370 REPORTS ON THE STATE OP SCIENCE. 



When k(s,{) is a bounded function, the convergence of the series 

 may be established by means of a theorem in determinants due to 

 Hadamard l which states that if A, A denote two determinants [a lh ], 

 [«;,,] whose constituents a ik> d lk are conjugate complex quantities such 

 that \a rli \ < p then 



A A < n" p in 



If the constituents are all real the maximum value of the determinant 

 when Sa 2 rt = 1 occurs when the quantities a rk are the elements of an 



r 



orthogonal matrix. 



The case in which k(s,t) becomes infinite for s = t like -. r has 



(s - ty 



been considered by Predholm, Hilbert, Korn, 2 Lalesco, 3 Goursat, 4 and 



Poincare. 5 Fredholin shows that the integral equation 



b 



/(*) = tt s ) ~ ^Us,t) f {t)dt 



may be replaced by another one in which the kernel is the n ih iterated 

 function and n may be chosen so that this function is bounded. Hilbert 

 shows that if « < ^, Fredholm's series may be used provided the function 

 k(s, s) is replaced by zero wherever it occurs. This result has been 

 extended by Poincare and Lalesco to the case in which u > \. Instead 

 of deriving D(A) from Fredholm's expansion 



b b 



j& = - \K{s,s)ds — \UJ,s,s)ds 



a a 



b 



— \ n - l [ Kn {s,s)ds 



a function D(A) is used such that =. is equivalent to the above ex- 



pansion when a suitable number of terms have been omitted. The new 

 representation of K(s,i) is of the form 



*- {s,t) D(\) 



The case in which the kernel is such that 



b 

 a 



exists and is uniformly convergent is considered by Levi. 6 



1 Bull, des Sciences Alathimatiques, 1893, 2nd ser., Bd. 17. Simplified proofs 

 have been given by Hilbert (Lectures on the Calculus of Variations') ; Winter Semester 

 (Gottingen, 1901-05); Wirtinger (Monatshefte fur Math, und Physik, 1907, vol.xviii., 

 p. 158); Fischer (Grnnert's Archiv, 1908, p. 32). 



2 Comptes rendus, 1907. a Ibid. 



* Ann. de Toulouse, 1908. * Acta Math., 1909, t. 33, pp. 57-86. 



• Rend. Lincei, 1907, 2nd seni. p. 604. 



