j^ (•'••//) 



1278 



d>i — ü 



We now liave 



b b 



ƒ |A(.r,v)r-"rfv = i 





(16«) 



f/y = 



=\ v{n)Ki'>\h) iuxjj) ^^'^"^ dii 



/>(»/ 



dK(xOi) C 



ón 



'I^C'w 



)K{ii,y)dy 



and this becomes O in consecinenee of (J6j and (16rï). As A(,t',v) i» 

 a continnons function of //, this |uo\es il to be zero for all values 

 of A' and // in {a,h). Hence the condition is necessary. 



hj.varnnle. Put L,=z - . The kernel /\(.r,y) will have the foi-in 

 ' dx' '' 



K{x,y) = F{x-^y) i- <P{x—y) 

 witli '/»(.'• — y) =r. f/>{j/~.v). We suppose {/=:(), h=[. We then ^et 

 h{x,y) = ({F(.r+1) -f ,fi{.r^])\lF'(yi-l)—<I*y- \)\ -- 



- \F'ix^l) ~- ip'(.c-~l)\[F(y^\) 4- 'ï»(y + l){) 

 - {\F{.>-) + 0{.v)\\F{y)-<t>'{y)\ - |F(.r)— '^''(^)l mv) ^ 'P(y)\) 

 In consequence of (3) A(.r,//) becomes zero; if we suppose (6) to 

 be satisfied (after substituting in that equation F for ƒ and '/' for 

 '/) the expression A(.r,y) also becomes zero. In botli cases the charac- 

 teristic functions are solutions of i,l), in agreement to what we have 

 found in § 2 and § 3. 



^ 5. We will now consider condition (14) more closelv. Putting 

 for brevity 



-.-- = P{^), ~—~ = Q(^), K(xM) = .%/•) K{.v,a) ^ T{xl 



0»J r.=zb 0»j y—,t 



it becomes 



p{h)\S[x)P{y)-T{x)S{y)\=p{a)\T{x)Q{y)-Q{x)T{y)] . . (14a) 



We first suppose p{b):^0, p{a)^0 alid we assume the existence 

 of two values y, and y, in {a,h), for which -S(y,) P(y,; — S{y^) P{y^) 7^ 0. 

 If for brevity we write P^, etc., P^, etc. instead of /^(//,), etc., /^(y^), 

 etc., equation (J 4a), after substitution of //, and //, for //, becomes 



p(b)PA-'>') - /.(6)8,P(.^-) = p(a)Q, Tix) - /Ha)'^\Q(-^-), 

 p(b)P,Six) — p{b)S,P{x) = p{a)QJix) - p(ar/\Q(x). 



Eliminating P{.r) or .S'(.r) we get 



