cylinder- Fun ctionen fortschreitenden 

 Entnichelungen, Leipzig (1881). 



3 Serie di Fourier, Pisa (1880). 4 Cours d' Analyse, t. 2, Paris (1894). 

 5 Math. Ami., 1908, Bd. 66, heft 1. 6 Dissertation Gottinycn, 1908. 



' Bend. Lined, October 1906, vol. xviii., 2nd ser.,1909. 



9 Froc. Lond. Math. Soc., 1908. Theory of Functions of a Real Variable Ch. 



• Jahresbericht der Deutsch. Math. Ver.. 1907, Bd. 16 ; Math. Ann., 1910, Bd. 68, 

 pp. 367-408. 



10 Mcmoire sur la thcoric des Ondes, 1815. " Ibid., 1816. 



12 Trans. Irish Acad., vol. 21 (1818). " Dissertation Kbnigsberg, 1891. 

 " Camb. Phil. Trans., vol. xxi. No. 1, pp. 1-48. 



13 Proc. Irish Acad., 1909, vol. xxvii., Sect. A. 



1'JIO. A A 



