( 407 ) 

 and ' '1 



N={ai-^)G:-G:-2G: 

 m = ~G: 



n = {a^^)G:~G;-G, 

 the further necessary conditions may be deduced from the equation 



1 4- J = 



where / ajid J represent the following polynomia of degree ;. — 1 



/—I 



1= :e{q,,n^ (jp_^ M) .fp 



/—I 

 . . ... . . , + -^ [j^^/^ + 0^ + 1) Qp+\] 't + l^'/J-i + P9/>\ ^«] ■'^^ 



;— I 



J= 2 (r^+i Gp' + .s^+i 6^^' 4- tpj^x Op) 



p=0 



;.— 2 



-{- X 2! {rp + 2 Gp" + .^^-|_2 Gp + <^_|_2 6?^;) 

 + •..., 



+ ,.—2 ^ (r^+;_i 6^;' 4- 6>+;,_l G; + tp^;,^, Gp) 



p=0 . :,,,.,. ..^;. ^,,,. . 



+ .t— 1 2 {rp^. Gp" + .v+' ^p' + tp+y Gp). 



p=0 



From this we may easily deduce that if ). = '2, the most general 

 differential equation of the second order possessing the property in 

 question is '; 



(^-«) i^-^ ■— + Vh ('^'-«) {^-^ + 2.r - « - I?] ^- + {t.^Jrh'^:)y=, 



where a, /?, t^ and ^j are arbitrary constants. 



When P. = 3 the most general equation may be ^vritten 



a A' ax 



J^{t,x'^t,x-\-t,)y^{) 



Here however the ten constants must satisf}- the following three 

 conditions 



., + («+.i) ., + («•^+«p?+/J-0 <, = -iQ, + («4-^) ^^ 



