384 REPORTS ON THE STATE OF SCIENCE 



for then 



^> m (s)x„(s)ds = U,tt)Mt)dt. 



A method of deriving a large class of orthogonal and biorthogonal 

 systems of functions from a single integral equation has been given by 

 the author. 1 



J. P. Gram 2 and E. Schmidt 3 have given a method of constructing 

 an orthogonal system of functions from a linearly independent set of 

 functions « B (s). It is assumed that 



»M S ) = ^21«l( S ) + *22 r ' 2 (s) 



i// 3 (s) = Xgaa^s) + X 8 2 a 2(s) + ^f'sOs) 



and th3 constants \ M are chosen so that the functions if/ n (s) form an 

 orthogonal system. 



It has already been noticed that orthogonal and biorthogonal systems 

 of functions occur as the solutions of linear differential and integral 

 equations ; they also occur in certain problems of the calculus of variations, 

 for instance, the problem of finding sets of functions f „(s) \p n (t), so that 



f f [* ( M) - \tm^ dsdi 



may be a minimum (Schmidt). 



The best known examples of orthogonal sets of functions are the 

 following : — 



!l 



If n , = 2h n \p n (s), then the functions 4/„{s) are orthogonal for the 



-I — fl ,1=0 



interval o < s < 1 (Abel 4 and Murphy 5 ). 



H Q p (s) = c*f sp (c-^) 



denotes the polynomial of Laguerre, 6 the functions 



form an orthogonal system for the interval (o < s < oo). 



" P ; ,(s) = (-l)^( C -^ 



denotes the polynomial introduced by Tschebyscheff 7 and Hermite, 8 the 



functions 



Us) = 4/o / , 



1 Cam b. Phil. Trans., vol xx. (1907), p. 281. = Crete's Journal, Bd. xciv. 



3 Dissertation Qottingm, 1005; Math. Ann., 1907. 



4 Memoires de mathrmallqves, par N. H. Abel (Paris, 1826), pp. 75-79. 



5 Trtwftr. Phil. Trans, 1833. 



6 (Uurres, t. 1, p. 428. » St. Petersburg Memoirs, 1860. 

 8 Comjrtes rcndus, 1864, t. 58, pp. 93, 266. 



