424 REPORTS ON THE STATE OP SCIENCE. 



1 



f(x) + X^(x,y)F{f(y)}dy = G(x) ... (1) 



o 



is unique for values of X whose moduli are sufficiently small, but the 

 equation appears to possess a ' band spectrum ' for values of X whose 

 moduli are greater than a certain number. 

 Equations of the type 



b 



f(x) = f(x) + X Ux,y)<j>(y)dy + Mg{x,y,z)f{y)i,{z)dydz + . (2) 



a a a 



are of considerable interest, partly on account of their connection with 

 Volterra's expansion theorem for functions which depend on other 

 functions and partly for their connection with non-linear differential 

 equations. There appears to be an intimate connection between equa- 

 tions of the forms (1) and (2) because a non-linear differential equation 

 may be reduced to either form by employing different artifices. For 

 instance the equation 



is reduced to the form 



% + 'V + rt«) = o 



1 



y(x) = ^G(x,t)[\{y(t)}* + g(t)]dt 



jo 



by using a Green's function of the differential equation \ = 0. It is 

 also reduced to the form 



1 1 



0(<r) = Xj \G(x,s)G(x,t)<l>(s)f(i)dsdt + g{x) 



o 



by the assumption 



i 



y(x) =[a{x,t)f(t)dt, 



o 



Examples of this type may of course be multiplied indefinitely. For 

 the purpose of studying equations of type (1) it is convenient to consider 

 some well-known equation such as the pendulum equation 



jli + k2 sin y = ° 



reducing it to an integral equation of type (1) by the first method. 



For a discussion of the properties of equations of type (2) we must 

 refer to Schmidt's memoir. 



