418 Mr. Horner on [June^ 



the arithmetical process for reducing an infinite decimal to con- 

 tinued fractions, and thence to a rational fraction, when possible, 

 may be applied to literal quantities also ; which he exemplifies 

 by reducing two easy recurring series. The general bearing 

 and chief use of the principle do not seem to have occurred to 

 him, and hence his remark is of little value. 



In the Memoirs of the AcacUmie des Sciences, 1772, I under- 

 stand Lagrange has proposed a method not very different from 

 this, for discovering from the known sum of a series whether the 

 latter is recurrent or not ; viz. by reducing the reciprocal of the 

 sum to continued fractions, extending each quotient as far as 

 hvo terms. Every object proposed by these distinguished writers 

 appears to be included in the following simple and general train 

 of deduction. 



The most convenient expression for a series whose sum is lo 

 be found or approximated by continued fractions is, 



(p= a— a a^ + a a^ a^ — a (iia^a^ ■\- &c (1) 



which is perfectly general, while it favours the fractional reduc- 

 tions that follow. Put V — p — a^,Q,~ q — a^p + a, o,, R = 

 r — a^ q + a, a.,p "' a^ a^a^, &c.^, q, r, &c. being indetermi- 

 nate, and we have 



1+P + Q + R+ .... ,c)s 



^ l+p+i/ + r + .... 



Make this = r-— \ ^^^n is 



1 + 9i 



1 + (p — a.^ + (q— a.p + (7.2 Cj) + ^, 



0. = — ■ • (It 



~' 1 + (p — o,) + (q — a,i> + a, «j) + 



Make this = --^, and let /3, = o, — «„ and generally /3„ = 

 «u+. — «i; then 



_ /3, + (g , p — g; m) + (g, g — g.. a-i p + 0^0.2 ai) + 



'^■' ~ ^1 + (p — a„) + (.q — a-i p + a„ a^) + 



Make this = -r^, and put y„ = 3,. + , - /3, = fl,.+, - a, ; then is 



•^3 



Yi + (Vi P — Ya "3) + (Vi g — Yii "iP + 73 "3 Oj) + 



^t + (18, p - $.2 flo) + (61 ?i - ftj a., p + ^3 a, 03) + 



Make this = ~^y and put 5; = a„ + „ (3, y,. + , — <7, /3„+ , 7j ; 

 then is 



J, + (S, p — S.J rij) + (?!,? — 82 «3 P + B3 03 04) + .... 



^*~' y, + (yi P - y^ "3) + (/i 5 - Yi "3 P + Y3 "s "4) + • • • • 

 Make this = -^; ands,. = y, 8,. + , - y„+. 8, ; then is 



Yi + Ts 



T=~ s, + (a, p - S.J 03) + (0, s - &! «3 p + 83 03 fl^) + .... * * 



Since the expression for ipj is of the same form as that for p^, 

 we now possess the law for continuing the series, the rest of 



