1277 



and if we substitute in this formula 



(fj{ii) = kJK 



i^'V) Y'.7"('')<^''N 



7/(^0- ^ 



'J 





(fj{ic)dx 



and otiier analogous expressions for v/fi^) and ff/irj), with // as I ho 

 vai-iahle o\' integration, we get 



b b 



''U — ''.)' 



'f.iU')'ri(y)<^-''(hi 



'I It 



— ;.' I \L{x,y]((j{.x^(fi{y)dxdy = 0. 



This proves that cij = cj;. By means of an orthogonal transforma- 

 tion it is always possible to find ;/ other functions t|?,(.r'), ...., i|j„(.r), 

 lineai-ly expressed in <p-^[.v), . . . ., (/'„{■!'), orthogonal and normalized, 

 but suoh, Ihat instead of (15) they satisfy equations of tlie form 



i.e. (8). This proves the condition A(,t%y) = to be sufficienl. 



We now suppose that we know a complete system of orthogonal 

 characleristic functions of K{x,i/) to consisi of solutions of (8). If 



f/',(.i'), . . . ., '(;{■>:), ■ • • • be that system and ƒ«, , iji the cor- 



i-esponding values of »/, it follows from f 12) and (18) that 



/; h 



^.,(pii'>') H^ !h'f{'i') = >■<• I A'(.?;,_(/) \l\,f(fi{ii) + Hi<f{y)\ dy—li' I L{x,ii)((;{y)dy 



a ' a 



and so, as (fd^) satisfies the differential equation, 





fpi (y) dy = 



for all values of /. Consequently A(.r,//) is a function of //, that is 

 orthogonal to all characteristic functions of A(, /•.?/) and this proves thai 



b 

 fA{.r,y)Kiy, z)dy -0 . . ' . . . . (16) 

 a 



for all values of x and : in (</,/>\ P'l-om this we get 



813 

 Proceedings Royal Acad. Amsterdam. Vol. XXI. 



