( 640 ) 



Mathematics. — "The quotient of two successive Bessel Functions" . 

 (2°"i paper). By Prof. W. Kapteyn. 



In our preceding paper we gave tlie value of the general coefficient 

 of the expansion 



liz) 



^y;^+./;^" +./>' + 



Now we wish to draw the attention to a couple of relations which 



exist between these coefficients. The first is obtained from a particular 



integral of the following ditferential equation of Riccati 



du 



«" + 2 r « + ^^ = (1) 



ch 



Puttina: 



2 (V + 1) + M, 

 this ditferential equation I'educes to 



dz 



+ «,^ + 2 (r + 1) u, + z^ = 0. 



Repeating this process, it is evident that the equation (1) is satisfied 

 by the continued fraction 



2(i> + 2)- 



2(i>-|-3)— etc. 



2(i'+l)- 



which represents tlie value of — 



Introducing therefore 



u = - /, z' - ./; z' - ./; c» - etc. 

 in the equation (1) we have 



acr-fi)/, = 1 



2 (1. + n + 1 ) f„ + , = ƒ, ƒ„ 4- /, ƒ„-■ + ...+/„/,. . . (/) 

 where n = 1, 2, 3 . . . . 



The second relation may be deduced from oui' former equation 



a,»+i rt," . . . a„' rt„+i fu+\ — (— If 



