
( 467) 
XXII.—New General Formule for the Transformation of Infinite Series into 
Continued Fractions. By THomas Murr, M A., F.R.S.E. 
(Read 7th February 1876.) 
In Crelle’s Journal, vol. x., STERN devotes fifteen pages (pp. 245-259) of his 
Theorie der Kettenbriiche to the examination and elucidation of the following 
problems :— 

Express (I.) 1 + Ayo + Aya’? + Aza? +... 
(II.) 1+ Aw + Aix? + Aa? +... 
1+ By + By’ + Bz? +... 
(IIL) 1 
1+ Bye + Bow’? + Bow? +... 
as continued fractions of the form 
ee? ge 
“a &, -F GF Gg.+ --- 
where “4, @, @3,... are independent of x. 
Express (IV.) 1 + Aiw + A,w’ + Ajx’? + ... as a continued fraction of the 
eee 2S 
1l+at+a,+a,+..-. 
A mode of solution is shown, and made use of in the case of particular series, 
but the general problems are left unsolved, STERN having tried in vain, as he 
himself says,* to discover the law of formation of the partial denominators 
G,, G, 43, .... of the continued fraction from the coefficients A,, A,, A;,.... of 
the series. 
The main result of the present paper is the discovery of this law. SrTERn’s 
mode of procedure is followed, but use is made of a more simple and compre- 
hensive notation, to which perhaps the success of the investigation is due. 
Problems (III.) and (IV.) being virtually the same, and being like (I.) parti- 
cular cases of (II.) we start with the second problem, where there is given 
1+ Ayw + Ayv? + Agv? +... 
1+ By + Bix’ + Bare +... a +— 
form 


x 
eG ee 
* “Hier scheint es vielmehr nothwendig zu sein, einen einfachen Ausdruck zu finden, welcher 
jeden Theilnenner a,, (oder jedes N,,, N,,,,) unmittelbar aus den Reihencoefficienten A,, Ay, ... finden 
lehrt. Der Verfasser hat sich vergebens bemiiht die Lésung dieser Aufgabe zu finden, vielleicht aber 
konnen die mitgetheilten Ausdriicke fiir N,,, N,,,,, welche wohl nirgendwo angegeben sind, darauf 
ftihren.”—Crelle’s Journal, vol. x. p. 257. 
VOL. XXVII. PART IV. 6H 
