( 310 ) 



-_^ hk k n{V) 



^ai O b <f{y) 



h=\ 



h=b—l 



'/lk\ 1 /■ 



and 



^ T,j,XPi/j]=-^-^^n2jtj {G) 



h—\ 

 Again the equation [F) supposes thai if we have 



k _ yt' 

 ~b~Y 

 no multiples of h' should be substituted for l> in I he snniniation at 

 the lefthand side. As for the equation (6^) this limitation is super- 

 fluous since the discontinuous function F{.v) vanishes for integer 

 values of x. 



As the solutions of the equation {F) and {G) seem in general to 

 present neither regularity nor symmetry, ^ve will proceed to consider 

 some particular cases. 



In the case h = 2, wq have at once 



1111 _ 



^Vo= -y + iT + ïö + ü ' 



1111 



^'^ 3 5 7 11 



Putting /^ = 3 and substituting k = 1 in {G) we find 

 11 1/3 



and since 



^'3, 1 4" ^'3, 2 = , 



we have 



I 1 1 1 _ 3|/3 



:/3.i = l— y + Y()~Ï3~Ï9 ~2^' 



12 11 __3|/3 



In the case h — Q we may apply [D). Thus we obtain relations 



_ 3^/2 

 ^6. 1 + ^'e. 5 = ^'2. 1 = 0, ï'e, 3= Ï3, 0=0, ï'e, 1 + ^'g, 4= ^3, i — -^, 



3^/3 



Te, 2 + Ï 6, 4 = :/^2, == , Te, •> + n, 5= ^'s, 2= - -^. 



Joining to these the equation resulting by substituting h = 6, 

 k = l in {G) 



