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Mathematics, -r- "On a special chus of homogeneous linear dif- 

 ferential equations of the second order" . By Prof. W. Kapteyn. 



The differential equation of Legendkk 



d^y dy 



.i(l-:^') ~ - 2.^ ;^ + n {n\-l) y = 



is satisfied by a' polynomiiim P„ {x) of the ?z*^ degree and by a 



function Qn (x) which may be reduced to the form 



1 



dz 



Qn (4 = j -^3-7 



— 1 



This function however is not determined for real values of the 

 variable in the interval — 1 to -|- 1 , the difference on both sides 

 of this line being 2/jr Fn {x). 



In analogy to this we have examined the question: to determine 

 all homogeneous linear differential equations of the second order of 

 the form 



_ d^ii dy 



dx^ dx 



where the coefficients are polynomia in ,x, which possess the property 

 that y^{x) being a first particular integral, the second integral may 

 be written 



. . CyMdr 

 y^ (■^') = I 



J .r, — z 



'J. 



where a and /? represent two real values, supposing moreover that 

 this integral has a meaning everywhere except on the line of dis- 

 continuity. 

 Let 



R{x)^kr^xP , S{x)^kH.,xP , T{x)-=.kt^,xP 

 Ü 



then we obtain firstly the conditions 



n {x) = (..-«) (.;-/?) r {x) = (x-a) (..-/?) V(),. xP 



Ü 



/— 2 

 .S (.7;) = H' {x) 4- (x—a) (x-^) 2 h,, XP. 







If now we put 



G,'=JzPy:'{z)dz , G;=JzPy,'{z)dz , G,, =j zP y^ (z) dz 



