1206 



wliich may be verified by the equalions (19) and (20). 

 10. We shall now show that the function 



where k is an arbitrary constant, may be developed in a conti- 

 nuous fraction. 



DiiFerentiating and eliminating k, the dilierential equation for 6, 

 takes the form 



do ( dz^ <^-A 



\'^ dx ^ dx ^ dx ' dx ) \ ^ dx ' " dx ) 



where 



Z^zzz Hn (.v) Z^^= Ln {x) 



According to (17) the coefficients of this equation may be written 

 dx dx- 



dVi dz^ dz^ dy.^ 



dx dx dx dx 



^.-P - ^1 -e + 3/.-T^ - ^1 4:' = 2«(//n-iZ.,.+i-//„+iA,-i) = -2"+2,,_,,,,x' 



2/1 ^' - 3/, ^ = 2 (n + 1) (ff,.+iL„ - if„i„+i) = 2"+2 (,,+ 1)/ ,x^ 

 air aa; 



thus 



— = 0' — 2;rö + 2(« + l) (23) 



dx 



Substituting 



2n 



(7 = 2 a; 



the function <t, satisfies an equation of the same kind viz. 



do, 



— ^ = (?^' — 2a'(7j + 2w. 



Substituting again 



2(^^-1) 

 a. = 2.r 



^, 

 the transformed equation is 



^ = (7,^ -2..Ö, + 2(«-l). 

 dx 



