402 REPORTS ON THE STATE OP SCIENCE, 



may be a minimum when u(x) satisfies the condition 



b 



i[u(x)] 2 dx = l 



a 



which is introduced to exclude the case of u(x) = 0. 



The linear differential equation corresponding to this isoporiinetric 

 problem is 



L(u) + \u = 



u. 



The function u(x) must satisfy' this equation and the conditions 

 already laid down, and this can only be done for certain values of A. 



Tins theory is developed in the Gottingen dissertations of Konig (1907) 

 and Cairns (1907). The latter considers a function which makes the 

 integral CI a minimum subject to the two conditions 



h b 



iu(x)g(x) = 0, L\x)dx = 1, 



and arrives at a differential equation 



L(«) + Xu + \'g(x) = 0. 



He shows that if r(.r,£) is the Green's function for the differential 

 equation when A' = 0, and 



6 



V (x) = j" r(*,*)flf(*)#, 



then the Green's function in the case when A is not zero is given by the 

 formula 



G(x,{) = T(x,i) - /**M0 

 v(t)g(t)dt. 



! 



This result is closely connected with a general theorem given by the 

 author. 1 



Konig discusses Hilbert's problem in connection with Jacobi's 

 criterion and the oscillation theorem. 



A fuller investigation has been given recently by Eichardson. 2 



20. Riemanris Problem. 



The work of Fuchs raised the question as to how far the branch 

 points and monodromic group belonging to a linear differential equation 

 can be considered as independent of one another. Eiemann endeavoured 

 to answer this question in 1857 by considering the problem of determining 



1 Cambr. Phil. Trans., 1907, vol. xx., pp. 281-290. 



2 Math, Ann., 1910, Bd. lxviii., p. 279. 



