ON THE THEORY OF INTEGRAL EQUATIONS. 415 



may be reduced ' to such a set of equations by using a system of 

 orthogonal functions ^> m {x) belonging to the interval (a,b). Putting 



b 



Jm— \fUU m tr)dx 



tn 



b 



<j>{x)^ m {x)dx 



h h 



{x,t)bjx)4>n{t)dxdt 



we find that the above equation may be replaced by 



CO 



fm = foil — X S *W*n 



!! = 1 

 CO 



J m *^ * mn I n 



n = l 



respectively, and so all the above results may be applied to them 

 directly; the transition from the quantities f m to the function a,(x) 

 being finally effected by means of the expansion theorem, or one of the 

 alternative methods suggested by Eiesz and Fischer. 



The theory of linear equation has been further developed by 

 E. H. Moore, 2 who comprises it in a form of general analysis. 



A report on the development and present state of the theory has 

 been published recently by Helge v. Koch. Comptc rendu du Concjrcs 

 des Mathcmaticiens tenu a Stockholm (1909). 



The solution of systems of linear equations in oo, 2 unknown quantities 

 by means of infinite determinants, is of very great importance in the 

 theory of linear differential equations of the type 3 



d ^ + 2k ^ + (9 + 29, cos 2t + 29 2 cos it + -) w = ¥(t) 



which occur in the lunar theory, and in many acoustical and mechanical 

 problems. The problem; of the lunar theory have been dealt with in 

 this way by Hill, Poincare, and others. 



The various mechanical acoustical and optical problems considered 

 by Lord Kayleigh 4 and A. Stephenson 5 possess the interesting feature 

 that a certain state of equilibrium of a mechanical system is rendered 

 unstable by the action of a periodic force which does not tend directly 



1 Hilbert, Goltingen Na.clrichtcn, 1906. Heft i. A similar method has been 

 developed by A. C. Dixon (Proc. Lond. Math. Sue, ser. 2, vol. vii., 1909). 



■ Pone Congress, 1908, vol ii., pp. 98-111 ; ' Lectures at the New Haven Mathe- 

 matical Colloquium," Yale Univ. Pres?, 1910. 



* A good account of the theory is given in Forsyth's Theory of Differential 

 Equations, vol. iv., ch. 8. 



4 On the maintenance of vibrations by forces of double frequency and on the 

 propagation of waves through a medium endowed with a periodic structure (Phil. 

 Mur/., 1887, vol. xxiv., pp. 145-59 ; Scientific Pajiers, vol. iii.). 



5 Quarterly Journal, 1900; Phil. Mag., 1907, vol. xiv., p. 115; ibid., 190S; 

 Manchester Memoirs, 1908, vol. Iii. 



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