( 547 ) 



Mathematics. — "The qaolient of tiro ■■<i(ccesi>ii\' Be.sse! Ftinction-f." 

 By Prof. W. Kaptevn. 



If /■■+'(-) 'T-iid /'(;) represent two successive Bessel Functions of 

 the lirst kind, the quotient may be expanded as follows: 

 /■'+i (z) 



liz) 



Of course this equation holds for all values of z within a circle 

 whose radius is equal to the modulus of the tirst root of the 

 equation I'\:) = 0, zero excepted. Euler and Jacobi have determined 

 the first coefficients of this expansion ; we wish to determine the 

 general coefiicient. 



Starting from the known dexelopment 



lis) 2(v + l) — — ^ z' 



^ ^ ' 2(v+3) — etc. 

 and putting 



z' = — .1; 2 (i' + p) = up 



the question reduces to the determination of the general coefficient 

 in the following equation: 



— , =/iA' —/,■»'+/,*•' — etc. 



«I +^ 



a-j -|- etc. 



Pn 



Let — stand for the approximating fractions of the continued 



'-In 



fraction in the first member, and let 



Qin+\ — I'o + 1', -e + 1', .(,•' + . . . + r„ .r» 

 Q-2n •= ft„ + /«, .'• + f', •'■' + . . . + J'« *•" 



Q.2n-^ = ;.„ + ;, ,,: + ;.^ ,,:' + . • . + -l„_l .^:''-l 



Q.,„_2 = K„ + X, ./• -(- X., .v:' + . . . + '>C„_i .I-"-' 



